风险管理与金融机构3e-john hull41734

1、Chapter 9Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 2012Value at Risk1The Question Being Asked in VaR“What loss level is such that we are X% confident it will not be exceeded in N business days?”Risk Management and Financial Institutions 3e, Chapter 9, Copyright

2、 John C. Hull 20122VaR and Regulatory CapitalRegulators base the capital they require banks to keep on VaRThe market-risk capital is k times the 10-day 99% VaR where k is at least 3.0Under Basel II, capital for credit risk and operational risk is based on a one-year 99.9% VaRRisk Management and Fina

3、ncial Institutions 3e, Chapter 9, Copyright John C. Hull 20123Advantages of VaRIt captures an important aspect of riskin a single numberIt is easy to understandIt asks the simple question: “How bad can things get?” Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 2012

4、4Example 9.1 (page 185)The gain from a portfolio during six month is normally distributed with mean $2 million and standard deviation $10 millionThe 1% point of the distribution of gains is 22.3310 or $21.3 millionThe VaR for the portfolio with a six month time horizon and a 99% confidence level is

5、$21.3 million. Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 20125Example 9.2 (page 186)All es between a loss of $50 million and a gain of $50 million are equally likely for a one-year projectThe VaR for a one-year time horizon and a 99% confidence level is $49 mil

6、lionRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 20126Examples 9.3 and 9.4 (page 186)A one-year project has a 98% chance of leading to a gain of $2 million, a 1.5% chance of a loss of $4 million, and a 0.5% chance of a loss of $10 millionThe VaR with a 99% confide

7、nce level is $4 million What if the confidence level is 99.9%?What if it is 99.5%?Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 20127Cumulative Loss Distribution for Examples 9.3 and 9.4 (Figure 9.3, page 186)Risk Management and Financial Institutions 3e, Chapter 9

8、, Copyright John C. Hull 20128VaR vs. Expected Shortfall VaR is the loss level that will not be exceeded with a specified probabilityExpected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss)Two portfolios with the same VaR can have ve

9、ry different expected shortfallsRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 20129Distributions with the Same VaR but Different Expected Shortfalls Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 2012VaRVaR10Coherent Risk Measures

10、(page 188)Define a coherent risk measure as the amount of cash that has to be added to a portfolio to make its risk acceptableProperties of coherent risk measureIf one portfolio always produces a worse e than another its risk measure should be greaterIf we add an amount of cash K to a portfolio its

11、risk measure should go down by KChanging the size of a portfolio by l should result in the risk measure being multiplied by lThe risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were mergedRisk Management and Financial

12、Institutions 3e, Chapter 9, Copyright John C. Hull 201211VaR vs Expected ShortfallVaR satisfies the first three conditions but not the fourth oneExpected shortfall satisfies all four conditions.Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201212Example 9.5 and 9.7

13、Each of two independent projects has a probability 0.98 of a loss of $1 million and 0.02 probability of a loss of $10 millionWhat is the 97.5% VaR for each project?What is the 97.5% expected shortfall for each project?What is the 97.5% VaR for the portfolio?What is the 97.5% expected shortfall for t

14、he portfolio?Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201213Examples 9.6 and 9.8A bank has two $10 million one-year loans. Possible es are as followsIf a default occurs, losses between 0% and 100% are equally likely. If a loan does not default, a profit of 0.2

15、 million is made.What is the 99% VaR and expected shortfall of each projectWhat is the 99% VaR and expected shortfall for the portfolioRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201214 eProbabilityNeither Loan Defaults97.5%Loan 1 defaults, loan 2 does not defaul

16、t1.25%Loan 2 defaults, loan 1 does not default1.25%Both loans default0.00%Spectral Risk MeasuresA spectral risk measure assigns weights to quantiles of the loss distributionVaR assigns all weight to Xth quantile of the loss distributionExpected shortfall assigns equal weight to all quantiles greater

17、 than the Xth quantileFor a coherent risk measure weights must be a non-decreasing function of the quantilesRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201215Normal Distribution AssumptionThe simplest assumption is that daily gains/losses are normally distributed

18、 and independent with mean zeroIt is then easy to calculate VaR from the standard deviation (1-day VaR=2.33s)The T-day VaR equals times the one-day VaRRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201216Independence Assumption in VaR Calculations (Equation 9.3, pag

19、e 193)When daily changes in a portfolio are identically distributed and independent the variance over T days is T times the variance over one dayWhen there is autocorrelation equal to r the multiplier is increased from T to Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C.

20、Hull 201217Impact of Autocorrelation: Ratio of T-day VaR to 1-day VaR (Table 9.1, page 193)Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 2012T=1T=2T=5T=10T=50T=250r=01.01.412.243.167.0715.81r=0.051.01.452.333.317.4316.62r=0.11.01.482.423.467.8017.47r=0.21.01.552.62

21、3.798.6219.3518Choice of VaR ParametersTime horizon should depend on how quickly portfolio can be unwound. Bank regulators in effect use 1-day for market risk and 1-year for credit/operational risk. Fund managers often use one monthConfidence level depends on objectives. Regulators use 99% for marke

22、t risk and 99.9% for credit/operational risk.A bank wanting to maintain a AA credit rating might use confidence levels as high as 99.97% for internal calculations.Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201219VaR Measures for a Portfolio where an amount xi is

23、 invested in the ith component of the portfolio (196)Marginal VaR: Incremental VaR: Incremental effect of the ith component on VaRComponent VaR:Risk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201220Properties of Component VaRThe component VaR is approximately the sam

24、e as the incremental VaRThe total VaR is the sum of the component VaRs (Eulers theorem)The component VaR therefore provides a sensible way of allocating VaR to different activitiesRisk Management and Financial Institutions 3e, Chapter 9, Copyright John C. Hull 201221Aggregating VaRsAn approximate approach that seems to works well iswhere VaRi is the VaR for the ith segment, VaRtotal is the total VaR, and rij is the coefficient of correlation between losses from the ith and jth segmentsRisk Management a