### Transcription of Lecture Note of Bus 41202, Spring 2011: Value at …

1 **Lecture** Note of **bus 41202** , **Spring** **2011** : **Value** at Risk, **expected** **shortfall** & Risk **management** Classification of Financial Risk 1. Credit risk 2. Market risk 3. Operational risk We start with the market risk, because more high-quality data are available easier to understand the idea applicable to other types of risk. What is **Value** at Risk (VaR)? a measure of minimum loss of a financial position within a certain period of time for a given (small) probability the amount a position could decline in a given period, associated with a given probability (or confidence level). A formal definition: time period given: t = `. loss in **Value** : L. 1. CDF of the loss F`(x). given (upper tail) probability: p VaR is defined as p = P r[L > VaR] = 1 F`(VaR). Quantile: xq is the 100qth quantile of the distribution F`(x) if q = F`(xq ), , q = P (L xq ).

2 And F`(.) is continuous. For discrete distribution, we have xq = min{x|P (L x) q}. Factors affect VaR: 1. the probability p. 2. the time horizon `. 3. the CDF F`(x). (or CDF of loss). 4. the mark-to-market **Value** of the position. Remark: Let R(L) be the risk associated with loss L. From a theoretical point, R(L) must possess the following basic properties: 1. Monotonicity: If L1 L2 for all possible outcomes, then R(L1) . R(L2). 2. 2. Sub-additivity: R(L1 + L2) R(L1) + R(L2) for any two port- folios. 3. Positive homogeneity: R(hL) = hR(L), where h > 0. 4. Translation invariance: R(L + a) = R(L) + a, where a is a positive real number. The sub-additivity is associated with risk diversification. The equal- ity holds when the two portfolios are perfectly positively correlated.

3 A risk measure is called coherent if it satisfies the above four prop- erties. Note. If the loss involved is normally distributed, then VaR is a coherent risk measure. The sub-additivity can be seen because ( 1 + 2)2 = 12 + 22 + 2 1 2. 12 + 22 + 2 a 2. = Var(L1 + L2), where 1 and 2 are the standard errors of L1 and L2, respectively, and is the correlation between L1 and L2. However, VaR fails to meet the sub-additive property under certain conditions. This is the reason that we shall also discuss **expected** **shortfall** or conditional VaR (CVaR), which is a coherent risk mea- sure. **expected** **shortfall** is the **expected** loss when the VaR is ex- ceeded. Some people call **expected** **shortfall** as Tail VaR (TVaR). or **expected** tail loss (ETL). In the insurance literature, **expected** 3.

4 **shortfall** is called Conditional Tail Expectation or Tail Conditional Expectation (TCE). Example of incoherent VaR. See the book of Klugman, Panjer and Willmot (2008, Wiley). Let Z denote a continuous loss random variable with the following CDF values FZ (1) = , FZ (90) = , FZ (100) = It is clear that (Z) = 90. Now, define loss variables X and Y. such that Z = X + Y , where .. Z, if Z 100. X= . 0, if Z > 100, .. 0, if Z 100. Y = . Z, if Z > 100. The CDF of X satisfies FX (1) = .91/.96 , FX (90) = , FX (100) = 1. Therefore, (X) 1. Turn to Y . The CDF of Y satisfies FY (0) = so that (Y ) = 0. Consequently, (X) + (Y ) = 1 < (Z). In what follows, we shall use log returns in the analysis (simple re- turns can also be used). Why use log returns? 4. log returns percentage changes.

5 VaR = **Value** (VaR of log return). Methods available for market risk 1. RiskMetrics 2. Econometric modeling 3. Empirical quantile 4. Traditional extreme **Value** theory (EVT). 5. EVT based on exceedance over a high threshold Data used in illustrations: Daily log returns of IBM stock span: July 3, 62 to Dec. 31, 98. size: 9190 points Position: long on $10 million. Note: For a long position, loss occurs at the left (or lower) tail of the returns. This is equivalent to using the right (or upper) tail if negative returns are used. RiskMetrics Developed by Morgan rt given Ft 1: N (0, t2). 5. t2 follows the special IGARCH(1,1) model t2 = t 1. 2 2. + (1 )rt 1 , 1 > > 0. VaR = t if p = . k-horizon: VaR[k] = kVaR. The square root of time rule Pros: simplicity and transparence Cons: model is not adequate Example: IBM data Model: rt = at, at = t t, 2.

6 T2 = t 1 + (1 )a2t 1. 2. Because r9190 = and 9190 = , 2. 9190 (1) = . For p = , VaR of rt = = VaR = $10, 000, 000 = $302, 500.. For p = , VaR of rt = = , and VaR = $426,500. **expected** **shortfall** . From the prior discussion, VaR is simply the 100(1 p)th quantile of the loss function, where p is the upper tail 6. probability. When an extreme loss occurs, VaR is exceeded, the actual loss can be much higher than VaR. To better quantify the loss and to employ a coherent risk measure, we consider the **expected** loss once the VaR is exceeded. Let q = 1 p. The **expected** **shortfall** (ES) is then ESq = E(L|L > VaRq ). For standard Normal distribution, we have ESq = f (VaRq )/p, where f (x) = 1 exp[ 12 x2], which is the probability density function of 2 . the standard normal distribution.

7 Consequently, the ES for a normal return N (0, t2) is f (VaRq ). ESq = t , p where t is the volatility and f (x) denotes the density function of N (0, 1). Therefore, for a N (0, t2) loss function, we have =. t. In general, for a normal distribution N ( , t2), the ES is f (VaRq ). ESq = + t . p **expected** **shortfall** can also be defined as the average VaR for small tail probabilities, 1Zp ES1 p = 0 VaR1 udu. p Econometric models rt = t + at given Ft 1. 7. t: a mean equation ( ). t2: a volatility model (Ch. 3 or 4). Pros: sound theory Cons: a bit complicated. IBM data: Case 1: Gaussian rt = 2 + at, at = t t t2 = + 1 + t2. 2. From r9189 = , r9190 = and 9190 = , we have 2. r 9190(1) = and 9190 (1) = If p = , then . = VaR = $10,000,000 = $287,700. If p = , then the quantile is.

8 = VaR = $409,738. 8. **expected** **shortfall** for normal distribution N ( t, t2). Following the prior discussion, we have f (VaRq ). ESq = t + t . p Case 2: Student-t5. rt = 2 + at, at = t t t2 = + 1 + t 1. 2.. 2. From the data, r9189 = , r9190 = and 9190 =. , we have 2. r 9190(1) = and 9190 (1) = If p = , then the quantile is r ..000367 ( 5/3) .0003386 = .000367 .0003386 = .028354. VaR = $10, 000, 000 = $283,520. If p = , the quantile is r . ( 5/3) = VaR = $475,943. Discussion: Effects of heavy-tails seen with p = Multiple step-ahead forecasts are needed. 9. Example (continued). 15-day horizon. r 9190[15] = and t[15] = . If p = , the quantile is = 15-day VaR = $10,000,000 = $1,039,191.. RiskMetrics: VaR = $287, 700 15 = $1,114,257. For standardized Student-t distribution with v degrees of freedom, the **expected** **shortfall** is given by x2q.

9 1 (v 2) +. ES1 p = f (xq |v) . , p v 1. where xq is the 100qth quantile of the distribution. If (Y )/ fol- lows a standardized Student-t distribution with v degrees of freedom, then (v 2) + x2q .. 1 . ES1 p = + f (xq |v) .. p v 1. Empirical quantile Sample of log returns: {rt|t = 1, , n}. Order statistics: r(1) r(2) r(n). r(i) as the ith order statistic of the sample. r(1) is the sample minimum r(n) the sample maximum. Idea: Use the empirical quantile to estimate the theoretical quantile of rt. 10. For a given probability p, what is the empirical quantile? If np = ` is an integer, then it is r(`). If np is not an integer, find the two neighboring integers `1 < np < `2. and use interpolation. The quantile is p2 p p p1. x p = r(`1) + r(` ). p 2 p1 p 2 p1 2. IBM data: n = 9190.

10 If p = , then np = 5% quantile is (r(459) + r(460))/2 = VaR = $216,030. If p = , then np = and the 1% quantile is p2 p1. x = r(91) + r(92). p2 p1 p2 p 1..00001 = ( ) + ( )..00011 VaR is $365,709. **expected** **shortfall** : 1 X n ESq = x(i)I[x(i) > x q ], Nq i=1. where Nq is the number of data that exceed the empirical quantile x q . In other words, ESq is the average of all the data that exceed the empirical 100qth quantile x 1 of the data. 11. > nibm=-ibm > quantile(nibm,c(.95,.99)). 95% 99%. <== R uses a slightly different method, but the results are close to those in the **Lecture** note, which are and , respectively. ## Compute **expected** **shortfall** . > idx=c(1:9190)[nibm > ]. > mean(nibm[idx]). [1] > idx=c(1:9190)[nibm > ]. > mean(nibm[idx]). [1] Extreme **Value** theory: Focus on the tail behavior of rt.