*This article is one in a long series which I hope will help explain the ins and outs of DFA – Dimensional Fund Advisors. NOTE: This is my interpretation and explanation only. For the final word, please refer to the DFA Canada Website.*

# The Capital Asset Pricing Model (CAPM)

As I mentioned before, this series on DFA is going to be long if you really want to understand what they are about. Further, there is some background knowledge that is required before we can truly make sense of what is going on. At the same time, I don’t want to go too far down some tangents so I am going to provide the coles notes on many topics.

To that end I thought it would be best to have a refresher on CAPM. CAPM stands for Capital Asset Pricing Model and the acronym is pronounced “Cap-ehm”. Basically CAPM is model that predicts what your expected return should be in your portfolio based on a few factors (actually just one factor). First let’s begin with some logic…

The “risk free” rate is equivalent to the T-bill rate (or the high interest savings account rate, if you prefer). This is basically the rate of return you can get on a portfolio without taking any risk. If you were to subject your portfolio to any risk, then you would expect to be compensated in the form of extra return, over and above the risk free rate. But the question then arises: how much extra return should you expect for each unit of risk? CAPM basically says that the *expected* return you get should be based on how much exposure you have to the market factor plus the risk free rate.

The “market factor” is also known as “the equity premium” or the extra return of stocks over the risk-free rate. So, to re-iterate, CAPM is saying that your exposure to the market factor explains your expected return on your portfolio.

# The Formula

Here is the CAPM formula:

E(Rp) = Rf + β(Rm – Rf)

Where:

E(Rp) = Expected return on the portfolio

Rf = The Risk Free Rate

β = Beta (Which is the measure of exposure to the market factor)

Rm = Return of the Market

Note: the term in brackets, (Rm – Rf), is “the market factor” (or “equity premium”)

Before we use some test numbers to see how this works, lets first put on our thinking hats. Let’s assume we are investing using an index tracking fund (like an ETF). Since our portfolio will move in tandem with the market, we have a β of 1. Let’s put this in the formula (and nothing else).

E(Rp) = Rf + 1(Rm – Rf)

So if we multiply (Rm – Rf) by 1 we will have (drum roll please)… (Rm – Rf). So this makes the equation as follows:

E(Rp) = Rf + Rm – Rf

You can see that the Rf terms cancel each other out, leaving:

E(Rp) = Rm

Which is what we want with an index fund. We want the return to equal the market return. But now let’s plug in some numbers for a non-index portfolio and see what happens. We’ll assume that the “risk free rate” is 3% since that is what a high interest rate savings account might yield. We’ll assume our portfolio tends to move up and down one and a half times as much as the market, therefore β is equal to 1.5. The return on the market is 10%. Now we can solve to see what CAPM says our expected return should be.

E(Rp) = 3% + 1.5(10% – 3%)

E(Rp) = 3% + 1.5(7%)

E(Rp) = 3% + 10.5%

E(Rp) = 13.5%

So in this case, CAPM says that if our portfolio is 1.5 times as volatile as the market, then for it to be a worthwhile investment, our portfolio needs to earn 13.5% versus the market’s 10% in order to be fairly compensated for taking on the extra risk over the risk free rate.

I’ll stop there for today, but will continue in the next post in the series to discuss what happens when the ACTUAL return of the portfolio does not match the Expected Return.

Michael James

I don’t believe it makes sense to use the T-bill rate in these calculations. The basis for the Sharpe ratio and the linear relationship between risk and return is that you can turn a low-risk, low-return investment into a high-risk, high-return investment with leverage. However, you can’t borrow at the T-bill rate. Also, maintaining the optimum amount of leverage requires trading which has the cost of commissions and spreads. My guess is that it would be more appropriate to use 5% or 6% in these calculations.

Preet

@Michael James – I agree that the risk free rate is not appropriate in the real world, especially for individual investors (it’s even questionable for some institutional money I would imagine). However, the CAPM assumes no transaction costs among other things.

Michael James

Institutional investors may have access to better rates for borrowing than individual investors, but they will still pay more than the risk-free rate because no lender would lend money to an entity for the same rate they could get by lending money to the government. I agree that CAPM assumes no transaction costs, but that is clearly just an approximation. To compensate for the error caused by higher borrowing rates and transaction costs, you can get quite good results by simply increasing the assumed risk-free rate in the risk-return calculations.

In your example, if we use a risk-free rate of 5%, then the return needed to compensate for the 1.5 times more risky portfolio drops to 12.5% instead of 13.5%. My guess is that this is a better estimate of the required return.

Preet

@Michael James – that seems like a good solution. But CAPM in the real world has many other imperfections too (like not accounting for tax, assuming everyone is rational, and everyone is a price-taker, etc). But even the other “improved” models suffer from some of the same criticisms. I think it would be safe to say that you could make a case for using 5% in the other models as well though.