Percival Hygiene has $10 million invested in long-term corporate bonds. The bond

Question

Percival Hygiene has $10 million invested in long-term corporate bonds. The bond portfolio's expected annual rate of return is 9%, and the annual standard deviation is 10%.

Amanda Reckonwith, Percival's financial adviser, recommends that Percival consider investing in an index fund that closely tracks the Standard and Poor's 500 index. The index has an expected return of 14%, and its standard deviation is 16%.

a. Suppose Percival shifts $4 million from his bond portfolio into the index fund. What would be his expected return and standard deviation? The correlation between the bond portfolio and bond fund is +0.3.

b. Suppose, instead, that Percival shifts all of his money into a combination of the index fund and Treasury bills. How much would he invest in Treasury bills and how much in the index fund to match the expected return of the combination of the bond portfolio and index fund described in part a? What is the standard deviation of this portfolio? The Treasury bill yield is 6%.

c. Which of these two portfolios, a and b, would you recommend? Why?

Explanation / Answer

rp = x1r1 + x2r2

rp = (0.5 ´ 0.09) + (0.5 ´ 0.14) = 0.115 = 11.5%

sP2 = x12s12 + 2x1x2s1s2r12 + x22s22

sP2 = (0.5)2(0.10)2 + 2(0.5) (0.5) (0.10) (0.16) (0.30) + (0.5)2(0.16)2

sP2 = 0.0113

sP = 0.106301=10.63%

By investing 50% in long term corporate bonds and 50% in index ,his expected return increases to 11.5% and the standard deviation of his portfolio remains 10%..

sP2 = x12s12 + 2x1x2s1s2r12 + x22s22

(0.10)2 = 0 + 0 + x22(0.16)2

x2 = 0.625 Þ x1 = 0.375

Further:

rp = x1r1 + x2r2

rp = (0.375 ´ 0.06) + (0.625 ´ 0.14) = 0.11 = 11.0%

Therefore, he can improve his expected rate of return without changing the risk of his portfolio.

We will recommend option B because the expected rate of return increases without increasing risk i.e. standard deviation.

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