Introduction and Summary

The purpose of this chapter is to discuss what is known as the “efficient market model of capital market theory,” the most significant part of which is the Capital Asset Pricing Model. The paper does not extend the model. It is a nontechnical summary of the papers by Treynor (1961),¹ Sharpe (1970), Lintner (1965), Black (1972), and others. Their writings require considerable familiarity with mathematical and econometric concepts, hence a more verbal exposition of the basic ideas may be helpful to some readers.

There are two appropriate caveats concerning the context of this chapter. They both follow from the fact that it is an exposition, and not an extension, of the thinking of the authors of capital market theory. The first is that this paper does not expose all of the work by all of these authors. It is not, therefore, a fully comprehensive exposition. In particular, completeness has been sacrificed in the treatment of some of the more advanced extensions and refinements of capital market theory, especially where considerable mathematical sophistication is involved. This incompleteness seems palatable on the grounds that the basic essentials should be exposed first. The second warning is that the authors are attempting to expose the basics of capital market theory and not to validate it per se. All theories are abstract simplifications of reality. No one theory encompasses all aspects of anything, nor is the most elaborate theory expected to go unchanged with the subsequent evolution of thought. Accordingly, all existing features of capital market theory do not inherently correspond to reality equally well. What the authors do assume, however, is that there is sufficient correspondence between reality and the extent of capital market theory exposed herein to warrant the attention of the financial community.

The chapter begins with a definition of the risk of investing in the capital markets and with a method of measuring return and risk. It is shown that risk can be decomposed into two parts: the systematic risk due to the response of any stock to changes in the market as a whole, and the specific risk attributable to a particular stock. It is shown how the market place compensates the investor for taking systematic risk with an appropriate expected return. It is also shown that specific risk is not compensated, since it can be essentially eliminated through diversification. It is shown that under the assumptions of borrowing and lending at a risk-free rate, the expected return is proportional to the systematic risk borne. The security market line, which represents the relationship between systematic risk and expected return, is derived. The concept of efficient portfolios is introduced. The capital market line is defined as the set of such combinations of risky and riskless assets that are superior to all other portfolios in terms of return and risk. It is shown that an investor's choice of a portfolio on the capital market line (or a well-diversified portfolio close to the capital market line) depends only on his attitude toward expected return and risk. A generalized form of the model is outlined for the case when borrowing at the risk-free rate is not possible.

As the name suggests, the efficient market model describes the equilibrium state of efficient capital markets. A perfectly efficient market is one in which new information is immediately and costlessly available to all investors and potential investors, and the cost of action (transaction costs, taxes, etc.) is zero. While actual capital markets do not conform exactly to “perfect” assumptions, empirical studies suggest that the main conclusions of the efficient market model hold very well in real markets. The theory of efficient markets represents the best description of capital markets available at present, and probably the only one that considers explicitly uncertainty and risk. Since it also offers important implications for investment decisions and portfolio construction, the efficient market model should command the attention of every practicing investor.

Risk, Risk Aversion, and Compensation

Since the future prices of common stocks are uncertain, the outcome of an investment in common stocks cannot be determined in advance, and inevitably carries some risk. Risk is a chance of loss, which can be thought of either as an actual capital loss or a failure in achieving the return that was expected. There can be no risk if the outcome is certain; the investor, by definition, achieves the return he has expected. The more uncertain an investment opportunity is, the larger is the chance of loss, or the risk. Risk is therefore intrinsic to the existence of uncertainty about the outcome of an investment, and a formal quantitative definition of risk can be based on the concept of uncertainty.

Some financial instruments exhibit fixed return characteristics. Government bonds, for instance, when held to maturity, leave virtually no uncertainty about the total amount of money that will be returned. In accordance with the concept of risk as uncertainty, we will call these instruments, when held to maturity, risk-free. Common stocks, on the other hand, are risky issues. They may result in high gains, but may lead to large losses as well. A wide distribution of possible outcomes exists over all future horizons, since they do not mature.

Within the class of common stocks, not all issues exhibit the same risk. Some firms—for example, electronics or airlines—are generally engaged in endeavors whose outcome is less certain than those of other firms. For an investor, it means that there is less certainty that the future price of such stock is not going to decline considerably in value. On the other hand, the relatively stable and predictable nature of a utility company's business implies less uncertainty about the future price of its stock.

This leads to the following definition of risk in a capital asset market: The risk of an issue is the degree to which future price of that issue can differ from its expected value. Stated in different terms, risk is the dispersion of future price changes. The next section will deal with the question of how the dispersion can be measured ex post and estimated ex ante. Before addressing this issue, however, it is useful to discuss some aspects of this definition of risk.

There are several implications of the risky character of common stocks that generally make investors prefer (other things being equal) less risky investments to more risky ones. These include the so-called gambler's ruin, the consumption effects, and the liquidity requirement effects.

The phenomenon known as gambler's ruin refers to a particular outcome of an investment under uncertainty. It consists of losing effectively all funds in an extremely unlucky turn of events, thus being excluded from further participation in the investment. Thereby, the possibility of recovering from such loss is eliminated. Although this is an extreme case and it is generally very improbable (particularly in the short run), the effect—if it occurs—is fatal.

Another reason for investors' reluctance to bear risk relates to their consumption requirements. Most investors expect to consume some part of their income through time, and preservation of their wealth thus demands achieving a certain level of gain. Although a highly risky investment opportunity may have the prospect of high gains (as well as large losses), it may not, therefore, be as attractive to a particular investor as an opportunity to obtain more stable returns.

An even more important factor concerns liquidity requirements: the need to be able to convert stock holdings into another form of asset, namely cash, at any moment. An investment in highly risky stocks has a considerable disadvantage in this respect relative to an investment in more stable stocks, not because it is difficult to convert to cash, but because high-risk assets may happen to be at low values when cash is required, quite apart from the longer-term prospects.

For these reasons, it can be expected that investors are generally risk-averse. That is, they prefer more stable holdings to less stable ones, other things being equal. If this is true, how can one explain the behavior of market participants who place their investments in very risky positions? What would induce investors to hold these securities? The efficient market model suggests that the market pays higher rates of return for more risky holdings; otherwise, investors could not be induced to hold these issues. That is, the expected value of return on investment is higher for more risky issues than it is for less risky ones. The principle of risk compensation has been well established empirically (cf. Black, Jensen, and Scholes (1972)). It will be discussed again in quantitative terms in the Capital Asset Pricing Model section.

Different investors have different investment objectives, different consumption and liquidity requirements, and different planning horizons; thus, the degree of risk aversion differs from one investor to another. This means that some investors will be more comfortable at low risk levels with attendant comparatively small expectation of return, while others would prefer positions at higher risk levels with the attendant higher expected compensation. The actions of market participants, trading off their individual requirements with each other, then result in the appropriate pricing of risky assets.

Every investor can find a position in the market that corresponds to the degree of risk he considers adequate, given the compensation for risk the market offers. To avoid unnecessary risk (risk that can be diversified away) and to choose a proper tradeoff between the risk and expected return, are some of the practical applications of the efficient market model.

Measurement of Risk and Return

Risk has been introduced in the previous section as the dispersion of future price changes. This ex ante uncertainty manifests itself ex post as volatility in price series. The price of a company engaged in a business of highly unpredictable nature is likely to vary considerably as the prospects and outcomes of business ventures evolve. The price of a stock subject to less uncertainty is bound to change less swiftly.

These observations suggest that the risk is measured ex post by the variability of price changes. Since it has been established empirically that the degree of volatility of price changes of any stock is a reasonably stable quantity (cf. Blume (1971)), it can also be taken as an estimate of the ex ante uncertainty, or risk. This is a concept of great importance: It allows quantitative measurement and anticipation of risk.

Risk is one of the essential concepts of capital market theory, and the theory would lose much of its impact if techniques to measure risk were not provided. When risk is measured by price fluctuation, it is necessary to know to what extent ex post volatility of price can be taken as an estimate of ex ante uncertainty (i.e., risk). This depends on two factors: instability of the parameters of price change distributions, and sampling errors in estimating the parameters from data. The latter source of error can be minimized by using enough data for sampling and by choosing appropriate statistical techniques. The former source of misestimation depends on the speed with which the parameters change. In most cases, the rate of change is moderate, and past dispersion is as good an estimate of future dispersion as an estimate obtained by other means. Therefore, ex post measurements of price volatility provide a usably accurate proxy of ex ante risks.

To measure risk directly as the variability of price changes is not quite convenient, since it would mean that risk depends on the level of the price—the higher the price in dollars per share, the higher the dispersion. To avoid this scaling problem, variability of the rates of return is used. Another reason for choosing rates of return rather than price changes is that rates of return include dividends.

The rate of return over a time period to t is defined as

where P_t is the price per share at time t, and is the (cash) dividends per share paid during the period to t. The rate of return is thus expressed as total return on a dollar invested at the beginning of the period. Rate of return on a portfolio is defined in analogy, with return on an individual issue, as the increase in total wealth during the period divided by the original wealth.

Under uncertainty, future rates of return behave as random variables and can, accordingly, be described by the characteristics of their probability distribution. The expected rate of return is the mean value of the distribution of future returns. The expected rate of return is thus the value around which future rates are expected to center. The expected rate is the most likely estimate of what the future rates will be.

The expected rate of return may be different from the average performance of the company in previous periods. If a company is believed to maintain the same a priori distribution of rates of return, however, the expected value can be estimated by the average rate of return in the previous period,

Any investor certainly prefers investment opportunities with higher expectations to those with lower expectations, if other things are equal. The expected return alone, however, does not fully characterize an investment opportunity. It is necessary to consider the deviation of possible future returns from the expected value, which brings us back to the concept of uncertainty, or risk.

Risk has been defined as the dispersion of the rates of return from their expected value. The larger the dispersion, the less sure the investor is that the expected performance will be attained. The dispersion can be measured by the variance, which is the mean squared deviation of the distribution of returns from the expected value.

The variance in rates of return measures the expected degree of fluctuation of future returns about their expectation—that is, the probable variability of future payments. It is therefore a convenient measure of the riskiness of an investment. Investors, being risk-averse, will generally prefer smaller variance to larger variance on their investments.

The variance (which is, like the expected return, an unobservable ex ante parameter of the distribution of future returns) can be estimated by the ex post sample variance of rates of return attained:

If the sample variance of past rates of return is used to estimate future dispersion, it must be assumed that the degree of fluctuation in the series of returns remains relatively stable over time. Stated differently: The variation in returns must be “usably” constant—a question that can be empirically tested.

The variance V is a quadratic measure of volatility. It is sometimes useful to consider its square root,

which is called standard deviation. Since these two measures of risk are so closely related, we will occasionally use one and occasionally the other in referring to risk. The difference between them is a matter of mathematical convenience, not economic significance.

The prices, and consequently the rates of return, of two common stocks or of a stock and the stock market as a whole do not move independently. When the market advances, a stock listed on that market is also likely to advance, and similarly for declines. This is due to the dependence among industries and among companies of the same industry. It is estimated that about 50 percent of the price fluctuation of a particular company can be explained by overall market movements, and some 10 percent by the fluctuation of that industry. Then, the remaining 40 percent fluctuation is that due to the characteristics of the individual company (cf. King (1966)).

Since the comovement of stocks plays an eminent role in constructing a portfolio (e.g., in diversification), it is necessary to introduce a measure of dependence between rates of return, in addition to measures of expected return and dispersion. A convenient measure of comovement is what statisticians call the covariance.

Covariance between two rates of return is defined as the mean value of the product of the deviations of the two rates from their respective expected values. A positive covariance means that the two issues are likely to move in the same direction (which is typically the case in the stock market). Negative covariance means that the two stocks tend to move in different directions.

On past rates of return, the covariance can be estimated by the sample covariance, which is defined as

where are the rates of return on two stocks, and are their respective averages.

It will be shown in a later section on the role of the portfolio that nearly all fluctuations of returns on a well-diversified portfolio are due to the average covariance of the issues with the market as a whole. The total dispersions of individual companies do not, therefore, add directly to form the dispersion of the portfolio. Rather, there are dispersion components of individual companies that cancel each other out. The result is that total portfolio dispersion is less than for any individual company. This is, of course, the desirable impact of diversification.

Efficient Market Hypothesis

Capital markets can be characterized by two important factors: divisibility, and liquidity. Divisibility means that real assets are divided into a large number of shares, which can be purchased by investors in arbitrary amounts. Therefore, an asset can be held in various proportions by a number of investors, and conversely, an investor can distribute his wealth among “shares” of many assets. Thus, divisibility in a marketplace permits diversification.

Liquidity means that each investor can easily and at relatively small expense trade his share in any real asset for shares of other assets, or convert his shares into cash, and vice versa. This implies that each investor can hold a portfolio of assets perceived to correspond best to his requirements. He may shift his holdings at any time when either his requirements or his perceptions of the characteristics of the assets change.

It is in the interest of each investor to collect information about the shares of real assets traded in capital markets. Such information allows the investor to evaluate the prospects of each investment opportunity, and therefore to invest in the portfolio with the most promising performance. The demand for this information generates the existence of various information channels expected to provide the investor with pertinent knowledge, such as periodic income statements and balance sheets of companies, stock prices, and volumes; there are also newspapers specializing in bringing and evaluating news relevant to investments, and reports from financial intermediaries who engage in estimating the prospects of corporations.

These channels are efficient in that information spreads rapidly, and each new piece of information quickly becomes public property. Because shares are both divisible and liquid, investors are able to react very quickly to perceived changes in the value of any company. This information-induced buying and selling affects the market price to the point where the price quickly corresponds to value again. Thus, information is rapidly discounted into the market price.

It thereby becomes very difficult for any individual investor to find a stock that is not priced correctly. In fact, it is hypothesized that the capital markets are very close to what is called an efficient market. In an efficient market, each common stock is, at any moment, priced fairly with respect to its value. This premise will be referred to as the efficient market hypothesis.

While it is very difficult to decide once and for all through empirical tests whether the capital markets are efficient, most empirical studies suggest that the principle of efficiency holds very closely. Only rarely does the performance record of an investor show results that can confidently be attributed to stock selection skill. More often than not, superior performances can be attributed to investors having systematically taken high risk (since the expected return on a high-risk portfolio exceeds the expected return of a market index).

In an efficient market, all currently available information about future prices is discounted in today's price. If the price were confidently expected to advance tomorrow by 10 percent, investors would buy the stock until that 10 percent expectation is arbitraged out of today's price. Hence tomorrow's expected price change is thereby reduced to zero. In this process of discounting information, today's price becomes an unbiased estimate of tomorrow's price. Strictly speaking, of course, this statement needs refinement, since the market exhibits a positive long-term trend. The principle of efficiency therefore asserts that today's price is an unbiased estimate of tomorrow's price, discounted by the expected long-term growth.

The process described in the preceding paragraph is called a martingale. Such a process implies that all information about future prices is already impounded in the current price. A special case of a martingale, with an additional assumption of independent distribution of price changes, is the well-known and frequently misunderstood random walk process.

The term random walk is often misinterpreted as implying that price changes “randomly,” that is, by chance alone and without any causal reasons. This is not what the efficient market hypothesis states. Prices change because the characteristics and prospects of the company or the general economy change, and because investors' perception and evaluation of these characteristics and prospects change. In other words, an investor's knowledge evolves with the continuous supply of new information and with the revision of old information. What the efficient market hypothesis does state, however, is that at any given moment in time, the next period price change is random with respect to the state of knowledge at this moment. Moreover, the hypothesis of efficiency asserts that the current price fully reflects the present state of knowledge in the sense that it is equal to the (discounted) mean value of the distribution of the next period price as given by the present state of knowledge.

To summarize, under the efficient market hypothesis all the information available at any given moment is discounted into the current market price. The market price at any moment is an unbiased estimate of the next period price. In an efficient market, no investor can expect consistently to obtain information not already discounted into the market price by actions of other investors. Consequently, no investor can consistently achieve abnormal returns (i.e., returns in excess of that paid for risk taking, as discussed later in this chapter). In an efficient market, benefits from security analysis cannot be expected to exceed the costs of trading.

Since most of the discussion thus far has dealt with price changes, a word about the role of dividends is in order. Capital market theory deals with total rates of return, generally without distinction between the capital appreciation component and the component due to dividend income. Empirical studies by Black and Scholes (1970) have established that, while for a taxpaying investor it may be important whether the return on his investment comes in the form of dividend or capital gains, it makes no difference for the equilibrium of the market as a whole.

The next two sections will deal with the Capital Asset Pricing Model. In these sections, it will be assumed that the (total) rates of return over nonoverlapping periods are independently distributed random variables, and that each investor tries to maximize his expected rate of return subject to consideration of risk, without regard to whether the rate of return is composed of dividends or of capital gains. The assumption here that investors are indifferent between capital gains and dividends is not a realistic assumption when the component returns are taxed differently. The usefulness of this assumption is only to simplify the analysis, and the conclusions reached are, in fact, affected unimportantly.

The Role of the Portfolio in Risk Reduction

It is intuitively obvious that “risk” of the volatility of future returns can be reduced by diversifying into several stocks rather than investing one's total wealth in a single stock. Let us, however, characterize this diversification phenomenon quantitatively.

Consider two different common stocks. Assume for simplicity that they both have the same expected rate of return, E. If a part of the wealth available for investment, call it , is allocated to one stock and the remaining part is invested in another, the expected rate of return, E_P, on this two-issue portfolio is a weighted average of the two expected returns, or

The expected return on the portfolio is thus equal to that of either stock, since we, of course, assumed them to be the same. The volatility of the two-issue portfolio, however, is less (as will be seen) than the volatility of either stock, if only we assume them to move together through time imperfectly.

According to a theorem in statistics, the variance of the portfolio is computed as:

In this equation, and are variances of the two stocks, respectively; is the covariance of their returns; and are their weights in the portfolio. The covariance term is crucial to the effect of diversification. If the two issues fluctuate in price independently of each other, then the covariance term is zero, and it is always possible to choose the relative proportions in such a way that the risk of the portfolio is smaller than that of either stock taken separately. This is due to the effect of squaring numbers less than one. For instance, when the volatility of both stocks is the same, , and the covariance is zero, the risk of a portfolio of equal investment in each stock is

Thus, the portfolio has a variance equal to only one half of the variance of either stock. Since the expected return on the portfolio is not reduced (i.e., as shown earlier, ), such a portfolio is clearly preferable to a single-issue portfolio for any investor who is averse to risk.

Typically, two stocks exhibit some positive co-movement; therefore, the covariance term cannot be realistically assumed to be zero. The reduction of risk in that case is not as large as if the two stocks were independent, but it always can be made smaller than the simple average risk of the two stocks. Hence, the amount of risk per unit of expected return can be decreased through diversification.

The foregoing illustration of a two-issue portfolio can be generalized to the case of any number of stocks in a portfolio. Assume for simplicity that the proportion of each of the n stocks in a portfolio is kept equal. Empirical studies conducted by Lorie and Fisher (1970) show the following relative risk reduction: If the average risk level of a single typical common stock is taken as the basis, the percentage reduction of risk for randomly selected portfolios with approximately the same expected return depends upon the number of issues as follows:

Thus, diversification can provide a substantial reduction of risk. By diversifying funds among a large number of issues, the chances of heavy losses are lowered, since a price decline in some stocks is likely to be offset by price appreciation of others.

Common stocks are usually positively correlated with each other, and it is thereby not possible to eliminate variance completely. A down movement of a set of stocks in a portfolio will be only partly compensated by an up movement of some other stocks in the portfolio. There is, therefore, a part of the total variance of the portfolio that is due to the positive covariance between stocks and can never be eliminated by diversification. This part is called the systematic risk and will be dealt with in more detail in the next section.

Let us now investigate more thoroughly the impact of diversification on the expected return of the portfolio. Suppose an investor wants to maximize his expected rate of return but does not want to accept more than a particular risk level, call it V₀. His unwillingness to take more risk might, for instance, be attributable to the fact that the investor wants to keep the chances of failing to meet certain expected financial obligations below a particular level. Of all possible combinations of all stocks in various proportions that have the risk level of V₀, there will be one particular combination that has the highest expected return. This combination represents the optimal portfolio for that investor, since it is superior to all other portfolios of the same risk.

This portfolio can be constructed by means of so-called quadratic programming, provided all the prerequisite estimates of expected returns, variables, and covariances can be obtained. While it is difficult (and expensive) in practice to obtain the exact solution to this problem, due to the large number of stocks involved, there are methods that lead to a portfolio reasonably close to the optimal portfolio.

There will be a portfolio with maximal expected return for each different risk level. These portfolios are called efficient portfolios. The set of efficient portfolios, or the efficient frontier, as it is sometimes called, is a set of portfolios superior to all other portfolios. They are superior since, for each level of risk, there is no other portfolio with higher expected return. It is also possible, and often useful, to refer to efficient portfolios as those portfolios possessing minimum risk for a given level of expected return.

An investor's choice of a portfolio from the efficient frontier is a matter of finding a suitable tradeoff between expected return and risk. This choice depends on the investor's consumption, fixed obligations, time horizon, and other factors (which can be, at least for theoretical purposes, summarized in a so-called “utility function”). Whatever these factors might be, a rational risk-averse investor will select an investment portfolio that is at, or close to, the efficient frontier.

From what has been discussed, it should be apparent that the expected rate of return on portfolios on the efficient frontier increases with increasing risk. This is in agreement with the principle of risk compensation, as mentioned before. In the next section, this principle will be formulated quantitatively in the context of the efficient market model.

The Capital Asset Pricing Model

The Capital Asset Pricing Model, or efficient market model, is usually derived under the assumption that there exists a riskless asset available for investment. The future return on this asset is not subject to uncertain fluctuations. It yields, therefore, a constant rate, R_F, called the risk-free rate. It is assumed, further, that any investor can borrow or lend as much as he desires at the risk-free rate. The assumption of unlimited borrowing at the risk-free rate has been properly viewed as unrealistic. In the next section, a generalized form of the efficient market model will be discussed, with this assumption relaxed. For the purposes of the present discussion, however, it will be retained.

In addition, two more assumptions will be made: First, that each investor is risk-averse; given his requirements in terms of expected wealth at the end of a period, each investor attempts to minimize the variance of his wealth at the end of the period. Second, that this period is the same for all investors, and all investors agree about the distribution of the end-of-period asset values. While the realism of the last assumption can again be disputed, it is important to note that it is not basically necessary to the model (cf. Lintner (1969)).

When an investor allocates part of his funds to the riskless asset and the remaining part to a portfolio of common stocks, the expected return on such holdings will be an average of the expected return on the (risky) common stock portfolio ER_P, and the risk-free rate R_F, weighted proportionally to the relative allocation. This can be expressed as

where x is the proportion of money invested at the risk-free rate, and ER is the expected rate of return on the total holdings. Since the variance of the risk-free asset is (by definition) zero, the variance V of the total holding is

where is the variance of the risky assets. Expressed in terms of standard deviations, this same relation is linear:

If an investor borrows some money at the risk-free rate R_F, the equations for ER and S will both still apply, with x now being a negative quantity expressing the proportion of borrowings to the investor's equity. Both the cases of lending and borrowing at the risk-free rate can be described as follows:

1. The expected rate of return in excess of the risk-free rate (the excess return) is proportional to the expected excess return on the common stock portfolio,
2. The risk (in terms of standard deviation) is proportional to the risk of the common stock portfolio,

These equations show that both expected return and risk are linear functions of the proportion x invested at the risk-free rate: If plotted in a coordinate system with expected return on the vertical axis and standard deviation on the horizontal axis (return-risk coordinates), the set of combinations of riskless asset and a common stock portfolio will be represented by a straight line with intercept equal to the risk-free rate.

Figure 22.1 gives a graphical representation of this relationship. The darkened area represents all possible portfolios of common stocks plotted in terms of their risk and return. The upper boundary of this set consists of portfolios with maximum expected return for a given level of risk, and is therefore the efficient frontier. The riskless asset, which has no risk, is represented by the point R_F on the vertical axis. Any combination of a common stock portfolio (point P) and the riskless asset is situated on a straight line determined by these two points (line R_FP). On this line, the points that fall between R_F and P are alternative portfolios, each of which consists of some lending at the risk-free rate; portfolios which consist of some borrowing (at risk-free rate and investing in portfolio P) are depicted on this line to the right of point P.

Any straight line going through the risk-free rate and a point at or below the efficient frontier represents an available set of investment opportunities. Since these lines differ only in slope, it is clear that there is one of them, namely the line tangential to the efficient frontier (line R_FM in Figure 22.1), with investment opportunities that dominate all the efficient frontier portfolios. That is, every point on the line segment R_FM lies above the efficient frontier and therefore represents investment opportunities with superior expected returns for each level of risk.

Thus, when the riskless asset is available for borrowing and lending, optimal investment opportunities are described as combinations of the risk-free asset and one particular portfolio of common stocks (point M in Figure 22.1). An investor who wants to take a relatively low-risk position would allocate his assets between the riskless asset and the risky portfolio M. An investor wishing to take more risk (with increased expectation of return, of course) will borrow at the risk-free rate and invest all available funds in the risky portfolio M.

This analysis therefore leads to the conclusion that the equity holdings of all investors should consist of a share of the same portfolio of common stocks, namely the one lying at point M on the efficient frontier. Thus, the selection of the risk level can be separated from the problem of an optimal combination of risky securities. This important result is known as the separation theorem, first attributed to Tobin (1958). The crucial conclusion thereby suggested is that every investor must resolve what risk level he is willing to assume, but need not select particular stocks nor be concerned with combining them into a portfolio.

If every investor's equity holdings are made up of a part of the same portfolio, it follows that this portfolio comprises all the shares outstanding of all the common stocks in the market. This portfolio is called the market portfolio. The separation theorem can therefore be stated as: Every investor should hold a combination of the riskless asset and the market portfolio.

Thus, point M in Figure 22.1 is the market portfolio. The line R_FM, which represents the dominant investment opportunities, is called the capital market line. Since the line goes through the point of the market portfolio's expected return ER_M and its risk S_M, the equation of the capital market line is

1

where ER and S are the expected rate of return and standard deviation, respectively, of any particular portfolio on the capital market line.

This equation expresses quantitatively the principle of risk compensation. In words, it can be stated: For a portfolio on the capital market line, the expected rate of return in excess of the risk-free rate is proportional to the risk of that portfolio.

The constant of proportionality

which represents the slope of the capital market line, is called the market price of risk. The price that one pays for his expected return is measured in risk, and hence the name “market price of risk.” Thus expected return in excess of R_F is given by the amount of risk taken, multiplied by the market price of risk.

Individual stocks and imperfectly diversified portfolios will all fall below the capital market line, thus demonstrating that the market does not compensate for unnecessary risk—that is, for risk that can be diversified away. To give exact meaning to this statement, it is necessary to define what is meant by unnecessary risk. For this purpose, it is useful to return to the earlier discussion about the efficient frontier.

It has been seen that the market portfolio is the tangent point of the efficient frontier with the line of highest slope in Figure 22.1. Now, it can be shown (with the help of some complicated mathematical techniques) that this tangent-point portfolio must be such that the covariance of return on every security with this portfolio is proportional to the expected return on that security (in excess of the risk-free rate).² The expected rate of return for security i must fulfill the equation

2

Here is the covariance of security i with the market portfolio, and K is a constant of proportionality. When combining securities into a portfolio, the expected return on the portfolio is equal to a weighted average of expected returns of individual securities, and similarly, the covariance of the portfolio with the market is a weighted average of the covariances of individual securities with the market portfolio. Consequently, the equation above holds for any portfolio as well as every individual stock. In particular, it holds also for the market portfolio, of which we must have

since the covariance of the market portfolio with itself is just the variance of the market.

The last equation allows us to identify this constant of proportionality:

Then, upon substituting for K into Eq. (2), the following relation emerges:

3

We can now compare Eq. (3), which holds for any security or portfolio, with Eq. (1), which holds only for the portfolios on the capital market line. It is seen that for portfolios below the capital market line, the market price of risk rewards only part of the total risk. The only part of the total risk-taking that has expected rate of return (reward) associated with it is the part:

This is the part of the total risk that is due to the covariance of that portfolio with the market portfolio. This part of the total risk is called the systematic risk.

The systematic risk never exceeds the total risk S_i. In fact, it is smaller than the total risk for all portfolios that are not on the capital market line. But for the efficient portfolios on the capital market line, the systematic risk is equal to the total risk and, consequently, all the risk of these portfolios is rewarded.

The portfolios whose risk consists of the systematic risk only (portfolios on the capital market line) are called perfectly diversified portfolios. Those portfolios whose total risk is composed of systematic risk and some additional specific risk are imperfectly diversified. These are, of course, the portfolios that lie below the capital market line.

To summarize, the total risk of a security or of a portfolio is composed of two parts: the systematic risk, which is due to the covariance of that security or portfolio to the market, and the specific risk, which is due to any volatility of the security or portfolio that is independent of market fluctuations. The market price of risk rewards investors only for the systematic risk they assume; no compensation is paid for bearing specific risk. The expected rate of return on a portfolio or single security is, then, solely a function of the systematic risk; the higher the systematic risk, the higher the expected return. Some portfolios, namely those on the capital market line, are perfectly correlated with the market portfolio and have no specific risk. Consequently, risk compensation applies to the total risk of such portfolios, and they represent investment opportunities superior to all other combinations of assets. These are the perfectly diversified portfolios.

The reason why only the systematic risk is compensated by appropriately higher expected return is that the systematic risk cannot be reduced by diversification, while the specific risk generally can. Specific risk can generally be reduced by diversification because it arises from fluctuations in the security price peculiar to that individual company, rather than from fluctuations in response to general market fluctuations.

Eq. (3) can be rewritten in a perhaps more familiar form:

Those familiar with least squares regression analysis will note at once that the term is the slope coefficient in the regression of the security's rate of return on the market rate of return. This coefficient is customarily denoted by β_i and referred to as beta of that security. Since

it is seen that beta of a security is merely its systematic risk expressed in units of market risk. For portfolios, beta is defined similarly as the covariance of that portfolio with the market, divided by the variance of the market portfolio itself. Since beta of the market portfolio is 1.0, beta is a suitable measure of relative riskiness. Portfolios whose betas are less than one have less systematic risk than the market as a whole; while those with betas greater than one have higher systematic risk.

Introducing β_i into Eq. (3), the following relation is obtained:

4

When this equation is plotted in expected return/beta coordinates, it will yield a straight line (line in Figure 22.2). This line is called the security market line. It is determined by the risk-free rate (where ) and the market expected rate of return for . Since beta is a measure of systematic risk only, all securities and all portfolios will be plotted along this line. This distinguishes the security market line from the capital market line, which depicts perfectly diversified portfolios only.

Eq. (4) represents one of the most important results of the Capital Asset Pricing Model. It states that expected rate of return of any security or portfolio is determined solely by the beta of that security or portfolio, thus promoting beta to the most important single characteristic of any security or portfolio. In addition, betas are easy to estimate through regression analysis, and consequently play a central role in portfolio construction and analysis.

Beta was defined as the systematic risk; the return whose expectation is defined by Eq. (4) can appropriately be named systematic return. If the market advances 10 percent during a period, a stock with beta of 2 will, on average, appreciate by 20 percent. Similarly, a market decline of 10 percent will cause the stock to drop 20 percent on average. On the other hand, stocks with beta of 0.5 will reflect market movements with a magnitude of only one half as large. For this reason, betas are sometimes called market sensitivities.

The difference between the total price movement of a stock and the component explained by the market movement multiplied by beta, is what we call the “specific risk.”

No applications of the Capital Asset Pricing Model to portfolio selection and management will be discussed in this paper. The reader interested in practical implications of the theory is referred to Wagner (1971).

Generalization of the Model

The principal conclusions of the capital asset pricing model previously developed are as follows:

1. In an efficient market, every investor should be expected to hold a combination of the riskless asset and the market portfolio. Such combinations of assets dominate any other alternatives in the sense that they are subject to less risk for the same level of expected return. The proportions to be invested in the riskless asset and in the market portfolio depend solely on each investor's tradeoff of risk and expected return.
2. In an efficient market, the expected return on each stock in excess of the risk-free rate is related only to its beta. A stock with a beta twice as high as another stock exhibits twice as high an expected excess return. The relationship of the expected excess return on all stocks with their betas is called the security market line. Mathematically, the relationship for any stock is described by:

5

The first of these two conclusions represents a normative rule; that is, it describes how a rational investor should behave in an efficient market. As such, it is not subject to direct empirical validation. The second conclusion, however, is of a descriptive nature; it predicts how an efficient market would appear if the assumptions of the model are fulfilled. The latter can therefore be tested empirically.

Of the studies that deal with empirical validation of the efficient market model, the work by Black, Jensen, and Scholes (1970) warrants attention for its rigorous character.³ The principal conclusion of this study is that while the relationship between expected excess return of a stock or portfolio and its systematic risk is linear, it is not directly proportional. Rather, the empirical security market line exhibits a positive intercept, and a slope that is flatter than that is predicted by Eq. (5). The empirically observed market line appears to conform to a model of the form

6

where γ is a positive quantity.

In intuitive terms, the observed relation (6) can be stated as follows: If the expost excess rates of return on a stock are regressed against the market excess rates of return in a simple regression model

7

estimates of the regression coefficients α, β can be obtained. The estimate of β is an estimate of the systematic risk (beta) as introduced in the previous section. The regression coefficient α can be called the abnormal return, or simply alpha. It is the additional rate of return left after the stock's rate of return is adjusted for its systematic risk by subtracting the factor from the total excess return.

Now, the simpler model (Eq. (5)) asserts that there should be no expected abnormal returns, or, more specifically, that

This implication can be seen by comparing Eqs. (5) and (7), bearing in mind that the expected value of the error term in the regression (e in Eq. (7)) is zero by design. However, in reality, we observe that

8

which can be seen when Eqs. (6) and (7) are compared.

Thus, empirical measurements imply that stocks (and portfolios) with systematic risk (beta) lower than that of the market portfolio exhibit a positive abnormal return, whereas stocks and portfolios with betas higher than that of the market show negative abnormal returns. That is, high-risk stocks are observed to return less than what is predicted by the simple theory, and the converse for low-risk stocks. Moreover, the lower the beta, the higher the alpha, and conversely. This empirical result is sometimes called the alpha effect.

Thus, the actual behavior of capital markets differs in an important aspect from that predicted by the simple efficient market model described in the section “The Capital Asset Pricing Model.” This, of course, casts some doubt on the validity of the assumptions on which this model is based. One questionable assumption is that each investor is able to borrow without limitation at a rate equal to the rate on the riskless asset. This, we know, does not correspond to actual behavior. Generally, borrowers pay more than the risk-free rate R_F. The question then naturally arises as to whether removing the unrealistic assumption of unlimited borrowing at the risk-free rate would produce a model that is in better agreement with empirical data than the Capital Asset Pricing Model.

Black (1972) investigated the market equilibrium under the assumption that there is no riskless asset, thereby preventing both borrowing at a risk-free rate and investing at a risk-free rate. Black has shown that ideally every investor holds a linear combination of the market portfolio and another portfolio that, though risky, possesses no market risk. This latter portfolio, which is called the zero-beta portfolio, is composed of long and short holdings in risky assets in such proportions that the systematic risk, or beta, of this portfolio is zero. The zero-beta portfolio (which, taken alone is not even an efficient portfolio in the sense developed previously in this chapter), takes on the general role previously played by the riskless asset in the Capital Asset Pricing Model. The expected rate of return on a security is still a linear function of the security's beta. The intercept of this relationship is the expected rate of return on the zero-beta portfolio. The security market line can thus be described by the equation

9

where ER_Z is the expected rate of return on the zero-beta portfolio. It is seen that this equilibrium equation is of the form of Eq. (6), and therefore consistent with the empirical results.

Vasicek (1971) dealt with the case when the riskless asset is available for investment, but investors cannot borrow at the risk-free rate. These assumptions correspond better to actual capital markets. This model is again based on each investor's minimizing the risk he assumes, while meeting his requirements in terms of expected return. Vasicek has shown that there exists a portfolio, called the tangent portfolio, which has the following properties: Every investor whose expected return requirements do not exceed the tangent portfolio's expected return, invests part of his assets into the riskless asset and the remaining part into the tangent portfolio. An investor with higher return requirements (which of course means taking more risk) will hold a combination of the tangent portfolio and the market portfolio.

The set of efficient portfolios is therefore composed of two parts: The first part consists of all combinations of the riskless asset and the tangent portfolio. The second part comprises all combinations of the tangent portfolio and a long position in the market portfolio. To select a portfolio satisfying his requirements, an investor can therefore separate the task of identifying suitable combinations of risky assets from the selection of risk level. The result can again be called the separation theorem.

The tangent portfolio itself is an efficient portfolio. It is a linear combination of the zero-beta portfolio and the market portfolio. Vasicek has shown that the tangent portfolio is that combination of assets with the highest expected excess return to standard deviation ratio of all combinations of risky assets.

In this model, the security market line is given by Eq. (9). Moreover, it is shown that the expected rate of return on the zero-beta portfolio is not smaller than the risk-free rate R_F,

Since the constant γ in Eq. (8) is equal to (as is readily seen by comparing Eqs. (6), (7), and (9)), it follows that

and therefore, the possible existence of the alpha effect is admissible in the generalized efficient market model, providing substantially better correspondence between the theoretical model and empirical tests.

This generalized capital asset pricing model then permits replacement of the two propositions at the beginning of this section by the following two:

1. In an efficient market, each investor holds either a combination of the riskless asset and the tangent portfolio, or a combination of the tangent portfolio and long holdings in the market portfolio. The choice between the two alternatives, and within each alternative, depends solely on the investor's attitude towards risk and expected return.
2. In an efficient market, the expected excess rate of return on each security or portfolio is linearly related to the systematic risk, or beta, of that security. The intercept of the security market line is equal to the expected excess return on the zero-beta portfolio, and is non-negative. The generalized model thus takes the form

The abnormal return α is related to beta by the relationship

Thus, stocks or portfolios with low betas may have positive alphas, and stocks or portfolios with high betas may have negative alphas. But since alphas also may be zero, as well as positive, the efficient market model discussed in this chapter is a special case of the more generalized efficient market model.

The descriptive aspect of this generalized efficient market model, as summarized in the second point, is consistent with empirical results. While the normative aspect of the first point is not necessarily met by all investors, it may be noted that it still corresponds reasonably well to the actual behavior of large institutional investors. Those investors who exhibit strong risk aversion often hold a combination of riskless assets and a low to medium risk portfolio of common stocks. The equity portion of their holdings can be interpreted as an attempt to obtain the tangent portfolio. With increased total risk, the riskless portion decreases and the risky part increases, probably without a considerable shift in the composition of the equity portion. Holdings of investors with even higher risk are often characterized by no riskless asset, while a portfolio of more risky common stocks is held in various proportions to the low-risk part.

In conclusion, it has been argued in this section that the capital asset pricing model can be generalized to correspond better to empirical observations by making the assumptions of the model more realistic. In particular, the possible existence of the alpha effect can be derived from the generalized model. While the more generalized models in this section certainly are not the final word in the theory of capital markets, the ability of the efficient market model to be generalized in various directions demonstrates the viability of the simplest form of the model.

Conclusion

The concluding remarks can appropriately contain some “so-what?” overtones, provided the reader is forewarned that we present these speculations without much support. We have, in fact, decided to exclude specific references to supporting materials since concluding remarks are hardly the appropriate point to introduce new evidence—especially since we have attempted, in this paper, to deal only with the concepts of the efficient market models and not with their empirical support. Having offered this warning, we now proceed to what we think are some plausible assertions.

If the efficient market model is to be applicable to real capital markets, and not idealized ones, it must be able to explain actual observed price changes. The beta coefficient in the model has been estimated by numerous investigators and found to be usefully stable and to be related in the predicted way to rate of return: the higher the beta, the higher the observed rate of return. This fact alone is sufficient to place the efficient market model in that rare class of theories that can be usefully employed. Given this fact, one may also expect that the efficient market model will be put to work in numerous ways (as, indeed, it has been) by practicing investors.

The efficient market model was developed under some simplifying assumptions concerning zero transactions costs and rapid information dissemination. However, the empirical findings to date tend to conform to the implications of the model, suggesting that these assumptions may be relaxed. Actually, it has been shown that the basic conclusions of the model hold under much more general assumptions—adding further confirmation to the empirical evidence. Thus, the reader should be cautious about rejecting the efficient market model for what he may correctively perceive to be unrealistic assumptions in the simplified model presented here.

The efficient market model is seen to be elegantly simple by many who have thoughtfully studied it. Less careful appraisals may tend to precipitate the view that a model containing considerably more variables is still waiting to be “discovered.” The present model, it may be argued, does not take certain obvious effects into consideration—such as market ability, industry, management, foreign competition, and government policies. Still, the model works remarkably well—suggesting that these effects do get imbedded in prices, the behavior of which empirical studies find to be so efficient. The model is, we assert, much richer than its simplicity may suggest.

The thoughtful investor may find it profitable to ask himself: “Unless I know the beta of my portfolio, what evidence do I have that the returns are not systematic, rather than specific, returns? And if they are indeed specific returns, what assurance do I have that the portfolio has earned them consistently enough to justify the extra risk incurred in departing from perfect diversification?”

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