1.The Capital Asset Pricing Model – The Sharp-Lintner-Mossin Model
The most celebrated idea of modern portfolio theory (MPT) is the Capital Asset Pricing Model (CAPM). William F. Sharpe, often cited as the originator of the Capital Asset Pricing Model, published his original analysis in an article entitled “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk,” in the September, 1964 edition of the Journal of Finance. Concurrently, John Lintner published his similar treatment of the CAPM model in “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” in the February, 1965 edition of the Review of Economic Statistics. Finally, Jan Mossin’s treatment of CAPM appeared, independently of Sharpe and Lintner in a January, 1966 issue of Econometrica in an article entitled, “Equilibrium in a Capital Asset Market.” The CAPM, developed independently by Sharpe, Lintner, and Mossin, uses Tobin's Theorem and applies it to what economists call a general equilibrium model. Tobin's Theorem is about what happens when a single individual investor faces the problem of choosing a best portfolio among all of those portfolios that the investor can afford. CAPM asks the question: if every investor acts like the investor in the Markowitz-Tobin world, what will be the prices of assets in equilibrium.
What do we mean by equilibrium? The term equilibrium means that (planned) supply equals (planned) demand. An investor is in equilibrium when he owns the assets he wants to own, given prices and expected returns, variances and covariances. If each investor is in equilbirum, then the reigning prices of each asset form a set of equilibrium prices. The Capital Asset Pricing Model is a theory about how assets are priced in equilibrium.
a.The Relationship Between the Return of An Asset and An Index -- A Preliminary Exercise
Suppose we have information about the rate of return of some particular asset over a rather lengthy period of time. We also have information about the rate of return of some group of assets, which, for simplicity, we shall refer to as an index. What can be said about the relationship between the return of the individual asset and the return of the index? Are they related and, if so, how? Suppose we have annual data on rates of return as in the following table:
Year / Asset / Index1970 / 5% / 8%
1971 / -2% / 1%
1972 / 12% / 10%
...... / ..... / ....
...... / ..... / .....
1990 / 8% / 13%
1991 / -5% / 6%
1992 / 2% / -4%
How can we take the data in the table and make statements about the relationship between the return of the particular asset and the return of the index? Suppose we plot the data as in the following diagram:
Each point in the diagram represents the return for one particular year in our table. The vertical coordinate is the rate of return of the asset and horizontal coordinate is the rate of return of the index. Notice that we have drawn a straight line in the diagram. We did that to suggest that perhaps the relationship between the returns is a linear relationship, i.e. the relationship can be described by a straight line. If the relationship is linear, what is it?
Suppose we try to find a line that best fits the data. What do we mean best fits the data? In order to answer this question, we need some way to describe the distance between the various points in the diagram and any straight line that we might draw in the diagram.
Suppose we begin with an arbitrary line defined as:
y = a + bx
Let y represent the return of the individual asset and let x represent the return of the index. If we have a specific value of x, say x1978, we can plug that value into the above equation and get a predicted value of y1978. This predicted value is not generally going to be identical to the value for y1978 that is inserted in our table or exhibited in our diagram. There will usually be a discrepancy, which we shall call an error and describe with the letter e1978.
This means we need to rewrite our equation to reflect the implicit potential error in the equation:
y = a + bx + e
The errors are really differences between some arbitrary line and the various points in the diagram. Notice the errors ei and ej in the following diagram:
If the errors were zero for each observation, that would mean that all the points in our diagram fall along the straight line. Otherwise the e's will be sometimes positive, sometimes negative, and sometimes zero. The positives and negatives might even tend to cancel each other out, even though individually the e's might be very large in absolute value. How well does all of our data fit some arbitrary line like y = a + bx. The e's measure how well the line fits. But, to avoid having e's cancel each other out, let us square the e's and then add them all up to get an idea how far from the line the typical point is.
Before we do all this, let us add to our notation. Let yt and xt represent the return to the asset and the return to the index in year t. We can now talk meaningfully about things like the sum of the yt's, the sum of the xt's, and their respective (sample) means and (sample) standard deviations. Let represent the sample mean of the xt's and represent the sample mean of the yt's.
Then:
and:
These are sample (of 23 observations) means not the true mean returns. We don't really know what the true mean return is. We can try to estimate the true mean by calculating the sample mean in this fashion. Similarly, let sx and sy be the sample standard deviations of the xt's and yt's. Then:
and:
Again, these are sample standard deviations, not true standard deviations. We don't have any way of really knowing the true standard deviation. The sample standard deviation can be thought of as an estimate of the true standard deviation.
Where is all of this headed? We want to find the line that best fits the data in our table and our diagram. Why not find the one that minimizes the squared errors? In other words find values of a and b that minimize the following sum:
Continuing to expand the interior of the parentheses:
Now, bringing the summation sign inside the parenthesis, gives:
With appropriate substitutions:
[A1]
where n ( = 23) represents the number of observations. So much for the algebra. To minimize the above sum, we need to differentiate the sum with respect to a and with respect to b and set the resulting equations equal to zero.
i.Minimize the Sum of Squared Errors
Mathematically, we intend to minimize the squared errors by appropriate choices for a and b.
We begin by differentiating the squared errors in respect to a and b and set the results equal to zero:
(1.)
(2.)
The first equation can be simplified to:
(1.)
The second equation after substituting in for a using (1.) becomes:
eliminating the 2's and simplifying:
Separately, consider the following expression and its expansion:
plugging this expression in where appropriate gives us:
(2.)
Again, separately consider the following expansion:
which means that:
Now, making a substitution of this last expression into (2.) gives us:
(2.)
Solving (2.) for b yields:
By inspection this expression equals the sample covariance divided by the sample variance of the index return:
[A2]
We have switched from b to to indicate that we have a very specific value of b in mind. We will do the same with a, converting it to . We conclude with the following two conditions that the best fitting line must satisfy:
(1.)
(2.)
where the 's are sample covariances and sample variances.
ii.We Have Found a Beta!
Beta, written symbolically as , is the single most widely used piece of jargon in finance theory.
What we have constructed is an empirical beta, not to be confused with the theoretical beta that plays such a prominent role in the Capital Asset Pricing Model. The beta of CAPM appears to be identical to the one that we have constructed here, but it is not identical. Later, when we speak of the theoretical beta in the CAPM context, we shall mean the covariance divided by variance, which seems the same as what we have here. But the covariance and variance in CAPM is the population covariance and variance, not the sample covariance and variance. The sample parameters could, of course, be thought of as estimates of the population parameters, but they are different concepts in principle. We can always calculate sample parameters. We cannot always necessarily calculate the population parameters, especially the population parameters that are relevant to finance theory. It is important to consistently maintain this distinction between sample parameters and population parameters.
What does our beta mean? Beta measures the relationship between an individual asset's return and the return of some index. Usually, the index is a basket of assets which typically will include the individual asset of interest. This is not necessary, though. The index can be any arbitrary index and beta will still attempt to measure the linear fit of the return of the asset to the return of the index.
Notice very carefully what beta is not. Often, when asked, investors say that beta measures the volatility (which always means variance in finance jargon) of return of an asset, usually a common stock. Such investors are confusing beta with variance. It is possible for an asset to have a very high variance, but bear no particular relationship to the index used in the beta calculation. In this situation, the asset return will be very volatile, but the beta might be zero. Similarly, it is possible to construct examples of assets with high betas (by suitable choice of index) but relatively low volatility (variance). We conclude that beta is not volatility or variance and need not be in any way related to volatility or variance.
I.The Startling Conclusions of the Capital Asset Pricing Model
The Capital Asset Pricing Model is a model that asks what happens when asset markets are in equilibrium. CAPM assumes the conditions of the Markowitz-Tobin model -- that investors are risk averse and that assets have known means, variances, and covariances with each other.
CAPM Conclusions:
for every asset i:
with:
where:
M is the market basket containing every asset Xi that is positively priced in equilibrium. The proportions, i's, of each asset in M, are the percentage of total market value of all assets represented by the total market value of asset i.
Ri is the return of asset i. RMis the return of the market basket,M. Rfis the return of the riskless asset. i,M is the covariance between the return of asset i and the return of the market basket M divided by the variance of the return of the market basket.
What do the CAPM conclusions mean? The first equation:
has come to be known as the Security Market Line. It says that the expected return net of the risk free rate for any asset is equal to its beta (with the market) times the expected return of the market net of the risk free rate.
Consider the expressions
=
Why are we subtracting the risk free rate, Rf, from these expected returns? Taking no risk at all will always earn us the risk free rate, Rf. Therefore, if we want higher returns we must take some risk. The return that we gain from taking risk is always relative to the risk free rate. We often call this difference between the expected return of a risky portfolio and the return from the riskless asset the risk premium. It shows the additional expected return that an asset must produce to compensate for its riskiness. One way to interpret the Security Market Line is the following: The risk premium earned by any asset is, on average, equal to beta times the risk premium earned by the total market portfolio. This is truly a revolutionary conclusion. Notice that the variance of the ith's asset's return plays no role in accounting for its expected return or its risk premium. Variance of an individual asset's return does not, in equilibrium, matter at all. What matters is the relationship between the return of an individual asset and the return of all assets taken together -- the market basket, M.
Another way of interpreting the security market line is that an asset's return depends upon its contribution to the return of the overall market portfolio return. No matter how volatile a stock's return might be its average return will depend solely, in equilibrium, upon its contribution to the total portfolio return of the entire market as assets. This is the startling conclusion of modern portfolio theory.
II.Diagramming the Security Market Line
We exhibit the Security Market Line diagrammatically:
The Security Market Line applies to any asset or portfolio in equilibrium. In particular, the market basket, which consists of all assets weighted by their relative market values, has a beta equal to one:
It is worth noting that the beta of a portfolio can be calculated by averaging the betas of the individual assets that comprise the portfolio. The weights used to average the betas are identical to the weights used in the construction of the portfolio. If a portfolio, P, is defined as:
then P's beta can be constructed as follows:
III.The Tobin Theorem Conclusion Re-emerges in CAPM
We have very quietly ignored a major conclusion of CAPM. The Tobin Theorem re-emerges. Each individual in CAPM owns a portfolio that consists of at most two assets: the riskless asset and the market basket portfolio. Borrowing and lending stories are exactly analogous to the Tobin Theorem. The only difference is that the E* portfolio of the Tobin Theorem becomes the market basket, M, in CAPM.
This conclusion, that individuals that wish to take on any risk at all, should own at least something of everything is not often emphasized in finance theory. I think some feel that this is an unwelcome consequence of CAPM. The spirit of this conclusion is the same as the Tobin Theorem. Diversification creates a very efficient mix of assets. Indeed all worthwhile assets are included in the diversification process. Only those assets of no market value are left out of the market basket, M.
Is this a reasonable conclusion? After all, we do not observe individuals buying a little bit of every asset. Or do we? Perhaps, this seems more sensible if we attack the problem from a slightly different point of view. Suppose one individual, in a CAPM world, owns three times as much GM stock as Eastman Kodak stock. What can you say about the relative ownership of GM and Eastman Kodak stock by other individuals in the same CAPM world? The answer is the ratio will be the same for all individuals who own any risky assets at all - three times as much GM stock as Eastman Kodak stock. Indeed it is obvious that according to CAPM it must be true the GM's total market value must be three times the total market value of Eastman Kodak. Are these reasonable conclusions? As opposed to what? If CAPM approximates reality, we should find that the ratios of risky asset holdings are similar across different individuals. This may well be the case of large asset groups like housing, common stocks, bonds, and so forth. This is an empirical question and a tough one, but it cannot be immediately dismissed as unreasonable.
The dramatic growth of broadly diversified mutual funds over the past fifty years may well be a testament to the Tobin Theorem. (The mutual fund industry was in its infancy when Tobin’s Theorem was first established, so that Tobin’s theorem anticipated the tremendous growth in the mutual fund industry that has since taken place). But, will we all end up owning the same portfolio?
The crucial assumption in CAPM is that all individuals (or institutions or whatever) see the same assets and agree on the means, variances and covariances. This clearly isn't true in the real world -- but it may, over time, be getting to be more true with the increased technology and improved communications that have been applied to financial markets. As we tend to view assets more nearly the same way, we may end up more nearly satisfying the Tobin Theorem -- mutual fund conclusion of the Capital Asset Pricing Model.
These remarks are intended to counter the idea that the Tobin Theorem flavor is a wholly unrealistic conclusion of CAPM. It may be that individuals perceive differences among assets in different ways through lack of common information. That lack of common information may dissipate as technology makes the same information available to more persons more quickly. CAPM may be forecasting a future where the Tobin Theorem holds more broadly for more investors.
i.The Capital Market Line
Expected Returns
M