MTH312 Assignment 1
1. An individual has the following utility function:
U(w) = 1000w - w2 , 0 < w < 500,
where w is wealth in $s. The individual has a current wealth of $200.
(a) Show that the individual is risk averse. [5]
(b) Show that U(w) exhibits increasing absolute risk aversion and increasing relative risk aversion. [5]
The individual is offered a chance to gamble for free. The outcomes of the gamble are distributed as follows: $200 gain (i.e. increase in wealth) with probability 10%, no change in wealth with probability 70%, and $100 loss with probability 20%.
(c) Determine whether the individual should accept this gamble. [5]
The individual is offered another chance to gamble for free. The outcomes of the gamble are distributed as follows: $200 gain with probability 50%, and $100 loss with probability 50%.
(d) Determine whether the individual should accept this gamble. [5]
The person organizing the second gamble now wants to charge participants an entry fee. [5]
[Total 25 marks]
2. An insurer holds an asset with an annual return, R, where R has probability density function f(r). The insurer assumes that R follows a Normal(µ,σ2 ) distribution. Consider the 95% Value at Risk (VaR) where 95% VaR = -t if P(R < t) = 5%.
(a) Derive an expression for the 95% VaR for R in terms of µ and σ 2 . [5]
The insurer defines, for any percentage level, Tail VaR as E[-R|R < -VaR]. (b) Show that the 95% Tail VaR for R can be expressed as:
5%/σ f0,1 (−1.64485) − µ.
[10]
[Hint: Iff0,1 (r) is the probability density function of the Standard Normal Distribution then rf0,1 (r) = - dr/df0,1 (r).]
Assume now that µ = 0.04 and σ 2 = 0.006.
(c) Calculate for R:
i. the 95% VaR.
ii. the 95% Tail VaR. [5]
(d) Set out the benefits and limitations of the VaR and Tail VaR metrics for monitoring risk. [5]
[Total 25 marks]
3. (a) Define, in your own words, the market price of risk in modern portfolio theory. [5]
A panel of investment experts has provided annual expected investment re- turns for three asset classes that vary depending on the state of the economy (recession, normal or bubble). These are shown in the table below. You may assume the risk-free annual rate of return is 2%.
|
|
Asset class
|
Probability
|
|
Property
|
Stock
|
Bonds
|
|
Recession
|
-1%
|
-2%
|
6%
|
0.1
|
|
Normal
|
4%
|
6%
|
2%
|
0.6
|
|
Bubble
|
8%
|
12%
|
5%
|
0.3
|
|
|
|
|
|
|
|
Market capitalisation ($ billion)
|
25
|
50
|
50
|
|
(b) Calculate the market price of risk. [10]
An analyst is looking at a portfolio (PA ) that offers a mean return, µA = 6%, and standard deviation of return, σA = 3.15%.
(c) Construct an efficient portfolio, PB , such that µB = µA and σB < σA , where µB is the mean return of PB and σB is the standard deviation of return of PB . [5]
(d) Construct an efficient portfolio, PC , such that σC = σA and µC > µA . [5] [Total 25 marks]
4. Consider the standard one-period binomial model, with S0 = 4, u = 2, d = 1/2, r = ln(5/4), so that q = 1 - q = 1/2. Consider a bank that has a long position in the European put written on the stock price S1 . The put expires at time one and has strike price K = 7.
(a) Determine V0 , the time-zero price of this put. [5]
At time zero, the bank owns this option, which ties up capital V0 . The bank wants to earn the continuously compounding interest rate 100ln(5/4)% on this capital until time one, i.e., without investing any more money, and regardless of how the coin tossing turns out, the bank wants to have (5/4)V0 at time one, after collecting the payoff from the option (if any) at time one.
(b) Specify how the bank’s trader should invest in the stock and money markets to accomplish this. [20]
[Total 25 marks]