Question 1 (Mean-variance optimization - 10 marks)
You have been approached by a client who is interested in investing in equities. After some
questionnaires, you discover that you client's utility function can be approximated by a
quadratic utility,
U(E[rq]; Var[rq]) = E[rq]
1
2
AVar[rq]; A > 0:
(a) Based on the monthly closing prices provided, estimate the expected return E[rp] and
variance Var[rp] of ASX200 per annum (p.a.). Note that you should rst compute the
continuously compounded returns by using ln(Pt+1=Pt), then compute the sample av-
erage and sample variance of the monthly returns. Finally, to annualize the estimates,
multiply the sample average and sample variance by 12. [1 mark]
(b) Given that the 1-year term deposit rate is 2.3% p.a. and the risk aversion coecient
is given by A = 4, suppose your client only invests in ASX200 and the 1-year term
deposit, compute the portfolio weights (w
p; 1w
p), expected return E[rq] and standard
deviation q for the optimal portfolio q. What is the maximum utility (or risk-adjusted
return) that can be achieved? Then, brie
y explain why is it not optimal to allocate
100% of capital to either the risk-free security or the equity index. [1 mark]
(c) Plot the capital allocation line (CAL) in the (;E[r]) space for in the range of 0%-
30%, you should also indicate the position of the optimal portfolio q. Furthermore,
on the same graph, also plot the indi erence curve that corresponds to the maximum
utility achieved by the optimal portfolio q. Based on the indi erence curve plotted,
brie
y explain why a higher utility cannot be attained by a portfolio di erent from q.
[2 marks]
(d) Based on the quarterly returns provided for the ASX200 and the Australian Semi-
Government Bond ETF, estimate the expected return and variance p.a. using the
same approach as in Part (a). Moreover, also estimate the correlation coecient.
Then, suppose your client only invests in ASX200 and the Bond EFT, by using the
Solver add-in in Excel, work out the portfolio weights (wE
;wD
), expected return E[rq]
and standard deviation q for the optimal portfolio q (assume A = 4). What is the
maximum utility that can be achieved? Brie
y explain why is it not optimal for any
risk-averse investor to allocate all her capital to debt (Hint: you may want to compute
the expected return of the minimum variance portfolio.). [2 marks]
(e) Plot the investment opportunity set (IOS) in the (; E[r]) space that represent all the
possible portfolios attainable by investing in ASX200 and the Bond ETF. You should
also indicate the positions of ASX200, the Bond ETF and the optimal portfolio q. On
the same graph, also plot the indi erence curve which corresponds to the maximum
utility that can be achieved by q. [2 marks]
(f) Suppose now your client invests in the ASX200, Bond ETF and the 1-year term deposit,
work out the portfolio weights (wE
;wD
), expected return E[rT ] and standard deviation
T for the tangent portfolio that maximizes the slope of the CAL when combined
with the risk-free security. Plot the CAL together with the IOS from Part (e) in the
(;E[r]) space, also indicate the positions of ASX200, Bond ETF and the tangent
portfolio T. Suppose your client has a target expected return of E[rq] = 0:06 p.a.,
what is the minimum standard deviation q? How do you construct this portfolio?
Finally, indicate this portfolio q on the graph. [2 marks]