2. Risk Management: Measuring Risk (Market Risk)

Correlation

*Correlation is a source of risk for certain types of options—for example, options on more than one underlying (when the correlations between the underlyings' returns constitute a risk variable).

HISTORICAL METHOD: • Actual daily prices from user specified period f. Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR and discuss the advantages and disadvantages of each*

Advantages: NON-PARAMETRIC (involves minimum probability assumptions) which means the user does not have to make assumptions about the type of the probability distributions (normal distribution, log normal distribution etc) Disadvantages: Relying completely on the events of the past • However this flaw is also present in the analytical method and Monte Carlo methods

VAR: • Advantages: g. *Discuss advantages and limitations of VAR and its extensions, including cash flow at risk, earnings at risk, and tail value at risk*

Advantages: o Quantify the potential loss in simple terms & easily understood by senior management o Regulatory bodies use VAR as a risk measure and require firms to use it in their reports o Versatility: many use VAR as a measure of capital at risk. Also can be used in risk budgeting process to allocate capital

The analytical or variance-covariance method f. *Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR*

Assumes that portfolio returns are normally distributed. Example: 5% VAR for a portfolio (i.e., VAR at a probability of 0.05), we would estimate its expected return and subtract 1.65 times its estimated standard deviation of returns

CASH FLOW AT RISK (CFAR) Earnings At Risk (Ear) g. *Discuss advantages and limitations of VAR and its extensions, including cash flow at risk, earnings at risk, and tail value at risk*

CFAR is the minimum cash flow loss that we expect to be exceeded with a given probability over a specified time period. EAR is defined analogously to CFAR but measures risk to accounting earnings *CFAR and EAR can be used when a company (or portfolio of assets) generates cash flows or profits but cannot be readily valued in a publicly traded market, or when the analyst's focus is on the risk to cash flow and earnings, for example, in a valuation. CFAR and EAR can complement VAR's perspective on risk.

5.2.3. The Historical Method Also sometimes called the HISTORICAL SIMULATION METHOD This term is somewhat misleading because the approach involves not a simulation of the past returns but rather what actually happened in the past f. *Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR*

Calculate returns for a given portfolio using actual daily prices from a user-specified period in the recent past, graphing these returns into a histogram. Example: The year examined here contains 248 returns. Having 5% of the returns in the distribution's lower tail would mean that about 12 return observations should be less than the VAR estimate

SECOND-ORDER MEASURES

Deal with the change in the price sensitivity of a financial instrument and include convexity for fixed-income portfolios and gamma for options. CONVEXITY measures how interest rate sensitivity changes with changes in interest rates. GAMMA measures the delta's sensitivity to a change in the underlying's value. Delta and gamma together capture first- and second-order effects of a change in the underlying. For options, TWO OTHER MAJOR FACTORS determine price: volatility and time to expiration, both first-order or primary effects.

Another method for scenario analysis scenarios based on HYPOTHETICAL EVENTS

Events that have NEVER HAPPENED in the markets or market outcomes to which we attach a small probability. These types of scenarios are very difficult to analyze and may generate confusing outcomes, so it is important to carefully craft hypothetical analyses if they are to generate information that adds value to the risk management processes.

PRIMARY SOURCES OF RISK first

For a stock or stock portfolio, BETA measures sensitivity to market movements and is a linear risk measure. For bonds, DURATION measures the sensitivity of a bond or bond portfolio to a small parallel shift in the yield curve and is a linear measure, as is DELTA for options, which measures an option's sensitivity to a small change in the value of its underlying. These measures all reflect the expected change in price of a financial instrument for a unit change in the value of another instrument.

Example of converting between time periods

For example, if the daily VAR is estimated at $100,000, the annual VAR will be This simple conversion of a shorter-term VAR to a longer-term VAR (or vice versa) does not work, however, if the average return is NOT ZERO. In these cases, one would have to convert the average return and standard deviation to the different time period and compute the VAR from the adjusted average and standard deviation.

5.5.2. Stressing Models

Given the difficulty in estimating the sensitivities of a portfolio's instruments to the scenarios we might design, ANOTHER APPROACH might be to use an existing model and apply shocks and perturbations to the model inputs in some mechanical way. This approach might be considered more scientific because it emphasizes a range of possibilities rather than a single set of scenarios, but it will be more computationally demanding

Discuss with Baba....

In addition, instruments such as bonds and most derivatives behave differently at different times in their lives, and any accurate historical VAR calculation must take this into account by adjusting current bond/derivative pricing parameters to simulate their current characteristics across the period of analysis. For example, a historical VAR calculation that goes back one year for a portfolio that contains bonds that mature in the year 2027 should actually use otherwise identical bonds maturing in 2026 as proxies; these bonds are the most accurate representations of the current risk profile because they would have presented themselves one year ago in time. When a company uses a different portfolio composition to calculate its historical VAR than the one it actually had in the past, it may be more appropriate to call the method a historical simulation.

Volatility for indexing

In some applications, such as INDEXING, volatility relative to a benchmark is paramount. In those cases, our focus should be on the volatility of the deviation of a portfolio's returns in excess of a stated benchmark portfolio's returns, known as ACTIVE RISK, TRACKING RISK, tracking error volatility, or by some simply as tracking error.

5.1. Measuring Market Risk

MARKET RISK refers to the exposure associated with actively traded financial instruments, typically those whose prices are exposed to the changes in interest rates, exchange rates, equity prices, commodity prices, or some combination thereof. The most common tool to MEASURE market risk is: Standard Deviation or Volatility. Volatility is often an adequate description of portfolio risk, particularly for those portfolios composed of instruments with LINEAR PAYOFFS.

5.5.2. Stressing Models: other models MAXIMUM LOSS OPTIMIZATION WORST-CASE SCENARIO ANALYSIS

MAXIMUM LOSS OPTIMIZATION: In which we would try to optimize mathematically the risk variable that will produce the maximum loss WORST-CASE SCENARIO ANALYSIS: in which we can examine the worst case that we actually expect to occur.

5.5. Stress Testing h. *Compare alternative types of stress testing and discuss advantages and disadvantages of each*

Managers often use stress testing to SUPPLEMENT VAR as a risk measure. The MAIN PURPOSE OF VAR analysis is to quantify potential losses under normal market conditions. Stress testing, by comparison, SEEKS TO IDENTIFY UNUSUAL CIRCUMSTANCES THAT COULD LEAD TO LOSSES IN EXCESS OF THOSE TYPICALLY EXPECTED. Two broad approaches exist in stress testing: 1. SCENARIO ANALYSIS 2. STRESSING MODELS

LEPTOKURTOSIS Distributions can deviate from normality because of SKEWNESS and KURTOSIS.

Many observed distributions of returns have an ABNORMALLY LARGE NUMBER OF EXTREME EVENTS. This quality is referred to in statistical parlance as LEPTOKURTOSIS but is more commonly called the property of FAT TAILS Using a normality assumption to estimate VAR for a portfolio that features fat tails could understate the actual magnitude and frequency of large losses. VAR would then fail at precisely what it is supposed to do: measure the risk associated with large losses.

5.4. Extensions and Supplements to VAR INCREMENTAL VAR g. *Discuss advantages and limitations of VAR and its extensions, including cash flow at risk, earnings at risk, and tail value at risk*

Measures the incremental effect of an asset on the VAR of a portfolio by measuring the difference between the portfolio's VAR while including a specified asset and the portfolio's VAR with that asset eliminated. We can also use IVAR to assess the incremental effect of a subdivision on an enterprise's overall VAR. Although IVAR gives an extremely limited picture of the asset's or portfolio's contribution to risk, it nonetheless provides useful information about how adding the asset will affect the portfolio's overall risk as reflected in its VAR

Theta

Option prices are also sensitive to changes in time to expiration, as measured by THETA, the change in price of an option associated with a one-day reduction in its time to expiration. Theta, like vega, is a risk that is associated exclusively with options.

5.2.5. "Surplus at Risk": VAR as It Applies to Pension Fund Portfolios

Pension funds face a slightly different set of challenges in the measurement of market exposures BC of the fact that the assets must fund pension obligations whose PV is itself subject to IR & other risks. Pension fund managers typically apply VAR methodologies not to their portfolio of assets but TO THE SURPLUS. To do so, they simply express their LIABILITY PORTFOLIO as a set of SHORT securities and calculate VAR on the NET POSITION. VAR handles this process quite elegantly, and once this adjustment is made, all three VAR methodologies can be applied to the task.

5.5.1. *Scenario Analysis* h. *Compare alternative types of stress testing and discuss advantages and disadvantages of each*

Process of evaluating a portfolio under different states of the world. Quite often it involves designing scenarios with deliberately large movements in the key variables that affect the values of a portfolio's assets and derivatives. ONE TYPE of scenario analysis, that of STYLIZED SCENARIOS, involves simulating a movement in at least one interest rate, exchange rate, stock price, or commodity price relevant to the portfolio

DELTA SOLUTION use of delta to estimate the option's price sensitivity for VAR purposes has led some to call the analytical method (or variance-covariance method) the DELTA-NORMAL METHOD.

Recall that delta expresses a LINEAR relationship between an option's price and the underlying price (i.e., Delta = Change in option price/Change in underlying). A linear relationship lends itself more easily to treatment with a normal distribution. That is, a normally distributed random variable remains normally distributed when multiplied by a constant. In this case, the constant is the delta. The change in the option price is assumed to equal the change in the underlying price multiplied by the delta. This trick converts the normal distribution for the return on the underlying into a normal distribution for the option return

Vega

Sensitivity to volatility is reflected in VEGA, the change in the price of an option for a change in the underlying's volatility. Most early option-pricing models (e.g., the Black-Scholes-Merton model) assume that volatility does not change over the life of an option, but in fact, volatility does generally change. Volatility changes are sometimes easy to observe in markets: Some days are far more volatile than others. Moreover, new information affecting the value of an underlying instrument, such as pending product announcements, will discernibly affect volatility. Because of their nonlinear payoff structure, options are typically very responsive to a change in volatility. Swaps, futures, and forwards with linear payoff functions are much less sensitive to changes in volatility.

SKEWNESS Distributions can deviate from normality because of SKEWNESS and KURTOSIS.

Skewness is a measure of a distribution's deviation from the perfect symmetry (the normal distribution has a skewness of zero). A positively skewed distribution is characterized by relatively many small losses and a few extreme gains and has a long tail on its right side. A negatively skewed distribution is characterized by relatively many small gains and a few extreme losses and has a long tail on its left side. When a distribution is positively or negatively skewed, the variance-covariance method of estimating VAR will be INACCURATE.

TAIL VALUE AT RISK (TVAR) Also known as conditional tail expectation *With some difficulty, VAR can be extended to handle CREDIT RISK, the risk that a counterparty will not pay what it owes g. *Discuss advantages and limitations of VAR and its extensions, including cash flow at risk, earnings at risk, and tail value at risk*

TVAR is defined as the VAR plus the EXPECTED LOSS IN EXCESS OF VAR, when such excess loss occurs. For example, given a 5 percent daily VAR, TVAR might be calculated as the average of the worst 5 percent of outcomes in a simulation.

The problem with options

The normal distribution assumption is inappropriate for portfolios that CONTAIN OPTIONS The return distributions of options portfolios are often far from normal (normal distribution has an unlimited upside and an unlimited downside) The distribution of returns for a call, put, covered call and protective put is HIGHLY SKEWED. One COMMON SOLUTION is to estimate the option's price sensitivity using its DELTA

SCENARIO ANALYSIS Potential problem... h. *Compare alternative types of stress testing and discuss advantages and disadvantages of each*

The results, of course, are only as good as implied by the accuracy of the scenarios devised. ONE PROBLEM with the STYLIZED SCENARIO APPROACH is that the shocks tend to be applied to variables in a sequential fashion. In reality, these shocks often HAPPEN AT THE SAME TIME, have much different CORRELATIONS THAN NORMAL, or have some causal relationship connecting them.

5.5.2. Stressing Models FACTOR PUSH

The simplest form of STRESSING MODEL is referred to as FACTOR PUSH. The basic idea of which to is to push the prices and risk factors of an underlying model in the most disadvantageous way and to work out the combined effect on the portfolio's value. This exercise might be appropriate for a wide range of models, including OPTION-PRICING MODELS such as Black-Scholes-Merton, MULTIFACTOR EQUITY RISK MODELS, and TERM STRUCTURE FACTOR models But factor push also has its limitations and difficulties, principally the enormous model risk that occurs in assuming the underlying model will function in an extreme risk climate.

Delta Solution (2)

The use of delta is appropriate only for SMALL CHANGES in the underlying. As an alternative, some users of the delta-normal method add the second-order effect, captured by GAMMA. Unfortunately, as these higher-order effects are added, the relationship between the option price and the underlying's price begins to approximate the true nonlinear relationship

Volatility for individual positions

The volatility associated with individual positions can be combined with other simple statistics, such as correlations, to form the building blocks for the portfolio-based risk management systems that have become the industry standard in recent years

Especial case: Expected return of ZERO Some approaches to estimating VAR using the analytical method assume an expected return of ZERO.

This assumption is generally thought to be ACCEPTABLE FOR *DAILY VAR* calculations because expected daily return will indeed tend to be close to zero. Because expected returns are typically positive for longer time horizons, shifting the distribution by assuming a zero expected return will result in a *larger projected loss*, so the VAR estimate will be *greater*. Therefore, this small adjustment offers a slightly more conservative result and avoids the problem of having to estimate the *expected return*, a task typically much harder than that of estimating associated *volatility*. Another advantage of this adjustment is that it makes it easier to *adjust the VAR* for a *different time period*. For example, if the daily VAR is estimated at $100,000, the annual VAR will be

VAR (official definition)

VALUE AT RISK (VAR) is an estimate of the loss (in money terms) that we EXPECT TO BE EXCEEDED with a GIVEN LEVEL OF PROBABILITY over a SPECIFIED TIME PERIOD. Implications: 1. VAR is an estimate of the loss that we expect to be exceeded (measures a minimum loss; actual loss may be much worse) 2.VAR is associated with a given probability (All else equal, if we lower the probability from 5% to 1%, the VAR will be LARGER bc we now are referring to a loss that we expect to be exceeded with only a 1% prob) 3. VAR has a time element and that as such, VARs cannot be compared directly unless they share the same time interval. *Potential losses over LONGER periods should be LARGER than those over SHORTER periods, but in most instances, longer time periods will not increase exposure in a linear fashion.

5.2. Value at Risk e. *Calculate and interpret value at risk (VAR) and explain its role in measuring overall and individual position market risk*

VAR is a probability-based measure of loss potential for a company, a fund, a portfolio, a transaction, or a strategy. It is usually expressed either as a percentage or in units of currency. Any position that exposes one to loss is potentially a candidate for VAR measurement. VAR is most widely and easily used to measure the loss from MARKET RISK, but it can also be used—subject to much greater complexity—to measure the loss from credit risk and other types of exposures.

5.2.1. Elements of Measuring Value at Risk

VAR may be implemented in several forms. You need to make a few decisions before calculating: a. probability level (1% vs 5%, leads to a more CONSERVATIVE VAR estimate) If LINEAR risk characteristics, the two probability levels will provide essentially identical information. If optionality or NON-LINEAR risks, may need to select the more conservative probability threshold b. selecting the time period over which to measure VAR (the longer the period, the greater the VAR number will be) c. choosing the specific approach to modeling the loss distribution (next slides)

5. Measuring Risk

We look at some techniques for measuring market risk and credit risk. Subsequently, we briefly survey some of the issues for measuring non-financial risk, a very difficult area but also a very topical one—particularly after the advent of the Basel II standards on risk management for international banks, which we will discuss.

ACTUAL EXTREME EVENTS that have occurred in the past Another approach to scenario analysis

We might want to put our portfolio through price movements that simulate for example the stock market crash of October 1987; This type of scenario analysis might be *particularly useful* if we think that the *occurrence of extreme market breaks *has a *higher probability* than that given by the probability model or historical time period being used in developing the VAR estimate.

5.2.4. The Monte Carlo Simulation Method f. *Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR*

When estimating VAR, we use Monte Carlo simulation to produce random portfolio returns. We then assemble these returns into a summary distribution from which we can determine at which level the lower 5 percent (or 1 percent, if preferred) of return outcomes occur. We then apply this figure to the portfolio value to obtain VAR.

MONTE CARLO METHOD: • Type of probability distribution for each variable • We can make any distributional assumption o Inappropriate to use normality assumption for some markets Derivatives for example f. Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR and discuss the advantages and disadvantages of each*

o Advantages: Is often the only practical means of generating the information needed to manage the risk (DOES NOT REQUIRE A NORMAL DISTRIBUTION) o Disadvantages: Extensive commitment of computer resources (tens of thousands of transactions)

VAR: • Limitations: g. *Discuss advantages and limitations of VAR and its extensions, including cash flow at risk, earnings at risk, and tail value at risk*

o Difficult to estimate; different methods can yield different results o VAR can also give a FALSE SENSE OF SECURITY o VAR underestimates the MAGNITUDE and FREQUENCY of worst returns This problem often derives from the erroneous assumptions in the model o Individual VAR does not generally aggregate to PORTFOLIO VAR o Fails to incorporate POSITIVE RESULSTS into risk profile, so gives an incomplete picture of overall risks • Users should apply BACKTESING to improve and test VAR results

ANALYTICAL OR VARIANCE COVARIANCE METHOD: Formula: VP x [mean return - 1.65/1.96/2.33 x standard deviation] 1.65 = used when asked for the 5% VAR value (5% in the lower tail and 5% in the upper tail, total 10%) 1.96 = used when asked for the 2.5% VAR value (1.25% in the lower tail, and 1.25% in the upper tail, total 5%) 2.33 = used when asked for the 1% VAR value (0.5% in the lower tail, and 1,25% in the upper tail, total 1%) f. Compare the analytical (variance-covariance), historical, and Monte Carlo methods for estimating VAR and discuss the advantages and disadvantages of each*

• Assumes normal distributions! o Advantages: Simple o Disadvantages: Relying on simplifying assumptions, including normality assumptions Normality assumptions is inappropriate for portfolios that contains OPTIONS There is no reason why the calculation has to have a *normal distribution*, but if we move away from the normality assumption, we *cannot rely on variance as a complete measure of risk*

A portfolio's exposure to losses because of market risk typically takes ONE OF TWO FORMS:

• Sensitivity to adverse movements in the value of a key variable in valuation (primary or first-order measures of risk) • Risk measures associated with changes in sensitivities (secondary or second-order measures of risk) Primary measures of risk often reflect linear elements in valuation relationships; secondary measures often take account of curvature in valuation relationships. Each asset class (e.g., bonds, foreign exchange, equities) has specific first- and second-order measures.