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????1 and ????2 , ????????1 ,????2 denotes the correlation coefficient, ????????1 ,????2 = ????????????(????1 , ????2 )/(????????1 ????????2 ).

Notation: 0.01 and 1% is the same.

QUESTION 1

A lottery ticket with random return R has the following properties.

P(R = ?100£) = 10%, P(R = ?20£) = 20%, P(R = 0£) = 10%,

P(R = 20£) = 45%, P(R = 100£) = 15%

(i) Calculate the following risk measures

(a) The downside semi-variance of return.

[3 marks]

(b) The shortfall probability based on a benchmark return of 10£.

[3 marks]

(c) The expected shortfall based on a benchmark return of 10£ conditional on a

shortfall occurring.

[2 marks]

(d) The 5% value at risk (VaR).

[3 marks]

(ii) Would an investor with utility function U (R) = ln(1000 + R) pay 8£ for the lottery

ticket described above?

[Total: 11 marks]

Page 1 of 6

QUESTION 2

(i) Show that the expected utility can be defined in terms of the mean and variance when

it is a quadratic function, i.e. when it is of the form

U (W ) = W ? bW 2

where b is a constant.

[4 marks]

(ii) (a) An actuarial trainee uses the function U 00 (W ) (the second derivative of the utility function) to measure the extent to which investors are risk-averse. Two investors exhibit risk-aversion and non-satiation and have utility functions U1 (W )

and U2 (W ) which appear below.

U (W )

U1 (W )

U2 (W )

W

Give one example of a possible function for each of U1 (W ) and U2 (W ) such that

the two investors have equal coefficients of absolute risk aversion for a given

wealth.

[2 marks]

(b) You are given the following information on projects A and B:

Return

Probability in Lottery A Probability in Lottery B

Zero

0.80

0.35

10 units

0.10

0.60

100 units

0.10

0.05

Determine whether either lottery offers a distribution of winnings that exhibits

first or second order stochastic dominance over the other.

[3 marks]

[Total: 9 marks]

Page 2 of 6

QUESTION 3

(i) The market consists of two securities A and B. Optimality of a portfolio is considered

via a mean-variance analysis.

(a) Assume that the returns RA and RB are perfectly positively correlated and that:

RA > RB , ?(RA ) = ?(RB )

What is the optimal portfolio if no short-selling is allowed?

[2 marks]

What would happen if we allow for short-selling?

[3 marks]

(b) Assume that the returns RA and RB are perfectly negatively correlated. Show

that there exists a portfolio P with zero risk (?P = 0).

[3 marks]

Is short-selling needed to obtain this portfolio?

[1 mark]

(ii) There are three securities with return, R1 , R2 and R3 , on the market. The expected

returns are:

R1 = 0.02, R2 = 0.03, R3 = 0.05

Also, the covariance matrix with entries

?

0.12

?

?ij = ?

?0.01

0.02

?ij = Cov(Ri , Rj ) is given by:

?

0.01 0.02

?

0.22 0.03?

?

2

0.03 0.3

Assume that short-selling is allowed. Calculate the composition of the minimumvariance portfolio which earns an expected return of 4.5%.

[3 marks]

[Total: 12 marks]

Page 3 of 6

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Explanation & Answer:

3 Questions

Tags:

Financial Economics

Mean Variance Analysis

Stochastic Dominance

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