1.   (a)   (i)  Bookwork.

(ii)  Follows from properties of expectation and the answer to part (a)(i). (b) The utility function is appropriate for

π

2

(c) The coefficient of absolute risk aversion is

A(w)   =   =  3

The coefficient of relative risk aversion is

R(w) = 3

(d)   (i) The value of the single payment is 2, 244.43.

(ii)  Repeating the same calculation with initial wealth ✩500,000 gives the premium as 1, 895.23.  This is lower than before as Alice has decreasing absolute risk- aversion (i.e. is less worried by losses as wealth increases).

2.   (a)   (i) The minimum variance problem is

min  36ω1(2) _ 41ω1 + 20

subject to: 2ω1 + 3(1 _ ω 1 ) = µ

(ii) The values of ω 1  and ω2  are

ω 1  = 41       ω2  = 31

(iii)  If the returns on the two risky assets were independent we would expect an increase in the variance. The portfolio variance would be

σ 2  = 15ω1(2) + 20(1 _ ω 1 )2  > 15ω1(2) + 20(1 _ ω 1 )2 + 2ω1 (1 _ ω 1 ) . (_ ).

(b)   (i)  Bookwork.

(ii) The  market  portfolio  is the only efficient combination of the  risky assets. Portfolios on the capital market line consist of a proportion in the risk-free asset and the remaining capital allocation to the market portfolio.

(c)   (i)  Bookwork.

(ii)  If βá!M   > 1 then Var(Rá ) > Var(RM ) and the random variables Rá  and RM  are positively correlated.

3.   (a) Any two from the list below.

-  Measures volatility of gains as well as losses.

-  Not well suited to asymmetric loss distributions.

-  Only measures spread around the mean loss and does not tell us anything about magnitude of extreme losses.

- Two loss distributions could have the same variance, but with different prob- abilities of extreme outcomes.

(b)   (i)

VaR0!99  = 23.3.

(ii)  Bookwork.

(iii) The expected shortfall gives more information than the VaR about the losses in the tail. Expected shortfall is a coherent risk measure, so in particular displays sub-additivity. Either of these reasons would be acceptable.

(c)  In the elementary one period market model we have 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) The price is x = 0.64.

(ii) The price of the digital call option will be half of the price of the European call option from this example since the random payoff is always equal to half the payoff from the European option in part (c)(i).  There is no arbitrage in the market, so the only price can be 0.32.

(d) We have:

(i) The set of all risk-neutral measures can be described as: ．, ．(/) 1 入2_λ 2(3入)   ．(、)  I0 < λ <  、．

The restrictions on λ follow from the requirements that 1 > qi  > 0 for all i = 1, 2, 3. (Equivalent descriptions are possible).

(ii) Since there exists at least one risk-neutral measure, the model is arbitrage-free. (iii) This is an attainable contingent claim because (1, _ ) is a replicating strategy.

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

1. TRUE

2.  FALSE.

3.  FALSE.

4. TRUE.

(b)   (i)  Bookwork.

(ii) F1  is a σ-algebra.  F2  is not a σ-algebra, the complement of {ω1 } is not in

F

2 .

(iii) X is F1-measurable.  It is constant on the sets {ω1 } and {ω2 , ω3 , ω4 } which

are a partition for F1 .

(c)   (i) All subsets of Ω in F are in the collection g . (ii)  Bookwork.

(iii) A filtration represent the gradual evolution of available information in mar- ket models.  We are able to distinguish different subsets of states, but not necessarily individual states.

(d) The filtration is as follows:

F0     =   {0, Ω}

F1     =   {0, {ω1 }, {ω2 , ω3 }, Ω}

F2     =   p(Ω).