Capital Asset Pricing Model (CAPM to shorten it) is a theoretical model that forecasts expected returns of investment assets, most often applied in the context of equities.

As any quantitative model, especially one that is 50 years old, it should be taken with a grain of salt.

However, up to this day, most discussions about pricing assets still start from CAPM, and many more modern models rely on it. Some even refer to it as a birth of asset pricing theory. I think it’s an important topic to be familiar with for anyone who wants to learn about investing or invest themselves.

Quoting Wikipedia [1]:

Despite it failing numerous empirical tests, and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton’s portfolio problem), the CAPM remains popular due to its simplicity and utility in a variety of situations.

## Introduction

Throughout this post, I’ll be using various terms which not all readers may be familiar with. If you’d like me to make to make a post about any of these, pleaselet me know!

• Risk-free rate - A rate of return of an investment where one can be certain to get a specified amount of money at the end of investment period. For example, if we can invest today 100$and we can be confident that we’ll get 101$ next year, then our risk-free rate is 1% per year. For practical reasons, the interest rate on a U.S. 3-month Treasury Bill is often considered a risk-free rate for U.S-based investors.
• Diversification - one of the most fundamental concepts in finance. When allocating your capital to multiple assets, you also distribute your risk. It is much less probable for prices of various assets to have a significant drop at the same time than a single one. If you invest half of your money in an oil refinery and the other half in solar panel producer, you know that whichever of these energy sectors performs better, your portfolio would do reasonably well. If you allocate everything you have to the loser, you will do significantly worse.
• Volatility - A measure of the riskiness of an investment. It’s calculated as a standard deviation of returns - the larger this value is, the larger in magnitude positive and negative returns of an asset. Usually quoted on an annual basis - standard deviation of a yearly return on given investment.
$\sigma = \sqrt{\frac{252}{N} \Sigma_{i=1}^{N} (x_i - x) }$

## Foundations

To start, let’s perform a thought experiment and imagine we live in a world where expected returns of all equity assets are known beforehand to every investor.

To make our example a bit more concrete, I’ve taken eight real-world companies with their stock returns in the period from 2006 until 2016 and calculated realized yearly returns and volatilities:

return volatility
Intel 10.5% 29.3%
Exxon Mobil 9.5% 24.8%
Ford 16.3% 45.5%
Goldman Sachs 14.4% 39.3%
Hasbro 19.0% 28.0%
Kellog 9.2% 17.7%
Marriott 15.7% 34.1%

We can take this table and plot it on the graph, to get a more visual overview of data:

We are an investor at the beginning of the year 2006, and we already know above numbers – we know the expected returns and volatilities. The question remains: what do we invest in?

One may say that the best investment is Hasbro - the highest expected return. Another one may choose Kellog - one with the lowest risk. 8% of annual return is still quite a lot.

But a definite answer to this question has already been given by Henry Markowitz in 1952[2]. It is known as Moden portfolio theory (And I’ll probably make a more detailed post about it in the future). To shorten a possible digression, it states that when we consider a set of all possible investment portfolios in the market (in our case portfolios of 8 stocks I’ve listed above), we have exactly one optimal investment portfolio for each level of risk tolerance of an investor.

These so-called “efficient” portfolios, can be plotted again on our graph to form a line called the “Efficient Frontier”.

Each point on this line is a portfolio composition we can invest in. We have a portfolio with a lower risk and higher return than Kellog and a more profitable portfolio for the same risk as Hasbro.

Now when we know a risk tolerance of an investor, we can find for him an optimal portfolio on an “Efficient Frontier”, so the investment problem is solved, right?

Not yet, because we still haven’t incorporated one important factor in our decision process: a risk-free rate.

In our theoretical model, an investor can freely lend or borrow money at risk-free rate – as a name suggests, without any risk involved. Although the actual rate changed significantly in the period of 2006-2016, an average was 1.02%, and that’s what I’ll use here for demonstration.

## Investing in a world with a risk-free rate

In a previous section, we were considering only portfolios with 100% of capital invested in stocks. But of course, we can allocate only a portion and put the rest safely stored in the risk-free investment, at a lower rate.

In the same way, if we are very risk-hungry, we can borrow money to buy more stocks.

What comes out, is that in such case there is only one portfolio on the “Efficient Frontier” that is an optimal investment - called the Market Portfolio. It is the one with the highest Sharpe ratio (return/risk ratio) on the blue line. Investors can scale this optimal portfolio to the level of their risk tolerance by borrowing or lending excess capital at the risk-free rate.

The new set of optimal investments is now a straight line called the Capital Market Line. We can clearly see on the graph that portfolios from Capital Market Line are better investments than the ones on the Efficient Frontier and that they intersect in a single point - the Market portfolio.

Let’s look at the returns of Market portfolio:

Not that bad overall.

We can compare it with Google returns:

And Ford returns:

One easy observation is that Market portfolio is the smoothest line going up.

## Investigating Market Portfolio

I think we should spend a bit more time to investigate Market portfolio. Basically what we observed so far is that this is the only risky investment in our thought experiment universe that a rational, profit-seeking and risk-averse investor should hold.

One interesting outcome of the fact that all investors hold the same portfolio is that the weight of each asset in that portfolio, by definition, must be the total value of all outstanding shares of the asset divided by the total market value of all assets.

Market portfolio is diversified - it consists of a combination of 8 stocks and has a high expected return - 15.58% with 19.62% annualized volatility.

Many people assign a special meaning to a Market portfolio and identify it as a marker of a state of the economy. Company shares move naturally up and down. The move can depend on missing or meeting revenue estimates, successful product launches, leadership changes, and various others. But the Market portfolio has a limited dependency on individual stock – in our case, it’s not entirely accurate because we consider only eight stocks. In the real world, we often find markets consisting of hundreds or thousands of companies. In other words, we can think that when a price of the Market portfolio goes down, it means that a sizeable part of stocks is going down, and consequently when it’s going up it means most of the stocks are doing pretty good.

Probably the most important property of a Market portfolio is that the risk it bears cannot be diversified any more. We cannot add any asset to the Market portfolio to make the risk lower for the same rate of return – that’s impossible by construction. We chose Market portfolio to be already the most diversified we can.

Sudden moves in the Market portfolio are usually associated with changes in government policy, macroeconomic forces or acts of nature. Such risk, common to all securities in the market, is commonly called a systematic risk[3]. We can decompose stock volatilities into a systematic risk and specific risk.

Some moves of an asset are because of broad-market moves and others happen because of company-specific events. Decomposing these two sources of return is a very useful technique.

Systematic Risk Specific Risk Total Risk
Google 20.3% 22.5% 30.3%
Intel 12.8% 26.3% 29.3%
Exxon Mobil 11.5% 22.0% 24.8%
Ford 20.5% 40.6% 45.5%
Goldman Sachs 18.1% 34.9% 39.3%
Hasbro 24.2% 14.1% 28.0%
Kellog 11.0% 13.9% 17.7%
Marriott 19.8% 27.8% 34.1%

What’s important is that systematic and specific risks are independent in the mathematical sense. Again, this happens so by construction, but it also has a nice interpretation – what’s happening specifically to one company doesn’t have anything to do with the broad economy.

We can see in the table, that volatilities don’t add up – independent risks add as a variance (volatility squared).

$\sigma_{total}^2 = \sigma_{systematic}^2 + \sigma_{specific}^2$

To take a closer look at one example, let’s see Marriott returns deconstructed on systematic and specific parts.

## Beta coefficient

As we have seen in the previous section, long-term performance of the stock came mostly from how the overall economy was doing, not from the company-specific events. One way to measure the sensitivity of stock price moves on the market portfolio performance is called the beta coefficient. It has a very simple interpretation: If a market portfolio moves up 1%, how much will my stock move?

In mathematical terms, beta is a ratio of stock systematic volatility divided by the volatility of the market portfolio.

$\beta = \frac{\sigma_{systematic}}{\sigma_{market}} = \frac{\textrm{Cov}(R_m, R_{asset})}{\textrm{Cov}(R_m, R_m)}$

Here are beta coefficients of selected stocks:

beta
Intel 0.65
Exxon Mobil 0.58
Ford 1.05
Goldman Sachs 0.92
Hasbro 1.23
Kellog 0.56
Marriott 1.01

One important detail is that we consider market returns in excess of risk-free rate here. It doesn’t matter that much over a short period when the effect of a risk-free rate is negligible. If the market portfolio moves 1% on a given day, that means Kellog moves 0.56% and, Hasbro moves 1.23%. Actual performance will differ but this is the systematic part, and the rest is the specific part.

If we are looking at the performance over a longer period – let’s say a year, then if a market portfolio returns 12%, that’s 1.02% risk-free rate return and 10.98% excess return. That means, Kellog will move $0.56 \times 10.98\% = 6.15\%$ as a result of market moves and the rest will be because of company-specific events (+ 1.02% because of risk-free rate).

## Capital Asset Pricing Model

Now, it is the time to go back from imaginary world to the reality.

In the real world, we don’t know the expected returns up front, but we are aware that current market prices are a consensus reached by all market participants.

What we assume, is that other investors are rational and invest in the optimal portfolio or close enough to it for practical reasons. We believe all assets are priced appropriately to the future expected price appreciation given all publicly available information.

If we know all company market capitalizations, we can calculate the capitalization-weighted stock index, right? In practice, that leads to of the most common criticisms of CAPM. For the thesis of the theorem to work, we would have to make a portfolio consisting of not only U.S. stocks, but all investable assets – world equities, bonds, real estate and human capital. That is a very hard if not an impossible exercise, so instead simplifications are being made.

In our case let’s take S&P 500 Index - a capitalization-weighted stock index of 500 biggest companies traded on NYSE or NASDAQ. We’ll treat it as our market portfolio or an approximation of it. It is quite hard to judge how close we are to the real market portfolio, though.

The good thing is that for S&P 500 we can get quite a long history of 29 years for a total return (including dividend payments) version of an index.

We can easily get the risk-free rate levels for the same period as well.

In that period, S&P 500 Index returned 8.08% per year in excess over risk-free rate with an annualized volatility of 17.68%. As these were calculated over long-term on a diversified index, I’ll treat them as persistent and assume they’re equal to future expected returns.

Once we have a market portfolio, we can calculate beta coefficients for all stocks we’re interested in. Let’s notice they’re slightly different than in the previous section as the market portfolio is different, but maybe there is a small similarity in the end?

beta
Intel 1.04
Exxon Mobil 0.95
Ford 1.28
Goldman Sachs 1.44
Hasbro 0.79
Kellog 0.50
Marriott 1.25

Given betas and market portfolio performance distribution, we can calculate expected stock returns.

To do that let's agree on a set of symbols we'll use.

\begin{align*} R_f &- \textrm{return on capital invested in the risk-free asset} \\ R_{syst} &- \textrm{systematic return of an asset} \\ R &- \textrm{total return of an asset} \\ R_{m} &- \textrm{return of a market portfolio} \\ R_{syst} - R_f &- \textrm{excess systematic return of an asset} \\ R_{m} - R_f &- \textrm{excess return of the market} \\ \beta &- \textrm{a beta coefficient of an asset} \end{align*}

From the definition of beta, we have:

$R_{syst} - R_f = \beta (R_{m} - R_f)$

And the whole thesis of CAPM becomes now a simple equation:

$\mathbb{E}\left(R\right) = \mathbb{E}(R_{syst}) = R_f + \beta (\mathbb{E}(R_m) - R_f)$

To rephrase:
An expected return of an asset is equal to an expected systematic return.

Since we already have a formula, why not use it to calculate the expected returns?
We have all the data we need in place.
I'll put the realized volatility in the table as well.

Expected Return Volatility
Intel 11.6% 29.3%
Exxon Mobil 10.9% 24.8%
Ford 13.5% 45.5%
Goldman Sachs 14.8% 39.3%
Hasbro 9.6% 28.0%
Kellog 7.2% 17.7%
Marriott 13.3% 34.1%

## CAPM - Interpretation

The CAPM formula may seem to be simple, but that should not be mistaken for lack of depth.

Firstly, let's ask ourselves a question: why would stocks have a return higher than a risk-free rate
in a perfectly efficient market? Simply reversing this argument seems to answer it:
If equities had no expected return above the risk-free rate, no one would invest in them.
The risk-free investment is preferable to a risky one with the same upside.

For the stock market to be a viable investment, there needs to be for an excess return above risk-free rate
that investors receive in exchange for taking a risk. It's called an equity risk premium.

It's this part of the CAPM equation:

$\beta (\mathbb{E}(R_m) - R_f)$

What’s interesting, is that investors are compensated only for the systematic risk they’re taking. The specific risk, because it can be diversified away, does not bring an additional premium.

The more systematic risk asset has, the more premium investor receives and the best investment is the market portfolio itself – an investment will all specific risk diversified away.

It’s important to note that all this time we were talking only about passive investing – buying and holding an asset through the period analyzed. Many money managers try to time the market, entering and exiting positions in securities trying to gain an additional excess return uncorrelated to the market portfolio. Such return is commonly called an alpha.

## Criticism and Problems

What we’ve discussed above is a theory. An intellectually coherent one, but like any other scientific theory, it needs to make multiple simplifications to model a complex real world. Not all simplifications are necessarily harming our predictive accuracy, but it’s an important step to look back and reevaluate any assumptions made.

• All investors try to maximize economic utility, are rational and risk averse– This is a relatively safe statement. While admittedly, it does not hold for all the investors individually, it probably does approximate pretty well the whole market. On the flipside, many investors take into consideration more risk metrics than just volatility, which can make their optimal portfolios look significantly different.
• Investors can lend and borrow freely at a risk-free rate, trade without transaction cost and short any security freely – Unfortunately, the real world does not satisfy this assumption.
• Investors have homogenous expectations, and all information is available at the same time to all – It is very much possible, that investors will have biases, use different sources of information and have significantly differing outlooks on future asset prices.
• The market portfolio consists of all assets in all markets, where each asset is weighted by its market capitalization – As we’ve already have discussed before, it’s prohibitively hard to build a true market portfolio.

Over the long-term, academics have tested this theory extensively, comparing vs. empirical results. Multiple teams consistently found this model unable to explain variation in asset returns well enough. Most notably, low beta stocks seem to have a higher expected return than CAPM would suggest and high beta stocks seem to have a lower expected return.

Nevertheless, CAPM has been an important stepping stone in the search for the theory of asset pricing.

## Closing remarks

Thank you very much for staying with me until the end. I hope you liked my article.

All the charts/figures above are based on real-world data that I’ve downloaded from Yahoo! Finance. As you have seen, I didn’t develop a thorough mathematical theory formally or present a detailed source code of all calculations. Instead, I’ve focused on telling a story, without getting lost in the details.

If you’d prefer for me to get more or less technical in my next posts, please let me know.