Problem The following data apply to Problem: A pension fund manager is consideri

Question

Problem The following data apply to Problem: 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 8%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is.10. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds?

Explanation / Answer

Answer)

First we need to find the proportion of stocks in optimal risky portfolio:

The formulae for which is give as :-

[(Ers – rf)* (St. dev of Bonds)^2 ] – [(Erb-rf)*covariance(Bond,stock)]

Divide by

(Ers – rf)* (St. dev of Bonds)^2 + (Erb-rf) )* (St. dev of Stock)^2 – [Ers-rf+Erb-rf]* covariance(Bond,stock)

Where,

Ers = Return of stock

Erb = Return of bond

Rf = risk free rate

Covariance (Bond,stock) = (St. dev of Bonds)* (St. dev of Stock)*correlation coefficient

When we put the values in the above formulae we get ,

Proportion of Stock in optimal risky portfolio as:- 0.4516

Proportion of Bonds in optimal risky portfolio as:- 0.5484

Now, we calculate the Return on the optimal risky portfolio

Erp =

(Weight of Stock * return on stock + weight of bond* return on bond )

= 0.090322581 + 0.065806452

= 0.156129032

St. dev of portfolio =[ (Weight of stock*St. dev of stock)^2 + (Weight of bond*St. dev of bond)^2 +( 2* Weight of bond*St. dev of bond* Weight of stock*St. dev of stock*correlation coefficient)]^1/2

= [0.018355879 + 0.006766389 + 0.002228928]^1/2

= [0.027351]^0.5

= 0.165382

Now we require a Cal portfolio of mean return 14%, the corresponding st. deviation is given as

Erc = rf + [(Erp-rf)/ St.dev of portfolio ]* St.dev of of Cal portfolio

Where Erc = 14% 0r 0.14

Erp = 0.156129032

Rf = 0.08

St. dev of optimal risky portfolio = 0.165382

Solving the above formulae for St. Deviation of the Cal portfolio

0.14 = 0.08 + [(0.156129032- 0.08)/0.165382 ]* St. Deviation of the Cal portfolio

Answer to part a)

St. Deviation of the Cal portfolio = (0.09-0.03) / (0.460322) = 0.130343419

Now to find the amount invested in T-bill we use the below formulae

Erc=Rf + y*(Erp – Rf) Where y is the amount invested in stocks and bond for a cal porfolio and 1-y is the amount invested in T-bills

0.14=0.08+y*(0.156129032-0.08)

Y = 0.7881356

Answer to b)

Amount invested in stocks = 0.7881356*0.4516= 0.355932203

Amount Invested in Bonds = 0.7881356*0.5484 = 0.43220339

Amount in T-bills = 1-y = 1-0.7881356 = 0.2118644

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