Question

Foe an introduction to Management Science 15 th edition chapter 8 problem 11; need step by step to do these problem?

Assume an investor has $50,000 to invest and wants to minimize the variance of his or her portfolio subject to a constraint that the portfolio returns a minimum of 10%.

Answer 1

1. Here we need to achieve a certain minimum return while minimizing the risk, or variance. The best model to this is Markowitz's mean variance portfolio model.

2. Let's denote our desired minimum return as R (=10% in this case).

Suppose we have in general all the investible assets accessible to us, i.e. we can trade any stock that we need. Let's assume that we choose n different assets in proportions (or weights) w_i, w₂, ...w_n, where

Also, the amount to be invested is also known (say A = 50,000), so we have

If the return of these n stocks (where returns can be defined as log(x1/x0), i.e. log of change in asset price since previous period) are indicated by r₁,r₂,....r_n, we have

  

3. Calculate the covariance of returns between the n assets. So, for n stocks, we get a n X n covariance matrix,

4. Using Markowitz theory, to minimize the variance of the portfolio while achieving a minimum level of return R, we need to find and invest in the set of asset weights that satisfy the following:

minimize (where denotes the return covariance matrix)

subject to , and , and

5. Solving the above optimization problem on a computer, we can get the different weights and hence the specific amounts to be invested in different assets to achieve at least the given level of return with least variance.

R and A should obviously be substituted with their actual values above while solving the problem for the weights.