Question

You are given a four security portfolio: Security Expected Return SD 0.06 0.10 0.12 0.17 0.09 0.14 0.18 0.22 Correlations 1 0.30 0.35 0.40 1 0.10 0.40 1 0.35 4 You have a target rate of return of 14.2%. Solve for the portfolio weights which minimize the portfolio standard deviation and achieves the target return. In this exercise the weights can be negative. What is the minimum standard deviation and what is the expected return?

Answer 1

We have E_{p =} W₁E₁ + W₂E₂ + ....... W_nE_n

Two asset portfolio, Standard Deviation =

$\inline \sigma$²_{p =} W²₁$\inline \sigma$²₁ + W²₂$\inline \sigma$²₂ + 2W₁W₂$\inline \sigma$₁$\inline \sigma$₂$\inline \rho$₁₂

Where E_p represents expected return on portfolio (p)

W = Weight of portfolio

$\inline \rho$ = Correlation

This question basically asks for combination of securities which with minimum risk provides return of 14.2% i.e. expected return. Here, we are given the details of 4 securities, each of its expected return and risk. Also provided in question that weights can be negative but this is not feasible, as negative investment can not be made.

Analysis of various combination of 2 securities in a portfolio provides the following:

Portfolio A : 40% in Security 2 and 60% in Security 4

E_p = (0.4 * 10%) + (0.6 * 17%) = 4 + 10.2 = 14.2%

$\inline \sigma$²_p = (0.4)² * (14)² + (0.6)² * (22)² + 2*0.4*0.6*14*22*0.4

= 31.36+174.24+37.8

= 243.4

$\inline \sigma$_p =  $\inline \sqrt{243.4}$ = 15.60

This exercise of various other weights can be done further with other securities also. Here the optimal solution is 40% of Security 2 and 60% of Security 4.

Individually Security 1 has least risk or standard deviation of 9% and expected return is 6%.