Ch14_ValueatRisk(金融工程.华东师范大学汤银才).ppt

Value at Risk Chapter 14 The Question Being Asked in Value at Risk (VaR) 揥hat loss level is such that we are X% confident it will not be exceeded in N business days?? Meaning is Probability (1-a) % a% Za VaR and Regulatory Capital Regulators require banks to keep capital for market risk equal to the average of VaR estimates for past 60 trading days using X=99 and N=10, times a multiplication factor. (Usually the multiplication factor equals 3) Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: 揌ow bad can things get?? Daily Volatilities In option pricing we express volatility as volatility per year In VaR calculations we express volatility as volatility per day Daily Volatility (continued) Strictly speaking we should define sday as the standard deviation of the continuously compounded return in one day In practice we assume that it is the standard deviation of the proportional change in one day IBM Example (p. 343) We have a position worth $10 million in IBM shares The volatility of IBM is 2% per day (about 32% per year) We use N=10 and X=99 IBM Example (continued) The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is IBM Example (continued) We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) We assume that the change in the value of the portfolio is normally distributed Since N(0.01)=-2.33, the VaR is ATT Example Consider a position of $5 million in ATT The daily volatility of ATT is 1% (approx 16% per year) The STD per 10 days is The VaR is Portfolio (p. 344) Now consider a portfolio consisting of both IBM and ATT Suppose that the correlation between the returns is 0.7 STD of Portfolio A standard result in statistics states that In this case sx = 632,456 and sY=158,114 and r = 0.7. The standard deviation of the chang