Risk
What is value at risk?
The downside risk on the tail of a normal distribution
How does VaR work?
A (normal) distribution's downside risk - the tail of the distribution - we need a probability distribution
What are the two variables a normal distribution uses to describe data?
Average/mean Variance
What are the benefits to credit risk monitoring?
Internal risk monitoring Evaluating credit risky securities
What are the strengths and weaknesses of normal VaR?
S: Only need two values in order to calculate the VaR at any CI or time W: assumed normal distribution
How can we use historical simulation?
Estimate returns distribution from recent history- easy with excel percentile() - this orders the daily returns by size and so we find the returns that are only exceeded 5% of the time
What is the Risk Management Irrelevance Proposition? (Not on Midsem)
"... a firm cannot create value by hedging risks when it costs the same for the firm to bear the risks directly than to pay capital markets to bear them" (Stulz 2003, p. 45). Removing Diversifiable Risk: is costly and lowers expected CFs, but not the required rate of return (lowers value). Also, shareholders can do this themselves. Removing Systematic Risk: reduces both expected CFs and the required rate of return (value unchanged). However, shareholders can do this themselves. This conclusion relies on the assumption of perfect capital markets. No transactions costs No taxes Equal access to information Rational Investors In this setting, the law-of-one price holds such that the return to shareholders for bearing the risk is the same as the cost to transfer the risk to capital markets. Hence, no value added! Put differently, shareholders can in mimic the firms risk management activity at no extra cost. Therefore, for risk management to be "relevant", we must identify important deviation from this that create value.
What is a VaR statement? What are the components?
"The 10 day, 95% VaR for our $100m portfolio is $3m." - value at risk - given probability - pre set horizon Value-at-Risk (VaR) is a measure of the maximum potential change in the value of a portfolio of instruments with a given probability over a pre-set horizon It shows feasible but large losses
Why might risk management adds value?
*Bankruptcy / Financial Distress Costs* - fees, management time, lost investment and associated costs. bankruptcy costs> risk management costs = value *Dead Weight Loss* - losses = can't fund positive NPV investment *Taxes* - risk management may increase the debt carrying capacity of the firm. Hence, the firm may acquire a greater tax shield from borrowing. This cannot be recreated by investors. *Debt Overhang* - owners accepting bad investments because it reduces the probability of bankruptcy. Positive NPV investments are rejected - owners are unwilling to accept a project because they need additional funding *Homemade Hedging* - possible for shareholders to recreate all of this risk management? Bad assumptions of irrelevance prop *Stakeholders* - relates to financial distress in terms of their willingness to invest
What are the 4 uses of VaR?
*Basis for capital holdings* - how much capital is needed to absorb potential loss - Recognises underlying asset diversification *Setting risk limits* - limits are set based on risk (relates to the amount of money that can be lost in a time-frame for a probability) - standard measure (comparable) *Benchmark Measure* - comparable across assets and functions - integrates market risk, credit risk... *Reporting tool* - VaR is often reported to investors
Explain the historical simulation method of calculating VaR
- constructing N scenarios of what might happen between today and tomorrow from N days of historical data. - for every day's return that you have, what would be the value of the portfolio with this return? - VaR is calculated from the probability distribution of portfolio value changes.
What is time varying variance?
- different periods have different volatility - we usually put more weight on recent observations - the two methods are: *Rolling Window* and *EWMA*
What are the benefits of the rolling window time varying volatility?
-Easy to implement and understand - just use a volatility estimate to get VaR -The extent of adaptability is based on the window length (changeable) - more adaptable than using the full sample
Define Risk
-degree of *uncertainty* of future returns - quantifiable likelihood of *loss* or less than expected returns You can have good returns as part of risk - what if you get high returns which leads to a sub-optimal allocation of resources
How can we improve the data fit from a normal distribution?
1) Use the empirical distribution (observed data) - it fits the data well but can be noisy 2) time varying volatility (risk changes with time - use short samples or EWMA) - less noisy than historical data but may not fit the extreme tails well
How can we calculate expected shortfall? Which method is better?
1. Historical Simulation 2. Normal Distribution - heavy tails may have more effect here "On the worst 1% of days, how badly do we expect it to be?" Average value
What are the criticisms of VaR?
1. It ignores the extreme tails - gives us a sense of losses but there is no information on the most extreme losses - A 95% VaR says nothing about the worst 5% of daily returns (13days in a year) - portfolios can have the same VaR but much higher risk on one - VaR is only a partial view of risk (summary) - we can supplement it with stress testing 2. VaR is not subadditive - The VaR of a portfolio can be more than the sum of the VaR of the individual components (due to correlations) - if you cap individual traders' risks but you won't have limited the total risk of the portfolio 3. VaR creates a false sense of security - exact number gives the impression of an exact result - VaR is only an estimate - There is model risk where your VaR method may be flawed - it is not a complete picture of risk
Outline three (3) criticisms of VaR.
1. VaR ignores the risk in the tails. - says nothing about the losses beyond the VaR. - These losses *will* happen and they will be large 2. VaR is not always sub-additive. - The VaR of the portfolio may be greater than the sum of the VaR of each asset - If we limit the risk of our two traders individually, we haven't necessarily limited the overall risk of the company. - VaR is sometimes sub-additive - depends on the distribution of the returns and since we don't know the true distribution, we don't know if this is a problem 3. Complacency. - you put all your faith in a model that may have some severe limitations. - Faith in high level mathematics and complex assumptions, - Managers / regulators who rely on these values have little understanding of how they were derived.
How does risk impact at the firm level?
1. affects value -the risk/return trade off 2. informs and regulates decision makers - are there enough returns/risk, should they hedge, expected losses, price changes and exposure, probability of default 3. behaviour - are managers incentives aligned with the owners
How do we calculate a portfolio VaR?
1. calculating the mean and STD of the portfolio 2. Historical simulation - based on returns of individual assets
What is the EWMA time varying volatility?
A recursive formula where the last piece of data is weighted 6%
How often should VaR be exceeded?
A 95% VaR should be exceeded 5% of the time on average
What is the cost of a bad VaR model?
A banks' assets have the capital charge determined from VaR Smaller VaR = smaller market risk charge = greater return on capital Consistently underestimating VaR increases penalty factor S A regulator will request corrective action if a VaR model is in the red zone
What is a swap?
A contract to exchange cash flows periodically for an agreed period of time.
What is a forward?
A forward is a contract made today to be performed at some set time in the future at a set price for goods of a particular quality delivered to a set location.
What was the orange county default?
Borrowed using short term floating rate bonds and invested in long term bonds Short term interest rate securities are less sensitive to shifts in interest rates Interest rates significantly rose -> short rates rose, so their debts was a little bit reduced -> long rates also rose, significantly reduced the value of their assets
What are the weaknesses of the Rolling Window time varying volatility?
The length of the window selected changes the accuracy - too short a window = estimate takes too long to change - too long a window = noisy estimate
Explain EWMA and two advantages of EWMA relative to other methods
ADV: Simplicity, doesn't need much data, accounts for time varying volatility, quick to adjust to current market conditions - DISCUSS and COMPARE
What is a future?
An futures contract is an agreement made today to exchange a specified asset at a specified price at a specified date in the future.
What are derivatives?
An instrument whose price is derived from another asset Often used to offset exposure or create exposure Major Types: Forward: an obligation to buy or sell Futures: an exchange traded forward that is marked-to-market Forward Rate Agreements (FRAs): Locks in future borrowing/lending rates Swaps: a bundle of forwards Options: the right but not the obligation to buy or sell
What is an option?
An option is the right (but not the obligation) to force a transaction to occur at some time in the future on terms and conditions agreed upon now.
What is the difference between a log return and an arithmetic return?
Arithmetic returns measure simple return whereas log returns measure continuously compounding returns
Why might it be useful to assume that returns follow a particular statistical distribution when trying to think about risk?
As a probability distribution essentially assigns probabilities to the various possible outcomes, this is a natural way to quantify risk. For the normal distribution, the variance parameter essentially governs risk under both distributions as there are only two parameters in this distribution and the mean is always approximately 0. Simple! The probability of gains and losses of the same magnitude are equal (the distribution is symmetric) and the probability of extreme events (defined relative to the variance) are always the same for the normal distribution. Summarises the risk of losses in two values.
Let's say that the annual returns on Stock X are normally distributed with an average return of 10% and a standard deviation of 50% (which implies a variance of 0.25), such that X ~ N( 0.1 , 0.25). What is the probability that the stock return is between than -40% and +60% in the next year?
As discussed in the lecture, what drives the probabilities of the normal distribution is how many standard deviations from the mean a given point of interest is. This is exactly what the z score from your introductory data analysis course measured. z=((y-μ))/σ This relates the our normally distributed data (e.g. - stock returns) back to the standard normal distribution which we understand very well. Stock X has a mean return of 0.1 and a standard deviation of 0.5. This means that -40% is one standard deviation below the mean, while 60% is one standard deviation above the mean. In the lecture we learnt that the probability of a normally distributed random variable falling within one standard deviation either side of the mean was 68%.
What is the market risk of bonds?
As the yield of bonds increases, bond prices fall - depends on how sensitive they are to shifts in yield (modified duration) Longer bonds = more sensitive Higher coupons = less sensitive Floating coupons = less sensitive
What is expected shortfall?
Average return you would expect given that the loss is greater that VaR - supplements VaR - subadditive - targets extreme risk outside VaR
Why was VaR adopted so widespread?
Because Basel II accord pushed for how banks should be regulated - VaR is the preferred method of quantifying market risk in order to determine capital levels - BIS recommended regulations
Explain how the confidence level and choice of time horizon affect VaR. Why are these choices so vital?
CI increases --> VaR increases - we are upgrading to the largest X% of losses time increases --> VaR increases - greater potential for large losses over larger time-frames - Going further into the tails makes it harder and harder to estimate VaR, as we have fewer observations to work with. - The appropriate time horizon is going to be related to how far we can forecast the distribution.
Can we apply VaR to more than one asset?
Collapse our portfolio into a 2D bell curve If all returns are jointly normally distributed then the portfolio returns are normally distributed - we just need to use a weighted mean and variance (incl correlation) This works because the portfolio returns are linear
How can we measure credit risk?
Credit ratings can apply to governments, firms and individual securities. There are guidelines which suggest the appropriate capital to hold against risk weighted assets i.e. on an A rated government hold 20% capital ($10m investment - hold $2m)
Define risk management
Defining a risk level that the firm wants, identifying the risk the firm has and using financial instruments to make those levels of risk meet
How can we use a time varying volatility?
EWMA - uses 94% previous data and 6% of the latest standard deviation This is adaptable to different portions
What is the normal VaR?
Estimate the mean and variance of returns from historical data and assume they're normally distributed
What is the rolling window version of time varying volatility?
Estimate volatility using only a small sample of recent data Equal weight on each piece of data from the last year, no weight on other observations
What is the heavy tails distribution?
Extreme losses are more likely in historical data than in the normal distribution
What is the simplest way to approximate bond returns?
Find the change in bond price from a 1 basis point change in interest rate - assume this sensitivity factor applies proportionally to any change in interest rate
How is value added through risk management? (irrelevance proposition rebuttal)
For value to exist, the return for the firm bearing risk must differ from the market's return. Dead Weight Loss - Risk leads to large losses and low cash flow and large borrowing costs. The firm must forego positive NPV projects Bankruptcy and Financial Distress Costs - fees, management time, lost investment (lawyers are expensive) Homemade Hedging - Is it really possible for shareholders to recreate all of this risk management ?? Given costs, inability, information asymmetry? Tax - smoothing, carrybacks and carryforwards, other tax shields Capital Structure - increases debt carrying capacity - tax benefits Stakeholders - relates to financial distress in terms of their willingness to invest Debt Overhang - accept negative NPV investments and reject positive NPV investments to increase firm equity, not firm value
How can we simulate returns?
Generate hypothetical returns with normal distribution we can very easily calculate prices from this Cons: no volatility clustering
What does each Basel Traffic Lights result mean?
Green zone = too conservative (overstate VaR) Yellow = good Red = aggressive (understate VaR)
Normal VaR vs Historical Simulation VaR
Historical = from a sample of returns, find the return that corresponds to the VaR percentile ADV: doesn't assume a distribution DIS: delayed response to changes in volatility and can be noisy in the tails Normal = assume the returns are normal, which only requires an estimation of mean and variance ADV: easy to calculate, easy to adjust for volatility DIS: assumes normal distribution, doesn't capture fat tails, not suited to complicated assets, requires a large number of correlations
What is Kurtosis?
How close a set of data is to being normally distributed
What are some other types of financial risk?
Liquidity Risk - the risk of not being able to liquidate a position in a timely manner at a reasonable price. Model Risk - results from a misspecified or inappropriate model being used to measure/quantify risk. Basis Risk - the risk that changes in the value of a hedging instrument imperfectly match changes in the original position
How can we choose a risk management policy based on the irrelevance proposition?
Many fund managers decide against hedging FX risk for similar reasons. Many firms don't hedge commodity risks. Firms tend to only hedge sizable risks due to the logic of these arguments.
What are the 3 main types of risk?
Market risk - loss due to changes in market factor e.g. price Credit risk - loss due to default Operational risk - loss due to human/system error Also: liquidity, model and basis risk
Why does society care about risk management?
Modern economies are interconnected - if one institution fails then this affects others - this leads into systematic risk
What are the different ways to calculate VaR for a bond portfolio? What are the benefits/disadvantages?
Monte carlo - precise but difficult (computationally expensive) Duration VaR - approximate but easy, only good over small interest rate changes - therefore bad at high CI's, less accurate over longer times because bond pricing is less linear
How can we calculate VaR for a portfolio of bonds?
Monte carlo - reprice the security from a sample - simulates random scenarios - simplified benchmark - precise but computationally demanding Find the best linear approximation (Duration VaR) - for small changes in IR, bond price is almost linear - sensitivity is captured by modified duration (proportional to slope of the straight line) - approximate but eary - good over small IR = bad for high CI - less linear over time = bad for long VaR
Is the VaR of a portfolio less than that of the individual assets added?
Mostly yes, due to diversification benefit where stocks are not perfectly positively correlation - however, this is not always true
What is a price?
Price at which an asset can be bought/sold May differ by the bid/ask spread
Identify and Discuss 2 factors that can influence credit risk
Probability of default - the probability that (maybe given certain market conditions) the entity you lent money too can no longer repay - what is the chance of this? - affected by market risk - affected by the firms financial stability - what about operational risk - liquidity.... - all of these factors will affect the probability of default Expected loss given default - if an entity defaults (they can't repay the full amount) - how much can they repay? Half? - this again might come down to how hard they are struggling
How is credit risk rated?
Reflects the probability of default over the short/long term Investment grade (AAA - BBB): AAA = capacity to meet commitments is extremely strong BBB = adequate protection parameters. Adverse economic conditions/changing circumstances are likely to lead to a weakened debt capacity to meet commitments Speculative Grade (BB-D): CC = highly vulnerable to non-payment. Haven't yet defaulted, but are expected to
Order the following bonds in terms of their interest rate sensitivity. a. 3-year, 10% annual coupon b. 3-year, 10% coupon paid semi-annually c. 10-year,0% annual coupon d. 10-year, 10% annual coupon
Relatively speaking, longer dated bonds, bonds with lower coupons and bonds with less frequent coupon payments are more sensitive to changes in yield. This can be measured with duration, however for the bonds specified, the sensitivities from least to greatest would be b < a < d < c The reason for this ordering is that b. and a. have the shorter maturities with b. paying more frequent coupons. For the longer dated bonds, d. pays the lager coupon so it would be less sensitive to yield changes than c.
Why do we use returns rather than prices in our histograms?
Return distribution conveys meaningful information More data will make the histogram look the same but with more detail (less noise)
Why do we use returns instead of price when predicting losses?
Returns are stationary so key statistics can apply to the whole series - stable avg over time - deviations somewhat stable We can easily convert risk to dollar values
"financial security returns have leptokurtic distributions. Therefore, calculating VaR using a normal distribution is a waste of time because the normal distribution doesn't allocate sufficient probability to extreme returns in the tails." Outline a simple correction to VaR that may appease (make them happier) this student but also provide two (2) reasons as to why normal VaR is useful.
Returns do have leptokurtic distribution = excess kurtosis (higher peak) and larger probability mass in the tails (fat-tails) relative to the normal distribution Adjustments to capture the 'fat-tails': 1. Time varying volatility e.g. EWMA model. This will allow the VaR to adjust for periods of high and low volatility. 2. Use a t-distribution. e.g. d.o.f < 30 = more probability mass in the tail than the normal distribution As to why normal VaR (NVaR) is useful. 1. Easy to use and it is easy to understand its limitations. 2. Allowing for time varying volatility while still using the normal distribution often works reasonable well for most equities. 3. Only requires estimation of the mean and standard deviation, where other distributions require other parameters.
What are the two components of VaR?
Risk Sensitivity
Why were risk measures before VaR bad?
Risk management was often not easy to understand Needed to convey meaningful information
What is credit risk?
Risk that the value of a portfolio changes due to credit quality of issuers/trading partners changing 1. Risk of suffering default and losing your claim to future repayments 2. Risk of losses through the downgrading of the counter-party to a loan e.g. probability of default increases, discount rate rises, can't sell the bond for as much We demand a higher interest rate for bonds with a lower credit rating = bond falls in value
What are the regulations placed on financial institutions' risk management?
Rules banks must follow to prevent financial crises Basel II Accords: - capital requirements - so they don't take on too much risk - quantifies potential losses - requires min holding of safe assets to ensure survival -in Australia, APRA
What are the 3 methods for calculating VaR?
normal method historical simulation method (monte carlo method) - helps calculate the first two with different sample spaces
The risk of equity returns can be measured with either standard deviation or beta. Discuss these two measures of risk and how they relate to each other.
Standard deviation = total risk, which includes systematic risk (market risk) and non-systematic risk (firm specific) Beta = systematic risk (market risk) In portfolio theory, all non-systematic risk can be diversified. However, this not the way we look at it in risk management - our assets are subject to both the systematic and non-systematic fluctuations, making the standard deviation a more appropriate measure of risk. If I hold a portfolio, I will either be interested in directly modelling the standard deviation of my portfolio returns or capturing the individual standard deviations and correlations, which would enable me to implicitly work out the standard deviation of my portfolio.
Let's say that the annual returns on Y are distributed Y ~ N( 0.05, 0.16). What is the probability that the stock return is less than -35% in the next year?
Stock Y has a mean return of 0.05 and a standard deviation of 0.4. This means that -35% is one standard deviation below the mean. In the lecture we learnt that the probability of a normally distributed random variable falling more than one standard deviation below the mean was 16%.
What are the Basel Traffic Lights?
Summarises the quality of a model: green = <4/250 days yellow = 5 to 9 in 250 days red = >10 in 250 days Calculate this for every day for the last 250 days - it is an accumulation of exceedences Anything in the red zone should be scrapped - regulator requests corrective action
What are the weaknesses with historical simulation VaR?
The 5% of worst losses are the 12 worst days or so - this can be noisy - whereas for the normal distribution, the standard deviation made all of the data indicate the worst performances so noise was less of a problem The choice of window length can also be a problem here - it is magnified in historical simulation because it is used to estimate the whole distribution rather than just the mean or variance (normal distribution will have less noise) You have to round the quantile - so be conservative or find the mid point
What are the strengths of the EWMA time varying volatility?
The adaptability of changing the % EWMA is very good Provides great estimates for most data
What is the market risk for bonds? Why does this make VaR tricky?
The market risk for bonds is interest rates -bond price and interest are not linearly related - complicated relationship
Explain the difference between market yield on the bond and yield implied by credit rating
The market views the bond as less risky - could be due to conservative risk measurement Bond yield = return that the market requires to be compensated for the risk associated with the bond- not the coupon Differences between yield and coupon are factored into bond price
How do we estimate VaR for an option?
The price of an option depends on the underlying stock price VaR for an options portfolio is calculated in the same ways: -monte carlo -approximations based on basis point changes
What is operational risk?
The risk of losses due to inadequate or failed internal processes, people and systems or from external events e.g. fraud (internal and external), IT failures, litigation, fire, flood They are difficult to quantify - little data, rare occurrences
What is market risk?
The risk to financial position due to changing values in the underlying component - potential impact on your holding due to daily changes in prices and rates stocks = change in price bonds = change in interest rate
What are the weaknesses of the EWMA time varying volatility?
The weighting on the last piece of data is important - too small % = estimate takes too long to change - too high % = noisy estimate
What is the problem with nonlinear pricing?
There are important deviations from the linear model that cannot be easily summarised - it would be inaccurate to use the linear model - even if returns are normally distributed a bond portfolio will not be
How should VaR exceedances be dispersed?
They should be randomly spread over time (not systematically spaced, not all together) - if your model isn't randomly spread it is not dealing with changing market conditions well
How can we measure market risk?
VaR - 95% sure our losses in the next 10 days will not exceed $3m Variance / Standard Deviation / Beta (stocks) Duration and Convexity (bonds) Delta and Gamma (options)
Explain the difference between VaR and ES
VaR = threshold in the returns distribution that will only be exceeded with a stated probability, ES = average return conditional on the return exceeding VaR. • ES is relative to a specific VaR. ES is not a Replacement for VaR. • ES complements VaR - it fills in information on the extreme tail • ES doesn't suffer from the subaddivity problem that VaR does
Should the return exceed VaR?
VaR is the max loss in a given level of probability If your VaR is never exceeded, it is too conservative
Explain Value-at-Risk (VaR).
Value-at-Risk measures the maximum potential loss for a given probability and investment horizon. For example, 95% of returns over the investment period of 1-day should not exceed the VaR (95%, 1-day). A numerical example is if VaR (95%, 1-day) = -2.00%, than there is a 5.00% probability of experiencing a loss greater than -2.00% over 1 day. Both the probability and investment can vary and the quality of VaR can be assessed based on the exceedances observed.
How can we measure the time varying correlations?
We measure the time varying variance and the covariance (using EWMA at each point in time)
What is a return?
When a price change is defined relative to an initial price over a Holding Period
What are time varying correlations?
Where the correlation of assets changes with time - this is important to capture because it changes the diversification benefit in VaR Correlations are not constant, they tend to increase in bad economic times - when things go bad they go bad for everyone
Can a big financial corporation take on too much risk?
While big corporations have economies of scale, the cost to bail them out is substantial Are they too big to fail? No We need to ensure risk is reduced and managed through regulation
What changes the price of a bond? Why is this relevant to VaR of a bond?
time (coupons) default risk Prices fall after coupons are paid and as time gets closer to the end of the bond - these fluctuations have nothing to do with risk We can't build our model on this as easily - bond prices change due to many factors We base our model on changes on interest rates
How can we time scale VaR?
variance scales linearly with time so assuming returns are uncorrelated (EMH), so we can simply take the root of the time to find the VaR for different periods
What are some of the dangers associated with calculating a stock's returns and their associated statistics straight from the stock's historical closing price?
• Dividends: the returns calculated from closing prices do not include any dividend distributions. The ex-dividend price will be negative usually. Return isn't negative because you've got a dividend • Stock Splits: If a stock splits 1 into 2, the stock's price will half. However, the investor's wealth is unaffected - they now own two stocks at half the price. If no adjustment is made the return is -50% but returns haven't actually changed Instead measure returns from an adjusted price series; The prices provided are artificial and can be different to the actual price