財(cái)務(wù)風(fēng)險(xiǎn)管理CH15課件

1、Chapter 15Market Risk VaR: Model-Building Approach1The Model-Building ApproachThe main alternative to historical simulation is to make assumptions about the probability distributions of the returns on the market variablesThis is known as the model building approach (or sometimes the variance-covaria

2、nce approach)2Microsoft Example (324)We have a position worth $10 million in Microsoft sharesThe volatility of Microsoft is 2% per day (about 32% per year)We use N=10 and X=993Microsoft Example continuedWe assume that the expected change in the value of the portfolio is zero (This is OK for short ti

3、me periods)We assume that the change in the value of the portfolio is normally distributedSince N(2.33)=0.01, the VaR is 5AT&T ExampleConsider a position of $5 million in AT&TThe daily volatility of AT&T is 1% (approx 16% per year)The SD per 10 days isThe VaR is6Portfolio (page 325)Now consider a po

4、rtfolio consisting of both Microsoft and AT&TSuppose that the correlation between the returns is 0.37S.D. of PortfolioA standard result in statistics states thatIn this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in the portfolio value in one day is therefore

5、220,2278The Linear ModelWe assumeThe daily change in the value of a portfolio is linearly related to the daily returns from market variablesThe returns from the market variables are normally distributed10Corresponding Result for Variance of Portfolio Value si is the daily volatility of the ith asset

6、 (i.e., SD of daily returns)sP is the SD of the change in the portfolio value per dayai =wi P is amount invested in ith asset 12Alternative Expressions for sP2page 32814Four Index Example Using Last 500 Days of Data to Estimate CovariancesEqual Weight EWMA : l=0.94One-day 99% VaR$217,757$471,02515Vo

7、latilities and Correlations Increased in Sept 2008DJIAFTSECACNikkeiEqual Weights1.111.421.401.38EWMA2.193.213.091.59CorrelationsVolatilities (% per day)16Alternatives for Handling Interest RatesDuration approach: Linear relation between DP and Dy but assumes parallel shifts)Cash flow mapping: Variab

8、les are zero-coupon bond prices with about 10 different maturitiesPrincipal components analysis: 2 or 3 independent shifts with their own volatilities17Handling Interest Rates: Cash Flow Mapping (333) We choose as market variables zero-coupon bond prices with standard maturities (1mm, 3mm, 6mm, 1yr,

9、 2yr, 5yr, 7yr, 10yr, 30yr)Suppose that the 5yr rate is 6% and the 7yr rate is 7% and we will receive a cash flow of $10,000 in 6.5 years.The volatilities per day of the 5yr and 7yr bonds are 0.50% and 0.58% respectively18Example continuedWe interpolate between the 0.5% volatility for the 5yr bond p

10、rice and the 0.58% volatility for the 7yr bond price to get 0.56% as the volatility for the 6.5yr bondWe allocate a of the PV to the 5yr bond and (1- a) of the PV to the 7yr bond20Example continuedSuppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6To match variancesTh

11、is gives a=0.07421Using a PCA to Calculate VaR (page 333 to 334)Suppose we calculatewhere f1 is the first factor and f2 is the second factorIf the SD of the factor scores are 17.55 and 4.77 the SD of DP is23When Linear Model Can be UsedPortfolio of stocksPortfolio of bondsForward contract on foreign

12、 currencyInterest-rate swap24Linear Model and Options continued As an approximationSimilarly when there are many underlying market variableswhere di is the delta of the portfolio with respect to the ith asset26ExampleConsider an investment in options on Microsoft and AT&T. Suppose the stock prices a

13、re 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectivelyAs an approximationwhere Dx1 and Dx2 are the percentage changes in the two stock prices 27But the Distribution of the Daily Return on an Option is not Normal The linear mo

14、del fails to capture skewness in the probability distribution of the portfolio value. 28Translation of Asset Price Change to Price Change for Long Call (Figure 15.2, page 337)Long CallAsset Price30Translation of Asset Price Change to Price Change for Short Call (Figure 15.3, page 338)Short CallAsset

15、 Price31Quadratic Model (340)For a portfolio dependent on a single asset price it is approximately true thatso thatMoments are32Quadratic Model continuedWhen there are a small number of underlying market variable moments can be calculated analytically from the delta/gamma approximationThe Cornish Fi

16、sher expansion can then be used to convert moments to fractilesHowever when the number of market variables becomes large this is no longer feasible33Monte Carlo Simulation (341)To calculate VaR using MC simulation weValue portfolio todaySample once from the multivariate distributions of the Dxi Use

17、the Dxi to determine market variables at end of one dayRevalue the portfolio at the end of day34Monte Carlo Simulation continuedCalculate DPRepeat many times to build up a probability distribution for DPVaR is the appropriate fractile of the distribution times square root of NFor example, with 1,000

18、 trial the 1 percentile is the 10th worst case.35Speeding up Calculations with the Partial Simulation ApproachUse the approximate delta/gamma relationship between DP and the Dxi to calculate the change in value of the portfolioThis is also a way of speeding up computations in the historical simulation approach36Alternative to Normal Distribution Assumption in Monte CarloIn a Monte Carlo simulation we can assume non-normal distributions for the xi (e.g., a multivariate t-distribution)Can also use a Gaussian or other copula m