Question

Suppose K1 and K2 have the following distribution:
Scenario Probability return K1 return K2
w(1)    0.3 -10% 10%
w(2)    0.4 0% 20%
w(3)    0.3 20% -10%

(a) Find the risk of the portfolio with w1 = 30% and w2 = 70%.
(b) Find the risk of the portfolio with w1 = 50% and w2 = 50%.
(c) Which of the portfolios above (in part (a) and (b)), has higher expected returns?

Answer 1

Probability(P)	Return(%)	P*Return	Deviation form expected return (D1)	PD^2
0.3	-10	-3	-13	50.70
0.4	0	0	-3	3.60
0.3	20	6	17	86.70

Expected Return = $\sum$P*Return

= -3+0+6

= 3%

Variance = $\sum$PD^2

= 50.7+3.6+86.7

= 141

Standard Deviation = $\sqrt$Variance

= $\sqrt$141

= 11.87%

Probability(P)	Return(%)	P*Return	Deviation form expected return (D2)	PD^2
0.3	10	3	2	1.20
0.4	20	8	12	57.60
0.3	-10	-3	-18	97.20

Expected Return = $\sum$P*Return

= 3+8-3

=8%

Variance = $\sum$PD^2

= 1.2+57.6+97.2

= 156

Standard Deviation = $\sqrt$Variance

= $\sqrt$156

= 12.49%

Probability(P)	Deviation (D1)	Deviation (D2)	P*D1*D2
0.3	-13	2	(7.8)
0.4	-3	12	(14.4)
0.3	17	-18	(91.8)

Covariance = $\sum$P*D1*D2

= -7.8-14.4-91.8

= -114

(a) Find the risk of the portfolio with w1 = 30% and w2 = 70%.

Standard Deviation of Portfolio = $\sqrt$(W1*SD1)^2+(W2*SD2)^2+(2*W1*W2*COV12)

= $\sqrt$(.3*11.87)^2+(.7*12.49)^2+(2*.3*.7*-114)

= $\sqrt$41.24077

= 6.42%

The return of a portfolio is the weighted average return of the securities which constitute the porfolio

Portfolio Return = .3*3+.7*8

= 6.50%

(b) Find the risk of the portfolio with w1 = 50% and w2 = 50%.

Standard Deviation of Portfolio = $\sqrt$(W1*SD1)^2+(W2*SD2)^2+(2*W1*W2*COV12)

= $\sqrt$(.5*11.87)^2+(.5*12.49)^2+(2*.5*.5*-114)

= $\sqrt$17.22425

= 4.15%

The return of a portfolio is the weighted average return of the securities which constitute the porfolio

Portfolio Return = .5*3+.5*8

= 5.50%

(c) Which of the portfolios above (in part (a) and (b)), has higher expected returns?

Portfolio in part a has higher expected return.