Question

You are the manager of a portfolio of risky securities. Your portfolio has an expected return (E(rP)) of 12% and a standard deviation (P) of 18%. The risk free rate (rf) is 6%. The following two clients want to invest some portions of their investment budget in your portfolio and the balance in the risk free asset: Client 1 needs an expected return of 10% from her complete portfolio. Client 2 needs a complete portfolio with a standard deviation of 20%. (i) What proportions should each investor invest in your risky portfolio and risk free asset? (ii) Which client is more risk averse?

Answer 1

A	B	C	D	E	F	G	H
2
3			Expected Return	Standard Deviation
4		Portfolio	12%	18%
5		Risk Free Asset	6%	0%
6
7		a)
8		Calculation of weights for client 1:
9		Expected return required by the client1	10%
10
11		Proportion invested in risky portfolio is w for client 1 then,
12		Expected return	=w*12%+(1-w)*6%
13		or
14		w*12%+(1-w)*6% = 10%
15
16		Solving the above equation,
17		w	0.67	=(D9-D5)/(D4-D5)
18
19		Hence for client 1
20		Weight in risky portfolio	0.67
21		Weight in risk free asset	0.33
22
23		Calculation of weights for client 2:
24		Standard deviation of complete portfolio required by the client1	20%
25
26		Proportion invested in risky portfolio is w for client 2 then,
27		Standard deviation of overall portfolio	=w*Standard deviation of risky portfolio
28		or
29		w*18% = 20%
30
31		Solving the above equation,
32		w	1.11	=D24/E4
33
34		Hence for client 2
35		Weight in risky portfolio	1.11
36		Weight in risk free asset	-0.11
37
38		ii)
39
40			Risky portfolio	Risk free asset
41		Expected Return	12%	6%
42		Standard Deviation	18%	0%
43		Weight for Client 1	0.67	0.33
44		Weight for Client 2	1.11	-0.11
45
46			Expected return	Standard deviation	Sharpe Ratio
47		Overall portfolio for client 1	10.00%	8.00%	50.00%	=(D47-E41)/E47
48		Overall portfolio for client 2	12.67%	20.00%	33.33%	=(D48-E41)/E48
49
50		Shape ratio shows the return required per unit increase risk.
51		Since the shape ratio for client 1 is higher than that for client2,
52		therefore client 1 required higher return per unit of increased risk than client 2.
53
54		Therefore client 1 is more risk-averse.
55

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