Optimal Portfolio Decisions. THe James Bond Fund is a mutual fund (open-end inve

Question

Optimal Portfolio Decisions. THe James Bond Fund is a mutual fund (open-end investment company) with an objective of maximizing income from a widely diversified corporate bond portfolio. The fund has a policy of remaining invested largely in a dviersified portfolio of investment-grade bonds. Investment-grade bonds have high investment quality and receive a rating of Baa or better by Moddy's, a bond-rating service. The fund's investment policy states that investment-grade bonds are to be emphazied, representing at least three time the amount of jun bond holdings. Junk Bonds pay high nominal returns byt have low investment quality, and they receive a rating of less than Baa from Moody's. To maintain the potential for high investor income, at least 20 percent of the fund's total portfolio must be invested in Junk bonds. Like many funds, the James Bond Fund cannot use leverage (or borrowing) to enhance investor returns. As a result, total bond investments cannot total more than 100 percent of the portfolio. Finally, the current expected return for investment-grade (I) bonds is 9 percent, and it is 12 percent for junk (J) bonds.

A. Using tche inequality form of the constraint conditions, set up and interpret the linear programming problem that the James Bond Fund would use to determine the optimal portfolio percentage holdings of Investment-grade (I) and junk (J) bonds. Also formulate the problem using the equality form ofthe constraint conditions. (Assume that the fund managers have decided to remain fully invested and therefore hold no cash at this time.)

B.Use a Graph to determine the optimal solution, and check your solution algebriacally. Fully interpret solution values.

C. Holding all else equal, how much would the expected return on junk bonds have to fall to alter the optimal investment policy determined in part B? Alternatively, how much would the return on investment-grade bonds have to rise before change in investment policy would be warranted?

D. In anticipation of a rapid increase in interest rates and a subsequent economic downturn, the investment committee has decided to minimize the funds exposure to bond price fluctuations. In adopting a defensive position, what is the maximum share of the portfolio that can be held in cash given the investment policies stated in the problem?

Explanation / Answer

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 5.5%. The probability distribution of the risky funds is as follows:
Stock fund (S) - Expected return - 15% Standard Deviation - 32%
Bond fund (B) Expected return - 9% Standard Deviation - 23%
The correlation between the fund returns is 0.15.
Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio
Portfolio invested in the stock %
Portfolio invested in the bond %
Expected return %
Standard deviation %
From the standard deviation and correlation we will develop Covariance Matrix
Bonds Stock
Bonds 23 x23 0.15 x 32 x 23
Stocks 0.15 x 32 x 23 32 X 32
Bonds Stock
Bonds 529 110.4
Stocks 110.4 1024
proportions of Stock in the optimal risky portfolio
Portfolio weight Stock= ^2B - COV(B,S)/^2S+ ^2B - 2COV(B,S)
Portfolio weight Stock= (529 -110.4)/(1024 + 529 - 2 x 110.4) 0.314217085
Portfolio weight Bond Wb = (1- 31.42%) 0.685782915
Expected Return of the portfolio = 15% x 31.42% + 9% x68.58% 0.108853025
Standard deviation of portfolio =sqrt (68.58%^2 x529 +31.42^2 x 1024 + 2 x 68.58% x 31.42 %x 110.4)
19.937 %

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