CAS Exam 9

Just like spot rates can be determined from forward rates, forward rates can be derived from spot rates:

(1 + fn) = [(1 + y of n)^n]/[(1 + y of (n-1))^(n-1)] Where: fn = rate that applies during the future interval "n"

Ratio of "PHSF on Loss Reserves" to "Earned Premium"

(Loss Reserves: Loss Incurred) x PLR

At the end of a RA DCF question finding UWP

(P-L-E)/P Note that nothing is discounted and taxes are not accounted for

The Sharpe ratio of an optimally constructed risky portfolio is calculated using the equation:

(SP)^2 = (SM)^2 + [αA/σ(eA)]^2 Where SM is the Sharpe ratio of the passive market index The term αA/σ(eA) is the information ratio: it represents the contribution of the active portfolio, when held at its optimal weight, to the Sharpe ratio. To maximize the Sharpe ratio of the portfolio, it is necessary to maximize the information ratio. In order to achieve this, the investment in each security needs to be in proportion to αi/σ^2(ei), while keeping the total investment in the active portfolio equal to w*A: w*i = w*Ax[αi/σ^2(ei)]/[∑αi/σ^2(ei)]

What form of Ferrari's equation is used for deciding whether to write more or less business?

(T/S) = (I/A) + (R/S)x(I/A + U/R)

The equation to determine the total variance (risk) of a portfolio of equally weighted assets is:

(σp)^2 = (1/n)σ¯^2 + [(n - 1)/n]Cov¯ where σ¯ if firm specific risk and Cov¯ is systematic risk

Selecting a trigger will involve a trade off between moral hazard & basis risk (the payment by the financial instrument may not be highly correlated with the insurer's actual losses):

*Moral Hazard* Party that is most concerned: Investors Appropriate Trigger: Industry Loss Index Disadv of using trigger to insurer: Higher basis risk *Basis Risk* Party that is most concerned: Sponsor Appropriate Trigger: Indemnity Disadv of using trigger to insurer: May need to reveal confidential info and May require more time to reach settlement due to length of the loss adjustment process Disadv of using trigger to investor: Potential for Moral Hazard and Needs to obtain info on the sponsor's portfolio, which may be difficult Note that: 1. parametric triggers have the lowest exposure to moral hazard, but highest exposure to basis risk. However, the basis risk can be reduced by selecting a geographic location where the insurer has a large concentration of insureds. 2. Modeled loss indices also have low morale hazard, and are much less exposed to basis risk. However, they are subject to model risk.

The Markowitz model (described in BKM7) has some problems:

- As the number of securities increases, the number of variables that need to be calculated/ estimated increase dramatically. - Due to the large number of required estimates, it is likely that some variables are estimated incorrectly. In this case, the model may produce nonsensical results

Cummins provides two methods to determine the cost of capital on a line

- Base it on the cost of capital of a monoline firm that writes the same line. (This is sometimes called the "pure play" technique) However, it will be difficult to find firms that write only the one line, and which have similar underwriting characteristics. - Perform regressions on the data of multiline insurers to derive cost of capital by line ("full information betas")

Some properties about the relationship between bonds and yields

- Bond prices & yield are inversely related - An increase in the yield produces a smaller price change than the same size decrease in yield - As the term of the bond increases, the price becomes more sensitive to yield changes - The sensitivity of bond prices to yield increases at a decreasing rate as the maturity increases - Lower coupon bonds are more sensitive to changes in yields - The sensitivity of bond prices to yield are inversely related to the yield

Note that bondholders are exposed to two sources of offsetting risk when interest rates change. When interest rates rise:

- Bond prices will fall (price risk) - Reinvested coupon income would grow (reinvestment risk) The reinvestment rate risk will offset the impact of the price risk.

Due to the problems of immunization, portfolio managers may consider alternatives:

- Cash Flow Matching: buy a zero which will make a payment that exactly matches the future cash obligation. - Dedication Strategy: this is cash flow matching over multiple periods. Purchase a combination of coupon paying bonds and/ or zeros to match a series of obligations. These strategies have pros and cons. Advantages include: > automatically immunizes the portfolio from changing interest rates > rebalancing will not be necessary The disadvantage is that the strategies are: > hard to implement because they impose strong constraints on the bonds that can be selected.

The contract between the bondholder and issuer is referred to as the bond indenture. This lists:

- Coupon rate - Maturity date - Par value

A swap is essentially a multiperiod extension of a forward contract. Examples of swaps include:

- Foreign exchange swap: exchange currencies on specific future dates (this can be compared to a forward contract, which involves exchange on one specific date) - Interest rate swap: exchange of cash flows at future dates based on the difference between fixed and floating rates at those dates

Income measures that can be used in the RAROC Calculation

- GAAP Net Income: calculated using GAAP accounting. This is convenient when RAROC will be used in management decisions - Statutory Net Income: calculated using Statutory accounting rules - IASB Fair Value: based on Fair value accounting. The advantage of this is that it removes the accounting biases of various accounting conventions - Economic Profit: the change in "economic value" of the firm over a period (assets are recorded at market value, and liabilities are discounted). There are a few problems with this measure: 1. ignores franchise value 2. may make less sense to management, as the economic values often do not reconcile to GAAP accounting 3. management may have difficulty justifying their decisions to external parties, as these parties only have access to statutory & GAAP accounting The calculations in Goldfarb's paper are based on the economic profit measure

Different parties are interested in different components of Ferrari's ROE equation

- Investors are interested in Return on Equity - Society is interested in Return on Assets - Regulators/ Actuaries are interested in Return on Sales

There are 2 ways to test models:

- Normative: tests model assumptions - Positive: examines the predictions

Spot rate and Short rater

- Spot rate: the interest rate that applies from time 0 to time t: it can be thought of as the yield to maturity on zero-coupon bonds - Short rate: interest rate that will apply during a future time interval Spot rates can be determined from the short rates. For example, the 2yr spot rate is equal to the product of the short rates that apply during each of the 2 years: (1 + y2)^2 = (1 + r1) * (1 + r2) Where: y2 = 2 yr spot rate rx = short rate for the year "x"

Duration is a measure of the average maturity of a financial instrument's cash flows. It is used in fixed income portfolio management for a number of purposes:

- Summary statistic of the effective average maturity - Helps immunize portfolios from interest rate risk - Measures interest rate sensitivity

Inverse floaters are similar to floating rate bonds, but the coupon falls when interest rates rise. These bonds therefore depreciate significantly when the rates rise:

- The coupon level falls - Each coupon is discounted by a greater magnitude

Properties of Duration

- The duration of a zero-coupon bond equals its time to maturity - Duration decreases as the coupon rate increases - Duration generally increases as the maturity increases - Duration increases as the yield to maturity reduces - The duration of a perpetuity = (1+y)/y

RBC Capital is the level of capital that must be held in order to avoid regulatory intervention. Cummins believes that it is not wise to use this basis to allocate capital because:

- The factors used in the derivation of the RBC are of questionable accuracy - The calculations are often based on book value instead of market - RBC ignores several important sources of risk However one advantage of this method of capital allocation is that it is relatively simple to calculate.

The franchise value presents the firm with a practical dilemma that has 2 components:

- The greater the franchise value of the firm, the more difficult it is to manage the interest rate risk - The firm could reduce the duration of invested assets to try offset the positive contribution of franchise value to the duration. However, this strategy may puzzle regulatory authorities/ rating agencies, as they only see the accounting numbers, and therefore do not account for the franchise value when analyzing the actions of the firm.

Difficulties involved with forming a bond index portfolio

- The index could consist of thousands of securities, which would make it difficult to purchase all in the correct proportions. To complicate things even further, many of the bonds would be thinly traded. - The composition of the bond index changes more often than that of the stock index. Bonds are dropped from the index as their maturities drop under a year, and also new bonds are added to the index as they are issued. Due to the difficulties, stratified sampling is utilized. This involves stratifying the bond market into several classes. The percentage of bonds in the index that fall into each class is then calculated. After this, the portfolio manager creates a portfolio with representation from each cell.

Credit Risk in the Swap Market: There has been concern about the credit risk associated with the swap market, especially because this market has being growing so rapidly. However, it is important to note that the credit risk is not as large as the volume of the aggregate notional principal would imply. This is because the exposure to credit risk is much lower than the actual notional principal:

- When the swap is initiated, it has a 0 NPV to each party. - The value can diverge from 0 after inception. For example, assume that the interest rates rise after an interest rate swap has been initiated. The floating rate payer therefore suffers a loss. If the floating rate payer did not pay the amount due, the fixed rate payer (or the dealer if applicable) would suffer a loss. For an interest rate swap, the credit loss would be based on the difference between the values of the fixed rate and floating rate obligations.

There are a few problems with immunization

- it is based on a measure of duration that makes the assumption that the yield curve is flat (constant discount rates for all terms). If this is not the case, each cash flow needs to be discounted at it's respective spot rate - immunizing the portfolio is only effective for parallel shifts in the yield curve - immunization is inappropriate in an inflationary environment

Market Structure assumptions are as follows:

-All assets are publicly traded & short positions are allowed. Investors can borrow/ lend at rf (the assumption that assets are tradable is necessary for investors to be able to derive identical input lists. The assumption about the interest rates is necessary for investors to derive the same tangency portfolio ) -All information is publicly available -No taxes (if investors had different tax rates, they would earn different after tax rates on the same stock, and could therefore derive different after tax optimal risky portfolios) -No transaction costs

Explain how a change in interest rates affects the modified duration of a bond.

-As interest increase, the more distant payments receive more discounting than the closer ones -The portion of the total PV (future payments) associated with the closer payments therefore increases, reducing the duration

Main conclusions of Cummins Capitol allocation paper

-EPD provides more information than VaR. Decisions should therefore focus on EPD (although it may be useful to calculate VaR too) - Using an allocation model that reflects the impact of diversification will lead to better decisions, as diversification does have value - The cost of capital that is allocated to a line is the spread cost (the excess cost of investing in an insurer as opposed to investing directly in the capital market) - Capital allocation must account for asset and liability risk, as well as the covariability between assets and liabilities. - The capital allocation should factor in the duration & maturity of liabilities - The decision making should drive the design of the data system (not the other way round). For example, annual data is insufficient to allocate capital appropriately. - Firms that allocate capital successfully will make better pricing, underwriting and entry/ exit decisions, creating value for the shareholders.

Identify two types of investment Income that may be excluded from rate calculations and defend the decision to exclude them.

-Investment Income on surplus: The surplus does not belong to the policyholders. In fact, the inclusion of investment income on surplus creates a situation in which an insurer with a large surplus relative to premium must charge lower rates than an otherwise equivalent insurer with less surplus. In other words, lower cost for more protection. -Investment Income in excess of the risk free rate: should the insurer engage in speculative investments resulting in the loss of policyholder supplied funds, the company cannot assess the insureds to make up the shortfall. Therefore the insureds are not exposed to investment risk, and should not receive a return higher than the risk free rate.

Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: Strong-Form Tests (Inside Information)

-Markets are not expected to be strong-form efficient: people with inside information should be able to make superior profits

What are two circumstances under which the Calendar Year Investment Income Offset Procedure is likely to suffer from distortion? Why do these circumstances cause distortion? Explain how the Present Value Offset Method overcomes the distortion in each case.

-Rapid growth/ Changes in reserve adequacy -This arises because data in prior calendar years comes from accident years stretching many years into the past. This data may therefore not be applicable to prospective ratemaking. -Since the offset method depends on the PLR, which is of a prospective nature, it is not affected by rapid growth. Changing reserve adequacy will impact this method, only if it affects the loss payout patterns.

An alternative argument to justify the movement of prices to the equilibrium level is the "risk return dominance" argument. There are some key differences between this and the arbitrage argument:

-Risk Return Dominance argument: if prices differ from the equilibrium level, many investors will make limited changes to their portfolios, depending on their degree of risk aversion. The culmination of the relatively small transactions of many investors produces sufficient volumes to move the market price. -Arbitrage argument: the investor who discovers the arbitrage opportunity will want to maximize his position in order to maximize profits. A lower number of investors are therefore needed to move prices back to equilibrium.

What is the key disadvantage of the Calendar Year Investment Income Offset Procedure? Why is this a problem? Explain how the Risk-Adjusted Discounted Cash Flow Model overcomes this issue.

-The CY Investment Income Offset procedure lacks economic theory supporting the calculation. -This is a problem as the provision is therefore hard to justify. -The risk adjusted discounted cash flow model is actually grounded in modern financial theory, as it attempts to directly calculate the fair premium, as opposed to relying heavily on user inputs/ assumptions.

Derivation of CAPM

-The contribution of an individual stock to the portfolio risk premium is: wi[E(ri) - rf] - The contribution of an individual stock to the portfolio variance is wiCov(ri, rM) The reward-to-risk ratio for the stock can therefore be calculated as: Ratio = Contribution to risk premium / Contribution to variance = [E(ri) - rf]/Cov(ri, rM) The reward-to-risk ratio for the market portfolio is known as the market price of risk: Market Price of Risk = [E(rM) - rf] / σM^2 Notice that the denominators of the reward-to-risk ratios are different: - The contribution of an individual asset to the variance of the portfolio variance is based on the asset's covariance to the market - If we look at the entire market, it is more appropriate to use the variance. Another way to look at this using the covariance of the market with itself, which is its variance.

There are 2 key implications of CAPM:

-The market portfolio is efficient -The premium on a risky asset is proportional to its beta

Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: SemiStrong-Form Tests (Market Anomalies)

-These test investigate whether publicly available information beyond the trading history can be used to generate abnormal returns. -Empirical results show that some easily available diagnostics can be used to predict abnormal returns (efficient market anomalies). -The efficient market anomalies include: Small-Firm-in-January Effect: small firms have historically generated superior returns, particularly in January. Neglected Firm Effect: Some firms are neglected by large investors, and so less information is available. A premium is required to compensate for the risk associated with less information. Note that this may explain the superior returns of small firms discussed above. Book-to-Market Ratios: High Book-to-Market firms have historically outperformed the rest of the market. Post-Earnings-Announcement Price Drift: The cumulative abnormal return of stocks has been shown to continue to increase even after the information about the event becomes public.

(AFTER an IRR question) Assume that regulators insist that the company charge premium rates 5% below what was calculated in Part A. i. Propose to upper management two modifications that the company could make to still earn the target 10% IRR for this line of business while charging no more than the regulated premium amount.

-investment more aggressively to get better returns -reduce fixed expenses

Discuss two reasons to support the discounting of policy holder supplied funds at a riskfree rate in the determination of an underwriting profit provision.

-the insured is not bearing the risk of the insurer's investments, and therefore should receive the risk free rate -profit greater than the risk free rate should belong to the owner's of the insurer, as a reward for putting their assets at risk

If can not borrow at rf, use a kinked CAL:

-the slope to the left of P will be unchanged from the regular CAL -the slope to the right of P=[E(rp)- rfB]/σP where rfB is the borrowing rate

After an immunization question, if they ask "Briefly describe two practical limitations to the immunization method in part a above."

-will need to rebalance as interest rates change -will need to rebalance as time elapses

*Defend* the following statement: "Credit rating agencies are to blame for the erosion of structured finance surrounding the events of 2007-2008."

1) Rated/Evaluated CDOs and deemed them "safe" 2) Rating agencies used the same scale as single name (corporate bond) ratings, which was misleading to investors 3) Rating agencies failed to appreciate the fragility of their models, especially correlation/systematic risk 4) Rating agencies did not consider systematic risk, which led to mispricing and enhanced demand especially for more Junior tranches 5) Rating agencies have inherent conflicts of interest because issuer pays for rating 6) Rating agencies failed to consider the risk/impact decreasing housing prices. 7) Rating agencies failed to consider the risk/impact decreasing underwriting standards.

*Dispute* the following statement: "Credit rating agencies are to blame for the erosion of structured finance surrounding the events of 2007-2008."

1) Regulators & Investors bought into the ratings - they did not do their own due diligence / did not seek to validate or understand risks 2) Regulators tied bank capital requirements to ratings 3) Investors did not consider whether ratings meant the same think for CDOs as they did for corporate bonds 4) Originating banks became fee driven, not concerned with underwriting risk as long as the mortgages could be sold 5) Investors only focused on ratings - were not concerned that little to no underwriting was being done by originators 6) Investment banks & rating agencies had perverse incentives to keep the machine running 7) Home buyers overextended themselve

List 2 differences between fixed assets of a manufacturer & surplus of an insurer

1. A manufacturer can objectively measure the needed assets, whereas the needed surplus is an estimate based on expected future development. 2. A manufacturer can divide the assets into product, but surplus cannot be divided into line.

3 advantages of Present Value Offset Method

1. Accounts for investment income in a simple manner 2. Not distorted by rapid growth / decline 3. No need to select target return or allocate surplus

List 6 Alternative Measures of Capital

1. Actual Committed Capital: The actual capital provided by the shareholders 2. Market Value of Equity: The market capitalization 3. Regulatory Required Capital 4. Rating Agency Required Capital: the amount of capital necessary to achieve a specic credit rating. 5. Economic Capital: the amount of capital necessary to provide the rm with a certain probability of achieving a specic objective over the time horizon. 6. Risk Capital: the amount of capital that needs to be provided by the shareholders to cover the risk that the liabilities may exceed the funds already provided

Capital measures that are not risk adjusted (Shouldn't be used in RAROC)

1. Actual Committed Capital: The actual capital provided by the shareholders 2. Market Value of Equity: The market capitalization

The capital of insurers will be invested in marketable securities. There are certain friction costs that reduce the return on this capital. Three sources of such costs are:

1. Agency & Informational costs: management may behave opportunistically, and therefore fail to achieve the owner's objective of value maximization 2. The current tax system results in double taxation: investing in securities via insurance companies produces lower after tax returns than purchasing the securities directly 3. Various regulations may force the insurer to hold inefficient investment portfolios (too much capital). Note that for most insurers, this generates no real cost, as they generally would hold more capital than the regulatory requirement anyway. The insurer needs to account for these costs when determining whether a line is earning the appropriate rate of return.

Quick rundown of Cummins various ways to allocate capital

1. Allocation by VaR: the appropriate amount of capital for each line is the amount which will equalize the exceedence probability among lines. 2. Insolvency Put / EPD: Capital can be allocated to achieve equivalent ratios of EPD to liabilities among lines. The advantage of this approach over VaR is that this approach reflects the severity of the losses. 3. Merton Perold Method: Unlike the prior methods, this method recognizes the impact of diversification. This tweaks the EPD method to reflect marginal impacts of a line on the capital required to achieve a certain EPD. 4. Myers-Read Method: This differs from the M-P method, because M-R looks at the impact of very small changes to the loss liabilities, as opposed to adding entire business units. It aims to equalize the marginal default values across lines of business.

List 4 methods that can be used to derive the aggregate distribution from the Frequency & Severity distributions:

1. Analytical Solution: generated based on the frequency & severity parameters 2. Numerical Method: numerical approximation 3. Approximation: based on the mean, variance (and possibly higher moments) of the collective risk model 4. Simulation

List 5 applications of risk adjusted performance metrics:

1. Assessing capital adequacy 2. Setting Risk management priorities 3. Evaluating alternative Risk Management strategies 4. Risk adjusted performance measurement 5. Insurance policy pricing

7 risks that surplus is meant to protect against

1. Asset risk: chance assets will depreciate 2. Pricing risk: chance that losses & expenses are ultimately greater than initially expected 3. Reserving risk: risk that reserves are insufficient 4. Asset-liability risk: changes in interest rates will affect the market value of assets differently to liabilities 5. Catastrophe risk 6. Reinsurance risk: risk that reinsurance recoverables wont be collected 7. Credit risk: risk that agents/insureds wont remit premium

3 ways to derive the thresholds that are used to compare to the risk measures

1. Bond default probabilities at a Selected Credit Rating Level: maintain enough capital that would result in a probability of default of the rm equal to the probability of default of a bond with a specific credit rating. 2. Managements risk preferences: a threshold based on the risk tolerance of management. 3. Arbitrary default probability: an arbitrary percentile that is relatively easy to measure.

Goldfarb has some issues with the popular methods of deriving the cost of risk capital variable that is used in calculating target RAROC when capital is committed for multiple periods. (CAPM and Fama French)

1. CAPM: The definition of risk used by CAPM is the systematic risk associated with an investment. However RAROC is based on a totally different type of risk: the difference between the cash flows expected value and the value in the tail. 2. The denominator of the RAROC calculation is understated , since it does not account for the franchise value. As a result, the RAROC value is overstated.

The M-P Method of capital allocation is conducted in two steps:

1. Calculate the risk capital required by firms that combine two of the business. (Here assuming 3 business units) 2. Calculate the marginal capital required when the excluded business is added to the two-business firms This marginal capital will be the amount allocated to business unit 3

List 4 techniques to aggregate the distributions of the different risks

1. Closed Form Solutions 2. Approximation Methods: Eg assume that all of the individual distributions are normal or lognormal, and then derive the parameters of the aggregate 3. Simulation Methods 4. Square Root Rule (described in Goldfarb)

(Goldfarb) Once the dependency has been estimated, the insurer can aggregate the distributions using one of several available techniques

1. Closed Form Solutions 2. Approximation Methods: Eg assume that all of the individual distributions are normal or lognormal, and then derive the parameters of the aggregate 3. Simulation Methods 4. Square Root Rule (described in Butsic)

Convertible bonds provide the bondholder the option to exchange the bond for a specified number of shares of the issuing firm.

1. Conversion ratio: the number of shares that each bond can be exchanged for 2. Market conversion value: current value of shares for which the bond can be exchanged 3. Conversion premium: excess of bond value over its conversion value Holders of convertible bonds benefit from price appreciation of the stock, and so the convertible bonds offer lower coupon rates.

2 advantages of the CY Investment Income Offset Procedure

1. Data easily obtained 2. Short & straight forward calculation

3 advantages of the CY Investment Income Offset Procedure

1. Data is easily obtained and verified 2. Since the figures are reported in filed documents, it is less likely that the insurer is making pessimistic projections in order to increase the profit provision 3. The calendar year investment portfolio yields are relatively stable

3 methods to quantify the dependency between risks

1. Empirical Analysis of Historical Data 2. Subjective Estimates 3. Explicit Factor Models: These link the variability of the risks to common factors

Once the insurer has derived the individual distributions of the above risks, it will then need to aggregate them. In order to do this the dependency between risks needs to be considered. Goldfarb proposes 3 methods to quantify the dependency

1. Empirical Analysis of Historical Data. Disadvantages include: -usually there is insufficient data to calculate the historical dependency -there is little insight as to how the dependencies will change during tail events 2. Subjective Estimates. Advantages include: -can account for the tail events -reflects the user's intuition Disadvantages include: -as the number of risk categories increases, the number of dependency parameters that need to be estimated increases exponentially. 3. Explicit Factor Models: These link the variability of the risks to common factors

3 examples of bases that can be used in the Rate of Return calculations

1. Equity 2. Assets 3. Sales

The single index can be used to derive the return of a security using the following steps:

1. Estimate the risk premium and risk of the market index via macroeconomic analysis 2. Calculate the beta and residual variance, σ^2(ei), of each security, using statistical analysis 3. Calculate the expected return of the security (based solely on the market return) 4. Calculate alpha using security analysis

3 reasons that historical catastrophe losses are not the best measure of potential future losses:

1. Events are rare 2. Exposures change over time 3. Severities change over time due to changes in the building materials & designs

Biases that cause information processing errors:

1. Forecasting Errors: This consists of 2 parts: -too much weight is assigned to recent experience -forecasts are too extreme given the actual level of uncertainty If a firm has had recent strong performance, the forecasts of future earnings may be too high. This will lead to an overstated P/E, resulting in poorer subsequent performance once investors realize their error (P/E effect) 2. Overconfidence: Many investors overestimate their abilities. 3. Conservatism: Investors are too slow to update their beliefs in response to new evidence. 4. Sample Size Neglect & Representativeness: Investors do not account for sample size, and therefore may infer a pattern based on too small a sample.

Behavioral Biases: Behavioral biases that cause suboptimal financial decisions include:

1. Framing: Decisions are often materially impacted by how the question is framed 2. Mental Accounting: Investors may segregate decisions. For example, they may segment their investment portfolios into pieces which go towards different goals, and they will have different risk attitudes to each piece. Rationally, it would be better to optimize the risk-return profile of the entire portfolio in aggregate. 3. Regret Avoidance: Investors who make decisions that turn out badly have more regret if it were an unconventional decision. 4. Affect: The "affect" is the good or bad feeling that investors may associate with investing in a stock. For example, stocks of firms that practice socially responsible policies may generate higher affect. If investors prefer these stocks, they may drive up the prices, reducing the expected return. In fact, the stocks ranked high on Fortune's survey of most admired companies have tended to produce lower risk adjusted returns. 5. Prospect Theory: This modifies the standard financial theory's definition of risk averse investors. -under standard theory, utility depends on the level of wealth -under prospect theory, utility depends on the changes in wealth. Again, there are diminishing returns of utility as wealth increases. In addition, it shows that investors become risk seeking as they begin to lose money

Limits to Arbitrage: Behavioral Finance makes the assumption that the actions of arbitrageurs are limited, and so price irrationalities can continue. Factors that limit their actions include:

1. Fundamental Risk: Their actions are actually not risk free, because the mispricings are not necessarily going to disappear. In fact, they can get worse 2. Implementation Costs: Arbitrageurs usually need to rely on short selling in order to exploit overpricing. However, there are often limitations to short selling: -short sellers may be given little notice to return the security that they borrowed to short sell -many investors are not permitted to short sell 3. Model Risk: It is possible that the prices are indeed valid. Instead, there may be issues with the arbitrageur's model that are causing it to falsely indicate a mispricing.

List 4 alternate measures of income that can be used in the RAROC calculation

1. GAAP Net Income: calculated using GAAP accounting 2. Statutory Net Income: calculated using Statutory accounting rules 3. IASB Fair Value: based on Fair value accounting 4. Economic Prot: the change in \economic value" of the rm over a period

3 advantages of the Risk Adjusted DCF Model

1. Great intuitive appeal 2. Grounded in modern financial theory 3. Not necessary to determine a target ROR

3 types of "real" costs to the firm from volatile cash flows

1. Higher expected bankruptcy costs: lawyers fees/ court fees/ interference from court on investment & operating decisions/ increased difficulty of raising funds 2. Higher expected payments to stakeholders to compensate for risk 3. Higher expected tax payments

Leading up to the financial crisis , What 4 characteristics of CMOs were biases against the investors:

1. Higher probability of default due to lower credit quality of borrowers. 2. Lower recovery values, because when the assets do need to be sold, they are often sold under financial pressure. 3. High level of default correlation due to pooling mortgages from similar geographic areas/ vintages. 4. Due to the CDO2 structure, the impact of errors in the estimates is magnified.

3 points to keep in mind if using bond default probabilities:

1. Historical vs Current estimates: historical default rates are more stable, but current market estimates of default rates better reflect current conditions 2. Different sources of default statistics may indicate dierent numbers, possibly because they use data from different periods 3. Time Horizon: tables often provide annual default probabilities. However, risk capital models are often based on default during the lifetime of the liabilities

2 reasons that insurers reinvestment rate is usually equal to the IRR

1. If a pricing model produces an IRR which is greater than the cost of capital, insurers can use this extra revenue to write more policies (and therefore effectively grow at the IRR rate). As long as it maintains its current underwriting standards, the "reinvestment rate" of revenue is equal to the IRR, and therefore the IRR assumptions hold. 2. Policies are usually priced using an underwriting profit provision which sets the IRR equal to the cost of capital. Again this would mean that the assumptions hold, since IRR = Cost of capital = Reinvestment rate

Since there is a lot of uncertainty in the estimation of losses, management has the ability to influence the results of the company reported in the financial statements, by either strengthening or weakening the reserves:

1. If the insurer is having a poor year, it may weaken the reserves, which will reduce the size of the underwriting loss, resulting in a higher profit. 2. On the other hand, when the insurer has a good year, it can strengthen the reserves (either because the reserves are currently weak, or to provide some cushion for future years). This adjustment will increase the incurred loss, so the insurer will have a good year instead of a great year. The current underwriting results will be affected by: 1. Current reserving decisions 2. Amortization of past reserving decisions Due to these various factors affecting the results, it is more difficult for an analyst examining the financials of the insurer to determine how they really are performing in a given year.

Ferrari mentions a few uses of the determination of the optimal cash structure:

1. If the public believes that the company is not insuring enough people, it may be able to justify this by demonstrating that the optimal capital structure calls for a relatively low reserve/ surplus ratio. 2. It can be seen if an overly aggressive investment portfolio is driving the low ratio of reserves: surplus from the optimal cash structure analysis 3. If the optimal structure requires a higher reserve to surplus ratio than the average company, this may suggest that the industry is overcapitilized

Risk Linked Securities can have 3 types of triggers that generate a payment

1. Indemnity triggers: the payouts are based on the sponsoring insurer's actual losses. 2. Index triggers: the payouts are based on an index of industry losses. Possible indices include: -industry loss indices: the estimated losses to the industry from an event -modeled loss indices: runs the parameters of the event through a model of a cat-modeling firm to generate either industry losses, or losses specific to the sponsoring insurer. -parametric indices: triggered by specified physical measures of an event (eg wind speed of a hurricane, or magnitude of an earthquake). 3. Hybrid triggers: these blend more than one trigger.

(Risk Linked Securities) In 1995, Nationwide issued $400M of contingent notes via a trust. The proceeds were invested in 10yr Treasuries, and the investors were provided a coupon of 220 basis points over the Treasuries. Nationwide has a "substitutability" option, where it could substitute surplus notes (debt securities that are treated by regulators as surplus) for the Treasuries during the 10yr period for any "business reason". There were two major problems:

1. Investors were exposed to the general business risks of the insurer (Nationwide could default on the notes) 2. Investors could withdraw the funds from the trust, in which case Nationwide would need to repay the trust. Due to these problems, contingent notes did not arise as a common risk linked security.

5 advantages of Frequency & Severity Models over Loss Ratio Distribution Models:

1. It is easier to account for growth in the volume of business 2. Inflation can be more accurately reflected 3. Changes in limit and deductibles can be more easily reflected 4. The impact of deductibles on frequency can be accounted for 5. The treatment of the split of loss between insured, insurer & reinsurer can be mutually consistent

3 problems with the guideline that an underwriting provision is adequate if it produces an adequate profit:

1. It is not obvious how to apply it in ratemaking 2. There is a question of what is an adequate total return 3. Ratemaking is conducted on a prospective policy year basis, but total return is measured on a calendar year basis

2 disadvantages of the CY Investment Income Offset Procedure

1. Lack of economic theory supporting the calculation 2. Distorted if large change in volume or reserve adequacy

3 problems of using fixed Profit Margin when pricing insurance policies

1. Lack of theoretical justification of the fixed margin 2. High interest rates suggest the fixed margins may be too low 3. Doesnt reflect increasing competitiveness of the insurance industry

The opportunity cost is affected by 2 factors:

1. Line of business: some lines have longer intervals before the losses are paid 2. Cash needed to support the infrastructure (eg the buildings/ furniture, etc) of the insurer. This cash can not be invested. Since most of the cash tied up in infrastructure was provided by the premiums of prior policyholders, and the current insureds are benefiting from these assets, they should not receive credit for investment income from the money invested in these.

List 3 methods that can be used to quantify the Underwriting Risk:

1. Loss Ratio Distribution Model 2. Frequency & Severity Model 3. Inference from Reserve Risk Models

3 categories of insurance underwriting risk

1. Loss reserves on prior policy years: potential adverse development 2. Underwriting Risk for the current policy year 3. Property catastrophe risk

There are a number of reasons why it is difficult to determine if the markets are truly efficient:

1. Magnitude Issue: Assume an investment manager of a large portfolio improves the performance of a huge portfolio by 0.1% per year. Even though this will be a big contribution in dollar terms, it is very small compared to the normal volatility of the market. It is therefore hard to assess how much the manager actually contributed. 2. Selection Bias Issue: Once an investment scheme becomes known by others, it will no longer generate abnormal returns. Therefore, only schemes that do not work will be reported. It is possible that techniques that do actually work exist, but are not being reported. 3. Lucky Event Issue: There are several cases of investors who have made huge investment returns over a period. However, this does not disprove EMH, because the number of investors is so large, just by chance, some have to make huge returns.

The text presents two examples where the value of the portfolio can be adversely impacted by changes in interest rates:

1. Many banks have a mismatch between assets & liabilities: -liabilities are usually deposits from customers and therefore short term -assets are often commercial loans or mortgages, which are much more long term Since the assets have a longer duration than the liabilities, the bank can suffer large losses if interest rates increase. 2. Pension funds may have a mismatch between the interest rate sensitivity of fund assets and liabilities. If the liabilities are more sensitive than assets, a drop in the interest rates will cause the liabilities to grow faster than the assets.

5 main categories of risk to which an insurer is exposed

1. Market risk 2. Credit Risk 3. Insurance Underwriting Risk 4. Operational Risk 5. Strategic Risk

3 sources of exposure to credit risk to the insurer

1. Marketable Securities/ Derivatives/ Swap Positions 2. Insureds Contingent Premiums & Deductibles Receivable 3. Reinsurance Recoveries

Trends & Corrections: A great deal of technical analysis has to do with uncovering trends. There are several approaches that try to profit from trends:

1. Moving Averages: This is a method to identify trends (if they exist). The moving average of a stock price is the price over an interval. If prices have been falling, the moving average will exceed the current price. If the market breaks through the moving average line from below, this is a bullish signal, as it is a sign of a shift from a falling trend to a rising trend. Another method briefly discussed is Elliott wave theory, which decomposes the actual movements into long-term and short term wave cycles. By identifying the long term waves, investors can buy when the long term direction is positive. 2. Relative Strength: This measures how the stock has performed relative to the industry, or the market as a whole. For example, the relative strength of Toyota can be measured by the movement of the ratio of the price of Toyota stock to the level of an auto industry index. Similarly, the performance of an industry can be compared to the performance of the market. 3. Breadth: This measures how widely the movement in the market index is reflected in the price movements of all stocks. There are many different ways to measure this. One example is the spread between the number of stocks that advance and decline. As the advances outnumber the declines by a wider margin, the market is viewed as being stronger.

2 components of PHSF

1. PHSF on UEPR 2. PHSF on Loss Reserves

4 features of "cooperative games with transferable utilities":

1. Participants that have benefits or costs to share 2. Opportunity to share the benefits/ costs due to the cooperation of all (or a subgroup of) the participants 3. Ability of the players to enter into negotiations 4. Conflicting player objectives, where each wants to maximize his/ her share of the benefits (or minimize the share of the costs)

(Goldfarb) Risk Measures: Insurers can derive the necessary amount of capital required to produce a specific risk measure. Three potential risk measures are:

1. Percentile Risk Measure: provides the capital required to achieve a certain prob of ruin (ruin is usually defined as the state where the assets are insufficient to meet the liabilities). 2. Conditional Tail Expectation (CTE): the mean loss of all the losses that exceed a certain percentile. It also is known as Tail VaR (TVaR) or Tail Conditional Expectation (TCE). 3. Expected Policyholder Deficit Ratio: discussed in Butsic. This metric is less arbitrary than CTE, which is based on a selected percentile.

Describe the 3 components of total risk that impact Loss Reserve Risk:

1. Process Risk: risk that actual results will deviate from their expected value due to the random variation inherent in the claim development process 2. Parameter Risk: risk that the actual expected value of the liability diers from the actuarys estimate of the expected value due to inaccurate parameters 3. Model Risk: risk that the actual expected value of the liability differs from the actuarys estimate of the expected value due to the use of the wrong model

2 ways to measure profit of an insurer

1. Profit 2. Rate of Return

2 situations when NPV and IRR may give different conclusions

1. Projects with unusual cash flows 2. Projects with budget constraints/ Mutually exclusive projects

Briefly describe the 5 types of underwriting profit provision that Robbin refers to:

1. Provision included in manual rates/ filings to change manual rates 2. Corporate target underwriting profit provision: this should be sufficient to generate an expected return similar to that provided by investments with similar risk 3. Breakeven underwriting profit provision: these provide a rate of return to stockholders equal to the rate of return on risk free investments. 4. Charged underwriting profit provision: the rate achieved after applying experience and schedule rating modifications, as well as other adjustments 5. Actual underwriting profit: these will differ from the charged provisions

Capital allocation can assist with value maximization: management can determine whether a line is producing sufficient return to cover its cost of capital. Various metrics can be used to determine if the value is being maximized, based on the allocated capital.

1. RAROC: Described in the Goldfarb paper. If RAROC is greater than the cost of capital, the insurer should continue to devote resources to the line. However, if it is less than the cost, the firm will have to take some action (eg tighten underwriting standards, reduce volume, etc). 2. EVA (Economic Value Added): This formula quantifies that value added to the firm from a given line: EVAi = Net Incomei - riCi Where: ri = cost of the capital for the line i. The above formula can be modified to be expressed in rate of return format. The subject of the equation becomes EVAOC (Economic Value Added on Capital): EVAOCi = Net Incomei/Ci - ri

According to Robbin, 2 ways in which profit provisions can be regulated

1. Rate of return approach: rates should be regulated to ensure that companies are able to achieve an adequate return. 2. Constrained free market theory: the premiums will move to an optimal level via market forces

Cummins compares the CAT bond to catastrophe reinsurance pricing. In order to do this, he defines the following terms that apply to catastrophe reinsurance: Rate on Line and Loss on Line

1. Rate on Line (ROL) = reinsurance premium / policy limit 2. Loss on Line (LOL) = expected loss / policy limit The ratio of ROL to LOL is comparable to the yield on catastrophe bonds. According to Cummins, empirical data indicates that according to this metric, the pricing of CAT bonds is similar to reinsurance. The advantages of CAT bonds are becoming stronger as investment banks gain experience with insurance linked securities, as they are reducing the transaction costs, and increasing the speed to market.

List 3 examples of the deteriorating quality of mortgages in the last decade:

1. Ratio of mortgage values to home prices increased 2. Increased use of second lien loans 3. Increased issuance of mortgages with low/ no documentation

Risk adjusted capital measures that can be used in the calculation of RAROC

1. Regulatory Required Capital 2. Rating Agency Required Capital: the amount of capital necessary to achieve a specific credit rating. 3. Economic Capital: the amount of capital necessary to provide the firm with a certain probability of achieving a specific objective over the time horizon. Objectives include: -Solvency Objective: that the firm can meet its existing obligations to policyholders -Capital Adequacy Objective: that the firm can continue to pay dividends/grow premiums/ maintain a certain financial strength 4. Risk Capital: the amount of capital that needs to be provided by the shareholders to cover the risk that the liabilities may exceed the funds already provided (in the form of loss reserves or premiums).

Determinants of Bond Safety (Profitability Ratios)

1. Return on assets = earnings before interest & taxes / total assets 2. Return on equity = net income / equity

Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: Weak-Form Tests (Patterns in Stock Returns)

1. Returns over short horizons -Looks at whether investors can use historic trends to earn abnormal profits over the short term, by measuring the serial correlation of stock market returns -Empirical results show only small correlations over short periods (with the exception that the sectors with the best and worst returns exhibit stronger correlations, which is known as the momentum effect) 2. Returns over long horizons -Similar to the prior test, but looks at the long term returns -Empirical results have shown a negative correlation over the long term: this may be explained by the fads hypothesis -if the negative correlation is correct, a contrarian investment strategy may be profitable

Arbitrage Pricing Theory (APT) is based on the following 3 assumptions:

1. Security returns can be described by a factor model 2. There are a sufficient number of securities to diversify away idiosyncratic risk 3. Well functioning securities markets do not allow for the persistence of arbitrage opportunities

5 examples of Model Construction Questions

1. Should surplus be included in the model? 2. How should the surplus requirement be determined? 3. How should risk be incorporated into the model? 4. It is better to use cash flows or incomes flows? 5. How to reflect income taxes?

Violations of the Law of One Price The limits to arbitrage described above have resulted in several cases where the Law of One Price has been violated. Examples include:

1. Siamese Twin Companies: In 1907, the Royal Dutch Petroleum and Shell Transport companies merged. They agreed to split the total profits on a 60/40 basis. With this structure, Royal Dutch shares should have sold for 150% the price of Shell's. However, the price relativity did deviate from this 150% level for extended periods. Arbitrageurs who tried to profit from this may have lost money because there were periods of time where the deviations actually grew in size. 2. Equity Carve-outs: In 1999, 3Com span off its Palm division. Each 3Com shareholder was to receive 1.5 shares of Palm. In the period between the initial trading of the Palm shares, and the spinoff, 3Com shares should have been trading for at least 1.5 times the value of Palm. Instead Palm traded for more than 3Com! Arbitrageurs could not exploit the mispricing by buying 3Com and selling Palm short, because no Palm shares were available: they had already all been short sold. 3. Closed-End funds: Many closed-end funds trade for a discount or premium to their net asset value. The authors mention that this however is not a true violation, as there are a few reasons why the two values can diverge. For example, the fund incurs expenses which will reduce the share price. In fact, it was proven that the premium of the fund over its Net Asset Value (NAV) can be derived with the following equation: (Price - NAV) / NAV = (α - ε) / (δ + ε - α) Where: dividend yield = δ; alpha = α; expense ratio = ε If the manager can generate a higher α than ε, the fund will sell at a premium

2 different objectives that can be used to derive the amount of Economic capital

1. Solvency Objective: that the firm can meet its existing obligations to policyholders 2. Capital Adequacy Objective: that the firm can continue to pay dividends/ grow premiums/ maintain a certain financial strength

List the 3 components (modules) of Catastrophe models:

1. Stochastic Module/Hazard Module: generate the events that can occur, including the location, intensity, etc 2. Damage (Vulnerability) Module: Derives the damage that would arise from an event, based on exposure information 3. Financial Analysis Module: Applies the insurance/ reinsurance terms to the losses to determine the financial impact to the insurer

2 rules of the allocation according to game theory:

1. The allocation methods must be renewal additive 2. Coalition should be stable (fair) This way, there would be no incentive for a player/ group of players to split from the group. Essentially, there should be no possibilities where are a subgroup is better on its own.

Behavioral Critique: Many economists disagree with Behavioral Finance. Reasons include:

1. The behavioral approach is too unstructured: it allows virtually any anomaly to be explained by a combination of irrationalities. These critics would have more faith in Behavioral Finance if it provides a single behavioral characteristic that explains a range of anomalies. 2. Some anomalies are inconsistent in their support for one irrationality vs another. 3. Selecting the wrong benchmark can produce an apparent abnormality.

CAT bonds are often issued to cover the high layers of exposure, which may be uninsured due to the following reasons:

1. The credit risk of the reinsurer is a major concern in events of this magnitude. CAT bonds are much safer as they are fully collateralized 2. The highest layers usually have the highest reinsurance profit margins. Investors are willing to accept lower spreads from CAT bonds because they offer diversification benefits: cat events usually have low correlations to investment returns Another advantage of CAT bonds over reinsurance is that multi year bonds are available,(unlike traditional reinsurance, which is usually only offered for a one year period). CAT bonds therefore shelter the sponsor from the cyclical price fluctuations of the reinsurance market. Note that the market does not like bonds longer than 5yrs, as they would like the opportunity to reprice the risk periodically.

TECHNICAL ANALYSIS & BEHAVIORAL FINANCE: Behavioral Finance suggests that in many cases that Technical Analysis will be successful. Behavioral Finance can explain:

1. The disposition effect, where investors hold on to losing investments, because they are reluctant to realize losses. This slows down the adjustment of the share price to the appropriate value. 2. The technical analysts' use of volume data: this arises due to the behavioral characteristic of overconfidence. Overconfident investors trade more, which produces a relationship between trading volume and market returns. 3. Market fundamentals can be disturbed by irrational factors or behavioral factors, sometimes labeled sentiment variables. It is possible for technical analysis to exploit the corrections.

In order to assist in the calculation of the optimum capital structure, the actuary should determine:

1. The expected relationship between I/A & U/R and resulting impact on the expected earnings stream of writing more business. 2. Increase in probability of unfavorable results due to the higher volatility, and resulting impact on insurer value due to the higher discount rates applied by the market.

There are two main differences between a sinking fund call and conventional bond call:

1. The firm can only repurchase a limited fraction of the bonds at the sinking fund call price (some indentures would contain a doubling option which allows repurchase of double the required bonds) 2. The call price of callable bonds usually exceeds par value, whereas the sinking fund call price is usually set at the par value

(Cummins) Issues involved with using VaR to allocate capitol

1. The firm may not have enough capital to achieve a certain exceedence probability 2. This approach does not consider the impact of diversification 3. This does not reflect the amount by which losses will exceed the resources in the event that the exceedence level is breached. One advantage though is that it is A consistent risk measure that can be used to compare between lines of business, companies, etc.

(Panning) Strategies a firm could implement to reduce its duration

1. The traditional way to do this would be to reduce the duration of the invested assets. (This only reduces the current economic value's contribution to duration) 2. An alternative strategy is to adopt a pricing strategy that reduces the sensitivity of the franchise value to interest rates. One such strategy is to alter the values of a & b. (used in the calculation of target return) Advantages to #2 include: Avoids rating agency and regulatory risk that are associated with managing the duration of the invested assets The disadvantages include: The desired combination of target return on surplus and target duration can be maintained only for small interest rate changes

There are a few points to keep in mind regarding franchise value:

1. The value is reflected in the firm's stock price 2. It is based on the present value of future cash flows to the firm, and is therefore exposed to interest rate risk 3. It is usually invisible to senior executives, and is therefore unmanaged

(Bodoff) 2 reasons that a loss would receive a higher allocation in the upper layers than in the lower layers:

1. There are a fewer events that pierce the layer 2. The layer of capital is wider because the incremental increase to severity of each event tends to increase in size.

robbin's IRR, ROE, and PVI/PVE Models determine the price of a policy by aiming to generate a return greater than the hurdle rate. He has the following goals for prices

1. They should be consistent and sensitive to risk. His models satisfy this as the rate of returns are sensitive to the impact of leverage. 2. They should reflect managements risk return preferences. To achieve this, the model is based on the theoretical amount of surplus as opposed to the actual level.

After Hurricane Andrew of 1992, the Chicago Board of Trade (CBOT) introduced catastrophe futures and later catastrophe put and call options. These however were later withdrawn due to lack of trading volume. There were several reasons for the low interest:

1. Thinness of the market (low number of buyers & sellers) 2. Possible counterparty risk on the occurrence of a major catastrophe 3. Insurer's wanting to avoid disrupting long term relationships with reinsurers 4. Excessive basis risk (discussed later)

Bodie Kane and Marcus account for privately held businesses by dividing them into 2 categories

1. Those with similar characteristics to traded assets: Owners of these businesses can still achieve diversification by reducing their portfolio holdings of similar traded assets, and so they will still essentially hold the market portfolio. There is therefore little impact to CAPM. 2. Those that do not have similar characteristics to traded assets: owners of these businesses would demand a portfolio of traded assets that hedge the risk of a typical private business. The price of these hedge assets will be bid up, reducing the expected return (relative to the systematic risk), which will cause them to appear to have a negative α according to the traditional CAPM. Securities that are highly correlated with the risk will appear to have a positive α.

Sentiment Indicators: Technical Analysts have also derived methods to measure market sentiment:

1. Trin Statistic: Market advances are considered to be stronger when they are accompanied by stronger trading volumes. Trin = [(Volume declining / Number declining)] / [(Vol. advancing / No. advancing)] Ratios over 1.0 are bearish. 2. Confidence Index: This is based on data from the bond market, under the assumption that the actions of bond traders reveal trends that will later follow in the stock market. Confidence Index = (Avg yield on top 10 rated corporate bonds) / (Avg yield on intermediate 10 rated corporate bonds) If bond traders are optimistic, they will require smaller default premiums, which will cause the ratio to approach 100% 3. Put/ Call Ratio: This equals the ratio of outstanding put options to outstanding call options. Since put options profit from falling markets, an increase in the ratio is a bearish sign.

2 approaches to derive the risk adjusted rate:

1. View the adjustment as a form of compensation to the insurer for placing its capital at risk in the insurance contract 2. Use CAPM

There are 3 versions of the EMH:

1. Weak form: stock prices reflect all information that can be derived from examining market data (past prices/volume/ etc) 2. Semistrong form: stock prices reflect all publicly available information about the firm's prospects 3. Strong form: stock prices reflect all information relevant to the firm, including that not publicly available (inside information)

2 examples of Parameter Selection Questions

1. What discount rate should be used? 2. What is the right target return?

There are some added complications to determining the optimal capital structure for insurers. Accounting for debt in the optimal capital structure for a non-insurer is simpler than accounting for reserves in the calculation for an insurer.

1. With debt, valuation analysts know that an increased level increases the risk and therefore results in demands by creditors for higher interest rates. 2. This relationship is not as straight forward for insurers. The earlier conclusion that increasing the reserves results in higher volatility to the insurer, is derived from holding everything else constant. In reality, a higher level of reserves implies a higher amount of premium written. It is possible that this higher level of premium will increase the diversification of the insurer's risk, and therefore have a negative impact on the variability of results.

2 advantages of using Subjective estimates to quantify dependency between risks

1. can account for the tail events 2. reflects the users intuition

3 reasons that the insurers credit risk exposure to reinsurance recoveries is unique

1. denition of "default": if the reinsurers credit rating gets downgraded below an investment grade level, it could enter a "death spiral" where its policyholders try to end their contracts, resulting in severe liquidity problems 2. substantial contingent exposure: the reinsurance recoverable at a given point in time may increase in the future due to adverse loss development 3. reinsurance credit risk is highly correlated with the underlying insurance risk

2 disadvantages of using bond default probabilities as a threshold for the risk measure

1. does not address which credit rating should be targeted 2. does not account for the risk of downgrade

3 major distinctions of the RA DCF models:

1. doesnt require the user to select a target return 2. The RA DCF model is not based on a particular accounting structure. It therefore has no way to reflect the conservative treatment of expenses in SAP accounting, nor can it reflect reserve discounting. 3. Surplus does not have a large impact on premium in this method.

2 problems of using ROE to regulate an insurer

1. forces regulator to focus on ROE instead of rate equity 2. surplus needs to be allocated

List 3 problems with the Economic Profit measure

1. ignores franchise value 2. may make less sense to management, as the economic values often do not reconcile to GAAP accounting 3. management may have diculty justifying their decisions to external parties, as these parties only have access to statutory & GAAP accounting

3 examples of changes to structure of industry if rates are inadequate

1. increasing size of residual market 2. Reducing degree of product diversity 3. Reduced innovation

List the available methods to allocate capital

1. proportional allocation based on a risk measure 2. incremental allocation 3. marginal allocation (Myers-Read) 4. co-measures approach

2 reasons that policyholders should only get credit for investment income from policyholder funds

1. shareholder funds do not belong to them 2. including shareholder funds will penalize high surplus insurers

(Empirical Evidence on Security Returns) There are several difficulties of the Expected Return Beta Relationship approach of testing CAPM, which may be causing the inconsistent results:

1. stock results are volatile, reducing the accuracy of the tests that rely on return 2. there are concerns about the validity of the tests: -the market index used is not the true market portfolio -due to the volatility, the betas measured in the first stage regression are subject to a large amount of sampling error -investors can't borrow at the risk free rate

List 2 assumptions for timing of surplus commitments/ release that an insurer can make

1. surplus is committed once a policy is written, and no longer needed when it expires 2. surplus is committed when UEPR is established, and declines as losses are paid

Some parties believe that the results of EMH tests are signs that the markets are not efficient. Others believe that the markets are indeed efficient, but that the results of the tests arise because:

1. the properties are proxies for fundamental determinants of risk 2. the properties arise just due to data mining. One supporting factor for this argument is that many of these anomalies disappear after discovered. One way to test for this is to see if the relationship holds in a different database.

2 advantages of using ROS

1. understandable 2. does not require surplus allocation

2 disadvantages of using Empirical analysis to quantify dependency between risks

1. usually there is insucient data to calculate the historical dependency 2. there is little insight as to how the dependencies will change during tail events

4 disadvantages of using managements risk preferences as a threshold for the risk measure used in RAROC calculations:

1. will be difficult to get management to articulate and agree on a threshold 2. the preferences of managers will often differ to the preferences of directors and shareholders 3. this does not factor in risk, which is an important consideration to compare to reward 4. managers will likely be focused on a number of issues that concern policyholders, and it may be hard to isolate the estimate of the threshold to just the probability of default

The way a practice question did the shapley method. Reinsurance load for the combined portolio was given (10,000). So were the variances of XYZ, ABC, and combined. (250,000, 1,000,000, 2,250,000)

10,000 = Multiplier * Var(combined) Multiplier = 10,000/2,250,000 = 0.00444 2,250,000 = 250,000 + 1,000,000 + 2Cov(XYZ, ABC) Cov(XYZ, ABC) = 500,000 XYZ = 0.00444 * (250,000 + 500,000) = 3,333 ABC = 10,000 - 3,333 = 6,667

Formula to allocate capital to loss event x(i) when dealing with discrete loss events:

= [x(α + j) − x(α)] × [Probability(x=x(i))/Probability(x>x(α))]

Recommend the risk management strategy for a company with very little debt, and almost no chance of default

> Since the low end of the tail never reaches the range where it imposes financial distress costs, there is no reason to hedge > It can also take bets if it has specialized information

(Butsic) To ensure equity for all parties to an insurance contract, the RBC method needs to satisfy several criteria:

> The standard needs to be the same for all types of insurers (personal vs commercial; primary versus reinsurers) > The RBC needs to be objectively determined (two insurers with the same risk exposure need to have the same RBC requirements) > The approach should be able to distinguish between items that differ materially in level of riskiness

List some problems with looking at the Insurance risks over only 1 year:

> There is little available methods/ data to estimate the timing of the recognition of adverse loss development > The change in value perspective for loss reserves will not be totally consistent with market risk, as it will most likely focus on the best estimate of the reserve. > The information that would result in the revaluation of the liabilities is often not available over the short term, and as a result, there will likely be minimal change in the liability valuation, despite potential signicant risk over the long term.

Two sources of potential value in active management of a bond portfolio

> interest rate forecasting: increase portfolio duration if rate declines are expected, and vice versa. > Identification of relative mispricing: eg the analyst may purchase a bond if he believes that the default premium of a bond is too high (implying that the bond is underpriced)

KAMP Re 2005 Ltd.

A $190 Million bond issued in July 2005. The first publicly acknowledged total loss of principal for a CAT bond. It paid out its entire principal to the sponsor as a result of Hurrican Katrina claims The smooth settlement of the Kamp Re bond established an important precedent in the market, showing that CAT bonds functioned as designed, with minimal confusion and controversy between the sponsor and investors. It reduced the overall uncertainty associated with the market place thus increasing investor and sponsor demand for these instruments

Protective covenants of bond indentures: Credit Risk and Collateralized Debt Obligations

A Collateralized debt obligation (CDO) is a tool that can reallocate credit risk. In order to create a CDO, a financial institution would create a separate entity (usually a Structured Investment Vehicle, or SIV) to buy a portfolio of bonds/ loans. These are pooled together, and then split into different tranches. Each tranche can be sold as a stand alone security. Each tranche has a different level of seniority. As the securities in the pool make their interest payments, the proceeds are distributed to the pay interest to each of the tranches in order of their seniority. In addition, the lower seniority tranches would suffer the default losses before the higher seniority. Due to the different priorities, each tranche would have a different exposure to credit risk. The lower tranches have the greatest credit risk but pay the highest coupon rates.

Protective covenants of bond indentures: Credit Default Swap

A credit default swap (CDS) is essentially an insurance policy on the default risk of a bond/ loan. CDS would typically be purchased by large bondholders. Note that an investor who holds a bond with a BB rating can purchase a CDS on the issuer to effectively raise the position to that of holding an AAA rated bond. CDS played a large role in the 2008 Financial Crisis. By August 2008, there were $63 trillion of swaps outstanding. This was incredible, especially given the fact that the US GDP in 2008 was "only" $14 trillion! When the subprime mortgage market collapsed, the obligation on these CDS' increased to an unimaginable level.

A credit default swap (CDS) is structured differently from a typical swap. Instead, it is an effectively insurance policy that is written on particular credit events. Bondholders may purchase CDS in order to transfer their credit risk exposure to the swap seller.

A difference from standard insurance is that the swap holder does not necessarily have to hold the bonds that underlie the CDS contract. As a result, the CDS can be used to speculate on the changes in the credit standing of the reference firms.

Mean-variance criterion:

A dominates B if: E(rA) ≥ E(rB) and σA ≤ σB

(Factors that could be used to supplement the market risk in a multifactor SML) Momentum

A fourth factor has been proposed to be included in the Fama French model: momentum. This may actually be what was causing the alpha that may have been apparent in the three factor model. It will be quite difficult to justify/ explain including this factor: it is hard to see what type of risk this would be acting as a proxy for.

Problems with ROE as measure of return: Problem One: Return on Equity VS Rate Equity

A problem of focusing on return on equity in rate regulation is that this forces the regulator to focus on return on equity, instead of rate equity. The return on equity can be distorted by the leverage of the insurer, and therefore the concept of rate equity may not apply.

Fair game

A risky investment with a risk premium of 0.

Problems with ROE as measure of return: Problem Two: Surplus Allocation

A second problem with ROE regulation is that surplus needs to be allocated to line and geographical area. This is an artificial allocation. This is because 100% of the insurer's surplus supports each risk → not just the portion of surplus allocated to the category that the risk falls in. Surplus allocation would result in the following insurers being treated the same: a multiline company with $100M of surplus, $1M of which is allocated to N Dakota auto insurance, a monoline auto insurer which only operates in N Dakota with $1M or surplus. The above shows that surplus allocation ignores the value of the unallocated surplus.

Catastrophe risk (CAT) bonds are most common type of risk linked security. These bonds are included in the class event-linked bonds¸ which provide payment on the occurrence of a specified event. Explain the process involved with cat bonds

A single purpose reinsurer (SPR) is created by the insurer (sponsor). The benefit of the SPR is that it shields the investors from the general business risk of the insurer. Other benefits include: - The financing costs for the issuer will be lower - The transaction is more transparent than a debt issue because the funds are held in a trust and released according to very specific criteria The SPR issues CAT bonds to investors, and invests the proceeds received in safe, short term securities. The CAT bonds contain a call option that is triggered by a specified catastrophic event: if the event occurs, the proceeds are transferred from the SPR to the insurer. The fixed returns from the investments are usually swapped for floating returns, in order to immunize the insurer and investor from interest rate risk.

The APT and CAPM

APT is based on a foundation of well diversified portfolios (resulting in 0 residual risk). However, even large portfolios have non-negligible residual risk. If residual risk is small, the APT SML should still approximate the risk premium. The deviations should be unbiased, and also uncorrelated with beta or the residual SD. However, if residual risk is sufficiently high, we cannot have full confidence in the APT. Despite the above, APT is still valuable. For example, it does not rely on CAPM's assumption that investors are mean-variance optimizers. Instead, we just need a small number of arbitrageurs that look for arbitrage opportunities. A big advantage of APT is that it does not require an all inclusive portfolio. As discussed, CAPM cannot be tested as it requires this unobservable true market portfolio.

(Butsic) Accounting Conventions & the Bias Problem

According to Butsic, basing RBC calculations on accounting book value is not appropriate, due to the inherent acct bias (current recorded value does not necessarily equal current realizable value). On the other hand, market value acct is based on the current realizable value. This is preferable for solvency assessment, as in the event of an insurer's failure, the items of the balance sheet are liquidated at the market rates.

(Theories that can explain the shape of the yield curve) Expectations Hypothesis

According to the Expectations Hypothesis, the forward rate equals the expected future short rate. Therefore an upward sloping yield curve is a signal that investors expect rate increases

Bodoff's alternate presentation of the vertical procedure integral is: Allocated Capital(x) = f(x)[integral from 0 to x[1/(1-F(y))]dy]. This provides some insights into the capital allocation of each loss event

According to this equation, the procedure of capital allocation by layer says that any loss event's allocated capital depends upon: 1. Probability of the event occurring (f(x)) 2. The severity of the loss event, or the extent to which the loss event penetrates layers of capital (The upper bound of the integration is x, the loss amount) 3. The loss event's inability to share the burden of its required capital with other loss events (the integral). We can think of this factor as the extent to which a loss event "sticks out" or is dissimilar in severity to other loss events

(Arbitrage Pricing Theory): Multifactor Model

Adding additional factors to the single-factor model produces a multifactor model. The advantage of this over the index model is that the varying sensitivities of securities to the different factors can be incorporated. An example of a multifactor model is a two-factor model: ri = E(ri) + βi1F1 + βi2F2 + ei Where βij is the sensitivity of the share's return to the jth factor

Equation for assets in the single period model:

Aj = UEPRj +XRSVj +LRSVj +Sj Where: UEPR = Unearned Premium XRSV = Statutory Expense Reserve LRSV = Loss Reserve

Describe the Horizontal procedure of capital allocation:

Allocate each layer of capital to the loss events that penetrate the layer.

Feldblum allocating surplus for the purpose of IRR: if surplus is committed when losses occur and released when they are paid. You are given the durations between occurence and payment for different lines.

Allocate surplus to lines proportional to their (Expected Loss Ratio x Annual Written Premium x Duration)

Describe the Vertical procedure of capital allocation:

Allocates capital to each loss event based upon the layers that it penetrates

Arbitrage Pricing Theory

An Arbitrage opportunity is the opportunity for an investor to earn a riskless profit without the need to make a net investment. Due to the actions of arbitrageurs, the Law of One Price exists: this states that two assets that are equivalent in all economically relevant aspects should have the same market price.

Advantage of using a broad index (like the S&P 500) for the single-index model

An advantage of using a broad index like the S&P500 is that a lot of data is available to estimate the parameters that are needed.

Goldfarb's risk measures are usually compared to a threshold level. There are several ways to derive this threshold: Arbitrary default probabilities

An arbitrary percentile that is relatively easy to measure. This way, the insurer does not have to estimate values at very low probability of occurrences, where a lot of uncertainty exists.

Riskiness Leverage Ratio

An arbitrary selection by management that allows their views towards risk to be incorporated.

Describe a scenario in which an investor would prefer a callable bond to a deferred callable bond.

An investor may prefer the callable bond if he does not think that the bond will decrease in the near future. If this turns out to be the case, the deferred component of the deferred callable bond has no value. These deferred callable bonds would therefore be inferior, as they presumably have a lower yield than a standard callable bond.

If the coupon rate is less than the market interest rate, the coupon alone would provide the investor with an insufficient return.

An investor would only purchase the bond if it is selling below par value, as it would offer a "built in" capital gain to offset the lower coupons.

In order to use the single index model in practice, we need estimates of the α, β and σi of each stock, in addition to RM and σm^2. This is significantly less than the number of estimates needed in the Markowitz model

Another advantage of the single index model over the Markowitz model is that the single index model allows for specialization of effort in security analysis: for example, it is possible for one group to specialize in the computer industry, and another in the auto industry, as there is no need for anyone to estimate the covariance between stocks from the two different industries.

The 4 factor model was also used to determine if there was consistency in mutual fund performance. There only appears to be minor persistence, but a lot of this is due to expenses and transaction costs.

Another test of the persistence in performance using the 4 factor model was conducted by Bollen & Busse. Mutual fund performance was ranked to decile. Performance during the following quarter was then measured. The exhibit below demonstrates that there is only small persistence that not sufficient to justify performance chasing.

What is the main difference between the arbitrage argument for restoring equilibrium prices and the risk-return dominance argument

Arb: it is the large actions of just a few investors that restore equilibrium Risk Return: based on the small actions of many investors. Since the arbitrage argument relies on changes by fewer investors, we can conclude that it is stronger

Describe the arbitrage argument for restoring equilibrium prices in comparison to the risk-return dominance argument for restoring equilibrium prices

Arbitrage: When an arbitrage opportunity exists, a few investors will make large changes to their portfolio to take advantage of the mispricing. This will very quickly restore equilibrium. Risk-return: When a security is mispriced many investors will make small changes to tilt the portfolio toward the mispriced security. The cumulative results of all these investors' actions will restore the equilibrium

Active vs Passive Portfolio Management

As mentioned above, according to EMH, only unique insight will produce profits. It takes a lot of effort and cost to gain this unique insight, and therefore it is usually only feasible for investors with large portfolios. For smaller investors, on the other hand, the costs will exceed any potential benefits, and they therefore will need to resort to alternatives: 1. Invest in mutual funds (pooling of several investor's portfolios) 2. Passive investment Advocates of EMH would recommend the latter, as they believe that active management should not produce excess returns. Note that passive investors will trade much less often than active because according to EMH, stocks are priced fairly, which means it is pointless to buy and sell frequently.

why risk pooling can increase risk for investment portfolio

As the investor increases the risk pooling to include n assets, both the Sharpe ratio and standard deviation will increase by n^0.5. Therefore, it can be seen that risk pooling does not reduce the level of risk. In order to reduce the risk, risk pooling alone is insufficient. The investor will also need to engage in risk sharing.

1 disadvantage of using Subjective estimates to quantify dependency between risks

As the number of risk categories increases, the number of dependency parameters that need to be estimated increases exponentially.

Summary of the Optimization Procedure

As we saw above, the optimal risky portfolio can be created using the above steps: 1. Calculate the ratio of each security of the active portfolio: wi0 = αi / σ^2(ei) 2. Scale the above weights so total will equal 1: wi = wi0 / ∑wi0 3. Calculate the alpha, beta & residual variance of the active portfolio: αA = ∑wiαi βA = ∑ wiβi σ^2(eA) = ∑ (wi^2)x(σ^2(ei)) 4. Calculate the weight of the active portfolio: (wA)^0 = [αA/(σ^2)(eA)]/[E(RM)/(σM)^2] w*A= [(wA)^0]/[1 + (1 - βA)(wA)^0] 5. Calculate the weights of the market and each security in the optimal portfolio w*M = 1 - w*A; and w*i = w*A wi 6. Calculate the risk premium and variance of the optimal portfolio: E(Rp) = (w*M + w*A βA) * E(RM) + w*A αA σP^2 = [(w*M + w*A βA)^2] x (σM)^2 + [w*A σ(eA)]^2

Insolvency Put Option/ EPD Ratio for Capitol Allocation

Assume that a firm has assets (A), and Liabilities (L) that are due in one period. At maturity, if: 1. A > L: the policyholders will receive the value of the liabilities, and the owners will keep the remaining assets. There will be no deficit to the policyholders. 2. A < L: the owners will default on the liabilities, and the policyholders will claim the assets. The deficit to the policyholders will equal L - A. The deficit to the policyholders is therefore equal to Max(0, L - A). This is equivalent to the payoff from a put option, and so the expected deficiency to the policyholder should be equal to the value of the put option In order to allocate capital, Cummins calculates the value of the policyholder's claim before the liabilities mature: Value = Le^(-rt) - P(A,L,r,τ,σ) Where: Le^(-rt) = the present value of the claim if the default risk were 0 P( ) = value of a put option (based on the parameters in parantheses). This is known as the insolvency put option, or expected policyholder deficit (EPD). Capital can be allocated to achieve equivalent ratios of EPD to liabilities among lines. The advantage of this approach over VaR is that this approach reflects the severity of the losses. The main problem is that it does not account for diversification.

Optimal Risky Portfolio of the Single Index Model

Assume that a portfolio manager has to construct an optimal risky portfolio from n actively researched firms, and a market index portfolio (S&P500). The input list includes estimates of: Risk premium of the S&P500 SD of the S&P500 Beta coefficients, stock residual values & alpha values of all n stocks The following equations can then be used: αp = ∑ wiαi βp = ∑ wiβi σ^2(ep) = ∑ (wi^2 x σ^2(ei) ) The manager will select weights that maximize the Sharpe ratio, E(Rp)/ σp, where: E(Rp) = ∑ wiαi + E(RM) * ∑ wiβi σp= [(σM)^2 (∑ wiβi)^2 + ∑ ((wi^2)x(σ^2)(ei))]^(1/2) The initial weight in the active portfolio is: (wA)^0 = [αA/(σ^2)(eA)]/[E(RM)/(σM)^2] the final weight is (i.e. not assuming beta=1): w*A= [(wA)^0]/[1 + (1 - βA)(wA)^0]

(BKM 8 Index Models) Tracking Portfolio

Assume that a portfolio manager has used the index model to derive an equation for the return of a security. She ideally would like to take advantage of her security analysis, while earning a return that is independent from the market return. To accomplish this, she can use a tracking portfolio, which will match the systematic component of P's return.

Panning's formula for the duration of Franchise Value

Assume that premium is selected to provide a target rate of return, k. Further assume that k is a function of the interest rates: k = a + b * y where y = spot rate corresponding to the firm's liabilities then D = [a - b + 1] / [(1 + y) x (a + by - y)] + 1 / [1 + y - cr]

Investors would usually not pay exactly the quoted bond price as there is interest that accrues between the coupon dates that is not included in the bond price. The sale price (also referred to as the invoice price or flat price) includes the stated price as well as the accrued interest.

Assuming semi annual coupons: Accrued interest = (Annual Coupon Payment/2) * (Days since last coupon payment/Days separating coupon payments)

Ferrari did mention one relationship in his paper: that U/P would decrease from higher P/S, due to the underwriting standards being loosened in order for the company to grow.

Balcarek argues that this relationship is not strong, as the insurer can avoid this decrease in U/P by ensuring that the current underwriting standards are maintained, while growing the business.

Methods used to quantify Underwriting risk: Loss Ratio distribution model

Based on an assumed distribution of loss ratios (combined with estimates of written premium during the risk horizon period). The model parameters used to generate the distribution can be based on either empirical or industry data. Note that the choice of distribution model is going to have a material impact on the amount of uncertainty recognized.

How is Investment Income derived in the Cash Flow Return Method

Based on surplus

How is Investment Income derived in the Risk Adjusted DCF Model

Based on surplus

According to Cummins the application of VaR as a capital allocation method requires very frequent data, but insurance prices and losses are not observed with sufficient frequency.

Because of this using tools like VaR requires an integration of the capital allocation methodology with data processing and information systems to ensure that pertinent and useful data are generated to provide inputs for VaR

Why does systematic risk make up such a large portion of the overall risk of structured products:

Because structured products pool together the risk from several assets, the non systematic risk is usually diversified away

Behavioral Critique

Behavioral finance differs from conventional financial theory in that it accounts for how real people make decisions, including the irrationalities that influence decision making. There are 2 categories of irrationalities: 1. Investors do not always process information correctly, and therefore derive incorrect probability distributions 2. Even when investors do generate a correct probability distribution, they still often make inconsistent/ suboptimal decisions due to their behavioral biases.

(BKM 8 Index Models) Predicting Betas

Betas tend to change over time. It may therefore make sense to create a forecasting model for Beta. One option is to use regression: Current β = a + b * (Past β) A more sophisticated approach could be to incorporate other financial variables that may impact β. For example, Current β = a + b1 * (Past β) + b2 * (Firm Size) + b3 * (Debt Ratio) Other potential variables that can be used include: - Variance of earnings - Variance of cash flow - Growth in earnings per share - Market capitalization - Dividend yield - Debt to asset ratio

The value of a bond is equal to the present value of the future cash flows. The discount rate exceeds the risk free rate due to bond specific characteristics like default risk, liquidity risk, tax attributes, call risk, etc Outline the formula of the cash flow payments including PV(annuity) formula for the coupon payments

Bond Value = Par value * (1 + r)^(-t) + Σ Coupon * (1 + r)^(-t) = Par value * (1 + r)^(-t) + Coupon * (1/r) * [1 − (1 + r)^(-T)]

Goldfarb's risk measures are usually compared to a threshold level. There are several ways to derive this threshold: Bond default probabilites

Bond default probabilities at a Selected Credit Rating Level: maintain enough capital that would result in a probability of default of the firm equal to the probability of default of a bond with a specific credit rating. This approach has some disadvantages: -does not address which credit rating should be targeted -does not account for the risk of downgrade With industry default rate data, there are several things to keep in mind: -Historical vs Current estimates: either historical default rates (which are more stable) or current market estimates of default rates (which better reflect current conditions) can be used -Sources of Historical Default Statistics: different sources of default statistics may indicate different numbers, possibly because they use data from different periods -Time Horizon: tables often provide annual default probabilities. However, risk capital models are often based on default during the lifetime of the liabilities

(Theories that can explain the shape of the yield curve) Liquidity Preference

Bond investors would ideally select a bond which matures around the time that they need the money. For example, - An investor who needs the money in the short term would prefer to purchase a short term bond. If they purchased a long term bond, they would be subject to interest rate risk. - Similarly, an investor who needs the money in the long term would prefer a long term bond. If they purchased a short term bond, they would be subject to reinvestment risk. Under Liq Pref, The shape of the yield curve will be influenced by the proportions of the different term investors. If the short term investors dominate the market (the most likely scenario), the forward rates will exceed the expected spot rate, which will produce a rising yield curve. On the other hand, if the long term investors dominate, the forward rates will be lower than the expected short rates.

Bond default risk is often referred to as credit risk. This is measured by firms such as Moody's Investor Services, Standard & Poor's Corporation, and Fitch Investors Service.

Bonds rated BBB or above (S&P and Fitch) or Baa or above (Moody's) are considered to be investment grade. The lower rated bonds are considered to be speculative grade/ junk bonds.

Calculation of EPD

Butsic first demonstrates the calculation of EPD by providing a scenario where an insurer has fixed assets, and variable losses (losses are exposed to risk). Using these criteria, the equation to determine the EPD is: DL = ∑ p(x) (x - A) for all x>A In order to compare the policyholder deficits of different insurers, the deficits can be adjusted to reflect the different exposure sizes. This can be accomplished by calculating the EPD ratio (d): the ratio of the deficit to the expected loss. We can also calculate the EPD for the scenario where the losses are fixed, and the assets are risky, using the equation: DA = ∑ q(y) (L - y) for all L>y

(Butsic) State the square root rule, used to approximate the capital need in the case of multiple risky assets and/ or liabilities:

C = [∑Ci^2 + ∑∑PijCiCj ]^(0.5) In the above equation, if the risky items appear on opposite sides of the balance sheet (eg an asset and liability), the correlation needs to be multiplied by -1 (see example below).

Extensions of the CAPM: CAPM with Non-Traded Assets & Labor Income

CAPM makes the assumption that all risky assets are traded. In reality, there are several assets not traded. 2 examples: - Human capital - Privately held businesses these can have a material impact on the equilibrium returns. Labor income (human capital) is a lot more difficult to hedge. One of the only options is for employees to avoid purchasing shares of their own employer, or companies from their own industry. As a result, securities offered by labor intensive firms will have a lower demand, and may appear to have a positive α according to the trad CAPM. An adjusted CAPM that accounts for labor income: E(Ri) = E(RM)x[Cov(Ri, RM) + (PH/PM)Cov(Ri, RH)]/[σM^2 + (PH/PM)Cov(RM, RH)] Where: PH = value of aggregate human capital PM = market value of traded assets the standard β is replaced by an adjusted β that also accounts for the cov with the portfolio of agg human cap. If Cov(Ri, RH) is positive, the adjusted β will be greater when the standard β is less than 1, and vice versa. Therefore, this model produces a SML that is less steep than the standard CAPM. This model may explain the negative alpha of high β securities indicated by some tests.

Even though both the SML & CML show the relationship between risk and return, there are differences between the two:

CML: graphs risk premiums of efficient portfolios (made up of the market and risk free assets) as a function of σ, because σ is the appropriate risk measure for portfolios SML: graphs risk premiums of individual assets as a function of β, because β is the appropriate risk measure for individual securities held as part of a well diversified portfolio

Equation for Combined Ratio, as presented by Robbin

CR = VR + [(1+c)L + FX]/P

How does the "Risk Adjusted DCF Model" determine the Profit Provision

Calculates a fair premium, and then derives a provision from that Fair premium =Risk adjusted present value of underwriting cash flows + present value of taxes.

How does the "Present Value Offset Method" determine the Profit Provision

Calculates an Investment Offset to the traditional provision that is based on the difference of PVs of the line being priced, and a short tailed reference line

Briefly describe the optimal strategy, given no constraints, for immunizing a liability. (Two options for bonds are given in this question)

Cash flow matching: match the cash flows of the assets and liabilities (Apparently you're supposed to ignore the bond information completely) The practical limitations of this are that its Very difficult to find the right combination of bonds that would produce an aggregate cash flow that would match that of the liability.

Historical catastrophe experience is not the best indication of potential future losses for a number of reasons, including: - Events are rare - Exposures change over time - Severities change over time due to changes in the building materials & designs

Catastrophe Models are therefore instead used. These often have several modules, including: - Stochastic Module/ Hazard Module: generate the events that can occur, including the location, intensity, etc. - Damage (Vulnerability) Module: Derives the damage that would arise from an event, based on exposure information - Financial Analysis Module: Applies the insurance/ reinsurance terms to the losses to determine the financial impact to the insurer

Swaps are a useful tool to fixed income managers. They enable the managers to quickly, cheaply and anonymously restructure the balance sheet.

Consider a corporation that had issued fixed rate debt. The firm now believes that the interest rates are likely to fall. Due to this view, it may prefer to convert this debt to floating rate. One option is to issue new floating rate debt, and use these proceeds to buy back the outstanding fixed rate debt. An easier alternative would be to convert the outstanding fixed rate debt into synthetic floating rate debt by entering into a swap to: receive a fixed interest rate (which offsets the fixed coupon rate obligation), and pay a floating rate

The CAPM can be used in capital allocation by calculating separate betas by line to determine the appropriate rate of return for each line.

Consider the scenario of an insurer that has two lines of business. The income can be derived with the following formula: I = rAA + r1P1 + r2P2 Where: rA = return on assets ri = rates on return on underwriting from line i Pi = premium from line i The β can be decomposed: βE = βA * (1 + k1 + k2) + β1s1 + β2s2 Where: βA = β of assets βi = β of insurance risk of line i (May be listed as beta of liabilities) ki = liability leverage ratio for line i = Li / E si = premium leverage ratio for line i = Pi / E Once the β of an individual line has been determined, the required rate of return can be calculated with the following equation: ri = -kirF + βi * (rM - rF) Then the combined ratio for the line can be calculated as CR = 1-ri

A company is considering two risk management strategies. It will either issue a CAT bond or enter into a catastrophe risk swap. The company's decision will be based on the following three concerns: • Counterparty default risk • Basis risk • Risk diversification Evaluate the company's two risk management strategies based on each concern and choose the better strategy for the company.

Counterparty default risk -CAT bond: risk limit is fully collateralized, so no credit risk -Risk swap: this is not prefunded and therefore the insurer is exposed to credit default risk. Basis risk For both, this would depend on the trigger selected -most CAT bonds are issued on an index trigger basis which exposes insurers to the risk that the actual loss is not perfectly correlated with the actual payoff. Therefore, basis risk exists. -Swap: trigger is usually predefined by the contract, and would usually be structured therefore to minimize basis risk (ie have payment correlated to loss). If it is instead based on an index trigger, there is no benefit over CAT bond (that also depends on index losses) Risk diversification: -CAT bond: this does not diversify risk, but instead is essentially equivalent to reinsurance (so basically the insurer is reducing the risk as opposed to diversifying) -Swap: this can allow the insurer to diversify risk, by pairing with an uncorrelated risk

Indifference Curves

Curves containing portfolios with equivalent utility levels. The optimal portfolio is located at the intersection point of the CAL, and the curve tangential to the CAL.

International bonds are often divided into two categories: Euro Bonds

Denominated in one currency (usually that of the country of the issuer), but sold in other markets (for example a USD denominated bond sold outside the US). These bonds include: -Eurodollar: denominated in USD -Euroyen: denominated in yen -Eurosterling: denominated in pounds

Why are insurers more likely than other companies to have positive IRRs less than the cost of capital

Deteriorating results are usually associated with an increase in reserves. This increase in reserves results in increasing investment income, which offsets the poor underwriting results.

Resource Allocation in relation to Efficient Markets

Deviations from market efficiency will generate a cost to everyone: inefficient resource allocation. This arises because securities are mispriced. One example is that firms with overpriced securities will be able to obtain capital too cheaply.

Protective covenants of bond indentures: Yield to Maturity and Default Risk

Due to default risk, the stated yield is the maximum possible yield to maturity that the bondholder can earn. If the financial condition of a firm deteriorates and it becomes likelier that the bonds will default, the bond price should fall, resulting in an increase to the promised yield to maturity (for someone who was to purchase the bond today). But the other hand, the expected yield to maturity should be much less affected. The bonds need to offer a default premium to compensate for the chance of default. This equals the difference between the yield of a corporate bond, and the yield of a comparable government bond (which has no credit risk).

Give the equations for E(L), Var(L) and Cov(L) used in Mango's exhibits

E(L) = Sum(i[Li ∗ pi]) Var(L)=Sum(i[L2i ∗pi∗(1−pi)] ) Cov(L,n) = Sum(i[Li ∗ni ∗pi ∗(1−pi)])

CAL Equation (Capital allocation line)

E(rc) = rf + σc([E(rp) - rf]/σp) Where c represents the complete portfolio and p represents risky assets The slope of the CAL is known as the Sharpe, or reward-to-volatility ratio [E(rp) - rf] / σp

(Markowitz Model) Once the investor has derived the weights to invest in the different assets, the expected return and variance of the optimal complete portfolio can be derived:

E(rp) = Σ wiE(ri) (σp)^2 = Σ Σ wiwjCov(ri, rj)

Equation for Economic Profit

EP = Prem - Expenses - Discounted losses + Investment return

In the "Risk Adjusted DCF Model", to what point in time are the Cash Flows discounted?

End of first year

A yield which is related to realized compound return is holding period return. This depends on the coupons received during the holding period, as well as the market price of the bond at the end of the holding period.

Example from Paper -A 30 year bond pays an annual coupon of $80 and sells at $1,000 -Assume that at the end of the year, the bond is worth $1,050 Holding period return = [80 + (1,050 - 1,000)] / 1,000 = 0.13

The realized compound return can be forecasted over specific holding periods. This process is called horizon analysis.

Example from paper: -You buy a 30 year 7.5% annual payment coupon bond for $980 and plan to hold for 20 years -You forecast that the yield to maturity will be 8% when it is sold -You also forecast that the reinvestment rate on coupons will be 6% What is the forecasted realized return? Forecasted sale price = 1,000 * 1.08^(-10) + 75 * (1 - 1.08^(-10)) / 0.08 = 966.45 Future value of coupon = [75 * (1 - 1.06^(-20)) / 0.06] * 1.06^(20) = 2,758.92 980 * (1 + r)^(20) = 966.45 + 2,758.92 r = 6.9%

The price appreciation from original issue discount (OID) bonds is treated by the IRS as an implicit interest payment. The IRS calculates a price appreciation schedule which applies even if the bond is not sold or does not mature until a future year.

Example from paper: -a 30 year zero is issued when the interest is 10% Initial price = 1,000 * 1.1^(-30) = 57.31 "IRS" Price in 1yr = 1,000 * 1.1^(-29) = 63.04 IRS Imputed Interest = 63.04 - 57.31 = 5.73 This amount is taxable. The imputed interest ignores the changes in the market interest rate. It is based on the initial rate when the bond was issued -assume that interest rates fall at time 1 to 9.9%, and the bond is sold Price = 1,000 * 1.099^(-29) = 64.72 The difference between 64.72 and 63.04 would be treated as capital gains income and taxed at the capital gains tax rate in addition to the 5.73 imputed interest. If the bond was not sold, that difference would instead be treated as unrealized gains.

Reason that IRR can be used for expected cash flows, whereas NPV should be used for actual cash flows

Expected cash flows do not usually have sign reversals, but actuals may.

Franchise value with varying cr ratio that drops to zero

F = [CR1/(1+y) + (CR1 x CR2)/(1+y)^2] x (P-E-L/(1+y))

Panning's Franchise Value formula

F = [P - E - L/ (1 + y)] * d / (1 - d) Where d = cr/ (1 + y) P is profit, E is expenses, L is losses, y is interest rate, cr is customer retention rate

Equation for Equity Flow:

F0 =I0 −Q0 =−Q0; SinceI0 =0 For j = 1,2,...,n, Fj =Ij − (Qj −Qj −1)=Ij −∆Qj −1

According to the interest rate parity relationship (that applies to the pricing of forward contracts), the forward forex swap price should be connected to the spot exchange rate E0 based on the formula:

F1 = E0(1 + rUS) / (1 + rUK); Where F1 and E0 are expressed in USD per pound This relationship should also apply to the one period swap since the swap is equivalent to the forward contract. Next consider a similar two period swap. This could be thought of as being equivalent to two forward contracts. We would need to consider the forward price of each of the two contracts: F1 = E0(1 + rUS) / (1 + rUK) F2 = E0[(1 + rUS) / (1 + rUK)]^2

(Ferrari) Reserves viewed as Non-Equity Capital

Ferrari rearranges his ROE equation into the following form: T/S = (I/A) + (R/S)(I/A + U/R) The insurer can use the above equation to determine whether or not to keep writing business. The investment income (I/A) can be thought of as the main source of return to the insurer. Just like in other industries, this return is modified by the amount of leverage (R/S). Leverage can either cause returns to increase or decrease. The direction depends on the relationship between I/A & U/R. - If the insurer has underwriting profits (positive U/R), the term in parenthesis is positive (assuming positive investment income), so it makes sense for the insurer to keep writing business - If the U/R is negative (underwriting loss), but the absolute value is less than I/A, the term in parenthesis is still positive, implying that is still profitable to keep writing business. - Finally, if U/R is negative and causes the term in parenthesis to become negative, at this point, it makes sense for the insurer to stop writing business, since writing business will reduce the ROE.

List & briefly describe the 2 markets of insurance transactions

Financial Market • transactions between shareholders & insurer • return to shareholders depends on the risk of investment Products Market • transactions between policyholders & insurer • premium depends on the supply/ demand of insurance

Insurance Leverage comparison to Financial Leverage

For companies in most industries, "leverage" results from taking out loans. However, to an insurer, "leverage" arises from the deferred nature of insurance liabilities. Reserves are essentially seen as loans from the policy holder. With the above interpretation, the term U/R (UW Profit/Reserves) (in the case of underwriting losses) can be thought of as an "interest cost" incurred by the firm to use the reserves, which were contributed by the policyholders.

Indexed bonds make payments that are based on a general price index, or the price of a specific commodity.

For example, Treasury Inflation Protected Securities (TIPS) are bonds issued by the US Treasury that are linked to the rate of inflation.

Firms can also use foreign exchange swaps to restructure their balance sheet.

For example, a British firm issues $10M US bonds that pay an 8% coupon rate (it issued the bond in the US to take advantage of the favorable interest rates). Assume that the firm prefers that its interest be denominated in British pounds. It can enter into a swap to exchange a certain number of pounds for $800K annually.

Financial leverage increases the volatility of results

For financial leverage this is because Equity will decrease with increases in Debt assuming a constant amount of capital. So Profit/Equity will be greater (more positive of more negative) with more leverage. Therefore there is a greater level of risk associated with an increased leverage.

(Risk Linked Securities) Cat Risk Swaps

For instance: Reinsurer A trades CA Earthquake exposure to Reinsurer B for Japan Quake Exposure The swap defines a specified amount of money that needs to be paid by each reinsurer in the occurrence of a specified event. If the swap is structured such that the two parties achieve parity (have equal expected losses), no money is exchanged at inception. Advantages include: 1. the (re)insurer reduces some of its core risk, and achieves diversification 2. lower transaction costs than some of the other securities Disadvantages include: 1. it is difficult to create a swap that achieves parity 2. can create more exposure to basis risk that some other types of contracts 3. not prefunded

A bond that is selling above the par value is referred to as a premium bond.

For these bonds, coupon > current yield > yield to maturity. This inequality is reversed for discount bonds (bonds selling below par value).

Equations for GBBOPk & GBEOPk:

GBBOPk =sum(Bj × (1+g)^(k−1−j)) GBEOPk = sum(Bj × (1+g)^(k−j))

Equation for GIEOPk

GIEOPk = sum( Ij × (1 + g)^(k−j))

Disadvantage of the Risk Adjusted DCF Model

Hard to determine beta

(Ferrari Interactions) U/P increasing leads to P/S increasing

Higher underwriting profit means that insurers can write more business.

Advantages and disadvantages to the opt risky portfolio of the single index model

However, the advantage of using the Optimal Risky Portfolio of the Single Index Model is that by using security analysis, he can hopefully identify assets with non zero alphas, and over/underweight these relative to the market in order to increase returns. The disadvantage of this is that it will introduce firm specific risk.

Calculating Invested Assets for Robin's IRR

IAj = Assetsj - Receivablesj Also Investment income = i x IA of j-1

IRR Questions. IF capital is allocated by line of business based on premium, the distribution back to equityholders will be at the end of the policy period. Then there may be a negative distribution to equity holders after the initial contribution is distributed if losses paid exceed the reserves

IF capital is allocated by line of business based on reserves, the distribution back to equityholders will be once all of the losses are paid

List a practical criticism of the IRR

If IRR >0, but less than the Cost of Capital, regulators may get the false impression that the insurer is profitable

Describe how companies use IRR to decide whether to undertake an investment.

If IRR >Opportunity Cost, project should be profitable

Yield to Call

If a bond is callable, the yield to call may be more relevant than the yield to maturity, especially if it is likely to be called. The same approach that is used to derive the yield to maturity can be used to calculate the yield to call, except that the time to maturity would be replaced by the time to call, and the par value would be replaced by the call price.

(information processing errors) Forecasting errors can lead to the P/E effect of a firm

If a firm has had recent strong performance, the forecasts of future earnings may be too high. This will lead to an overstated P/E, resulting in poorer subsequent performance once investors realize their error (P/E effect)

Problem of IRR for projects with unusual cash flows

If cash flow patterns change between inflow and outflow more than once, there may be two positive roots to the IRR

If rates are inadequate, insurers will: 1. Tighten underwriting standards 2. Reduce current volume Therefore the regulators can get an idea of whether or not the current rate of return is appropriate by examining the structure of the industry.

If insurers perceive that there is less opportunity to achieve a reasonable rate of return, regulators should be able to see the following occur in the market: 1. Increasing size of residual market 2. Reducing degree of product diversity 3. Reduced innovation.

Dealing with difficulty 1 of Event Studies: Leakage

If leakage of information occurs, prices may begin to move before the event. If this is the case, it may be more appropriate to measure the impact of the event by referring to the cumulative abnormal return, beginning at a time period before the actual event is made public.

Role of Portfolio Management in Efficient Market

If markets are indeed efficient, an argument can be made that portfolio management is not necessary, as stocks are fairly priced. However, portfolio management can actually still be beneficial. The text cites three uses of portfolio management: 1. Diversification: selects a diversification strategy to eliminate firm-specific risk 2. Reflects tax considerations of the individual investor 3. Adjusts portfolio to reflect the unique risk profile of the investor

When determining the optimal capital structure, one factor that actuaries have to account for is the volatility of the investment earnings stream

If the investment earnings stream is very volatile, insurers should reduce the leverage from reserves, in order to prevent the total risk from getting too large.

Describe a scenario in which an investor would prefer a callable bond to a regular bond.

If the investor believes that the interest rates are going to be stable or to increase, it would make sense to purchase the callable bond. There is little chance that the bond will be recalled. The investor will be able to earn the higher coupons.

Calculating Convexity of a bond

If the rates are compounded "k" times per annum: Convexity = [1/(P(1+y/k)^2)]∑[(CFt x n(n+1))/((k^2) x (1 + y/k)^n)] Where: -"y" is the annual rate, compounded k times per annum -"n" is the period at which each cash flow is made (eg if rates are compounded semiannually: "n=1" refers to the cash flow at time 6mnths, "n=2" refers to the cash flow at time 1yr, and so on). -P is the sum of the PV of the future cash flows

Describe the market conditions that would cause a risk-adjusted discount rate to be negative.

If the return on the market is negative enough (compared to the risk free rate), it could produce a negative risk adjusted discount rate.

Non normal returns and optimal portfolio allocation

If the returns are more heavy tailed than a normal distribution would imply, it may be more appropriate to reduce the allocation to the risky portfolio than you would under the assumption of normal returns

Protective covenants of bond indentures: Subordination of further debt

If there were no restrictions, the risk would be that after the investor purchases a bond, the firm borrows a huge amount of money (thus increasing the risk of financial difficulty). To protect the bondholder, subordination clauses exist, which: - restrict the amount of additional borrowing - require that the additional debt may be subordinated in priority to existing debt.

Immunization techniques are strategies used to protect the portfolio from interest rate fluctuations.

Immunization is the process where an investor creates a portfolio with duration equal to the investment horizon. In this case, the price risk and reinvestment risk will cancel out.

(Arbitrage Pricing Theory): Multifactor SML: Factor portfolios

In order to calculate the risk premiums for the risk factors, a factor portfolio can be used. This is a portfolio designed to have a β of 1 to the factor for which the risk premium is being measured, and a β of 0 on all other factors.

(Kreps Examples of Leverage Models) Risk neutral

In this case the Leverage is constant, and so the risk load is 0. According to Kreps this is appropriate if the range of x where f(x) is significant is small compared to the available capital.

(Kreps Examples of Leverage Models) Variance approach

In this case the whole distribution is relevant. L(x) = (Beta/S)(x-mu) Where Beta is a dimensionless constant variable for overall scaling

EURODOLLAR EXAMPLE: Consider a Eurodollar contract that has a quoted settlement price of F0 = 99.71. The way the contract works is that the participants in the contract would negotiate over the interest rate ("contract rate"), and the price would be set at "100 - contract rate"

In this case, because the futures price is 99.71, the contract rate must be 100 - 99.71 = 0.29%. The final futures price at the maturity date will be set as FT = 100 - LIBORT. The profit to the contract buyer would be: FT - F0 = (100 - LIBORT) - (100 - contract rate) = contract rate - LIBORT Assume that the contract multiplier is $1 million, and that it is based on a quarterly rate. Then for every 1% increase in LIBOR, the quarterly rate would increase by 0.25%, and the profit to the buyer would decrease by: 0.01 * 0.25 * 1M = $25

Homer and Liebowitz characterized portfolio rebalancing strategies into one of four types of bond swaps: Pure yield Pickup Swap

Increase returns by moving to a higher yield bond. If the yield curve is upward sloping, this could involve moving to a longer term bond. Note that this strategy does not exploit a perceived mispricing. It will subject the investor to interest rate risk.

Individual behavior assumptions of CAPM

Individual behavior assumptions are as follows: 1. Investors are rational mean-variance optimizers (investors are only concerned about mean and variance, and are not concerned about the correlation of the asset) returns with inflation/ prices of consumption items 2. Their planning horizon is a single period: (similar to above, longer periods would result in extra-market risk factors. i.e. a change in interest rates may decrease investor's income.) 3. Investors use identical input lists (due to the market assumption that all info is public)

Delineate the financial transactions that take place between the following counterparties to a CAT bond with a single-purpose reinsurer: • Insurer • Single-purpose reinsurer • Investors • Trust account

Insurer: -pays premiums to the SPR -receives funds from the SPR if a catastrophe occurs SPR: -sells a bond to investors, and invests the funds in a highly rated investment. -collects premiums from the insurer and will pay the insurer a sum of money if a catastrophe occurs -returns money (plus additional return) to the investors if catastrophe doesn't occur Investors: -provide the initial funds to the SPR (via purchase of the cat bonds) -receive interest payments from the SPR, and will receive principal at maturity if not catastrophe occurs Trust account: -SPR maintains the funds from the investors in this trust account until they need to be paid out

-Assume that a single factor market exists, where the well diversified portfolio, M, represents the market factor, F -Assume that we have identified a well diversified portfolio P, that has a positive alpha How would you construct a zero beta portfolio, Z from P and M.

Invest weight wp in P: βZ = wpβp + (1 - wp)βM = 0 -For this to be zero beta, wp = 1 / (1 - βp) wM = 1- wp = -βp / (1 - βp) Portfolio Z is riskless, as there is no residual risk (the component portfolios are well diversified), and the systematic risk is 0 (it was constructed to have zero beta). αZ = wpαp + (1 - wp)αM = wpαp Since the beta of Z is 0, E(RZ) = wpαp = αp / (1 - βp) The risk premium of Z must be 0, as Z is riskless. Otherwise it would be possible to make an arbitrage profit. For example: - If βp < 1, borrow money and invest the proceeds in Z, to make a risk free profit with zero net investment - Similarly if βp > 1, sell Z short, and invest the proceeds at the risk free rate

Insurers prefer to use an SPR when issuing CAT Bonds to capture the tax and accounting benefits associated with traditional reinsurance

Investors prefer SPRs to isolate the risk of their investment from the general business and insolvency risks of the insurer. Thus creating an investment that is a "pure play" in catastrophic risk.

International bonds are often divided into two categories: Foreign Bonds

Issued by a borrower from a country other than the one where the bond is sold. It is denominated in the currency of the country in which it is marketed (for example, a German firm issues a USD denominated bond in the US). These bonds include: -Yankee bonds: sold in US -Samurai bonds: sold in Japan -Bulldog bonds: sold in UK

Explain why the Marginal Variance method is not renewal additive

It double counts the covariance, and therefore overstates the risk load in renewal

Effective Duration

It is difficult to calculate the duration of callable bonds using the normal equations, as the future cash flows are unknown. Instead, an alternate measure of duration, the effective duration, can be used. Effective duration = -(ΔP/P) / Δr

Disadvantage of using Profit to rate an insurer, compared to using Rate of Return

It is difficult to use "Profit" to compare companies if we don't have other information

Problem with quantifying the Market Risk over a longer term period:

It is much more difficult to derive the parameters. Also the investment strategy is likely to change in response to market movements

Explain why the Marginal Surplus method is not renewal additive

It is sub additive, and therefore understates the risk load in renewal

Disadvantage of single index model

It oversimplifies the true uncertainty: it simply divides the uncertainty into micro vs macro risk. It ignores things like the correlation between security returns. It also ignores industry events: factors that impact many firms within an industry, without materially impacting the overall economy. In the case where several stocks have correlations not accounted for by the M factor, it may be more appropriate to use a multi index model Therefore if the universe of securities from which we will construct a portfolio is small, it is possible that the two models produce different optimal portfolios: the optimal portfolio constructed by the single model will be inferior, because of the correlations that it ignores.

List an advantage of the PVI/PVE method over the IRR approach:

It uses a market based rate as the reinvestment rate

Formulae for L(x) and R for SVaR (semi-variance) method:

L(x) = (β/S)(x−μ)Θ(x−μ) R = (β/S) x integral(dxf(x)(x-u)^2) from u to inf With this measure, the risk loads are only non-zero for results that are greater than the mean. The thought here is that risk is only relevant for the bad results.

Formula for L(x) for Proportional Excess:

L(x) = h(x)Θ[x−(u+∆)]/(x-u) here the allocation for any outcome is pro-rata to its contribution to the excess over the mean

Formula for L(x) TVaR method:

L(x) = Θ(x−xq )/(1−q) Where: • q is a management chosen percentage (e.g. q = 99%) • xq is the value of x where the cumulative distribution of X = q : [F(xq) = q] • Θ(x) is a step function: it equals 0 for a negative argument, and 1 for positive. So the Leverage ratio is 0 up to a point and then constant

Formula for L(x) for Mean Downside Deviation method:

L(x) = β Θ(x−μ)/(1−F (μ)) Keeps believes that this is the most natural naive measure, as it essentially assigns capital to bad outcomes depending on how bad they are.

Formula for L(x) VaR method:

L(x) = δ(x−xq )/f(xq) This function is 0 everywhere except close to 0 and it integrates to 1. Kreps proves that under this measure C = xq, therefore only xq is relevant. The shape of the loss distribution doesn't matter. Recall capital C = mu + R

Formulae for L(x) and R for Variance method:

L(x)= (β/S)(x−μ) R = (β/S) x integral(dxf(x)(x-u)^2) from 0 to inf R = (β/S) x integral(dF(xk-uk)) x sum(xj-u)

(Risk Linked Securities) Industry Loss Warranties (ILW)

Later in the paper, Cummins discusses the possibility that the risk linked securities may not be treated as reinsurance by regulators. An ILW is one type of security that does not suffer from this problem. This security has dual-triggers that need to be satisfied for the loss to be paid: 1. retention trigger: based on the incurred loss of the insurer 2. warranty trigger: based on an industry wide loss index ILWs have two categories of payments that can be made: 1. binary trigger: full payment is made once both triggers are satisfied 2. pro rata trigger: payoff depends on the magnitude by which the loss exceeds the warranty

Capital Requirements for single line vs. multiple independent lines

Less capital per line will be needed to achieve the same EPD ratio for an insurer that has two lines of business, than for an insurer that only has one line.

(Single-index model) regression equation for the premium Ri(t)

Letting the market index be M, the authors construct the regression equation for the premium, Ri(t): Ri(t) = αi + (βi)RM(t) + ei (t) Where α is the expected excess return of the security when the market excess return is 0. α will arise if the stock is underpriced. Taking the expected value of the above equation: E(Ri) = αi + (βi)E(RM) As can be seen above, the components of the risk premium include: - Systematic risk premium, (βi)xE(RM) - Nonmarket premium, αi

Macaulay's duration

Macaulay's duration is a weighted average of the lengths of time to future cash flows: D = Σ t wt Where: wt = [CFt / (1 + y)^t] / ∑[CFt / (1 + y)^t] CFt = Cash flows at time t This can be used to determine the impact to the bond price when the yield changes, using the following equation: ΔP/P = -D [Δ(1+y)/(1+y)]

List one issue with the IRR method:

Makes the assumption that the cash flows are reinvested at the IRR, which may differ from the market rate.

Goldfarb's risk measures are usually compared to a threshold level. There are several ways to derive this threshold: Management's risk preferences

Management's risk preferences: a threshold based on the risk tolerance of management. This also has disadvantages: -will be difficult to get management to articulate and agree on a threshold -the preferences of managers will often differ to the preferences of directors and shareholders -this does not factor in risk, which is an important consideration to compare to reward -managers will likely be focused on a number of issues that concern policyholders (eg ratings downgrade, weakened financial position), and it may be hard to isolate the estimate of the threshold to just the probability of default

Equilibrium

Market prices should change to eliminate arbitrage opportunities: an investor that identifies an arbitrage opportunity will take a large position in the security that will drive the price up or down until the opportunity disappears. Prices are said to have reached an equilibrium level.

Briefly describe the inconsistent time horizons that firms use when assessing the aggregate risk profile:

Market risk is often based on a one year period Insurance risks are based on the ultimate liability

To determine to the degree to which diversification benefits a firm, risk needs to be segmented into 2 categories:

Market/ Systematic/ Nondiversifiable Risk: the risk that can not be diversified away Unique/ Firm-Specific/ Nonsystematic/ Diversifiable: the portion of the risk that can be eliminated via diversification

Determinants of Bond Safety (Liquidity Ratios)

Measures the insurer's ability to pay bills with its most liquid assets. 1. Current ratio = current assets / current liabilities 2. Quick ratio = current assets excluding inventories / current liabilities

Modified Duration D*

Modified duration (D*), which is based on Macaulay's duration, is the measure of duration that is used in practice. D* = D / (1 + y) (Note that if the interest rates are continuous, Modified duration equals Macaulay's duration; D* = D). The equation showing the impact of changing yields on bond prices becomes: ΔP/P = -(D*)Δy

Disadvantage of Cash Flow Return Method

Not clear what type of profit is being measured, as cash flows do not have the same timing as GAAP income

Deferred callable bonds are similar to callable bonds, except that these have an initial period of call protection, which is a time period in which the bonds are not callable

Note that a regular callable bond selling at a deep discount from the call price would have implicit call protection, as it is very unlikely that the price will hit the par value in the near future.

Floating rate bonds provide interest payments that are based on the current market rates. There is therefore less interest rate risk: as interest rates rise, the increase in interest offsets the higher discounting rate

Note that an interest rate risk related to the change in the issuer's financial condition still exists because the yield spread is fixed: if the financial condition deteriorates, the investors would demand a greater yield premium than what the security offers.

(CAT bonds) Principle protected trenches

Note that some bonds have principal protected tranches, where the return of principal is guaranteed. If an event does occur, it only impacts the interest & spread payments, as well as the timing of the principal repayment of this tranche. These tranches are quite rare, as they do not provide as much protection to the sponsor.

When immunizing, we need to keep in mind that duration is not constant

Once the duration changes, the portfolio will no longer be fully immunized. In order to keep it immunized, it is necessary to rebalance the portfolio: rebalancing involves realigning the portfolio to produce a duration equal to the horizon length.

Mortgage-backed derivatives, which have been created from MBS, are available to help investors manage interest rate risk.

One example of a mortgaged-back derivative is a collateralized mortgage obligation (CMO). This segments the cash flow stream from MBS into different tranches. The different tranches have different levels of risk: for example the lower tranches have their principal repaid before the higher tranches. Therefore, the different tranches have different effective durations. (Investors can therefore use these investments to modify their portfolio duration).

Problems with ROE as measure of return: Regulating ROE is often a complicated method of regulating ROS

One method that regulators use to assign surplus is with target ratios of premium to surplus (eg looking at the premium of a specific category and then allocating surplus to that category based off the target ratio). The problem with this is that the ROE regulation actually becomes return on sales regulation. Regulating return on equity is just a complicated way of regulating return on sales (complicated because it is not direct, unlike focusing directly on sales).

Extensions of the CAPM: Zero-Beta CAPM

One of the interesting properties of efficient frontier portfolios is that each portfolio on the efficient frontier has a portfolio (the zero-beta portfolio) on the inefficient side of the minimum variance frontier with which it is uncorrelated. Use if investors face borrowing restrictions of the risk free asset: E(ri) = E(rZ) + βi[E(rM) - E(rZ)] where Z is the zero-beta portfolio for M. Investors who face borrowing restrictions will invest more heavily in high beta stocks, and less in low beta. The price of high beta stocks will therefore increase. The SML in this case will be smaller than the SML of the regular CAPM. The risk premium on the market portfolio is smaller, as the expected return on the zero beta portfolio will exceed the risk free rate. As a result, there is less reward for bearing risk.

Mortgagors often sell their mortgage loans to federal agencies, which then go on to combine several and resell as Mortgage Backed Securities (MBS).

One of the risks to investors of MBS is that the homeowner has the right to prepay the loan at any time. They may choose to do this if interest rates reduce, as they can take out a loan at a lower rate, and use the proceeds to pay off the original loan. MBS can therefore be viewed as being callable, just like the callable bonds that were discussed earlier.

Dissemination of Info on Bonds as an impediment to the risk linked securities market

One problem is that securities regulations discourage releasing information about private placements (which would include catastrophe bonds). There is therefore a lack of info about these bonds, impeding the market developments. Prospectuses of privately placed bonds can only be issued to "accredited investors", which mainly consists of institutional investors & high net worth individuals. This discourages research on the bonds. Cummins believes that the rules should be changed to allow the sponsors to distribute the prospectuses to researchers Cummins also suggests that regulators in certain jurisdictions mandate catastrophe loss reporting of events with industry losses exceeding a certain threshold. Another suggestion that he made was that regulators account for reinsurance credit quality in regulatory capital calculations. This will improve insurance solvency regulation, and also encourage the use of the insurance linked securities market.

Many people believe that the Expectations Theory is correct, and therefore we can use the term structure to determine the expectations of investors.

One problem with this approach is that in addition to expectations, the term structure may be impacted by other factors, like the liquidity premium. The impact of these additional factors will need to be removed. Despite the fact that the yield curve is affected by other factors, it can still provide signs about the future business cycle: - if the curve has a steep positive rise, it is still likely that the market expects an expansion in economic activity - if the curve is sloping downward, it is likely that interest rates are expected to decline A rough approach to derive the expected future spot rates would be to assume that the liquidity premiums are constant. There are two issues with this: - It is very difficult to accurately estimate the liquidity premium - There is no reason to believe that the liquidity premium is constant

Equation for P in the Risk Adjusted Discounted Cash Flow Model (RA DCF):

P = L/(1+rA) + X/(1+rf) + (TI x rf x (P-X+S))/(1+rf) + TUx(P-X)/(1+rf) - TUxL/(1+rA) Where: P = premium L = loss X = expense rA = risk adjusted rate < rf

Equation for Premium, as presented by Robbin

P = [(1+c)L+FX]/[ 1−(VR+U)]

Panning's calculations of Premium and Current Economic Value under simple assumptions

P = [S x (k - y) + L] / (1 + y) + E and C = S + P - E - L/ (1 + y) - P = premium - E = expenses - L = loss & LAE - y = risk free rate - S = surplus - k = target return on surplus - cr = client retention - F = franchise value - C = current economic value: PV of surplus & business already written

Briefly explain what the insurance exposure measures

P/S: the amount of surplus that backs each dollar of premium written.

Premium Formula used in Robbin's Present Value Offset Method of calculating UW profit provision

P=PVL+FX+VR * P+U0 * P + L*(1-PVx0) PVL = (Sum of discounted payment patterns for evaluated line) x (Loss Provision) PVx0 = Present value of losses for the reference line U0=UWP before adjusting for investment income

Alternate way of doing present value offset method to find Combined ratio from exam question

PLR = (PV(Losses) / PVx) / P U = 0.05 - PLR * (PVx0 - PVx) plug U into equation for P P = ((PV(Losses) / PVx) + FX) / (1 - VR - U) = ((PV(Losses) / PVx) + FX) / (1 - VR - [0.05 - PLR * (PVx0 - PVx)]) Then solve for P Use P to solve for PLR Use PLR to solve for U CR = 1 - U

Advantage of Cash Flow Return Method

PV of Cash Flows is how most people think about underwriting profits

Equation for PVI/PVE:

PVI/PVE = (1+rI)×sum( Ij wL^j)/[sum(Qj wQ^j)] This discounts income to the end of the first year

Investment for the Long Run

People believe that extending investments across time will reduce risk due to possible diversification benefits (time diversification). We can think of extending an investment horizon for an additional period to be the same as adding an additional risky asset to a pool. For example, is a 2yr investment safer than a 1yr investment? Holding the portfolio for the 2nd year simply results in more risk (risk pooling). Therefore this time diversification is not true diversification. In order to increase the Sharpe ratio, while still maintaining the same level of risk, the 2yr investor will need to halve the amount invested in the risky asset over the 2yrs.

Insurers are earning investment income on their total surplus. This investment income needs to be segmented into investment income from policyholder funds and shareholder funds.

Premium credit should only be given for the investment income of policyholders funds. The policyholders should not receive the benefit from the investment of surplus: - surplus is not owned by the policyholders, but rather the owners of the insurer - including surplus will penalize high surplus insurers, as they will need to charge lower premiums

How does the "PV Cash Flow Return Model" determine the Profit Provision

Produce a PV of total cash flow (discounted at investment ROR) equal to the PV of changes in equity (discounted at target ROR).

Equation to derive Qj (required GAAP equity) from Sj (required surplus):

Q0 = S0 + DAC Qj = Sj,forj = 1,2,...,n Where: DAC = Deferred acquisition costs (this will fall to 0 after a year)

Equation for RAROC

RAROC = Income/Risk-Adjusted Capital

Regulators focus on rate of return (as opposed to profit) when looking at the return of insurers.

Rate regulation is prospective and the methods described by McClenahan are meant to be applied on a prospective basis. If applied retrospectively, keep in mind that a single year of experience is not sufficient to assess the true profitability of the business.

Determinants of Bond Safety (Coverage Ratios)

Ratio of company earnings to fixed costs. Low/ falling ratios may signify possible cash flow difficulties. Examples include: 1. Times interest earned ratio = ratio of earnings before interest payments and taxes to interest obligations 2. Fixed charge coverage ratio = ratio of earnings to all fixed cash obligations (including lease payments, sinking fund payments (described below), and interest obligations)

Determinants of Bond Safety (Leverage Ratios)

Ratio of debt to equity. If this is too high, the firm may be unable to satisfy its obligations.

Return on Sales.

Regulating Return on Sales specifies an adequate profit margin as a percentage of premium. This is the same as the concept of markup, which is commonly used in other industries. Return on Sales is true rate regulation (as opposed to rate of return regulation) because it does not depend on the relationship between premium and equity. Mclenehan prefers Return on Sales, due to the Three major problems of Return on Equity. (other cards)

When deriving surplus required, some insurers rely only on true insurance risk. Give an example of a type of policy that will therefore have understated surplus required

Retro/ Excess/ LDD

Equation for Risk Load of x in buildup Marginal Surplus method where x is written first

Risk Load = (yz/(1+y))* [StDev(x)]

Equation for Risk Load of x in Renewal Marginal Surplus method

Risk Load = (yz/(1+y))* [StDev(x+y) - StDev(x)]

(End of a Goldfarb question) Decide whether the insurer should allocate its total estimated risk capital or its actual book value to the risk sources. Give two reasons for your decision.

Risk capital: -the book value is just the value of capital at a particular point in time. It is not a prospective value that is helpful in predicting performance. -actual capital is a volatile number that is based on past underwriting performance Alternate answer: Book capital: -Risk capital is highly dependant on the selected risk metric -Since book value is available in financial statements, it is more reliable and readily available

What discount rate should the opportunity cost be calculated at? Why?

Risk free rate: the policyholder is not exposed to insurer's investment risk

List some source of risk that may call for a generic management risk load:

Risk of not making plan/ Risk of serious deviation from plan/ Risk of not meeting the investor analysts expectations/ Risk of a downgrade from the rating agencies/ Risk of triggering a regulatory notice/ Risk of going into receivership/ Risk of not getting a bonus

Risk Sharing

Risk sharing involves taking a fixed amount of risk, and sharing it among several investors. This way, the investor can benefit from the higher Sharpe ratio achieved from risk pooling, without having to increase the exposure to risk.

State the form of Riskiness Leverage models:

Rk = integral(dF(xk-uk)L(x)); x=sum(xk) R = integral((xk-uk)L(x)) =integral(f(x)(xk-uk)L(x)dx) where f (x) is the density function for x

In one question: "Calculate the insurance leverage factor" Things were calculated differently

S = "statutory capital and surplus" + "Equity in the UEPR" R = Total Assets - S Leverage = (1+R/S)

The separation principle dictates that there are 2 steps in portfolio selection:

Selection of the optimal risky portfolio: this step is purely technical Allocation between risk free vs risky assets: this will depend on the level of risk aversion

Which of the methods discussed by Mango are renewal additive

Shapely Value/ Covariance Share

Forward Rates definition

Short rates that will apply during a future time period are subject to change, and therefore we can think of the them as being expectations of the actual rates that will apply during the future. These future expected short rates are referred to as "forward rates".

Disadvantage of the CY Investment Income Offset Procedure

Since CY results are retrospective, they may not be totally applicable to prospective ratemaking

(BKM 11) What are Event Studies and the difficulties associated with them?

Since price changes reflect new information, it is possible to determine the importance of an event by measuring the resulting price changes (this is known as an event study). There are two difficulties associated with these studies: 1. the stock price may respond to a wide range of economic news in addition to the specific event 2. information about the event may be leaked prior to the actual event.

Another method to calculate the duration of the franchise value is to weigh together the durations of the components of franchise value by their Present Value

So [PV(Premium)xD(Premium) + PV(Losses)xD(Losses)+PV(Expenses)xD(Expenses)]/(Sum of PV) PVxD is known as the dollar duration Assuming perpetual retention: PV(Premium) = P x (d/(1-d)) PV(Expense) = E x (d/(1-d)) PV(Loss) = L x (d/(1-d))

Tax Issues impeding the risk linked securities market

Some are concerned about the tax treatment of these securities. However, according to the experts, they do not pose tax problems on the sponsors. There are no income, corporate or other significant taxes in offshore jurisdictions that impact CAT bonds. Sponsors have been deducting the premium payments on the bonds for income tax purposes, which is consistent with the treatment of reinsurance premiums.

Protective covenants of bond indentures: Collateral

Some bonds may be supported by collateral. Examples of bonds and associated collateral include: 1. Mortgage bond: property 2. Collateral trust bond: other securities 3. Equipment obligation bond: equipment (this would typically be offered by firms like railroads that use equipment that is standard, so the equipment should be fairly liquid) General debenture bonds on the other hand are unsecured (no collateral exists). There is therefore more credit risk, the level of which depends on the earning power of the insurer. Due to the higher credit risk, these bonds would have to offer higher yields than general debentures.

Regulatory/ Accounting Issues impeding the risk linked securities market

Some people believe that due to regulations, CAT bonds have mainly been issued offshore, and this fact may threaten their attractiveness. However, issuing the bonds offshore has a number of advantages: 1. lower transaction costs 2. off-shore jurisdictions have demonstrated that they can perform very well in issuing and settling the securities Others are concerned about the uncertain regulatory/ accounting treatment of nonindemnity CAT bonds due to the basis risk and potential use as a speculative instrument. However, the market has come up with a number of different methods to circumvent these concerns: 1. base the payment on narrowly defined geographic indices. 2. dual-trigger contracts (where the insurer can not collect more than its net loss)

Balcarek warns the reader to be very careful when using Ferrari's formulae. In particular, the equation for reserves viewed as non-equity capital: T/S = (I/A) + (R/S)(I/A + U/R)

Some people may conclude from this that the premium volume must be expanded as long as I/A remains above U/R. This conclusion ignores the additional risk that arises from writing more premium. Ferrari's conclusions are derived by focusing on certain variables in the equation, and holding everything else constant. However, in reality, the variables do interact with eachother. Incorporating these interrelationships into the formulae would result in more accuracy

In the "PV Cash Flow Return Model", to what point in time are the cash flows discounted?

Start of the first year

Mutual Fund and analyst Performance: Stock Market Analysts

Stock market analysts typically work for brokers, and as a result, are usually positive in their assessments. Therefore, in order to assess their assessments, we need to look at either the relative strength of a recommendation compared to other analysts, or alternatively the change in recommendation of an analyst.

One argument supporting the lack of persistence in mutual fund returns is that funds that have good performance will attract new funds, driving the alpha down, due to the additional cost & complexity of managing the new funds.

Studies have also been conducted on bond funds. These indicate that the bond funds underperform the passive indexes by a magnitude roughly equal to expenses. Also, there is no persistence in performance. This would suggest that the bond market is efficient.

Determinants of Bond Safety (Altman Z)

Studies have been conducted to determine if financial ratios can be used to determine default risk. One of the best known studies is by Edward Altman, who derived the following equation to distinguish between failing and nonfailing firms: Z = 3.1 * (EBIT / Total Assets) + 1.0 * (Sales / Assets) + 0.42 * (Shareholder's equity / Total liabilities) + 0.85 * (Retained earnings / Total assets) + 0.72 * (Working capital / Total assets) Altman found that the following ranges apply: - Z < 1.23: vulnerability to bankruptcy - 1.23 < Z < 2.90: no clear relationship - Z > 2.90: safe

(Ferrari) Return on Equity Basic Equation

T/S = (I/A)(1+R/S)+(U/P)(P/S) Where: T = Total after-tax return to the insurer I = Investment gain or loss after tax A = Total Assets R = Reserves & other liabilities U = Underwriting profit after tax P = Premium S = Equity T/S represents ROE I/A represents ROA (1+R/S) is insurance leverage factor U/P is return on sales P/S is insurance exposure

Equation Target Econ Profit after calculating target RAROC when capital is committed for multiple periods

Target Econ Profit = C1 x Target Raroc Where C1 is the capital at beginning of period 1

Equation for target RAROC when capital is committed for multiple periods

Target RAROC = R[∑[Ci(1+r)^(-i)]/C1] Ci = Beginning risk capital for period i R = cost of risk capital (may be target return on capital in question) r = investment income

Technical Analysis is the search for predictable patterns of stock prices, which can be used to derive a profit from trading.

Technical analysts/ Chartists rely on a sluggish response of the stock price to fundamental factors, during which time the trend can be exploited. Examples of metrics include "resistance levels" (levels above prices should not rise) and "support levels" (levels below which prices should not fall). Technical analysis can only be successful if markets are not efficient.

The APT Portfolio Optimization in a Single Index Market (Treynor Black)

The APT indicates how to generate infinite profits if the risk premium of a well diversified portfolio differs from the SML. However, if the arbitrage portfolio is not perfectly well diversified, increasing the size of an investment to take advantage of an arbitrage opportunity will also increase the risk of the arbitrage position. Consider an investor who faces a single factor market, and who has identified an underpriced asset (or portfolio). The investor could adopt the Optimization Procedure discussed in BKM8 to construct an optimal risky portfolio. This procedure is said to follow a Treynor-Black (T-B) model. In that T-B model, if the residual risk of a portfolio was 0, the position in the portfolio would go to infinity. This procedure therefore has the same implication as APT: if there is no residual risk, you will take up to an infinite position. On the other hand, if residual risk does exist, the T-B model produces an optimal risky portfolio with a specific investment in the portfolio, which reflects the risk. APT however will ignore the level of residual risk. Therefore, the T-B has more flexibility than APT in reflecting the residual risk that may exist.

What is the most common problem with the Asset Liability Analysis conducted by most firms according to Panning

The Asset Liability analysis conducted by most firms does not account for the franchise value of the firm: the franchise value represents the economic value of the profits that will be earned in the future years from the future renewals of the current policies

Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is an equation that shows the relationship between the risk and expected return of a security in equilibrium. It assumes that all investors optimized their portfolios using the Markowitz procedure, where they construct an efficient frontier based on all available risky assets, and identify an efficient risky portfolio P. Each investor faces an identical investable universe, and also uses the same input list to construct the efficient frontier. As a result, they would select the same weights for each risky asset. In addition to this, because the market portfolio is made up of the identical risky portfolios, it too will have the same weights. Thus, each investor will hold the market portfolio. This means that the CAL will also be the CML.

When to use MP versus MR method of capital allocation

The MP is appropriate when adding businesses to the firm, while M-R is appropriate for the firm's normal operations.

Reason that IRR and NPV analyses may give different results for mutually exclusive projects

The NPV of different projects can vary based on different discount rate (cost of capital) assumptions. Projects that defer income to later periods will be less favorable under higher interest rate assumptions because the positive cash flows receive more discounting. Meanwhile comparing the IRR's of different projects will not depend on discount rate

Security Market Line

The SML graphs the relationship between β & E(r) According to CAPM, the expected return at β=1 is equal to rM. The slope of the SML is rM - rf. If an asset is fairly priced, it will lie on the SML. An underpriced asset (positive α) will lie above the SML.

Extensions to the CAPM: CAPM with Liquidity Adjustments

The Standard CAPM ignores liquidity costs of trading like the Bid-ask spread/ Price impact. In reality, the security price should be discounted to reflect any illiquidity. The size of this discount will increase as trading costs increase. However, this increase in discount will not be proportional to the increase in the trading costs, due to the clientele effect: frequent traders tend to hold more liquid assets, and long term traders tend to hold less liquid assets. Investors are also concerned about liquidity risk (risk of unanticipated changes in liquidity). They are therefore going to demand additional return if there is a possibility that the stock will lose liquidity at a time when it is needed. The CAPM can be adjusted to include a liquidity beta: this reflects the sensitivity of the return of the security to changes in market liquidity.

(Krep's Different forms of risk load formulae) expressing risk load as an integral over risk load density: integral(rld(x) dx) where rld(x)=f(x) (x-mu) L(x)

The advantage of this formula is that it shows which outcomes contribute most to the risk load

EPD Ratio with Continuous Probability Distributions

The appropriate formulae for the EPD when the distribution of losses or assets is continuous are as follows: DL = ∫{from A to ∞} (x - A)p(x)dx DA = ∫{from 0 to L} (L - y)q(y)dy

Issues with CAPM assumptions that make extensions of the CAPM necessary

The assumption about no restrictions on short sales is one of the most problematic. In reality, short positions are not as easy to take as long positions: 1. There is no cap on the liability of short positions. A large short position will require significant collateral 2. There is a limited supply of stocks that can be borrowed by short sellers. 3. Many investment companies are prohibited from short sales. In addition, several countries restrict short sales. Short sales are vital to prevent prices from rising to unsustainable levels. The most unrealistic assumptions of CAPM are that: 1. All assets trade 2. No transaction costs AND single period horizon These assumptions generate the major challenges to CAPM.

Contribution of Stock to the Market Portfolio Variance

The authors derive the formula for the contribution of an individual stock to the Market Portfolio variance. Assume that the portfolio consists of n stocks. We are focusing on one specific stock, "i". The contribution of a stock to the variance of the market portfolio is: = wi [w1Cov(r1,ri) + w2Cov(r2,ri) + ... + wiCov(ri,ri) + ... + wnCov(rn,ri)] We know that ∑wyry = rM, so we can simplify the equation above: = wi [Cov(rM,ri)]

Protective covenants of bond indentures: sinking funds

The bond issuer has to pay its par value at maturity. This is a large cash commitment, and therefore the issuer may have difficulty meeting this obligation. The issuer therefore needs to establish a sinking fund to spread this burden over several years. This may operate in one of two ways: 1. The firm may repurchase a portion of the outstanding bonds in the open market annually 2. The firm may purchase a fraction of the bonds at a special call price that is associated with the sinking fund provision. It has the option to purchase these at either the market price or the sinking fund price (whichever is lower). The bonds selected for the call would be chosen at random (in order to be fair to the bondholders).

Bodoff's alternate presentation of the vertical procedure integral shows how to describe the disutility or "pain" given a loss event x: Allocated Capital(x) = *r* f(x)[integral from 0 to x[1/(1-F(y))]dy]. where r = required % rate of return on capital

The cost of capital of an event given the loss event is then: *r*[integral from 0 to x[1/(1-F(y))]dy]. And the total cost given the event is the above equation plus x The disutility function given loss event x after taking into account it's premium's contribution to capital is: x + *(r/(1+r))*[integral from 0 to x[1/(1-F(y))]dy - x]

Protective covenants of bond indentures: Dividend Restrictions

The covenant would limit the dividends that the firm can pay. This would force the firm to retain assets rather than pay them out to the stockholders. An example of a typical restriction disallows dividend payments if the cumulative dividends paid out since inception exceeds the cumulative retained earnings plus proceeds from stock sales.

Swaps usually involve a swap dealer (e.g. a bank), who acts as a financial intermediary in the swap. This dealer is willing to assume the opposite position from each party of the swap.

The dealer would charge a bid-ask spread in order to earn a profit. It is however exposed to the credit risk that one of the parties to the swap may default

Preferred stocks are technically equity, but are often grouped with bonds. This is because preferred stocks promise to pay a specified stream of dividends (just like bonds promise to pay a stream of coupon payments).

The difference is that if the preferred stock issuer is unable to pay the dividend, it does not enter corporate bankruptcy, but rather the dividends owed cumulate (to be paid in the future).

Bodoffs interpretation of the statement that investors demand that the insurer hold capital based on the VaR (Value at Risk) at the 99th percentile:

The firm holds capital to pay for the loss level at the 99th percentile, but not for loss events that are greater or less than this level: capital is allocated only to the components that contribute to this particular scenario.

What is the alternate interpretation of the 99th percentile capital requirement that Bodoff provides:

The firm holds enough capital even for the 99th percentile loss. The key difference between this and the prior definition is that this also considers losses at lower percentiles.

(Markowitz Model) The minimum-variance frontier is made up of the portfolios that have the lowest variance for each level of expected return.

The global minimum-variance portfolio is the single portfolio with the lowest variance.

Ten thousand policy years of losses are simulated using the data above. Explain whether a company should or should not expect to achieve its target return on equity using the risk loads calculated by the marginal surplus method.

The insurer should not expect to achieve the target return, as the MS method understates the risk loads due to the subadditive nature of the standard deviations In the case of the MV method, the insurer should expect to overachieve its target return (assuming it still writes both accounts) as the MV double counts the variance, resulting in a risk load that is too high.

Opportunity cost from a policyholder viewpoint

The investment income from policyholder funds should be recognized in the premium calculations. This arises from the fact that premiums are paid by the policyholder before the losses and expenses are paid by the insurer. The policyholder is therefore potentially losing out on some investment income, and should receive credit for this.

Bond term structure example: -assume that the current 1yr spot rate is 5%, and the expected short rate for the following year is 6% -the investor has a time frame of 1yr -two options that the investor has are: 1. purchase a 1yr bond that pays out $1,000 at maturity. The price of the bond is $1,000/1.05 = $952.38 2. purchase a 2yr bond that pays out $1,060 at maturity. Sell this in 1yr. Based on the expected future spot rate of 6%, the bond should be worth $1,060/1.06 = $1,000 in one year. Purchase price now is $952.38

The investor is subject to risk under the second option. If the actual spot rate at time 1 exceeds 6%, the bond will be worth less than $1,000, and he therefore may have insufficient money. Since the 2yr bond would subject him to additional risk, it would only make sense to purchase it if it offers a higher expected return (initial price is less than $952.38) to compensate. This higher rate of return is known as the *liquidity premium*. This liquidity premium will cause the forward rate to exceed the future expected spot rate.

CAPM and the Academic World

The market portfolio upon which CAPM is based is impossible to construct. To test the SML equation [E(Ri) = βiRM], we can use a regression of the excess returns of a sample of stocks against their betas, over a period t: Ri,t = λ0 + λ1βi + λ2σ^2ei + ηi,t CAPM predicts: - λ0 = 0: the average alpha = 0 - λ1 = RM: the slope of the SML equals the market risk premium - λ2 = 0: σ^2ei doesn't earn a risk premium The beta and resid variance for each stock needs to be estimated from a time series of stock returns. The problem with this is that these parameters will be estimated with large errors, which may be correlated. As a result, there may be a downward bias in λ1, and an upward bias in λ0. We may therefore reject CAPM even if it is valid. To demonstrate this, Miller & Scholes simulated rates of returns that followed the CAPM equation, but the regression test still indicated that CAPM did not hold! Another issue is that alpha, beta and the residual covariance are likely to be time varying. Again, the regression techniques, which do not recognize this, are more likely to lead to a rejection of the model.

Bonds are usually sold in denominations of $1,000, but the bid/ ask prices are quoted as a percentage of par value.

The minimum price increment (tick size) for a bond is 1/128.

The mutual fund theorem

The mutual fund theorem states that if all investors would hold a common risky portfolio, they would not object if all the stocks in the market were replaced with shares of a single mutual fund holding the market portfolio.

In a portion invested in risky portfolio question, if y* is greater than 1

The optimal complete portfolio requires more invested in the optimal risky portfolio than the investor has, so the investor must borrow funds at the risk-free rate which will then be invested in the optimal risky portfolio.

optimal risky portfolio

The optimal risky portfolio is the available portfolio that has the highest Sharpe ratio: this would be the portfolio formed where the CAL is tangential to the portfolio opportunity set (the tangency portfolio). Note that the optimal risky portfolio consists of risky assets only (no risk free assets).

Why PV Offset method may have a much larger profit provision than calendar year investment income offset method

The part a traditional provision is made up of underwriting income and the investment gain from PHSF. We therefore need to exclude ALL of the investment gain from PHSF to derive the underwriting gain that is required. On the other hand, the part b provision says that the traditional provision is made up exclusively of underwriting gains, assuming that the line earns the investment income of the reference line. If it earns more or less income, the underwriting provision can be reduced/ increased respectively, to account for the difference. There is therefore a much smaller adjustment (reduction) for investment income in part b, resulting in a higher underwriting provision

Risk Tolerance & Asset Allocation

The portfolio that maximizes utility is the optimal portfolio, C. The optimal ratio, y*: y* = (E(rp) - rf)/A(σ of p)^2 (This is the amount to invest in the riskier asset)

Portfolio risk distribution vs number of stocks for the single index model

The portion of the portfolio risk from the firm-specific sources reduces as the number of stocks in the portfolio increases.

(Butsic) Insurance as a going Concern

The prior sections assumed that the insurer was in a run off situation: it did not consider the risk from policies becoming effective in the future. However, there is a lot of risk associated with the new policies. (In fact, rapid growth has caused several insolvencies in the past). Insurers can calibrate the capital need periodically to maintain a constant EPD ratio. Butsic introduces some more notation: C1 = Year end capital P = Added premium LP = Losses from added premium (assumed to be incurred at end of period, and paid at the end of the following period) r = Annual return on assets g = Annual change in the value of the liabilities The Assets and Losses as of the end of the period are: A1 = (A0 + P) x (1 + r) L1 = L0 x (1 + g) + LP Assume that the premium is related to the initial capital, based on a factor p: P = pC0 = pcL0 Assume that the incurred loss ratio is "b": LP = bP = bpcL0 Butsic mentions that the equation for c as of the end of the year is: c1 = c x (1 + p) + [1 + c x (1 + p)] x r - pcb - g Using the above equation, we can derive the capital, c, required to produce a specific expected value of c1. The equation above is based on three variables that make up most of the risk that the insurer faces: 1. Asset risk, r 2. Reserve risk, g 3. Incurred loss risk, b

Problems with Fama French

The problem with the Fama French and other models derived from empirical evidence is that none of the factors can be identified as hedging a specific significant source of uncertainty. It is possible that the patterns observed were purely due to chance. Fama & French however argue that these variables have successfully predicted returns in alternate time periods, as well as other markets internationally.

Profit on a Eurodollar contract

The profit on the contract is proportional to the difference between the LIBOR at the contract maturity and the LIBOR at the inception date.

For a bond The nominal rate of return is based on both the interest and the price appreciation.

The real return adjusts the nominal rate of return to be net of inflation.

Realized Compound Return vs Yield to Maturity

The realized compound return may differ from the Yield to Maturity in situations where the reinvestment rate differs from the Yield to Maturity. When reinvestment rates can change the investor will actually not earn the yield to maturity. Because the future interest rates are uncertain, the future reinvestment rate is unknown. The realized return cannot therefore be calculated in advance

No-Arbitrage Equation of the APT

The return on a well diversified portfolio is given by the following equation: E(RP) = αP + βPxE(RM) The action of arbitrageurs will send the α of a portfolio to 0. Therefore, E(RP) = βPxE(RM) This is known as the APT SML. It can be seen that the APT SML is the same as the CAPM SML, due to the no-arbitrage requirement of the APT Consider the following graph, which depicts the relationship between F and the return for 2 portfolios, A & B. A & B each have betas of 1, but have different expected returns: It is impossible for A & B to coexist, as this would produce an arbitrage opportunity: Portfolio A is clearly dominant to Portfolio B. An investor for example could sell short $1M of B, and buy $1M of A. This will produce a riskless payoff of: $1M * (0.1 + 1 * F) - $1M * (0.08 + 1 * F) = $1M * 0.02 = $20,000

Capital Market Line

The risky portfolio P can be determined by either an active or passive strategy: Active: determined with security analysis Passive: a benchmark portfolio is selected (therefore security analysis is unnecessary). There are certain advantages to using a passive approach: significantly cheaper than an active strategy Free rider benefit: because some investors are implementing the active strategy, mispricings should disappear. Therefore, most assets should be fairly priced at any given point in time. The Capital Market Line is simply the CAL that uses a passive portfolio as the risky portfolio.

Extensions of the CAPM: Multiperiod CAPM

The single period assumption of CAPM can also be relaxed. Instead assume that investors will optimize their consumption/ investments over their lifetime. The appropriate equation in this case is the Intertemporal CAPM (ICAPM). The authors first consider the simple case where: 1. the only type of risk is the uncertainty about portfolio returns 2. investment opportunities are constant over time They state that under this scenario, the ICAPM has the same equation as the single period CAPM. Assuming there are K sources of extramarket risk and K associated hedge portfolios, the ICAPM becomes: E(Ri) = βiME(RM) + ΣβikE(Rk) Where βik is the β on the kth hedge portfolio

(Arbitrage Pricing Theory): Single Factor Model

The single-factor model, introduced in BKM8, can be used to derive the return for a stock, based on its sensitivity to a single factor, F: ri = E(ri) + βiF + ei Where: F = deviation of a factor from its expected value βi = sensitivity of the security return to the factor

Define "renewal additivity"

The sum of renewal risk loads of each risk is equal to the risk load for the aggregate portfolio.

Terminal value given amount A rate R and n years compounded m times per annum

The terminal value (value at maturity) is: A * (1 + R/m)^(mn) A*e^(Rn)

yield curve (aka pure yield curve)

The yield curve is a graph that shows the relationship between the interest rate and the time to maturity. The curve is generally upward sloping, but can deviate from this. There are 2 theories that can explain the shape of the yield curve: - Expectations Hypothesis - Liquidity Preference

(Butsic) Market Inefficiency in relation to RBC Requirements

Theoretically, regulators should not need to set minimum capital requirements, as if markets are perfectly efficient, customers would adjust the premium that they are willing to pay to reflect the possibility that the claims will not be fully paid. In this case, insurers could potentially reduce the capital to target customers who prefer lower priced policies (at the cost of increased risk). However, realistically, most consumers are unable to properly assess the financial condition of the insurer. As a result, regulators need to impose solvency protection.

If one believed markets to be totally efficient, they also believe that research effort is not justified. This however is not correct...

There are many anomalies in prices that indicate that searching for underpriced securities can be profitable. That being said, the evidence does indicate that it is very unlikely that there is consistent superior investment strategy.

From an insurance perspective, we can reduce risk by selling shares of the insurer to investors. Assuming that the total risk per investor remains constant, the Sharpe ratio will rise as the number of policies written increases. There are some problems with this approach though:

There are some disadvantages of managing a very large firm. These disadvantages will put pressure on the profit margins The impact of any error when estimating the risk of the insured will be compounded over many policies

Sinking funds are designed to protect the bondholders, but can actually hurt them. This is because the firm would choose to buy back discount bonds at the market price, but would buy back the premium bonds at par.

There is a type of bond issue that does not require a sinking fund: the serial bond issue, in which the firm issues bonds with staggered maturity dates. A sinking fund is not as necessary here because the repayment burden is spread. The main disadvantage is that the bonds with different maturities are not interchangeable, thus reducing the liquidity of the issue.

(Risk Linked Securities) Sidecars

These are special purpose vehicles formed by insurers or reinsurers to provide additional capacity to write reinsurance. They usually accept retrocessions exclusively from a single reinsurer. The reinsurer will receive commissions for the premium ceded to the sidecar. Sidecars are mainly capitalized by private investors, but can be funded by insurers and reinsurers as well. Advantages include: 1. transactions are usually off-balance sheet, and therefore the sidecars can be used to improve the reinsurer's leverage 2. can be formed quickly with minimal documentation/ administration costs

Risk linked securities

These enable insurance risk to be transferred to the capital market, providing the insurance market with additional capacity: catastrophes that are large relative to the resources of insurers will often still be small relative to the size of capital markets.

Myers- Read method of capitol allocation

This (M-R) differs from the M-P method, because M-R looks at the impact of very small changes to the loss liabilities, as opposed to adding entire business units. It aims to equalize the marginal default values across lines of business. Another difference between M-R and M-P is that M-R allocates 100% of the capital. Therefore, M-R avoids the problem of how to deal with the excess capital.

Shapely Value

This Value is the average marginal variance from all different combinations in which a new account can be added to a portfolio Shapley Value for Y= Average(Var(Y)+(Var(X+Y)-Var(X))) Multiply Shapley Value by the multiplier to get the risk load. In a build up scenario the shapely of the FIRST account is just the variance The Shapely Value method allocates the mutual covariance equally between the accounts by splitting it up evenly.

One optimal complete portfolio question showed a reward to volatility that was implied to be for the complete portfolio but had no label. You were supposed to assume it was for the risky portfolio.

This assumption made it easier to find Volatility since they did not provide correlation info in the question.

Goldfarb risk sources: Insurance UW risk

This consists of 3 categories: 1. Loss reserves on prior policy years: potential adverse development 2. Underwriting Risk for the current policy year: 3. Property catastrophe risk

Efficient Market Hypothesis (EMH)

This dictates that stock prices should reflect all available information. New information relevant to the stock will cause the price to change. Since this new information is unpredictable, the stock price must move unpredictably. This process of unpredictable stock movements is called the random walk process.

(Krep's Different forms of risk load formulae) integral(dx f(x) r(x)) where r(x) = (x-mu)L(x)

This expresses the risk load as the probability weighted average of risk loads over outcomes of the total loss. This riskiness leverage factor would reflect the fact that all dollars are not equally risky. For instance those that trigger regulatory or analyst tests would be more risky.

Methods used to quantify Underwriting risk: Inference from Reserve Risk Models

This involves inferring the risk from new business based on the Reserve Risk Models. It's important to recognize that the Reserve Risk Models reflect the amount of risk conditional on information generated after the policy was written, and are therefore not totally applicable in this circumstance. Unconditional models need to be generated from these conditional models.

(Butsic) Probability of ruin

This is a commonly used diagnostic to determine the amount of capital necessary to provide the company with an adequate level of protection. This measure has the disadvantage that it does not account for the severity of a loss. Butsic prefers an alternate measure that does account for the severity, the expected policyholder deficit.

The dividends on preferred stocks are not considered tax deductible expenses to the issuer (unlike interest payments on bonds).

This is a disadvantage to the issuer. However, there is a tax advantage to the holder: only 30% of dividends received are taxable.

Extensions of the CAPM: Consumption Based CAPM

This is based on the assumption that in a period, investors need to allocate the current wealth between consumption today; and savings/ investment to support future consumption. The optimal mix would result in the utility from an additional dollar of consumption today equal to utility associated with the future consumption generated from the investment of that dollar. Investors will value the additional income from the savings more during tough economic times (with limited consumption opportunities). Therefore assets that have a positive covariance with consumption growth (those that have a higher payoff when consumption is already high) are viewed as being riskier. The equilibrium risk premium for these assets will be higher. We can construct the CCAPM: E(Ri) = βiCRPC Where Portfolio C is a "consumption tracking portfolio", the portfolio with the highest correlation with consumption growth. Note that unlike in CAPM, the beta of the market portfolio in CCAPM is not necessarily equal to 1. Instead, it can be substantially greater than 1. This model is fairly similar to CAPM. But it has the disadvantages that the consumption growth figures are published infrequently, and are measured with significant error. Despite this, empirical evidence indicates that this model is more successful at explaining returns than the standard CAPM

Insurance Leverage, much like financial leverage, increases the volatility of results

This is because Equity = Invested Assets - Reserve Liabilities, so with increased Reserves the insurer will have lower equity so Profit/Equity will be greater (more positive or more negative)

Bond Convexity: According to the equation ΔP/P = -(D*)Δy the graph of the change in bond price to change in yield should be a straight line. However, the true relationship is actually not linear.

This is because the equation is just an approximation that works better for small changes in yield. The actual graph of bond prices is convex. As a result, the approximation always understates the bond value: - it understates the increase when yields fall - it overstates the losses when yields rise. Investors like bonds with higher convexity, because as convexity increases, a bond appreciates more when yields fall, and depreciates less when yields rise. Accounting for this convexity will improve the accuracy of the approximation: ΔP/P = -(D*)Δy + 0.5Convexity(Δy)^2

The calculation of the opportunity cost (of the policyholder) should be made at the risk free rate (although the insurer is probably earning a return other than the risk free rate when investing the money).

This is because the policyholder is not exposed to any of the risk of the insurer's investment: if the insurer speculates and loses money, the policyholders do not have to provide for this shortfall.

Calculate the illiquidity premium for stock A based on the risk free rate, E(rM), E(rA), and BetaA

This is equal to E(rA) minus the expected return implied by the CAPM equation

Problems with using the new money after tax yield when calculating the profit with the present value offset method

This is not necessarily equal to the portfolio's actual yield, so it makes the provision harder to support.

The Yield to Maturity is the interest rate that produces a PV of bond payments equal to its price.

This is the average return that the investor would earn if he purchases the bond now and holds it till maturity. Note that if someone mentions "yield", they are most likely referring to the yield to maturity. Another measure of yield current yield = annual coupon / bond price

Goldfarb risk sources: Credit Risk

This is the risk of loss due to credit events (eg counterparty default, changes in counterparty rating etc). The firm can be impacted via several sources of exposure: > Marketable Securities/ Derivatives/ Swap Positions: the firm's investments in these securities may lose value from credit events. > Insured's Contingent Premiums & Deductibles Receivable: eg loss sensitive premium adjustments, deductibles, etc. The insurer is exposed to the risk that the insured may not pay the balance that it owes. > Reinsurance Recoveries: this is the most difficult to estimate, due to three unique aspects: -definition of "default": this should be modified to reflect the fact that if the reinsurer's credit rating gets downgraded below an investment grade level, it could enter a "death spiral" where it's policyholders try to end their contracts, resulting in severe liquidity problems. As a result, the reinsurer may settle claims for less than 100% of the full amount. -substantial contingent exposure: the reinsurance recoverable at a given point in time may increase in the future due to adverse loss development, thereby causing the credit risk exposure to increase as well -reinsurance credit risk is highly correlated with the underlying insurance risk Note that this also encompasses "partial default" (settling at less than full recovery, due to disputes with the reinsurer)

Goldfarb risk sources: Market risk

This is the risk to the firm's current investments from changes in market variables (eg equity indices, interest rates, foreign exchange, etc). This is typically calculated in the financial industry, where it is based on a very short horizon (10 days or less). We will however assume that the insurer will calculate it over a 1yr horizon, which will make it easier to aggregate with the other risks to which it is exposed.

Merton and Perold argue that a full allocation of capital will lead the firm to reject projects that would add to it's market value.

This is why some of the firms capital will be allocated to the corporate level rather than any of the firms lines of business under M-P method.

Part of an IRR question: At the beginning of the policy period $100,000 of capital must be committed and will be released at the end of year three

This means 100,000 goes into required surplus, not contribution by equityholders

Asset Liability Management

This measures the impact of changing interest rates on the economic value of a firm

The first publicly acknowledged total loss of principal from a CAT bond

This occurred in 2005 (although there may have been earlier losses that were not publicly announced). Kamp Re had to pay the principal due to losses from Hurricane Katrina. The "bright side" of this loss however was that settlement was smooth, demonstrating that CAT bonds can function as designed

On the Run Yield Curve

This plots the yield as a function of recently issued coupon bonds selling at or near par value.

Callable bonds are bonds that are issued with call provisions that allow the insurer to repurchase the bond at a specific call price before the maturity date. Issuers will typically repurchase the bonds if the interest rates fall, as they can replace them with new bonds that have lower coupons.

This process is called refunding. This option is disadvantageous to the bondholders as they potentially will have to forfeit the bond at a favorable time. To compensate the bondholders for this, the callable bonds need to offer higher coupons.

(Ferrari Interactions) P/S increasing leads to I/A decreasing

This property is being driven by two factors: 1. A higher portion of the surplus is from current business, and is therefore in the form of either cash or agent's balances. This can not be invested. 2. A higher ratio of P/S results in more risk to Owners Equity. To compensate, insurers will need to follow more conservative investment policies.

Problems with Multiperiod CAPM (ICAPM)

This scenario is not very realistic. Over the long run, additional sources of risk are likely to arise, which will have an impact on the ICAPM. In particular, two types of risk considered: Changes in the parameters that describe investment opportunities: 1. some assets have higher returns during periods where economic parameters change adversely. Since the returns on these assets will counter the adverse impact of the parameters on the remainder of the portfolio, investors will bid up the price of these assets, thus reducing the expected return. 2. Changes in the prices of consumption goods: similar to the point above, investors will bid up prices of assets that hedge the increase in prices of consumption goods. Both of the assets described above that will be bid up are known as "hedge portfolios".

Fundamental Analysis

This type of analysis uses the fundamentals of a firm to determine the appropriate price. Fundamental analysts try to gain insight into the future performance of the firm that is not yet reflected by the market. Fundamentals examined include: 1. Earnings & dividend prospects 2. Future interest rate expectations 3. Risks According to EMH, fundamental analysis will usually not work. The exception is that it may be successful for investors who have a unique insight.

(Arbitrage Pricing Theory): Fama French Three Factor Model

This uses firm characteristics that seem to be a proxy for exposure to systematic risk: characteristics that have done a reasonable job of predicting average returns. Rit = αi + βiMxRMt + βiSMBxSMBt + βiHMLxHMLt + eit where: SMB = Small Minus Big; the return of a portfolio of small stocks in excess of the return of a portfolio of large stocks HML = High Minus Low; the return of a portfolio of high book to market ratio stocks in excess of the return of a portfolio of low book to market ratio stocks The market index is included as well. The purpose of its inclusion is to account for the systematic risk from macroeconomic factors. Empirical data has indicated that firm size and book to market ratio have predicted deviations from CAPM indications. They therefore may be acting as proxies for other fundamental factors. For example, high book to market ratio stocks are more likely to be in financial distress, and small stocks may be more sensitive to changes in business conditions.

Loss Reserve Risk is impacted by three components of total risk

This will be impacted by the three components of total risk: 1. Process Risk: risk that actual results will deviate from their expected value due to the random variation inherent in the claim development process 2. Parameter Risk: risk that the actual expected value of the liability differs from the actuary's estimate of the expected value due to inaccurate parameters 3. Model Risk: risk that the actual expected value of the liability differs from the actuary's estimate of the expected value due to the use of the wrong model

Asset backed securities (ABS)

Those where income from a specified group of assets is used to service the debt. A common type of ABS is a mortgage backed security (MBS).

Tracking portfolios need to have the same Beta as the security

To achieve this beta, the tracking portfolio would need to have a levered position in the S&P500: 1) β units in the S&P500 2) -(β - 1 )in T bills This will have an alpha of 0, since it is made of just the S&P500 and T bills The manager would purchase the security, and short sell the tracking portfolio. The manager is therefore earning the 4% alpha, without being exposed to the market risk. Note that the residual risk, eP, does still apply.

Dealing with difficulty 1 of Event Studies: market movement

To segment the movement that is due to the event, we need to calculate a proxy for what the return would have been in absence of the event (the benchmark). The movement specifically due to the event, the abnormal return, can then be calculated: Abnormal return = Actual return - Benchmark return There are several methods to calculate the benchmark return. Examples include: 1. Market return 2. Return of the stock implied by CAPM 3. Return of the stock implied by index model (et is the abnormal return)

Treasury notes: original maturity between 1 to 10 years

Treasury bonds: original maturity between 10 to 30 years

Equation for Underwriting Profit Provision in terms of CR

U = 1 - CR

Equation for Adjusted Underwriting Profit Provision under CY Investment Income Offset Procedure

U=U0−iAFIT x PHSF

Zero Coupon Bonds & Treasury Strips

US Treasury Bills are short term zero coupon instruments. The Treasury also issues coupon bonds. Longer term zero coupon bonds can be formed from these Treasury bonds. The bond dealer who purchases the Treasury bond would request the Treasury to break down the coupons into independent securities (each security would be a claim to one of the payments from the original bond). Each of the new securities would be assigned a CUSIP number (a security identifier) by the Committee on Uniform Securities Identification Procedures. This program of stripping the coupons is called STRIPS (Separate Trading of Registered Interest and Principal of Securities). The new zero coupon securities are called Treasury STRIPS.

Equation for underwriting income during the jth accounting period:

Uj =EPj −ILj −IXj Where: U = Underwriting Gain EP = Earned Premium IL = Incurred Loss IX = Incurred Underwriting & General Expense

SWAP DEALER EXAMPLE: Consider Company A, which has issued a 7% coupon fixed rate bond that it wishes to convert into a synthetic floating rate debt. It may enter into a swap with a dealer, who would assume a position in the swap to pay LIBOR and receive the fixed rate. Assume that the dealer also enters into a swap with Company B, which has issued a floating rate bond tied to LIBOR that it wishes to convert to fixed rate debt. In the swap with Company B, the dealer would pay fixed rate and receive LIBOR. After aggregating these two positions, the dealer will be in a net position where it is immune to the interest rate changes.

Under this arrangement illustrated in the exhibit above, Company A's net payment = 0.07 + (LIBOR - 0.0695) = LIBOR + 0.0005 It has transformed its fixed rate debt into synthetic floating rate. Company B's net payment = LIBOR + (0.0705 - LIBOR) = 0.0705 It has transformed its floating rate debt into synthetic fixed rate Dealer Bid ask spread = (0.0705 - LIBOR) - (LIBOR - 0.0695) = 0.001

In PV Cash Flow Return Model, Cash Flow =

Underwriting Cash Flow + Investment Income - Tax

The CAS Statements mention that the underwriting profit & contingencies provision needs to provide for an appropriate after tax return.

Unfortunately, there is no common view of what defines "appropriate". This can be seen for example in the US, where different jurisdictions have different opinions about an excessive provision.

(Risk Linked Securities) Catastophic Equity Puts (Cat-E-Puts)

Unlike the previous instruments, Cat-E-Puts are options (not asset backed securities). The insurer purchases the Put from the writer, and in return, receives the right (option) to issue preferred stock to the writer at a specified price on the occurrence of a specified event. Advantages include: 1. the insurer will be able to raise equity after a catastrophe, when its stock price is likely to be depressed. 2. lower transaction costs than CAT bonds, as no need to form a SPR. Disadvantages include: 1. not collateralized, so exposes the insurer to credit risk 2. if the insurer issues preferred stock, the value of the existing shares will be diluted

Merton-Perold method of capitol allocation

Unlike the prior methods, this method recognizes the impact of diversification. It is based on the concept of risk capital, which is the smallest amount that needs to be invested to insure the value of a firm's net assets against a loss in value relative to the risk free investment of those net assets. To derive the required amount of risk capital, Cummins tweaks the EPD methodology discussed earlier: in this case, he is interested in the marginal impacts of a line on the capital required to achieve a certain EPD. One interesting observation here is that this Merton Perold method allocates less than 100% of the risk capital. One reason is that allocating all the capital would cause the firm to reject projects that would add to its market value.

Main Utility function

Utility is a measurement of "benefit" U = E(r) - 0.5Aσ2 where A is the investor's risk aversion This produces the certainty equivalent rate: the rate that a risk free investment would need to offer to provide the same level of utility as the investment being analyzed.

VaR for Capital Allocation

VaR analysis can be used to calculate the exceedence probability (ε), which is the probability that the actual losses will exceed expected losses plus allocated capital. P [Lossi > E(Lossi) + Ci] = ε i the appropriate amount of capital for each line is the amount which will equalize the exceedence probability among lines

Fully explain the differences between Value at Risk and Expected Policyholder Deficit, including a description of how these metrics may be used to drive decisions regarding the capital levels for a company

VaR calculates the loss associated with a given confidence level. The VaR analysis can be used to derive the exceedance probabilities: P [Lossi / E(Lossi)> 1 + Ci/E(Lossi)] = ε i Capital can then be allocated to produce equal exceedance probabilities among lines. -EPD calculates the expected value of the difference between the obligation owed to the claimant and the amount actually paid -this differs from VaR because it is more impacted by the potential severity of the tail losses -EPD can be used to allocate capital by ensuring that the EPD ratios are equal among lines

The total economic value is the sum of the current economic value and the franchise value.

We can consider this to be the same as the total market value (market capitalization):

Bodoff's derivative of the alternate presentation of the vertical procedure integral is: d/dx{AC(x)} = f(x)/(1-F(x)) + f'(x)[integral from 0 to x[1/(1-F(y))]dy].

We can understand this formula as saying that as the loss amount under x under consideration increases, 2 factors simultaneously affect the allocated capital: 1) The AC increases to the extent that the loss amount receives allocation from an additional layer of capital based upon conditional probability (f(x)/(1-F(x)) 2) The AC changes to the extent that the loss amount is less likely to occur and thus receives a lower allocation on the lower layers of capital (the rest of the formula) Two observations about these factors: 1) Usually f'(x) is negative so item 2 is usually negative 2) When dealing with the simulation of output of n discrete events, each discrete event has likelihood of 1/n and thus is equally likely; therefore the AC to each larger event increases only with respect to factor #1, whereas factor #2 will equal zero. (Factor 1 is like boom new layer, more AC. Factor 2 is like "this event being unlikely will reduce its AC")

Estimating the Single Index Model

We can use regressions to estimate the parameters of the Single Index Model. The resulting equation is called the Security Characteristic Line (SCL). Based on the regression output, we can test the hypothesis that alpha is 0. In order to reject this hypothesis, the magnitude of alpha would need to be large enough for it to be deemed economically significant. alpha would also need to be statistically significant. To reject this, we would require an absolute value of t greater than 2 Note that even if the alpha value is shown to be both economically & statistically significant, it is not appropriate to rely on it to forecast a future period, as alpha values do not persist over time. We can also test the hypothesis that beta equals 0

Impact of Diversification

We can use the single factor model to derive the return of a portfolio comprising of n stocks: rP = E(rP) + βPF + eP The authors study the impact on eP as n increases. They show that: σ^2(ep) = (1/n) σ^2(ei) As n increases, the variance of eP approaches 0. Based on this, and the knowledge that the expected value of eP is 0, we can conclude that after diversification, the realized value of eP will virtually be 0. The single factor model becomes: rP = E(rP) + βPF Because the nonfactor risk can be diversified away, the security will only receive a risk premium for the factor risk.

Convexity of callable bonds: The convexity of a callable bond is different to that of a regular bond, because the callability option places a ceiling on the price to which the bond can rise.

When the yield falls enough, the value of the bond is compressed to the call price. The shape of the curve at this region is said to have negative convexity. This negative convexity is unfavorable to investors, as an increase in interest rates produces a larger price decline than the price increase produced by an equivalent decrease in interest rates.

Briefly describe which of the two bonds above should be used to manage the interest rate risk of the liability that needs to be repaid. (This question includes mod D and convexity info for the two bonds)

Whichever bond has a modified duration that is closest to that of the liability

Mutual Fund and analyst Performance: Mutual Fund Managers

While there is no evidence that managers can consistently beat the market, there is some evidence that indicates better managers in a period tend to be better managers in following periods. The EMH tests can be modified to adjust for exposure to systematic risk factors. One option is to look at the risk adjusted returns (in excess of the CAPM prediction). However, the market index may not be an adequate benchmark: the managers may have significant holdings in small firms, whereas the market index would be dominated by large firms. A common benchmark used today is a 4 factor model, that uses the 3 Fama French factors, in addition to a momentum factor (a portfolio that is constructed dependant on the prior year's return). Again, the funds have not been shown to exceed this benchmark.

"Calculate the amounts that should be invested in each of Security A and Security B in order to immunize the obligation from interest rate fluctuations."

With these questions you need to find the actual dollar amount to be invested in each, not just the percentage of the portfolio in each asset. You need to take (Percentage) x (PV(Liabilities)).

If you believe in the liquidity preference theory and you believe the liquidity premium is 1% what is the expectations of the price that a coupon bond will sell for in one year?

[(1+s2)^2]/(1+s1) - 1% = (1+ true expected rate in the future) Price of bond = (face value + coupon)/(1+true expected rate in the future) Where si is the expected spot rate on an i year zero The 1% is whatever the liquidity premium is for the problem

Duration of a Par Value Annual Coupon Bond with unknown Coupon Rate for n years

[(1+y)/y] [1-(1/(1+y))^n]

(CAPM) In equilibrium, all securities should have the same reward-to-risk ratio, and therefore the reward-to-risk ratio for an individual stock should equal the market price of risk:

[E(ri) - rf] / Cov(ri, rM) = [E(rM) - rf] / σM^2 E(ri) = rf + βi[E(rM) - rf] Where βi = [Cov(ri, rM) / σM^2] We can use the above formula to prove that βM = 1: βM = [Cov(rM, rM) / σM^2] = [σM^2 / σM^2] = 1

Ratio of "PHSF on UEPR" to "Earned Premium"

[UEPR(1 - Prepaid expense %) - Premium Receivable] / EP

Alternate form of Franchise Value

[cr x S x (a+(b-1)y)] / [(1+y) x (1+y-cr)]

Equation for Multiplier in the Marginal Variance method

[yz/(1+y)]/[SD(L+n)]

single-index model

a version of the single-factor model, where the return on an index (eg S&P500) is used as a proxy for the common factor (m)

asset allocation vs security selection

asset allocation: the allocation of a complete portfolio to the various asset categories. Within each category of assets, investors can select specific securities in order to try increase return: this is known as security selection.

conditional betas

betas that are conditional on the state of the economy

Methods used to quantify Underwriting risk: Frequency & Severity Models

construct an aggregate distribution based on separate frequency and severity models. This requires sufficiently detailed claim data. It has a few advantages compared to the Loss Reserve Distribution Models: 1. It is easier to account for growth in the volume of business 2. Inflation can be more accurately reflected 3. Changes in limit and deductibles can be more easily reflected 4. The impact of deductibles on frequency can be accounted for 5. The treatment of the split of loss between insured, insurer & reinsurer can be mutually consistent Several different methods can be used to generate the aggregate distribution: 1. Analytical Solution: generated based on the frequency & severity parameters 2. Numerical Method: numerical approximation 3. Approximation: based on the mean, variance (and possibly higher moments) of the collective risk model 4. Simulation: allows for complex policy structures with minimal mathematical complexity, although a large number of iterations are necessary to produce stable results

Homer and Liebowitz characterized portfolio rebalancing strategies into one of four types of bond swaps: Intermarket Spread Swap

could be pursued if the investor believes that the yield spread between two sectors is temporarily out of line. For example, if the yield spread between a 10yr Treasury & a 10yr Baa corporate bond is 3%, but the historical spread was only 2%, the investor may want to sell Treasuries and purchase corporates. If the spread narrows, the corporates will outperform the Treasuries. However, it is important to first investigate whether there is a reason behind the increase in the yield spread (eg the default premium of corporate bonds may have increased because the market expects a severe recession).

EPD Ratio for a Normal Dist:

dL = kφ(-c/k) - cΦ(-c/k) dA = [1/(1-cA)]x[kAφ(-cA/kA) - cAΦ(-cA/kA)] Where: k = CV of losses kA = CV of assets c = capital / loss ratio cA = capital / asset ratio Φ( ) = cumulative standard normal distribution φ( ) = standard normal density function Butsic mentions that for the loss risk and asset risk, the probabilities of ruin are Φ(-c/k) and Φ(-cA/kA) respectively.

EPD Ratio for a lognormal Dist:

dL = Φ(a) - (1 + c)Φ(a - k) dA = Φ(b) - Φ(b - kA)/(1 - cA) Where: a = (k/2) - (ln(1 + c)/k) b = (kA/2) - (ln(1 - cA)/kA) Butsic mentions that the capital requirement for losses under the lognormal distribution is higher than the requirement of a normal distribution, where the difference increases as the coefficient of variation increases. This is because the probability of large losses is higher under lognormal. If assets follow a lognormal distribution, less capital is required than under the normal case, because under lognormal assets can not be negative.

Homer and Liebowitz characterized portfolio rebalancing strategies into one of four types of bond swaps: Rate Anticipation Swap

eg move to longer duration bonds if you forecast that rates will decrease

When deriving surplus required, some insurers do not distinguish between policy form. Give an example of a type of policy that will therefore have understated surplus required

excess

Homer and Liebowitz characterized portfolio rebalancing strategies into one of four types of bond swaps: Substitution Swap

exchange of a bond for an almost identical substitute (same coupon, maturity, quality, call features, etc). The reason to engage in this is due to a belief that the market has temporarily mispriced the two bonds. For example, a 20yr 6% coupon Toyota bond may be at a value that would provide a yield to maturity of 6.05%. A 20yr 6% coupon Honda bond may yield 6.15%. Assuming that the two bonds have the same credit rating, investors may believe that the Honda bond is more attractive. However, the higher return may exist because the Honda bond is actually riskier.

Single Factor Model

first consider the return of a security. This return can be divided into 2 pieces: an expected and an uncertain component: ri = E(ri) + ei where ei has a mean of 0 and a standard deviation of σi. Next, assume that the security returns are joint normally distributed (being driven by one factor, "m"). We can modify the above equation to assume that the uncertainty comes from two sources: - Uncertainty about m. This influences all securities. This factor m generates the correlation across securities. The following equations assume that the sensitivity of a stock to m is βi (this recognizes that some firms are more sensitive to changes in the economy). - Firm specific uncertainty (ei) m & ei are uncorrelated (ei is independent to shocks that impact the entire economy). This produces the single-factor model: ri = E(ri) + βim + ei where m has a mean of 0 and a standard deviation of σm. The equations for var and cov of the securities are derived from the single factor model equation: σi^2 = (βi^2)x(σm^2) + (σ^2)x(ei) Cov(ri, rj) = Cov(βim + ei, βjm + ej) = βiβj(σm)^2

(Ferrari Interactions) U/P increasing leads to I/A increasing

higher underwriting profit means that the insurer can engage in more aggressive investments.

Provide the Horizontal procedure equation for capital allocation:

integral( from 0 to V aR(99%) ){integral(from y to inf){f(x)/[(1-F(y))]dxdy}}

Provide the Vertical procedure equation for capital allocation:

integral( from x(0%) to inf ){integral(from 0 to min(x,VaR(99%)){[f(x)/(1-F(y))]dydx}}

Investors in CAT bonds are insulated from credit risk because the bonds are fully colateralized with the collateral held in trust

jsky

(Homer and Liebowitz four swaps) In the intermarket spread swap and substition swap the investor believes that the relationship is only temporarily out of alignment. The period of realignment is called the workout period.

jsyk

A put bond is similar to a callable bond, but gives the option to "retire" the bond to the bondholder.

jsyk

Another type of swap that can be used (outside of Homer and Liebowitz's four) is a tax swap, which is a swap that exploits a tax advantage. For example, the investor can swap from a bond that has decreased in price in to receive the tax benefit from the capital loss.

jsyk

As investors become more risk averse their utility curves become more steep. They require a greater amount of expected return for additional levels of risk.

jsyk

If a bond question does not include information about compounding frequency you should assume it compounds semi-annually

jsyk

If all bonds are perfectly correlated then the probability of default for all tranches is the probability of default for just one bond (or any other asset)

jsyk

If calculating the combined ratio, you want to use undiscounted loss amounts.

jsyk

Most bonds are traded over-the-counter (ie not traded in a formal exchange like the NYSE).

jsyk

Note that weak and semistrong form EMH tests require the level of risk to be reflected. As a result, if a particular portfolio is shown to generate higher returns, this could actually be caused by an inappropriate risk adjustment technique, as opposed to inefficient markets. In fact, the risk adjustment techniques are based on more questionable assumptions than the EMH.

jsyk

On a Ferrari question: when talking about increase volatility make sure to mention the potential increase in earnings as well as the increased discount rate.

jsyk

The duration of the total economic value is a PV weighted average of the franchise value, and the current economic value durations

jsyk

The multi-year terms of most CAT bonds allow sponsors to spread the fixed costs of issuing the bonds over a multi-year period, reducing costs on an annualized basis

jsyk

The principal advantage of ILWs are that they are treated as reinsurance for regulatory purposes

jsyk

The process of separating the search for alpha from the choice of market exposure is known as alpha transport.

jsyk

To convert a fixed rate portfolio to a floating rate portfolio, a portfolio manager would need to enter into a "pay fixed/ receive floating" swap.

jsyk

Under the liquidity preference theory, investors will tend to choose shorter term investments if the yield is the same as the larger term yield. For investor liabilities, the opposite is true; borrowers prefer longer terms if yields are the same.

jsyk

Efficient Diversification

lowest risk level for a given return

Marginal Variance Method of calculating a risk load

r = λ * Marginal Variance Where λ = [yz/(1+y)]/[ SD(L + n) ] = λ [Var(L + n) - Var(L)] = λ [Var(L) + Var(n) + 2Cov(L,n) - Var(L)] = λ [Var(n) + 2Cov(L,n)] The term [Var(n) + 2Cov(L,n)] will be (VAR (X+Y) - VAR(X)) in a renewal scenario and VAR(X) in a buildup scenario assuming X is the first account. If not first it will always be (VAR (X+Y) - VAR(X))

When deriving surplus required, some insurers do not distinguish between policy form. Give an example of a type of policy that will therefore have overstated surplus required

retro

Equation to determine the capital via the Myers Read equation

si = s - ( dp/ds )^(-1)( dp/dσ)[(σiL - σL^2) - (σiV - σLV )]/σ

Equation to derive the IRR on Equity Flows:

sum(Fj×(1+y)^−j) =0

Goldfarb's in depth RAROC equation

target RAROC = [(Original Premium + Risk Margin - Expenses) * (1 + Expected Invested Income) - PV Expected Claims] / Allocated Capital

Optimum Capital Structure

the mix of Owners' Equity & Liabilities which maximizes the value of the firm Two factors which affect the value of the firm are: 1. Expected earnings stream 2. Rate at which this stream is discounted by the market The mix of Owners Equity & Liabilities affect both of these factors are therefore has an impact on the value of the firm:

Determinants of Bond Safety (Cash flow to debt Ratio)

total cash flow / outstanding debt

Equation to derive the weights of the optimal risky portfolio

wD = [E(RD)σE^2 - E(RE)Cov(rD, rE)]/[E(RD)(σE)^2 + E(RE)(σD)^2 - [E(RD) + E(RE)]Cov(rD, rE)] Note that the equation refers to excess returns (excess over the risk free rate), "R", instead of the actual returns, "r".

The minimum variance portfolio is the portfolio with the lowest variance that can be constructed from assets with a certain level of correlation.

wMIN(D) = [(σE)^2 - Cov(rD, rE)]/[(σE)^2 + (σD)^2 - 2Cov(rD, rE)] As long as ρ < σD/σE, the minimum variance portfolio will consist of both bonds and stocks. If ρ > σD/σE, the minimum variance portfolio will consist exclusively of bonds (the lower variance asset).

(single index model) The equation for portfolio variance (assuming a portfolio of equally weighted securities):

σP^2 = βP^2 + σm^2 + σ^2(eP) where σ^2(eP) = (1/n)σ¯^2(e) As the equation above shows, the firm specific risk approaches 0 as the number of securities increase.

Advantages of capital allocation by percentile layer:

• Allocates capital to the entire range of loss events (rather than just the most extreme events) • Tends to allocate more capital to events that are more likely • Tends to allocate significantly more capital to events that are more severe • Eliminates the need to select an arbitrary percentile level to use as a basis for allocating capital, as it is based on all relevant percentile thresholds • Always allocates 100% of the capital • Provides a method to allocate capital by layer

Regulators preferences for the riskiness leverage:

• Be 0 until the capital is seriously impacted • Not decrease (for excess that significantly exceeds capital) because of the risk to the state guaranty fund

Managements desired properties of the riskiness leverage ratio:

• Be a down side measure • Be roughly constant for excess that is small compared to capital • Become much larger for excess that significantly impacts capital • Reduces to 0 (or at least doesnt increase) for excess that significantly exceeds capital

Reasons why most CAT bonds are issued offshore

• Favorable regulatory treatment • Low issuance cost and high levels of expertise in the issuance of risk-linked securities • Effective handling of issuance and settlement • Non-indemnity CAT bonds are not treated as reinsurance • Insurance regulation in the United Stated seems generally inflexible and intrusive compared to other jurisdictions

What is the difference between "IRR" and "Opportunity Cost"

• IRR: rate which sets the NPV of cash flows to zero. • Opportunity Cost: the investment return that the providers of capital could earn from an alternate investment

List the desirable qualities for an allocatable risk load formula:

• It can be able to be allocated down to any level • The risk load of any sum of random variables should equal the sum of the risk loads allocated individually • The same additive formula can be used to calculate the risk load for any subgroup or group of groups

Explain how the Marginal Variance, Shapely Value and Covariance Share methods allocate the mutual covariance upon renewal

• Regular Marginal Variance: allocates the entire mutual covariance to each account • Shapely: allocates an equal amount of the covariance to each account • Covariance Share: allocates a user specified portion of the covariance to each account

List 2 goals that Robbin has for prices:

• Should be consistent & sensitive to risk. • Should reflect managements risk return preferences. To achieve this, the model is based on the theoretical amount of surplus as opposed to the actual level.

(Mango Game Theory) Factors to consider when deciding how much to allocate to a member of the group:

• Stability/ incentive to split from the group • Bargaining power • Marginal impact to the groups characteristic function value

Difference between a sub-additive and super-additive characteristic function: (game theory)

• Sub-additive: Σv(Xi) > v (Xi elements in unison) • Super-additive: Σv(Xi) < v (Xi elements in unison)