Market Risk and Value at Risk

1. Market Riskand Value at Risk Finance 129

2. Market Risk • Macroeconomic changes can create uncertainty in the earnings of the Financial institutions trading portfolio. • Important because of the increased emphasis on income generated by the trading portfolio. • The trading portfolio (Very liquid i.e. equities, bonds, derivatives, foreign exchange) is not the same as the investment portfolio (illiquid ie loans, deposits, long term capital).

3. Importance of Market Risk Measurement • Management information – Provides info on the risk exposure taken by traders • Setting Limits – Allows management to limit positions taken by traders • Resource Allocation – Identifying the risk and return characteristics of positions • Performance Evaluation – trader compensation –did high return just mean high risk? • Regulation – May be used in some cases to determine capital requirements

4. Measuring Market Risk • The impact of market risk is difficult to measure since it combines many sources of risk. • Intuitively all of the measures of risk can be combined into one number representing the aggregate risk • One way to measure this would be to use a measure called the value at risk.

5. Value at Risk • Value at Risk measures the market value that may be lost given a change in the market (for example, a change in interest rates). that may occur with a corresponding probability • We are going to apply this to look at market risk.

6. Position A Position B Payout Prob Payout Prob -100 0.04 -100 0.04 0 0.96 0 0.96 VaR at 95% confidence level 0 VaR at 95% confidence level 0 A Simple Example From Dowd, Kevin 2002

7. A second simple example • Assume you own a 10% coupon bond that makes semi annual payments with 5 years until maturity with a YTM of 9%. • The current value of the bond is then 1039.56 • Assume that you believe that the most the yield will increase in the next day is .2%. The new value of the bond is 1031.50 • The difference would represent the value at risk.

8. VAR • The value at risk therefore depends upon the price volatility of the bond. • Where should the interest rate assumption come from? • historical evidence on the possible change in interest rates.

9. Calculating VaR • Three main methods • Variance – Covariance (parametric) • Historical • Monte Carlo Simulation • All measures rely on estimates of the distribution of possible returns and the correlation among different asset classes.

10. Variance / Covariance Method • Assumes that returns are normally distributed. • Using the characteristics of the normal distribution it is possible to calculate the chance of a loss and probable size of the loss.

11. Probability • Cardano 1565 and Pascal 1654 • Pascal was asked to explain how to divide up the winnings in a game of chance that was interrupted. • Developed the idea of a frequency distribution of possible outcomes. This slide and the next few based in part on Jorion, 1997

12. An example • Assume that you are playing a game based on the roll of two “fair” dice. • Each one has six possible sides that may land face up, each face has a separate number, 1 to 6. • The total number of dice combinations is 36, the probability that any combination of the two dice occurs is 1/36

13. Example continued • The total number shown on the dice ranges from 2 to 12. Therefore there are a total of 12 possible numbers that may occur as part of the 36 possible outcomes. • A frequency distribution summarizes the frequency that any number occurs. • The probability that any number occurs is based upon the frequency that a given number may occur.

14. Establishing the distribution • Let x be the random variable under consideration, in this case the total number shown on the two dice following each role. • The distribution establishes the frequency each possible outcome occurs and therefore the probability that it will occur.

15. Discrete Distribution Value 2 3 4 5 6 7 8 9 10 11 12 (x i) Freq 1 2 3 4 5 6 5 4 3 2 1 (n i) Prob 1 2 3 4 5 6 5 4 3 2 1 (p i) 36 36 36 36 36 36 36 36 36 36 36

16. Cumulative Distribution • The cumulative distribution represents the summation of the probabilities. • The number 2 occurs 1/36 of the time, the number 3 occurs 2/36 of the time. • Therefore a number equal to 3 or less will occur 3/36 of the time.

17. Cumulative Distribution Value2 3 4 5 6 7 8 9 10 11 12 Prob 1 2 3 4 5 6 5 4 3 2 1 (p i) 36 36 36 36 36 36 36 36 36 36 36 Cdf 1 3 6 10 15 21 26 30 33 35 36 36 36 36 36 36 36 36 36 36 36 36

18. Probability Distribution Function (pdf) • The probabilities form a pdf. The sum of the probabilities must sum to 1. • The distribution can be characterized by two variables, its mean and standard deviation

19. Mean • The mean is simply the expected value from rolling the dice, this is calculated by multiplying the probabilities by the possible outcomes (values). • In this case it is also the value with the highest frequency (mode)

20. Standard Deviation • The variance of the random variable is defined as: • The standard deviation is defined as the square root of the variance.

21. Using the example in VaR • Assume that the return on your assets is determined by the number which occurs following the roll of the dice. • If a 7 occurs, assume that the return for that day is equal to 0. If the number is less than 7 a loss of 10% occurs for each number less than 7 (a 6 results in a 10% loss, a 5 results in a 20% loss etc.) • Similarly if the number is above 7 a gain of 10% occurs.

22. Discrete Distribution Value 2 3 4 5 6 7 8 9 10 11 12 (x i) Return -50% -40% -30% -20% -10% 0 10% 20% 30% 40% 50% (n i) Prob 1 2 3 4 5 6 5 4 3 2 1 (p i) 36 36 36 36 36 36 36 36 36 36 36

23. VaR • Assume you want to estimate the possible loss that you might incur with a given probability. • Given the discrete dist, the most you might lose is 50% of the value of your portfolio. • VaR combines this idea with a given probability.

24. VaR • Assume that you want to know the largest loss that may occur in 95% of the rolls. • A 50% loss occurs 1/36 = 2.77% 0f the time. This implies that 1-.027 =.9722 or 97.22% of the rolls will not result in a loss of greater than 40%. • A 40% or greater loss occurs in 3/36=8.33% of rolls or 91.67% of the rolls will not result in a loss greater than 30%

25. Continuous time • The previous example assumed that there were a set number of possible outcomes. • It is more likely to think of a continuous set of possible payoffs. • In this case let the probability density function be represented by the function f(x)

26. Discrete vs. Continuous • Previously we had the sum of the probabilities equal to 1. This is still the case, however the summation is now represented as an integral from negative infinity to positive infinity. Discrete Continuous

27. Discrete vs. Continuous • The expected value of X is then found using the same principle as before, the sum of the products of X and the respective probabilities Discrete Continuous

28. Discrete vs. Continuous • The variance of X is then found using the same principle as before. Discrete Continuous

29. Combining Random Variables • One of the keys to measuring market risk is the ability to combine the impact of changes in different variables into one measure, the value at risk. • First, lets look at a new random variable, that is the transformation of the original random variable X. • Let Y=a+bX where a and b are fixed parameters.

30. Linear Combination • The expected value of Y is then found using the same principle as before, the sum of the products of Y and the respective probabilities

31. Linear Combinations • We can substitute since Y=a+bX, then simplify by rearranging

32. Variance • Similarly the variance can be found

33. Standard Deviation • Given the variance it is easy to see that the standard deviation will be

34. Combinations of Random Variables • No let Y be the linear combination of two random variables X1 and X2 the probability density function (pdf) is now f(x1,x2) • The marginal distribution presents the distribution as based upon one variable for example.

35. Expectations

36. Variance • Similarly the variance can be reduced

37. A special case • If the two random variables are independent then the covariance will reduce to zero which implies that V(X1+X2) = V(X1)+V(X2) • However this is only the case if the variables are independent – implying hat there is no gain from diversification of holding the two variables.

38. The Normal Distribution • For many populations of observations as the number of independent draws increases, the population will converge to a smooth normal distribution. • The normal distribution can be characterized by its mean (the location) and variance (spread) N(m,s2). • The distribution function is

39. Standard Normal Distribution • The function can be calculated for various values of mean and variance, however the process is simplified by looking at a standard normal distribution with mean of 0 and variance of 1.

40. Standard Normal Distribution • Standard Normal Distributions are symmetric around the mean. The values of the distribution are based off of the number of standard deviations from the mean. • One standard deviation from the mean produces a confidence interval of roughly 68.26% of the observations.

41. Prob Ranges for Normal Dist. 68.26% 95.46% 99.74%

42. An Example • Lets define X as a function of a standard normal variable e (in other words e is N(0,1)) X= m + es • We showed earlier that • Therefore

43. Variance • We showed that the variance was equal to • Therefore

44. An Example • Assume that we know that the movements in an exchange rate are normally distributed with mean of 1% and volatility of 12%. • Given that approximately 95% of the distribution is within 2 standard deviations of the mean it is easy to approximate the highest and lowest return with 95% confidence XMIN = 1% - 2(12%) = -23% XMAX = 1% + 2(12%) = +25%

45. One sided values • Similarly you can find the standard deviation that represents a one sided distribution. • Given that 95.46% of the distribution lies between -2 and +2 standard deviations of the mean, it implies that (100% - 95.46)/2 = 2.27% of the distribution is in each tail. • This shows that 95.46% + 2.27% = 97.73% of the distribution is to the right of this point.

46. VaR • Given the last slide it is easy to see that you would be 97.73% confident that the loss would not exceed -23%.

47. Continuous Time • Let q represent quantile such that the area to the right of q represents a given probability of occurrence. • In our example above -2.00 would produce a probability of 97.73% for the standard normal distribution

48. VAR A second example • Assume that the mean yield change on a bond was zero basis points and that the standard deviation of the change was 10 Bp or 0.001 • Given that 90% of the area under the normal distribution is within 1.65 standard deviationson either side of the mean (in other words between mean-1.65s and mean +1.65s) • There is only a 5% chance that the level of interest rates would increase or decrease by more than 0 + 1.65(0.001) or 16.5 Bp

49. Price change associated with 16.5Bp change. • You could directly calculate the price change, by changing the yield to maturity by 16.5 Bp. • Given the duration of the bond you also could calculate an estimate based upon duration.

50. Example 2 • Assume we own seven year zero coupon bonds with a face value of $1,631,483.00 with a yield of 7.243% • Today’s Market Value $1,631,483/(1.07243)7=$1,000,000 • If rates increase to 7.408 the market value is $1,631,483/(1.07408)7 = $989,295.75 • Which is a value decrease of $10,704.25