Question

An investment company intends to invest a given amount of money in three stocks.

The means and standard deviations of annual returns are as follows:

Stock	Mean	Standard Deviation
1	0.14	0.20
2	0.11	0.25
3	0.15	0.08

Stock Coorelation among annual returns
Stocks 1 and 2	0.5
Stocks 1 and 3	0.8
Stocks 2 and 3	0.1

Construct the efficient frontier for portfolios of these stocks. Please explain the involved steps in your modeling and construction.

Answer 1

An efficient frontier represents the set of efficient portfolios that will give the highest return at each level of risk or the lowest risk for each level of return. A portfolio is efficient if there is no alternative with:

• Higher expected return with same level of risk
• Same expected return with lower level of risk
• Higher expected return for lower level of risk

Let’s take a portfolio of two assets and see how we can build the efficient frontier in excel. Let’s say we have two securities, 1 and 2, with the data in given question,

We can combine these two assets to form a portfolio. In the portfolio, we can combine the two assets with different weights for each asset to create an infinite number of portfolios having different risk-return profiles. For example, if we take 50% of each asset, the expected return and risk of the portfolio will be as follows:

E(R) = 0.50*14% + 0.50*11% = 13% or 0.13

σ = Sqrt(0.20^2*0.5^2+0.25^2*0.5^2+2*(0.5)*0.5*0.5*0.2*0.25) = 20% or 0.20

Using the above formula let us pool up the data under 7 different proportions as shown in the excel below. Then the risk and return of these portfolios can be plotted on the XY scatter graph with return on Y axis and risk on X axis. The graph looks as follows and is called the efficient frontier.