Download Economics 872

Economics 872
Milne 2010
The final exam will have three questions based on these problems.
1. Consider a 3 date binomial tree with two asset price processes. One asset is a riskless
bond with a constant interest rate of r and the other a risky asset with a constant volatility.
(a) Sketch the tree.
(b) Set out the mathematical problem for an agent with initial wealth, W0 and invests in
the two assets at time 0, rebalances at time 1, and consumes at time 2. Assume the
agent’s objective is E( ln (W2 ) ). Set out the first order conditions for a maximum and
derive the martingale pricing condition.
(c) Is the martingale measure unique and is it independent of the preferences of the
agent? Explain.
(d) Now assume that in the down state at t = 1 there is a possibility of a Flu epidemic.
i.e. assume that the down state is split into two down states, one with SARS and one
without. Draw the tree, and repeat the analysis of parts (b),(c). What has changed?
Explain. Hint: are markets complete? Could the markets be made complete? How?
(e) Assume that the Flu epidemic can be in the office of the agent but cannot occur
anywhere else. When it breaks out, the agent suspends trading. Show how that alters
the agent’s problem and solution.
(f) If the agent can spend an amount K to avoid any impact on trading by the Flu
epidemic, then how would the agent compute whether it would be worthwhile to
spend K?
(g) If the agent is a firm and two executives argue over the probability of a Flu epidemic,
and whether to invest in K, show how they can they can be both “rational”
optimizers. What are the crucial parameters, and what is the real issue?
2. (a) Set up a simple two date model of a bank. Assume costs of trading at the first date,
random costs at the second date, and different buying and selling costs for the
securities.
(b) Show that if there are no costs, and if there are differential buying and selling
prices, the bank would want to take an unbounded position in any asset.
(c) If there are costs of trading of trading at the first date, but none at the second date,
how does that modify your answer in (b)?
(d) If there are random operational risks (costs) at the second date, no costs at the first
date, and buying and selling prices of assets are equal, then give an example
where the operational risks can be hedged; and an example when they cannot.
(e) From case in (d), provide the first order conditions for a utility maximizing
solution. Explain the intuition.
3. You observe a star trader in the Kingston Bank, Sun Tzu, who for the last year has
recorded above normal returns using the Sharp ratio. Sun has had big bonuses and
boasts that he has a great trading strategy. As a risk manager, what should you do to
investigate Sun’s claim?
Hint 1: Recall Baring’s Nick Leeson.
Hint 2: Recall Jorion’s discussion of the multiperiod returns to writing out of the
money puts.
4. Consider an equally-weighted portfolio of two risky securities. The RM dept is using
a binomial model to model the returns of these risk assets.
Return to Asset 1
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
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.
5. 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 for 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 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? Explain with an example.