# Financial Economics and Mean Variance Analysis Questions

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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]
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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