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 rate of 8%. The probability distribution of the risky funds is as follows:

Expected Standard Deviation Return Stock fund (S) Bond fund (B) 17% 30% 22 11 The correlation between the fund returns is 0.10 a-1. What are the investment proportions in the minimum-variance portfolio of the two risky funds. (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.) Portfolio invested in the stock Portfolio invested in the bond a-2. What is the expected value and standard deviation of its rate of return? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.) Rate of Return Expected return Standard deviation

Answer 1

Given information:

stock fund return rs = 0.17

stock fund standard deviation $\sigma_s$ = 0.3

Bond fund return rb = 0.11

Bond fund standard deviation $\sigma_b$ = 0.22

correlation $\rho$ = 0.1

Part a-1:

Find the investment proportions (weights) of the risky funds in the portfolio.

Assume "w" is the stock fund weight in the portfolio. Since the sum of weights =1, the bond fund weight will be (1-w)

Expected return of the portfolio = weight of stock fund * returns of stock fund + weight of bond fund * returns of bond fund

E (portfolio) = w * 0.17 + (1-w) *0.11 = 0.06 w +0.11

Variance (portfolio) = w² * $\sigma_s$ ² + (1-w)² * $\sigma_b$ ² + 2 *  $\rho$ * w * (1-w) * $\sigma_s$ * $\sigma_b$

Substitute the given values:

Variance = w² * 0.3² + (1-w)² * 0.22² + 2 * 0.1 * w * (1-w) * 0.3 * 0.22

= 0.09 * w² + (1-2w+w² ) *0.0484 +0.0132 (w - w² )

= 0.1252 * w² - 0.0836*w +0.0484

In order to find minimum variance portfolio, we will use calculus concepts here. determine the derivative of variance V with respect to weight w and equate it to 0.

dV/dw = 0.1252 * 2 w - 0.0836 = 0

w = 0.0836 / (0.1252*2) = 0.3339

Now take the second derivative and if this is positive, then from the rules of basic calculus, the variance is at its minimum.

d²v/dw² = 0.1252 * 2 dw/dw = 0.1252 * 2 = 0.2504 > 0

Since the second derivative is positive, the variance is at its minimum.

Answer:

Portfolio invested in stock = 0.3339

Portfolio invested in bond = 1- 0.3339 = 0.6661

Part a-2

Substitute the value of w in the expected return and standard deviation formula

E (portfolio) = w * 0.17 + (1-w) *0.11 = 0.06 w +0.11 =0.06 * 0.3339 + 0.11 = 0.13 = 13%

Variance = 0.1252 * w² - 0.0836*w +0.0484 = = 0.1252 * 0.3339² - 0.0836*0.3339 +0.0484 = 0.0344

Standard deviation = square root (0.0344) = 0.1856

Answer:

Expected return = 13%

Standard deviation = 18.56%