1.   (a)   (i) State the expected utility theorem.                                                  [2 marks]

(ii)  For the following utility functions,

U1 (w) = w2      and   U2 (w) = w2 - 

show that for any two portfolios W and W˜ these utility functions always result in the same preferences.                                                                   [3 marks]

(b) Assuming that an investor is non-satiated and risk-averse, in what subset of [0, 2T] must w be in so that U (w) = sin(w) is an appropriate utility function?   [4 marks]

(c)  For an investor with utility function, U (w) = w   where w  > 0, calculate the coefficient of absolute risk aversion and the coefficient of relative risk

aversion.                                                                                                 [4 marks]

(d) Alice believes that her car would cost $12,500 to replace if it was stolen or damaged. Based on crime statistics for the area she lives in, she believes that the probability of her car being stolen or damaged is 0.15.

(i) Alice’s utility function is given by  U (w)  =  ln(w) for w  >  0 and she has $35,000 in the bank. Calculate how much Alice would be prepared to pay (in a single payment) to insure her car against theft or damage.            [4 marks]

(ii)  Repeat the calculation in (d)(i) but now assume that Alice has $500,000 in

the bank. Give a very short explanation of your answer.                   [3 marks]

Total   [20 marks]

2.   (a)  For a market with two risky assets which have returns R1  and R2  respectively, we have the following information

E[R1] = µ 1  = 2,    E[R2] = µ2  = 3,    Var(R1 ) = σ1(2)  = 15,    Var(R2 ) = σ2(2)  = 20.

Moreover, we are given that the covariance of R1  and R2  is -  .  There are no other assets in the market.

(i) State the minimum variance problem for the market described above.   [4 marks]

(ii)  For the global minimum variance portfolio, find the proportion of wealth to invest in each asset, the expected return and the variance of return.   [8 marks]

(iii) Without further calculation, explain how you would expect the variance of return calculated in (a)(ii) to change if the two risky assets were                   independent.                                                                                    [2 marks]

(b)  Now suppose that a money market account (i.e. a risk-free asset) is added to the market model described in part (a).

(i) State the equation of the Capital Market Line, being careful to explain all of the notation you use.                                                                       [4 marks] (ii)  Describe the market portfolio and how it features in portfolios on the Capital Market Line.                                                                                    [2 marks]

(c)  Consider a particular risky asset with return Ra . Let RM  denote the return on the market portfolio.

(i) State the definition of βa,M , the “beta” of the risky asset.              [2 marks] (ii)  If the beta of the risky asset, βa,M , is greater than 1, how does Var(Ra )

compare with Var(RM ) and what can we say about the correlation coefficient

of returns on the market portfolio and the risky asset?                     [3 marks] Total   [25 marks]

3.   (a) State two disadvantages of using the standard deviation of returns as a risk         measure.                                                                                                 [2 marks]

(b)   (i) Suppose that the daily return, R, on a portfolio is normally distributed with expected value E[R] = 0 and standard deviation σ = 10.  Calculate VaR0.99 showing all of your working. You are given that for a standard normal random

variable, Z , P (Z < -2.33) = 0.01.                                                 [4 marks]

(ii) State the definition of the expected shortfall for a continuous loss distribution explaining any notation you use.                                                      [3 marks] (iii) State one advantage the expected shortfall has over the value at risk.   [1 mark]

(c)  In the elementary one period market model we have state space Ω = {H, T}, one risky asset S and a money market account with effective interest rate r = 3%.

A  European call option on the  risky asset  has strike  price  K  =  13.   You are given that S1 (H) = 15, S1 (T) = 8 and S0  = 10.

(i) Assuming no arbitrage, derive from first principles the option price.   [6 marks] (ii) The payoff, X, on a digital European put option is given by

X = { 0(1)

if K > S1

otherwise.

Without further calculation, state the price of the digital call option and briefly justify your answer.                                                                          [3 marks]

(d)  In the general single period market model with Ω = {ω1 , ω2 , ω3 }, one risky asset, S, and a money market account we have S0  = 4 for the risky asset. Moreover, the effective rate of interest on the money market account is 0% and at time t = 1 we have:

(i)  Calculate all risk-neutral probability measures for this model.           [7 marks]

(ii) State if the model is arbitrage-free.                                                  [2 marks]

(iii) A large bank has designed an investment product with payoff X at time t = 1. Given,

[4 marks] Total   [30 marks]

4.   (a) State whether the following are TRUE or FALSE in the context of the general single period market model.

1.  If the trading strategy (x, φ) is an arbitrage in some market model with finite

E[Gˆ (x, φ)] > 0

where Gˆ (x, φ) is the discounted gains process.

2.  If a market model is arbitrage-free, then all contingent claims have unique risk-neutral price.

3.  In complete market models, it is possible to find a contingent claim which does not have a replicating trading strategy.

4.  In an arbitrage-free and complete market model, there exists only one possible risk-neutral probability measure.

[4 marks] (b) Suppose a multi-period market model has a finite state space, Ω = {ω1 , ω2 , . . . , ωk }.

(i) State the definition of a σ-algebra on Ω .                                          [3 marks]

Now suppose that we have a state space Ω = {ω1 , ω2 , ω3 , ω4 }.

(ii)  For each of the collections of subsets of Ω = {ω1 , ω2 , ω3 , ω4 } set out below,

state whether it is a σ-algebra or not. If it is not, explain why.

r1     =   {0, {ω1 }, {ω2 , ω3 , ω4 }, Ω}

r2     =   {0, {ω1 }, {ω1 , ω2 }, {ω3 , ω4 }, Ω}

(iii)  Let X : Ω - R be a contingent claim such that

X(ω) =

State whether or not X is r1-measurable? Justify your answer.

(c)   (i)  Let r and g be σ-algebras on a set Ω . State what r c g means. (ii) What is a filtration (rt ) of σ-algebras on a set Ω .

(iii)  Provide an explanation of what filtrations are used to represent in multi-period market models.

[5 marks]

(d) A two-period market has a single risky asset and Ω = {ω1 , ω2 , ω3 }. The risky asset prices are as follows

Calculate the filtration generated by the asset prices.

[7 marks] Total   [25 marks]