Economics 872 Risk Management
Frank Milne Spring 2011
Theory Exercise Sheet #1
You should read Jorion Chs 4-9.
- Consider an equally-weighted portfolio of two risky securities. The RM dept is using a two date binomial model to model the returns of these risk assets. The price today for each asset is $1.50
Total $ Return to Asset 1 Total $ Return to Asset 2
State 1 2 1 Prob(state 1) = 0.5
State 2 1 2 Prob(state 2) = 0.5
(a)Compute the Mean, Standard Deviation and Correlation Coefficient for the two securities.
(b) Compute the Mean and Standard Deviation for the return on the equally weighted portfolio.
(c) What is the 99% VaR for the portfolio. What does your result mean? Explain.
(d) If there is a riskless asset, then can you relate the return on the portfolio to the riskless rate?
Now the RM Dept is worrying about a recession in the next period, and posits a possible downward shock in a third state that will destroy the value of both assets. The returns and probabilities are:
Return to Asset 1 Return to Asset 2
State 1 2 1 Prob(state 1) = 0.45
State 2 1 2 Prob(state 2) = 0.45
State 3 0 0 Prob (state 3) = 0.1
Redo the calculations in (a)-(d) and explain the difference in the results from the previous results.
- Frank’s Bank has a portfolio of two major funds, A and B.
The mean monthly, return for the two funds are 1.5% and 2.2% respectively.
The standard deviation of monthly returns for the two funds are 0.9% and 1.2% respectively. The correlation between the two funds is 0.13. Given the bank holds 70% in A and 30% in B, and returns are assumed to be bivariate normal, then compute the monthly VaR at 99% level, given that the portfolio is $100 million.
How critical is the assumption of normality in the distribution?
- Frank’s bank has issued a European derivative security that expires in one month’s
time. The derivative payoff at the end of the month is a known function C(S), where
S is the value of a stock index. Assume that we can write the stock index price
Process for the next month as: S = S..t + . S.B, where the time interval is one
month, is the monthly mean return, is the volatility and B is a binomial process
which has a prob ½ of an upward move of +1 and the prob ½ of a downward move
of -1. Assume the existence of a riskless rate of r for the month.
(a)Can you perfectly hedge the derivative? Explain.
(b)Can you price the derivative by absence of arbitrage arguments?
(c)Assume now that the derivative expires in two months time. Use the price process to derive the possible prices for the index in two months. Then given that you rebalance your portfolio of borrowing or lending and the index each month, then can you hedge and price the derivative?
(d)Given that you have written the call option and can hedge the position, what is the VaR on the portfolio of the stock index, bond and call option?
- Af’s Bank assumes a more complex stochastic process than Frank’s bank. They issued the same derivative, but assume that the Stock index process has a normal error term ie. S = S..t + . S.W, where W is a standardized normal distribution.
(a)Can Af’s bank hedge the derivative using the stock and bond, given that they
can only trade at the beginning of the month?
(b)Will they make gains and losses? Explain.
(c)If they could trade frequently during the month (say every day), could they reduce their losses (and gains)? Assume that price process stays in the same form, except that the time intervals and parameters are now for a day.
(d)How would you go about calculating the VaR on the net position of the stock, bond and call option? Does your answer differ from question 3.(d)? Explain.