Suppose you invest $25,000 in an S&P 500 Index fund (S&P fund) and $15,000 in a

Question

Suppose you invest $25,000 in an S&P 500 Index fund (S&P fund) and $15,000 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?

Explanation / Answer

Weight of S&P500, w1 = 25,000 / (25,000 + 15,000) = 62.5%, Weight of bond fund, w2 = 1 - 62.5% = 37.5%

Expected Return = w1 x R1 + w2 x R2 = 62.5% x 8% + 37.5% x 4% = 6.5%

Std. Dev. = [(w1 x SD1)^2 + (w2 x SD2)^2 + (2 x w1 x w2 x SD1 x SD2 x corr12)]^(1/2)

= ((62.5% x 18%)^2 + (37.5% x 7%)^2 + (2 x 62.5% x 37.5% x 18% x 7% x 0.4))^(1/2)

= 12.53%

Sharpe Ratio = (Rp - Rf) / SDp

For Portfolio, SR = (6.5% - 2%) / 12.53% = 0.359

For S&P fund, SR = (8% - 2%) / 18% = 0.333

Your portfolio has higher Sharpe Ratio due to the benefit of lower correlation between S&P fund and Bond fund.

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