The capital asset pricing model was the biggest innovation in 20th century finance. Understanding this model is fairly easy at the intuitive level. The technical details, however, can be challenging.

The purpose of this note is to present the simplest form of the portfolio model that is the basis for the CAPM. This reading is purely optional. You should only continue if you’re really interested in mathematical details.

## Preliminaries

We need some basic definitions of variables. Here’s a table that describes most of them.

Mathematically, here are the definitions of what’s described above.

The covariance as defined above is the sample covariance.

## The Basic Portfolio Model

The idea behind the portfolio model is simple. I’ll refer to E(*R _{p}*) and

*σ*as the expected portfolio return and portfolio risk as measured by the standard deviation of the rate of return.. An investor wants to structure a portfolio in one of two (equivalent) ways:

_{p}- Maximize expected portfolio return subject to a maximum amount of risk the investor is willing to assume.
- Minimize portfolio risk subject to a minimum desired rate of return.

If you’re familiar with mathematical programming, you’ll quickly figure out that these two problems are duals of each other. That means the solutions to the two problems will be the same for any solution for E(*R _{p}*) and

*σ*. So the problem looks like this:

_{p}

is the maximum portfolio variance the investor can tolerate. The equivalent dual problem is:

is the investor’s minimum acceptable portfolio return.

In both cases it’s easy to show that the inequality constraint can be changed to an equality. The logic is straightforward. Risk and return are inversely related. Therefore, choosing a value for that is less than implies that expected return is not being maximized. A similar argument applies to the dual model.

Before solving this for the general case, a simple two-asset portfolio will serve as a useful example. Let’s solve this problem:

## Let’s Not Solve That

Even a two-security model is very difficult to solve analytically. So I won’t bother. Instead I’ll simply say that Excel’s Solver plug-in is a very powerful tool for solving constrained optimization problems. And, luckily for me, Samir Khan has done the heavy lifting at http://investexcel.net/financial-modeling-analysis-excel/. (The workbook is on page 15. Here’s a direct link: http://investexcel.net/page/15/. Look for “Mean-Variance Portfolio Optimization with Excel.”)

The method used by Samir is elegant and simple. He uses three stocks and the rate of return on T-bills. I changed the T-bill rate and the label to the ten year Treasury note since we have ten years of security data. I also changed the risk-free return to 0.02.

Samir calculates the variance-covariance matrix using Excel’s COVARIANCE( ) function. He makes good use of semi-fixed cell references to save time. His method is simple. He’s solving the problem by minimizing the portfolio variance subject to a constraint on the rate of return. The objective variables are the percentages of each security in the portfolio. Naturally, those percentages have to add up to 1.00. Here’s what the Excel workbook looks like:

And here’s his Solver setup:

Anyone interested in understanding the basic Markowitz model should visit Samir’s website. There’s a ton of other useful material there (including options pricing and real-time data links for U.S. stocks.