Exercise 10

Exercise Session 10, Nov 24th ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva Exercise 1 Assume a one-year Treasury bond o¤ ers a safe annual rate of return of 10%. You are about to lend $1 million to a risky company. The loan maturity is one year. (a) What is your expected rate of return if you decide to charge 10%, but the company pays $750.000 in case of default (fails to pay all that it has promised), which happens with probability of 50%, otherwise the company is solvent (pays back the principal plus interest). (b) What is the interest rate you need to charge to break even. Exercise 2 Show that for any random variable X, with mean ance 2 ; > 0 and any constant k > 0; P (jX j
and …nite vari1 k2 :

Exercise 3 The monthly return of AMERICATEL a company traded at the NASDAQ, has been observed for a long period of time and tabulated at the last day of the month. It is found to have mean 12 and standard deviation of 2. What is the probability that in the next observation the return is greater than 8 but less than 16 ? x2

Exercise 4 Consider the pdf de…ned by f (x) = p12 e 2 , x 2 ( 1; 1):Show (a) It is well de…ned pdf function. (b) Suppose the stock return you have invested follows the above distribution. Find the expected value for your stock return. (c) What is the variance and the skewness of your stock return? Exercise 5 Assume a random variable X follows exponential distribution with e x; x 0 pdf: f (x; ) = : 0 x<0 (a) Compute cdf, mean , variance and E(euX ). (b) Show X is memoryless w.r.t t; which means P (X > s + t j x > t) = P (X > s); for all s; t 0. Exercise 6 Assume after you graduate from HEC, you …nd a job in UBS as a trader. Since you have no inside information about the market movement, you need to update your belief according to the latest buy or sell transaction. Suppose investors in stock markets can be classi…ed by two di¤ erent group of people: rational investors and crazy investors with probability 70% and 30%. The action of crazy investors does not convey any information. Crazy investors would buy or sell stocks with 50% possibility. Rational investors would buy stocks only when they receive a good signal. Assume the probability for rational investors who would buy stocks is 60%. Could you identify the probability of rational investor based on a buy transaction? Exercise 7 Exercise Session 9, Q2,Q4.

1