1. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook Financial Risk Management

2. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Chapter 2 Quantitative Analysis Fundamentals of Probability Following P. Jorion 2001 Financial Risk Manager Handbook

3. Ch. 2, HandbookZvi Wiener slide 3 Random Variables Values, probabilities. Distribution function, cumulative probability. Example: a die with 6 faces.

4. Ch. 2, HandbookZvi Wiener slide 4 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

5. Ch. 2, HandbookZvi Wiener slide 5 Random Variables Probability density function of a random variable X has the following properties

6. Ch. 2, HandbookZvi Wiener slide 6 Multivariate Distribution Functions Joint distribution function Joint density - f 12 (u 1,u 2 )

7. Ch. 2, HandbookZvi Wiener slide 7 Independent variables Credit exposure in a swap depends on two random variables: default and exposure. If the two variables are independent one can construct the distribution of the credit loss easily.

8. Ch. 2, HandbookZvi Wiener slide 8 Conditioning Marginal density Conditional density

9. Ch. 2, HandbookZvi Wiener slide 9 Moments Mean = Average = Expected value Variance

10. Ch. 2, HandbookZvi Wiener slide 10 Its meaning... Skewness (non-symmetry) Kurtosis (fat tails)

11. Ch. 2, HandbookZvi Wiener slide 11 Main properties

12. Ch. 2, HandbookZvi Wiener slide 12 Portfolio of Random Variables

13. Ch. 2, HandbookZvi Wiener slide 13 Portfolio of Random Variables

14. Ch. 2, HandbookZvi Wiener slide 14 Product of Random Variables Credit loss derives from the product of the probability of default and the loss given default. When X 1 and X 2 are independent

15. Ch. 2, HandbookZvi Wiener slide 15 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%.

16. Ch. 2, HandbookZvi Wiener slide 16 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

17. Ch. 2, HandbookZvi Wiener slide 17 Quantile Quantile (loss/profit x with probability c) 50% quantile is called median Very useful in VaR definition.

18. Ch. 2, HandbookZvi Wiener slide 18 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

19. Ch. 2, HandbookZvi Wiener slide 19 FRM-99, Question 11

20. Ch. 2, HandbookZvi Wiener slide 20 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

21. Ch. 2, HandbookZvi Wiener slide 21 FRM-99, Question 21

22. Ch. 2, HandbookZvi Wiener slide 22 Uniform Distribution Uniform distribution defined over a range of values a  x  b.

23. Ch. 2, HandbookZvi Wiener slide 23 Uniform Distribution abab 1

24. Ch. 2, HandbookZvi Wiener slide 24 Normal Distribution Is defined by its mean and variance. Cumulative is denoted by N(x).

25. Ch. 2, HandbookZvi Wiener slide 25 Normal Distribution 66% of events lie between -1 and 1 95% of events lie between -2 and 2

26. Ch. 2, HandbookZvi Wiener slide 26 Normal Distribution

27. Ch. 2, HandbookZvi Wiener slide 27 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

28. Ch. 2, HandbookZvi Wiener slide 28 Central Limit Theorem The mean of n independent and identically distributed variables converges to a normal distribution as n increases.

29. Ch. 2, HandbookZvi Wiener slide 29 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!

30. Ch. 2, HandbookZvi Wiener slide 30 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.

31. Ch. 2, HandbookZvi Wiener slide 31 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.

32. Ch. 2, HandbookZvi Wiener slide 32 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.

33. Ch. 2, HandbookZvi Wiener slide 33 Binomial Distribution Discrete random variable with density function: For large n it can be approximated by a normal.

34. Ch. 2, HandbookZvi Wiener slide 34 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!

35. Ch. 2, HandbookZvi Wiener slide 35 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

36. Ch. 2, HandbookZvi Wiener slide 36 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

37. Ch. 2, HandbookZvi Wiener slide 37 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

38. Ch. 2, HandbookZvi Wiener slide 38 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

39. Ch. 2, HandbookZvi Wiener slide 39 FRM-98, Question 10

40. Ch. 2, HandbookZvi Wiener slide 40 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

41. Ch. 2, HandbookZvi Wiener slide 41 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

42. Ch. 2, HandbookZvi Wiener slide 42 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.

43. Ch. 2, HandbookZvi Wiener slide 43 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