2. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Zvi Wiener mswiener@mscc.huji.ac.il 02-588-3049 Financial Risk Management

3. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Following P. Jorion, Value at Risk, McGraw-Hill Chapter 4 Measuring Financial Risk Financial Risk Management

4. Zvi WienerVaR-PJorion-Ch 4-6 slide 3 Risks Measures Durationbonds, futures, fixed income Convexitybonds Betadiversified portfolio SigmaFX, undiversified portfolio Deltaoptions Gammaoptions Risk is measured by short term volatility

5. Zvi WienerVaR-PJorion-Ch 4-6 slide 4 Basic Statistics Certainty and uncertainty Probabilities, distribution, PDF, CDF Mean, variance Multivariable distributions Covariance, correlation, beta Quantile

6. Zvi WienerVaR-PJorion-Ch 4-6 slide 5 A100 km.B 100 km/hr 50 km/hr 1 – 1002 – 503 – 50 (100+50+50)/3 = 66.67 km/hr.

7. Zvi WienerVaR-PJorion-Ch 4-6 slide 6 1.+40% 2.+10% 3.-50% 4.+20% 1.-2% 2.+1% 3.-1% 4.+1% 1.4*1.1*0.5*1.2 = 0.924 0.98*1.01*0.99*1.01 = 0.9897

8. Zvi WienerVaR-PJorion-Ch 4-6 slide 7 Probabilities Certainty Uncertainty Probabilities

9. Zvi WienerVaR-PJorion-Ch 4-6 slide 8 Probabilities Mean Variance

10. Zvi WienerVaR-PJorion-Ch 4-6 slide 9 Probabilities 1234512345 0.2 0.3 0.1 20% 30% 10%

11. Zvi WienerVaR-PJorion-Ch 4-6 slide 10 Probabilities 1234512345 0.2 0.3 0.1

12. Zvi WienerVaR-PJorion-Ch 4-6 slide 11 Probabilities

13. Zvi WienerVaR-PJorion-Ch 4-6 slide 12 Probabilities

14. Zvi WienerVaR-PJorion-Ch 4-6 slide 13 Sample Estimates Sometimes one can use weights

15. Zvi WienerVaR-PJorion-Ch 4-6 slide 14 Normal Distribution N( ,  )

16. Zvi WienerVaR-PJorion-Ch 4-6 slide 15 Normal Distribution N( ,  )  

17. Zvi WienerVaR-PJorion-Ch 4-6 slide 16 Normal Distribution  quantile 1%

18. Zvi WienerVaR-PJorion-Ch 4-6 slide 17 Lognormal Distribution

19. Zvi WienerVaR-PJorion-Ch 4-6 slide 18 Covariance Shows how two random variables are connected For example: independent move together move in opposite directions covariance(X,Y) =

20. Zvi WienerVaR-PJorion-Ch 4-6 slide 19 Correlation -1    1  = 0 independent  = 1 perfectly positively correlated  = -1 perfectly negatively correlated

21. Zvi WienerVaR-PJorion-Ch 4-6 slide 20 Properties

22. Zvi WienerVaR-PJorion-Ch 4-6 slide 21 Time Aggregation Assuming normality

23. Zvi WienerVaR-PJorion-Ch 4-6 slide 22 Time Aggregation Assume that yearly parameters of CPI are: mean = 5%, standard deviation (SD) = 2%. Then daily mean and SD of CPI changes are:

24. Zvi WienerVaR-PJorion-Ch 4-6 slide 23 Portfolio  2 (A+B) =  2 (A) +  2 (B) + 2  (A)  (B)    rfrf A B

25. Zvi WienerVaR-PJorion-Ch 4-6 slide 24 $ £ ¥  $¥  £¥  £$ $  £¥ £  $¥ ¥  $£

26. Zvi WienerVaR-PJorion-Ch 4-6 slide 25   12 11 22 John Zerolis "Triangulating Risk", Risk v.9 n.12, Dec. 1996

27. Zvi WienerVaR-PJorion-Ch 4-6 slide 26 Example We will receive n dollars where n is determined by a die. What would be a fair price for participation in this game?

28. Zvi WienerVaR-PJorion-Ch 4-6 slide 27 Example 1 ScoreProbability 11/6 21/6 31/6 41/6 51/6 61/6 Fair price is 3.5 NIS. Assume that we can play the game for 3 NIS only.

29. Zvi WienerVaR-PJorion-Ch 4-6 slide 28 Example If there is a pair of dice the mean is doubled. What is the probability to gain $5?

30. Zvi WienerVaR-PJorion-Ch 4-6 slide 29 Example 1,12,13,14,15,16,1 1,22,23,24,25,26,2 1,32,33,34,35,36,3 1,42,43,44,45,46,4 1,52,53,54,55,56,5 1,62,63,64,65,66,6 All combinations: 36 combinations with equal probabilities

31. Zvi WienerVaR-PJorion-Ch 4-6 slide 30 Example 1,12,13,14,15,16,1 1,22,23,24,25,26,2 1,32,33,34,35,36,3 1,42,43,44,45,46,4 1,52,53,54,55,56,5 1,62,63,64,65,66,6 All combinations: 4 out of 36 give $5, probability = 1/9

32. Zvi WienerVaR-PJorion-Ch 4-6 slide 31 All combinations: 1 out of 9 give $5, probability = 1/9 Additional information: the first die gives 4. 1,12,13,14,15,16,1 1,22,23,24,25,26,2 1,32,33,34,35,36,3 1,42,43,44,45,46,4 1,52,53,54,55,56,5 1,62,63,64,65,66,6

33. Zvi WienerVaR-PJorion-Ch 4-6 slide 32 All combinations: 4 out of 24 give $5, probability = 1/6 Additional information: the first die gives  4. 1,12,13,14,15,16,1 1,22,23,24,25,26,2 1,32,33,34,35,36,3 1,42,43,44,45,46,4 1,52,53,54,55,56,5 1,62,63,64,65,66,6

34. Zvi WienerVaR-PJorion-Ch 4-6 slide 33 Example 1 -2 -1 0 1 2 3

35. Zvi WienerVaR-PJorion-Ch 4-6 slide 34 Example 1 123456we pay 12345676 NIS. 2345678 3456789 45678910 567891011 6789101112

36. Zvi WienerVaR-PJorion-Ch 4-6 slide 35 P&L 123456 1-4-3-2-101 2-3-2-1012 3-2-10123 4-101234 5012345 6123456

37. Zvi WienerVaR-PJorion-Ch 4-6 slide 36 Example 1 (2 cubes)

38. Zvi WienerVaR-PJorion-Ch 4-6 slide 37 Example 1 (5 cubes)

39. Zvi WienerVaR-PJorion-Ch 4-6 slide 38 Random Variables Values, probabilities. Distribution function, cumulative probability. Example: a die with 6 faces.

40. Zvi WienerVaR-PJorion-Ch 4-6 slide 39 Random Variables Distribution function of a random variable X F(x) = P(X  x) - the probability of x or less. If X is discrete then If X is continuous then Note that

41. Zvi WienerVaR-PJorion-Ch 4-6 slide 40 Random Variables Probability density function of a random variable X has the following properties

42. Zvi WienerVaR-PJorion-Ch 4-6 slide 41 Moments Mean = Average = Expected value Variance

43. Zvi WienerVaR-PJorion-Ch 4-6 slide 42 Its meaning... Skewness (non-symmetry) Kurtosis (fat tails)

44. Zvi WienerVaR-PJorion-Ch 4-6 slide 43 Main properties

45. Zvi WienerVaR-PJorion-Ch 4-6 slide 44 Portfolio of Random Variables

46. Zvi WienerVaR-PJorion-Ch 4-6 slide 45 Transformation of Random Variables Consider a zero coupon bond If r=6% and T=10 years, V = $55.84, we wish to estimate the probability that the bond price falls below $50. This corresponds to the yield 7.178%.

47. Zvi WienerVaR-PJorion-Ch 4-6 slide 46 The probability of this event can be derived from the distribution of yields. Assume that yields change are normally distributed with mean zero and volatility 0.8%. Then the probability of this change is 7.06% Example

48. Zvi WienerVaR-PJorion-Ch 4-6 slide 47 Quantile Quantile (loss/profit x with probability c) 50% quantile is called median Very useful in VaR definition.

49. Zvi WienerVaR-PJorion-Ch 4-6 slide 48 FRM-99, Question 11 X and Y are random variables each of which follows a standard normal distribution with cov(X,Y)=0.4. What is the variance of (5X+2Y)? A. 11.0 B. 29.0 C. 29.4 D. 37.0

50. Zvi WienerVaR-PJorion-Ch 4-6 slide 49 FRM-99, Question 11

51. Zvi WienerVaR-PJorion-Ch 4-6 slide 50 FRM-99, Question 21 The covariance between A and B is 5. The correlation between A and B is 0.5. If the variance of A is 12, what is the variance of B? A. 10.00 B. 2.89 C. 8.33 D. 14.40

52. Zvi WienerVaR-PJorion-Ch 4-6 slide 51 FRM-99, Question 21

53. Zvi WienerVaR-PJorion-Ch 4-6 slide 52 Uniform Distribution Uniform distribution defined over a range of values a  x  b.

54. Zvi WienerVaR-PJorion-Ch 4-6 slide 53 Uniform Distribution abab 1

55. Zvi WienerVaR-PJorion-Ch 4-6 slide 54 Normal Distribution Is defined by its mean and variance. Cumulative is denoted by N(x).

56. Zvi WienerVaR-PJorion-Ch 4-6 slide 55 Normal Distribution 66% of events lie between -1 and 1 95% of events lie between -2 and 2

57. Zvi WienerVaR-PJorion-Ch 4-6 slide 56 Normal Distribution

58. Zvi WienerVaR-PJorion-Ch 4-6 slide 57 Normal Distribution symmetric around the mean mean = median skewness = 0 kurtosis = 3 linear combination of normal is normal 99.99 99.90 99 97.72 97.5 95 90 84.13 50 3.715 3.09 2.326 2.000 1.96 1.645 1.282 1 0

59. Zvi WienerVaR-PJorion-Ch 4-6 slide 58 Central Limit Theorem The mean of n independent and identically distributed variables converges to a normal distribution as n increases.

60. Zvi WienerVaR-PJorion-Ch 4-6 slide 59 Lognormal Distribution The normal distribution is often used for rate of return. Y is lognormally distributed if X=lnY is normally distributed. No negative values!

61. Zvi WienerVaR-PJorion-Ch 4-6 slide 60 Lognormal Distribution If r is the expected value of the lognormal variable X, the mean of the associated normal variable is r-0.5  2.

62. Zvi WienerVaR-PJorion-Ch 4-6 slide 61 Student t Distribution Arises in hypothesis testing, as it describes the distribution of the ratio of the estimated coefficient to its standard error. k - degrees of freedom.

63. Zvi WienerVaR-PJorion-Ch 4-6 slide 62 Student t Distribution As k increases t-distribution tends to the normal one. This distribution is symmetrical with mean zero and variance (k>2) The t-distribution is fatter than the normal one.

64. Zvi WienerVaR-PJorion-Ch 4-6 slide 63 Binomial Distribution Discrete random variable with density function: For large n it can be approximated by a normal.

65. Zvi WienerVaR-PJorion-Ch 4-6 slide 64 FRM-99, Question 12 For a standard normal distribution, what is the approximate area under the cumulative distribution function between the values -1 and 1? A. 50% B. 66% C. 75% D. 95% Error!

66. Zvi WienerVaR-PJorion-Ch 4-6 slide 65 FRM-99, Question 13 What is the kurtosis of a normal distribution? A. 0 B. can not be determined, since it depends on the variance of the particular normal distribution. C. 2 D. 3

67. Zvi WienerVaR-PJorion-Ch 4-6 slide 66 FRM-99, Question 16 If a distribution with the same variance as a normal distribution has kurtosis greater than 3, which of the following is TRUE? A. It has fatter tails than normal distribution B. It has thinner tails than normal distribution C. It has the same tail fatness as normal D. can not be determined from the information provided

68. Zvi WienerVaR-PJorion-Ch 4-6 slide 67 FRM-99, Question 5 Which of the following statements best characterizes the relationship between normal and lognormal distributions? A. The lognormal distribution is logarithm of the normal distribution. B. If ln(X) is lognormally distributed, then X is normally distributed. C. If X is lognormally distributed, then ln(X) is normally distributed. D. The two distributions have nothing in common

69. Zvi WienerVaR-PJorion-Ch 4-6 slide 68 FRM-98, Question 10 For a lognormal variable x, we know that ln(x) has a normal distribution with a mean of zero and a standard deviation of 0.2, what is the expected value of x? A. 0.98 B. 1.00 C. 1.02 D. 1.20

70. Zvi WienerVaR-PJorion-Ch 4-6 slide 69 FRM-98, Question 10

71. Zvi WienerVaR-PJorion-Ch 4-6 slide 70 FRM-98, Question 16 Which of the following statements are true? I. The sum of normal variables is also normal II. The product of normal variables is normal III. The sum of lognormal variables is lognormal IV. The product of lognormal variables is lognormal A. I and II B. II and III C. III and IV D. I and IV

72. Zvi WienerVaR-PJorion-Ch 4-6 slide 71 FRM-99, Question 22 Which of the following exhibits positively skewed distribution? I. Normal distribution II. Lognormal distribution III. The returns of being short a put option IV. The returns of being long a call option A. II only B. III only C. II and IV only D. I, III and IV only

73. Zvi WienerVaR-PJorion-Ch 4-6 slide 72 FRM-99, Question 22 C. The lognormal distribution has a long right tail, since the left tail is cut off at zero. Long positions in options have limited downsize, but large potential upside, hence a positive skewness.

74. Zvi WienerVaR-PJorion-Ch 4-6 slide 73 FRM-99, Question 3 It is often said that distributions of returns from financial instruments are leptokurtotic. For such distributions, which of the following comparisons with a normal distribution of the same mean and variance MUST hold? A. The skew of the leptokurtotic distribution is greater B. The kurtosis of the leptokurtotic distribution is greater C. The skew of the leptokurtotic distribution is smaller D. The kurtosis of the leptokurtotic distribution is smaller

75. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Following P. Jorion, Value at Risk, McGraw-Hill Chapter 5 Computing Value at Risk Financial Risk Management

76. Zvi WienerVaR-PJorion-Ch 4-6 slide 75

77. Zvi WienerVaR-PJorion-Ch 4-6 slide 76 Lunch Breakfast $2$4 $5$7$9 $11$13$15 50%  = $11  = ?? 50%

78. Zvi WienerVaR-PJorion-Ch 4-6 slide 77 Correlation  =+1 $2$4 $5$7$9 $11$13$15 Lunch Breakfast 50%  = $11  = $450%

79. Zvi WienerVaR-PJorion-Ch 4-6 slide 78 Correlation  =-1 $2$4 $5$7$9 $11$13$15 Lunch Breakfast 50%  = $11  = $250%

80. Zvi WienerVaR-PJorion-Ch 4-6 slide 79 Correlation  =0 $2$4 $5$7$9 $11$13$15 Lunch Breakfast 50%  = $11  = $3.1650%

81. Zvi WienerVaR-PJorion-Ch 4-6 slide 80 How to measure VaR Historical Simulations Variance-Covariance Monte Carlo Analytical Methods Parametric versus non-parametric approaches

82. Zvi WienerVaR-PJorion-Ch 4-6 slide 81 Historical Simulations Fix current portfolio. Pretend that market changes are similar to those observed in the past. Calculate P&L (profit-loss). Find the lowest quantile.

83. Zvi WienerVaR-PJorion-Ch 4-6 slide 82 Example 4.00 4.20 4.10 4.15 Assume we have $1 and our main currency is SHEKEL. Today $1=4.30. Historical data: 4.30*4.20/4.00 = 4.515 4.30*4.20/4.20 = 4.30 4.30*4.10/4.20 = 4.198 4.30*4.15/4.10 = 4.352 P&L 0.215 0 -0.112 0.052

84. Zvi WienerVaR-PJorion-Ch 4-6 slide 83 today USD NIS 2003 100 -120 2004 200 100 2005-300 -20 2006 20 30

85. Zvi WienerVaR-PJorion-Ch 4-6 slide 84 today Changes in IR USD: +1%+1% +1% +1% NIS: +1% 0% -1% -1%

86. Zvi WienerVaR-PJorion-Ch 4-6 slide 85 Returns year 1% of worst cases

87. Zvi WienerVaR-PJorion-Ch 4-6 slide 86 Profit/Loss VaR 1% VaR 1%

88. Zvi WienerVaR-PJorion-Ch 4-6 slide 87 Variance Covariance Means and covariances of market factors Mean and standard deviation of the portfolio Delta or Delta-Gamma approximation VaR 1% =  P – 2.33  P Based on the normality assumption!

89. Zvi WienerVaR-PJorion-Ch 4-6 slide 88  Variance-Covariance 2.33   -2.33  1%

90. Zvi WienerVaR-PJorion-Ch 4-6 slide 89 Monte Carlo

91. Zvi WienerVaR-PJorion-Ch 4-6 slide 90 Monte Carlo Distribution of market factors Simulation of a large number of events P&L for each scenario Order the results VaR = lowest quantile

92. Zvi WienerVaR-PJorion-Ch 4-6 slide 91 Monte Carlo Simulation

93. Zvi WienerVaR-PJorion-Ch 4-6 slide 92 Weights Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential. See RiskMetrics Technical Document for details.

94. Zvi WienerVaR-PJorion-Ch 4-6 slide 93 Stock Portfolio Single risk factor or multiple factors Degree of diversification Tracking error Rare events

95. Zvi WienerVaR-PJorion-Ch 4-6 slide 94 Bond Portfolio Duration Convexity Partial duration Key rate duration OAS, OAD Principal component analysis

96. Zvi WienerVaR-PJorion-Ch 4-6 slide 95 Options and other derivatives Greeks Full valuation Credit and legal aspects Collateral as a cushion Hedging strategies Liquidity aspects

97. Zvi WienerVaR-PJorion-Ch 4-6 slide 96 Credit Portfolio rating, scoring credit derivatives reinsurance probability of default recovery ratio

98. Zvi WienerVaR-PJorion-Ch 4-6 slide 97 Credit Rating and Default Rates RatingDefault frequency 1 year10 years Aaa0.02%1.49% Aa0.05%3.24% A0.09%5.65% Baa0.17%10.50% Ba0.77%21.24% B2.32%37.98%

99. Zvi WienerVaR-PJorion-Ch 4-6 slide 98 Returns Past spot rates S 0, S 1, S 2,…, S t. We need to estimate S t+1. Random variable Alternatively we can do

100. Zvi WienerVaR-PJorion-Ch 4-6 slide 99 Independent returns A very important question is whether a sequence of observations can be viewed as independent. If so, one could assume that it is drawn from a known distribution and then one can estimate parameters. In an efficient market returns on traded assets are independent.

101. Zvi WienerVaR-PJorion-Ch 4-6 slide 100 Random Walk We could consider that the observations r t are independent draws from the same distribution N( ,  2 ). They are called i.i.d. = independently and identically distributed. An extension of this model is a non-stationary environment. Often fat tails are observed.

102. Zvi WienerVaR-PJorion-Ch 4-6 slide 101 Time Aggregation

103. Zvi WienerVaR-PJorion-Ch 4-6 slide 102 Time Aggregation

104. Zvi WienerVaR-PJorion-Ch 4-6 slide 103 FRM-99, Question 4 Random walk assumes that returns from one time period are statistically independent from another period. This implies: A. Returns on 2 time periods can not be equal. B. Returns on 2 time periods are uncorrelated. C. Knowledge of the returns from one period does not help in predicting returns from another period D. Both b and c.

105. Zvi WienerVaR-PJorion-Ch 4-6 slide 104 FRM-99, Question 14 Suppose returns are uncorrelated over time. You are given that the volatility over 2 days is 1.2%. What is the volatility over 20 days? A. 0.38% B. 1.2% C. 3.79% D. 12.0%

106. Zvi WienerVaR-PJorion-Ch 4-6 slide 105 FRM-99, Question 14

107. Zvi WienerVaR-PJorion-Ch 4-6 slide 106 FRM-98, Question 7 Assume an asset price variance increases linearly with time. Suppose the expected asset price volatility for the next 2 months is 15% (annualized), and for the 1 month that follows, the expected volatility is 35% (annualized). What is the average expected volatility over the next 3 months? A. 22% B. 24% C. 25% D. 35%

108. Zvi WienerVaR-PJorion-Ch 4-6 slide 107 FRM-98, Question 7

109. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Following P. Jorion, Value at Risk, McGraw-Hill Chapter 6 Backtesting VaR Models Financial Risk Management

110. Zvi WienerVaR-PJorion-Ch 4-6 slide 109 Backtesting Verification of Risk Management models. Comparison if the model’s forecast VaR with the actual outcome - P&L. Exception occurs when actual loss exceeds VaR. After exception - explanation and action.

111. Zvi WienerVaR-PJorion-Ch 4-6 slide 110 Backtesting Green zone - up to 4 exceptions Yellow zone - 5-9 exceptions Red zone - 10 exceptions or more OK increasing k intervention

112. Zvi WienerVaR-PJorion-Ch 4-6 slide 111 Probability of Multiple Exceptions Each period the probability of exception is 1%, then after 250 business days the probability that there will be 0 exceptions is General formula of binomial distribution is

113. Zvi WienerVaR-PJorion-Ch 4-6 slide 112 The End

114. Zvi WienerVaR-PJorion-Ch 4-6 slide 113 FRM-00, Question 93 A fund manages an equity portfolio worth $50M with a beta of 1.8. Assume that there exists an index call option contract with a delta of 0.623 and a value of $0.5M. How many options contracts are needed to hedge the portfolio? A. 169 B. 289 C. 306 D. 321

115. Zvi WienerVaR-PJorion-Ch 4-6 slide 114 FRM-00, Question 93 The optimal hedge ratio is N = -1.8  $50,000,000/(0.623  $500,000)=289

116. Zvi WienerVaR-PJorion-Ch 4-6 slide 115 VaR system Risk factors Historical data Model Distribution of risk factors VaR method Portfolio positions Mapping Exposures VaR