A pension fund manager is considering three mutual funds. The first is a stock f

Question

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 sure rate of 5.5%. The probability distributions of the risky funds are:

The correlation between the fund returns is 0.15.

Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Standard deviation             %

b-1. What is the proportion invested in the T-bill fund? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Proportion invested in the T-bill fund             %

b-2. What is the proportion invested in each of the two risky funds? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Expected Return Standard Deviation Stock fund (S) 15 % 32 % Bond fund (B) 9 % 23 %

Explanation / Answer

First we need to find the proportion od 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:- 0.6466

Proportion of Bonds in optimal risky portfolio as:- 0.3534

Now, we calculate the Return on the optimal risky portfolio

Erp =

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

= 0.096994132 + 0.031803521 = 0.128797653

= 0.128797653

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

= [0.042816224 + 0.006605734 + 0.005045288]^1/2

= [0.054467]^0.5

= 0.233382

Now we require a Cal portfolio of mean return 12%, the corresponding st. deviation is given as

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

Where Erc = 12% 0r 0.12

Erp = 0.128797653

Rf = 0.055

St. dev of optimal risky portfolio = 0.233382

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

0.12 = 0.055 + [(0.128797653- 0.055)/ 0.233382]* St. Deviation of the Cal portfolio

Answer to part a)

St. Deviation of the Cal portfolio = (0.12-0.055) / (0.316209449) = 0.205559955

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

0.12=0.055+y*(0.128797653-0.055)

Y = 0.8807868

Answer to b-1 & B-2)

Amount invested in stocks = 0.8807868*0.6466 = 0.569541021

Amount Invested in Bonds = 0.8807868*0.3534 = 0.311245801

Amount in T-bills = 1-y = 1-0.8807868 = 0.1192132

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