Question

3 Question 3 In a market are listed two risky assets whose returns are described by the following parameters HA=0.01. MB = 0.07, 01 = 0.2 and op = 0.12. The correlation among the securities is constant and equal to p=0.1. 1. Derive the equation for the frontier 2. Derive the minimum variance portfolio and the equation for the efficient frontier 3. Let's add a risk free asset among the possible investments with return r = 0.03 and derive the equation for the frontier

Answer 1

Since there are multiple questions only Question 3 is answered below:

Given information

$\mu_A = 0.01, \sigma_A = 0.2$

$\mu_B = 0.07, \sigma_B = 0.12$

$\rho$ = 0.1

1. Derive the equation of the frontier:

Frontier equation comprise two equations, a)expected return of the portfolio and b)variance of the portfolio

Let us assume the weight of stock A in the portfolio as w. Then the weight of stock B is (1-w).

The efficient frontier is a combination of the expected return and variance equation for various weight combinations of A and B. The graph plots expected return on y-axis and variance on x-axis for various values of w between 0 and 1.

Expected return of portfolio = w * $\mu_A$ + (1-w) * $\mu_B$ = w * 0.2 + (1-w) * 0.07

Variance of the portfolio = $w^2*\sigma_A^2 +(1-w)^2 *\sigma_B^2 +2*\rho * w*(1-w)*\sigma_A*\sigma_B$

=

= 0,0496 w² - 0.024w +0.0144

2. Determine the minimum variance portfolio

To determine minimum variance portfolio, take the first derivative of variance (V) with respect to weight w and equate it to zero to determine w. This is a basic rule of calculus. Subsequently, take the second derivative of V and if this is positive then a minimum does exist.

V = 0,0496 w² - 0.024w +0.0144

dV/dw = 0.0496 * 2w - 0.024

Now equate this to 0

After rearranging w on left hand side and everything else on the right hand side we get,

w = 0.024/(0.0496*2) = 0.2419

Equation for the efficient frontier:

Expected return = w * 0.2 + (1-w) * 0.07 = 0.2419*0.2+(1-0.2419 )*0.07 = 0.1014

Variance = 0,0496*0.2419²-0.024*0.2419+0.0144 = 0.0115

Part 3:

We will solve this problem similar to part 2. This time instead of the two stocks A and B, we have the portfolio and risk free asset.

E(return) = w * R_P + (1 − w)R_f = w *0.01014 + (1-w) * 0.03

Variance = $w^2*\sigma_P^2$ . Since risk free rate has a zero standard deviation and correlation with the portfolio is zero all the other terms are zero.

Variance = w² * 0.0115

Equation of the frontier has two parts:

E(return) = w *0.01014 + (1-w) * 0.03

Variance = w² * 0.0115

No, this is not the capital allocation line (CAL). CAL is line whose slope is sharpe ratio of the portfolio and is a combination of risk free asset and risky portfolio.Equation of CAL:

E(Rc) - E(Rf) = $\sigma_C$ * sharpe ratio of the portfolio

Complete portfolio C is the porfolio formed by combining the risky portfolio P and the risk free asset.

sharpe ratio of portfolio = Excess return / standard deviation of portfolio = [E(RP)- E(RF) ] / $\sigma_P$