Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Value at Risk Chapter 18

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull The Question Being Asked in VaR “What loss level is such that we are X % confident it will not be exceeded in N business days?”

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull VaR and Regulatory Capital Regulators base the capital they require banks to keep on VaR The market-risk capital is k times the 10- day 99% VaR where k is at least 3.0

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull VaR vs. C-VaR (See Figures 18.1 and 18.2) VaR is the loss level that will not be exceeded with a specified probability C-VaR is the expected loss given that the loss is greater than the VaR level Although C-VaR is theoretically more appealing than VaR, it is not widely used

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: “How bad can things get?”

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Historical Simulation (See Table 18.1 and 18.2) Create a database of the daily movements in all market variables. The first simulation trial assumes that the percentage changes in all market variables are as on the first day The second simulation trial assumes that the percentage changes in all market variables are as on the second day and so on

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Historical Simulation continued Suppose we use m days of historical data Let v i be the value of a market variable on day i There are m- 1 simulation trials The i th trial assumes that the value of the market variable tomorrow (i.e., on day m +1) is

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull The Model-Building Approach The main alternative to historical simulation is to make assumptions about the probability distributions of return on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically This is known as the model building approach or the variance-covariance approach

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Daily Volatilities In option pricing we express volatility as volatility per year In VaR calculations we express volatility as volatility per day

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Daily Volatility continued Strictly speaking we should define  day as the standard deviation of the continuously compounded return in one day In practice we assume that it is the standard deviation of the percentage change in one day

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Microsoft Example We have a position worth $10 million in Microsoft shares The volatility of Microsoft is 2% per day (about 32% per year) We use N = 10 and X = 99

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Microsoft 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

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Microsoft 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(–2.33)=0.01, the VaR is

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull AT&T Example Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx 16% per year) The S.D per 10 days is The VaR is

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Portfolio (See Trading Note 18.1) Now consider a portfolio consisting of both Microsoft and AT&T Suppose that the correlation between the returns is 0.3

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull S.D. of Portfolio A standard result in statistics states that In this case  X = 200,000 and  Y = 50,000 and  = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull VaR for Portfolio The 10-day 99% VaR for the portfolio is The benefits of diversification are (1,473, ,405)–1,622,657=$219,369 What is the incremental effect of the AT&T holding on VaR?

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull The Linear Model We assume The daily change in the value of a portfolio is linearly related to the daily returns from market variables The returns from the market variables are normally distributed

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull The General Linear Model continued (equations 18.1 and 18.2)

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Handling Interest Rates We do not want to define every bond as a different market variable We therefore choose as assets zero-coupon bonds with standard maturities: 1-month, 3 months, 1 year, 2 years, 5 years, 7 years, 10 years, and 30 years Cash flows from instruments in the portfolio are mapped to bonds with the standard maturities

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull When Linear Model Can be Used Portfolio of stocks Portfolio of bonds Forward contract on foreign currency Interest-rate swap

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull The Linear Model and Options Consider a portfolio of options dependent on a single stock price, S. Define and

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Linear Model and Options continued (equations 18.3 and 18.4) As an approximation Similar when there are many underlying market variables where  i is the delta of the portfolio with respect to the i th asset

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Example Consider an investment in options on Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively As an approximation where  x 1 and  x 2 are the percentage changes in the two stock prices

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Skewness (See Figures 18.3, 18.4, and 18.5) The linear model fails to capture skewness in the probability distribution of the portfolio value.

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Quadratic Model For a portfolio dependent on a single stock price where  is the gamma of the portfolio. This becomes

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Usual Approach to Estimating Volatility (equation 18.6) Define  n as the volatility per day between day n -1 and day n, as estimated at end of day n -1 Define S i as the value of market variable at end of day i Define u i = ln( S i /S i-1 ) The usual estimate of volatility from m observations is:

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Simplifications (equations 18.7 and 18.8) Define u i as (S i –S i-1 )/S i-1 Assume that the mean value of u i is zero Replace m–1 by m This gives

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Weighting Scheme Instead of assigning equal weights to the observations we can set

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull EWMA Model (equation 18.10) In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time This leads to

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Attractions of EWMA Relatively little data needs to be stored We need only remember the current estimate of the variance rate and the most recent observation on the market variable Tracks volatility changes RiskMetrics uses = 0.94 for daily volatility forecasting

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Correlations Define u i =(U i -U i-1 )/U i-1 and v i =(V i -V i-1 )/V i-1 Also  u,n : daily vol of U calculated on day n -1  v,n : daily vol of V calculated on day n -1 cov n : covariance calculated on day n -1 cov n =  n  u,n  v,n where  n on day n-1

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Correlations continued ( equation 18.12) Using the EWMA cov n = cov n -1 +(1- ) u n -1 v n -1

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Comparison of Approaches Model building approach has the disadvantage that it assumes that market variables have a multivariate normal distribution Historical simulation is computationally slower and cannot easily incorporate volatility updating schemes

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Back-Testing Tests how well VaR estimates would have performed in the past We could ask the question: How often was the loss greater than the 99%/10 day VaR?

Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Stress Testing This involves testing how well a portfolio would perform under some of the most extreme market moves seen in the last 10 to 20 years