Objectives: Discuss the rationale for risk-measurement and the definition of Value at Risk (VaR).

1. Risk Measurement Using Value at Risk • Objectives: • Discuss the rationale for risk-measurement and the definition of Value at Risk (VaR). • How to implementing VaR when portfolio returns are assumed to be normally distributed. • The use of historical back simulation to calculate VaR. • Understand how VaR can be applied to set banks’ risk-based capital requirements.

2. Measuring Downside Risk • Recall that risk management can add value to a financial intermediary (FI) because it mitigates dead-weight costs of financial distress and/or bankruptcy and it also reduces the value of taxes paid. • To implement a FI’s risk management program, FI managers need to be able to measure the FI’s risk. A popular approach to measuring the downside risk of a portfolio position in assets and liabilities which was developed by J.P. Morgan (now J.P. Morgan Chase) is called Value at Risk (VaR). • VaR quantifies the size and probability of a portfolio loss. The portfolio can be the entire FI’s assets and liabilities. In this case, VaR measures a loss to the FI’s net worth or capital. VaR is used to set risk-based capital requirements for large international banks under the Basel II proposals.

3. A risk-measurement approach, such as VaR, can also be applied to the portfolio position of a single trader or single type of asset (e.g., FX or fixed-income). In this context, it can be used to • set limits on a trader’s portfolio positions. • decide what FI activities provide the best trade-off of risk for expected return. • evaluate a trader or activity’s performance based on their risk exposure. Compensation can be determined not only by profits earned but also by VaR. • For a given probability, p, and a given future investment horizon, h days, VaR is defined as the loss in value that has a probability p of being exceeded over the next h days, assuming that the portfolio position is not changed over the investment horizon.

4. Example: Suppose that the loss in portfolio value that has a one percent probability of being exceeded over the next 10 days is estimated to be $1 million. Then, $1 million is the portfolio’s VaR. • VaR depends on the firm’s or portfolio’s distribution of value, which, in turn, depends on the depends the firm’s or portfolio’s assets, liabilities, and derivative positions. • We can graphically illustrate VaR using the probability distribution for a portfolio’s % return. For example, suppose that over some given time horizon, h days, a particular portfolio’s % return is estimated to have the following cummulated probability distribution function.

5. Cumulative Distribution Function of Portfolio Return Probability Probability that % return < p p, % return -0.23 • We see that there is a 5 % probability of the portfolio returning less than -23%. If, say, the portfolio’s initial value is $1 m, then VaR(p=5%,h days) = 0.23x$1m = $230,000.

6. VaR with Normally Distributed Portfolio Returns • One approach to computing VaR is to assume that the portfolio’s returns are normally distributed. Let the portfolio’s random rate of return over a period of h days be . Its probability density function is Probability density function 5 % of area under curve 1 % of area under curve

7. This implies that there is a 5 % probability of a return less than • and a 1 % probability of a return less than • VaR is usually calculated over a measurement horizon of a small number of days. For this short horizon, a portfolio’s standard deviation is typically much greater than its expected return. Hence, the practice is to ignore the expected return and set • If this is done, then we have • VaR(p=5 %, h days) =1.65x x(Portfolio Value) • VaR(p=1 %, h days) =2.33x x(Portfolio Value) • Example: a portfolio’s 1 day standard deviation is 10%, and its initial value is $1m, then VaR(p=1%,1 day) =2.33x(0.10)x1m = $233,000.

8. What should be the time horizon (h days) over which to calculate VaR? If a FI can measure its risk and change it once a day, a one-day VaR is most useful. This would be relevant when a FI’s portfolio consists of liquid securities that can be bought or sold quickly. • However, if a FI holds a portfolio of illiquid assets that cannot be sold quickly, a longer horizon would be relevant. The FI should choose the VaR’s h to be the number of days over which it could change its portfolio. • If the return on a portfolio is estimated to have a one-day standard deviation of , then, assuming the portfolio’s composition stays the same over h days, its h-day standard deviation can be estimated as

9. Example: A bank holds a $20 m. portfolio of syndicated loans that would likely take 5 days to arrange for a sale. The daily standard deviation of the portfolio’s value is 0.3 %. Therefore • Suppose a portfolio consists of n different assets. Its standard deviation depends on the standard deviations and correlations of the individual assets composing the portfolio. • Let i be the proportion of the portfolio’s total value that is invested in asset i, and let i asset i’s standard deviation of return. Further, let ij be the correlation between the returns on asset i and asset j. Then the portfolio return’s variance is

10. Example: A portfolio has three assets held in proportions 1 = 0.2, 2 = 0.5, and 3 = 0.3. The assets’ h-day standard deviations are 1 = 0.3, 2 = 0.2, and 3 = 0.4. Their correlations are 12 = 0.1, 13 = 0.6, 23 = -0.1. The portfolio’s h-day return standard deviation is then

11. Therefore, the portfolio return’s standard deviation is • If the three-asset portfolio was initially worth $50m, then, for example, VaR(p=5%, h days) =1.65x0.188x$50 m = $15.5 m. • Under the approach outlined thus far, implementing VaR for a portfolio of many different assets requires estimates of each asset return’s standard deviation and the correlations between all of the assets’ returns. • A consulting firm, RiskMetrics™, provides daily estimates of standard deviations and correlations for different types of assets in many different countries. Of course, an individual FI could compute these estimates on its own using historical data.

12. VaR Using Back Simulation • Rather than assuming a portfolio’s return is a normal distribution, other methods to computing VaR are used. • The historic or back simulation approach uses the historical returns on the individual assets contained in the FI’s current portfolio. It then simulates what would have been the losses on the current portfolio if this portfolio had been held during the historical period, say the last 250 or 500 trading days. • Specifically, this approach involves the following steps: • Consider each of the asset/liability positions in the FI’s current portfolio. Suppose, as before, that there are n different assets and i is the proportion of the portfolio’s total value that is invested in asset i. Obtain data on the returns of these assets and liabilities for, say, the last 500 trading days.

13. Let rit is the return on asset/liability i on prior day t. Then if the portfolio had been held on that day, its return would have been • Next, rank the portfolio returns, Rt, for previous t = 1, … , 500 days from the lowest return to the highest. Let R5be the fifth worst return over the last 500 days and let R25 be the 25th worst return over the last 500 days. Most likely, both of these returns are negative (losses). • Then we would compute VaR as

14. A similar procedure using historical returns over, say, h= 5 day intervals could be used to calculate VaR(p, h=5 days). • The advantage of this historical simulation approach is that it uses the sample frequency (histogram) of actual returns. There is evidence that empirical distributions of asset returns display large losses more frequently than would be predicted by the thin-tailed normal distribution, and the back simulation method could account for this. • One disadvantage is that the historical period may not be representative of the near future. It may have been an unusually quiet (low volatility) period, and volatility is now likely to be greater. One correction for this is to give more recent observations a greater weight.

15. Risk-Based Capital Requirements Using VaR • VaR is the basis for setting minimum capital (net worth) requirements for large international banks. • In 1996, an amendment to the 1988 Basel Capital Accord created a rule for bank capital requirements to cover their liquid securities (non-loan) portfolios, so-called “trading accounts:” • Required capital for day t+1 = • where SRt is an additional capital to cover idiosyncratic risks. The terms in brackets are the bank’s current VaR estimate and an average of VaR estimates over the last 60 days. • The multiplier St depends on the accuracy of the bank’s VaR model.

16. St is computed by “back-testing” the bank’s VaR model estimates over the last 250 days. If the bank’s daily trading portfolio losses exceeded VaR(p=1%, 1 day) on x days over the last 250 days, then • Thus, a bank with a less accurate internal VaR model has a higher multiplier, St , and must have more capital.