1、Chapter 12Risk Management and Financial Institutions 2e, Chapter 12, Copyright John C. Hull 20091Market Risk VaR: Historical Simulation Approach12.1 The MethodologylHistorical simulation involves using past data as a guide to what will happen in the future.lSuppose that we want to calculate VaR for

2、a portfolio using a one-day time horizon, a 99% confidence level, and 501 days of data.lThe time horizon and confidence level are those typically used for a market risk VaR calculation; 501 is a popular choice for the number of days of data used because it leads to 500 scenarios being created.The st

3、eps of historical simulation approachlThe first step is to identify the market variables affecting the portfolio, these will typically be exchange rates, equity prices, interest rates and so on.lData are then collected on movements in these market variables over the most recent 501 days. this provid

4、ed 500 alternative scenarios for what can happen between today and tomorrow.lFor each scenario, the dollar change in the value of the portfolio between today and tomorrow is calculated. This defines a probability distribution for daily loss( gains are negative losses) in the value of our portfolio.l

5、The 99th percentile of the distribution can be estimated as the fifth-worst loss( there are alternatives here, a case can be made for using the fifth-worst loss, the sixth-worst loss, or an average of the two).lThe estimate of VaR is the loss when we are at this 99th percentile point.Algebraically e

6、xpression of the approachlDefine vi as the value of a market variable on day i and suppose that today is day n. the ith scenario in the historical simulation approach assumes that the value of the market variable tomorrow will be: Value under ith scenario=vn*vi/vi-1 (12.1)Illustration lTo illustrate

7、 the calculations underlying the approach, suppose that an investor owns, on September 25, 2008, a portfolio worth $10 million consisting of investments in four stock indices: the Dow Jones Industrial Average(DJIA) in the US, the FTSE 100 in the UK, the CAC 40 in France, and the Nikkei 225 in Japan.

8、 對表對表12.312.3的相關說明的相關說明（市場變量在（市場變量在2008年年9月月26日對于選定情形的取值）日對于選定情形的取值）l情形1假定9月25日至26日的市場價格百分比變化等同于2006年8月7日至8月8日之間市場價格的百分比變化。情形1是市場變量2008年9月26日的一種取值形式。l情形2假定9月25日至26日的市場價格百分比變化等同于2006年8月8日至8月9日之間市場價格的百分比變化，情形2也是市場變量在2008年9月26日的一種取值形式。l以此類推，情形i假定9月25日至9月26日的市場價格百分比變化等同于第i-1天與第i天的百分比變化(1i500),情形i定義了市場變量

9、在2008年9月26日的一種取值形式。情形情形1 1下組合資產價值的計算下組合資產價值的計算情形情形1 1下組合的收益為下組合的收益為21.50221.502千美元，即千美元，即-21.502-21.502千美元的損失。千美元的損失。結論：結論：1 1天展望期天展望期99%99%的置信水平下，的置信水平下，VaRVaR為為247 571247 571美元。美元。l因為1天展望期99%置信水平下的VaR為247 571美元，所以10天展望期99%置信水平下的VaR等于： 10*247571 782889美元12.2 Accuracy (page 184-185) Suppose that x i

10、s the qth quantile of the loss distribution when it is estimated from n observations. The standard error of x iswhere f(x) is an estimate of the probability density of the loss at the qth quantile calculated by assuming a probability distribution for the lossnqqxf)1 ()(1Risk Management and Financial

11、 Institutions 2e, Chapter 12, Copyright John C. Hull 200915Example 12.1 (page 184)lWe estimate the 0.01-quantile from 500 observations as $25 million lWe estimate f(x) by approximating the actual empirical distribution with a normal distribution mean zero and standard deviation $10 millionlThe 0.01 quantile of the approximating distribution is NORMINV(0.01,0,10) = 23.26 and the value of f(x) is NORMDIST(23.26,0,10,FALSE)=0.0027lThe estimate of the standard error is therefore Risk Management and Financial Institutions 2e, Chapter 12, Copyrigh