Suppose you invest $15,000 in an S&P 500 Index fund (S&P fund) and $10,000 in a...

Suppose you invest $15,000 in an S&P 500 Index fund (S&P fund) and $10,000 in a total bond market fund (Bond fund). The expected returns of the S&P and Bond funds are 8% and 4%, respectively. The standard deviations of the S&P and Bond funds are 18% and 7% respectively. The correlation between the two funds is 0.40. The risk-free rate is 2%. What is the expected return on your portfolio? What is the standard deviation on your portfolio? What are the Sharpe ratios for your portfolio and the S&P fund? Why is it not surprising that your portfolio has a higher Sharpe ratio than either the S&P fund or Bond fund?

Solutions

Expert Solution

Total Portfolio value = Value of S&p + Value of Bond fund

=15000+10000

=25000

Weight of S&p = Value of S&p/Total Portfolio Value

= 15000/25000

=0.6

Weight of Bond fund = Value of Bond fund/Total Portfolio Value

= 10000/25000

=0.4

Expected return%=	Wt S&PReturn S&P+Wt Bond fundReturn Bond fund
Expected return%=	0.60.08+0.40.04
Expected return%=	6.4
Variance	=( w2Aσ2(RA) + w2Bσ2(RB) + 2(wA)(wB)Cor(RA, RB)σ(RA)*σ(RB))
Variance	=0.6^20.18^2+0.4^20.07^2+20.60.40.180.07*0.4
Variance	0.01487
Standard deviation=	(variance)^0.5
Standard deviation=	12.19%

Portfolio

Sharpe ratio(reward to variability)
=(Return-risk free rate)/std dev
=(6.4-2)/12.19
=0.36

S&P

Sharpe ratio(reward to variability)
=(Return-risk free rate)/std dev
=(8-2)/18
=0.33

Portfolio sharpe ratio is higher than that of S&P because std dev for portfolio is lower because of diversification with bond fund

jeff jeffy answered 11 months ago

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