,

CHAPTER 6

PORTFOLIO RISK AND RETURN: PART II

Vijay Singal, CFA

Blacksburg, VA, U.S.A.

LEARNING OUTCOMES

After completing this chapter, you will be able to do the following:

• Discuss the implications of combining a risk-free asset with a portfolio of risky assets.
• Explain and interpret the capital allocation line (CAL) and the capital market line (CML).
• Explain systematic and nonsystematic risk and why an investor should not expect to receive additional return for bearing nonsystematic risk.
• Explain return-generating models (including the market model) and their uses.
• Calculate and interpret beta.
• Explain the capital asset pricing model (CAPM) including the required assumptions, and the security market line (SML).
• Calculate and interpret the expected return of an asset using the CAPM.
• Illustrate applications of the CAPM and the SML.

1. INTRODUCTION

Our objective in this chapter is to identify the optimal risky portfolio for all investors by using the capital asset pricing model (CAPM). The foundation of this chapter is the computation of risk and return of a portfolio and the role that correlation plays in diversifying portfolio risk and arriving at the efficient frontier. The efficient frontier and the capital allocation line consist of portfolios that are generally acceptable to all investors. By combining an investor’s individual indifference curves with the market-determined capital allocation line, we are able to illustrate that the only optimal risky portfolio for an investor is the portfolio of all risky assets (i.e., the market).

Additionally, we discuss the capital market line, a special case of the capital allocation line that is used for passive investor portfolios. We also differentiate between systematic and nonsystematic risk, and explain why investors are compensated for bearing systematic risk but receive no compensation for bearing nonsystematic risk. We discuss in detail the CAPM, which is a simple model for estimating asset returns based only on the asset’s systematic risk. Finally, we illustrate how the CAPM allows security selection to build an optimal portfolio for an investor by changing the asset mix beyond a passive market portfolio.

The chapter is organized as follows. In Section 2, we discuss the consequences of combining a risk-free asset with the market portfolio and provide an interpretation of the capital market line. Section 3 decomposes total risk into systematic and nonsystematic risk and discusses the characteristics of and differences between the two kinds of risk. We also introduce return-generating models, including the single-index model, and illustrate the calculation of beta by using formulas and graphically by using the security characteristic line. In Section 4, we introduce the capital asset pricing model and the security market line. We discuss many applications of the CAPM and the SML throughout the chapter, including the use of expected return in making capital budgeting decisions, the evaluation of portfolios using the CAPM’s risk-adjusted return as the benchmark, security selection, and determining whether adding a new security to the current portfolio is appropriate. Our focus on the CAPM does not suggest that the CAPM is the only viable asset-pricing model. Although the CAPM is an excellent starting point, more advanced readings expand on these discussions and extend the analysis to other models that account for multiple explanatory factors. A preview of a number of these models is given in Section 5. Finally, in Section 6 we conclude the chapter and provide a summary.

2. CAPITAL MARKET THEORY

You have learned how to combine a risk-free asset with one risky asset and with many risky assets to create a capital allocation line. In this section, we will expand our discussion of multiple risky assets and consider a special case of the capital allocation line, called the capital market line. While discussing the capital market line, we will define the market and its role in passive portfolio management. Using these concepts, we will illustrate how leveraged portfolios can enhance both risk and return.

2.1. Portfolio of Risk-Free and Risky Assets

Although investors desire an asset that produces the highest return and carries the lowest risk, such an asset does not exist. As the risk–return capital market theory illustrates, one must assume higher risk in order to earn a higher return. We can improve an investor’s portfolio, however, by expanding the opportunity set of risky assets because this allows the investor to choose a superior mix of assets.

Similarly, an investor’s portfolio improves if a risk-free asset is added to the mix. In other words, a combination of the risk-free asset and a risky asset can result in a better risk–return trade-off than an investment in only one type of asset because the risk-free asset has zero correlation with the risky asset. The combination is called the capital allocation line (and is depicted in Exhibit 6-2). Superimposing an investor’s indifference curves on the capital allocation line will lead to the optimal investor portfolio.

Investors with different levels of risk aversion will choose different portfolios. Highly risk-averse investors choose to invest most of their wealth in the risk-free asset and earn low returns because they are not willing to assume higher levels of risk. Less risk-averse investors, in contrast, invest more of their wealth in the risky asset, which is expected to yield a higher return. Obviously, the higher return cannot come without higher risk, but the less risk-averse investor is willing to accept the additional risk.

2.1.1. Combining a Risk-Free Asset with a Portfolio of Risky Assets

We can extend the analysis of one risky asset to a portfolio of risky assets. For convenience, assume that the portfolio contains all available risky assets,1 although an investor may not wish to include all of these assets in the portfolio because of the investor’s specific preferences. If an asset is not included in the portfolio, its weight will be zero. The risk–return characteristics of a portfolio of N risky assets are given by the following equations:

The expected return on the portfolio, E(Rp), is the weighted average of the expected returns of individual assets, where wi is the fractional weight in asset i and Ri is the expected return of asset i. The risk of the portfolio (σp), however, depends on the weights of the individual assets, the risk of the individual assets, and their interrelationships. The covariance between assets i and j, Cov(i, j), is a statistical measure of the interrelationship between each pair of assets in the portfolio and can be expressed as follows, where ρij is the correlation between assets i and j and σi is the risk of asset i:

Note from the following equation that the correlation of an asset with itself is 1; therefore:

By substituting the preceding expressions for covariance, we can rewrite the portfolio variance equation as

The suggestion that portfolios have lower risk than the assets they contain may seem counterintuitive. These portfolios can be constructed, however, as long as the assets in the portfolio are not perfectly correlated. As an illustration of the effect of asset weights on portfolio characteristics, consider a simple two-asset portfolio with zero weights in all other assets. Assume that Asset 1 has a return of 10 percent and a standard deviation (risk) of 20 percent. Asset 2 has a return of 5 percent and a standard deviation (risk) of 10 percent. Furthermore, the correlation between the two assets is zero. Exhibit 6-1 shows risks and returns for Portfolio X with a weight of 25 percent in Asset 1 and 75 percent in Asset 2, Portfolio Y with a weight of 50 percent in each of the two assets, and Portfolio Z with a weight of 75 percent in Asset 1 and 25 percent in Asset 2.

EXHIBIT 6-1 Portfolio Risk and Return

From this example we observe that the three portfolios are quite different in terms of their risk and return. Portfolio X has a 6.25 percent return and only 9.01 percent standard deviation, whereas the standard deviation of Portfolio Z is more than two-thirds higher (15.21 percent), although the return is only slightly more than one-third higher (8.75 percent). These portfolios may become even more dissimilar as other assets are added to the mix.

Consider three portfolios of risky assets, A, B, and C, as in Exhibit 6-2, that may have been presented to a representative investor by three different investment advisers. Each portfolio is combined with the risk-free asset to create three capital allocation lines, CAL(A), CAL(B), and CAL(C). The exhibit shows that Portfolio C is superior to the other two portfolios because it has a greater expected return for any given level of risk. As a result, an investor will choose the portfolio that lies on the capital allocation line for Portfolio C. The combination of the risk-free asset and the risky Portfolio C that is selected for an investor depends on the investor’s degree of risk aversion.

2.1.2. Does a Unique Optimal Risky Portfolio Exist?

We assume that all investors have the same economic expectation and thus have the same expectations of prices, cash flows, and other investment characteristics. This assumption is referred to as homogeneity of expectations. Given these investment characteristics, everyone goes through the same calculations and should arrive at the same optimal risky portfolio. Therefore, assuming homogeneous expectations, only one optimal portfolio exists. If investors have different expectations, however, they might arrive at different optimal risky portfolios. To illustrate, we begin with an expression for the price of an asset:

where CFt is the cash flow at the end of period t and rt is the discount rate or the required rate of return for that asset for period t. Period t refers to all periods beginning from now until the asset ceases to exist at the end of time T. Because the current time is the end of period 0, which is the same as the beginning of period 1, there are (T + 1) cash flows and (T + 1) required rates of return. These conditions are based on the assumption that a cash flow, such as an initial investment, can occur now (t = 0). Ordinarily, however, CF0 is zero.

We use the formula for the price of an asset to estimate the intrinsic value of an asset. For ease of reference, assume that the asset we are valuing is a share of HSBC Holdings (parent of HSBC Bank), a British company that also trades on the Hong Kong Stock Exchange. In the case of corporate stock, there is no expiration date, so T could be extremely large, meaning we will need to estimate a large number of cash flows and rates of return. Fortunately, the denominator reduces the importance of distant cash flows, so it may be sufficient to estimate, say, 20 annual cash flows and 20 rates of returns. How much will HSBC earn next year and the year after next? What will the banking sector look like in five years’ time? Different analysts and investors will have their own estimates that may be quite different from one another. Also, as we delve further into the future, more serious issues in estimating future revenue, expenses, and growth rates arise. Therefore, to assume that cash flow estimates for HSBC will vary among these investors is reasonable. In addition to the numerator (cash flows), it is also necessary to estimate the denominator, the required rates of return. We know that riskier companies will require higher returns because risk and return are positively correlated. HSBC stock is riskier than a risk-free asset, but by how much? And what should the compensation for that additional risk be? Again, it is evident that different analysts will view the riskiness of HSBC differently and, therefore, arrive at different required rates of return.

HSBC closed at HK$89.40 on 31 December 2009 on the Hong Kong Stock Exchange. The traded price represents the value that a marginal investor attaches to a share of HSBC, say, corresponding to Analyst A’s expectation. Analyst B may think that the price should be HK$50, however, and Analyst C may think that the price should be HK$150. Given a current price of HK$89.40, the expected returns of HSBC are quite different for the three analysts. Analyst B, who believes the price should be HK$50, concludes that HSBC is overvalued and may assign a weight of zero to HSBC in the recommended portfolio even though the market capitalization of HSBC is in excess of HK$1 trillion. In contrast, Analyst C, with a valuation of HK$150, thinks HSBC is undervalued and will significantly overweight HSBC in a portfolio.

Our discussion illustrates that analysts can arrive at different valuations that necessitate the assignment of different asset weights in a portfolio. Given the existence of many asset classes and numerous assets in each asset class, one can visualize that each investor will have his or her own optimal risky portfolio depending on his or her assumptions underlying the valuation computations. Therefore, market participants will have their own and possibly different optimal risky portfolios.

If investors have different valuations of assets, then the construction of a unique optimal risky portfolio is not possible. If we make a simplifying assumption of homogeneity in investor expectations, we will have a single optimal risky portfolio as previously mentioned. Even if investors have different expectations, market prices are a proxy of what the marginal, informed investor expects, and the market portfolio becomes the base case, the benchmark, or the reference portfolio that other portfolios can be judged against. For HSBC, the market price is HK$89.40 per share and the market capitalization is HK$1.08 trillion. In constructing the market portfolio, HSBC’s weight in the market portfolio will be equal to its market value divided by the value of all other assets included in the market portfolio.

2.2. The Capital Market Line

In the previous section, we discussed how the risk-free asset could be combined with a risky portfolio to create a capital allocation line. In this section, we discuss a specific CAL that uses the market portfolio as the optimal risky portfolio and is known as the capital market line. We also discuss the significance of the market portfolio and applications of the capital market line.

2.2.1. Passive and Active Portfolios

In the previous subsection, we arrived at three possible valuations for each share of HSBC: HK$50, HK$89.40, and HK$150. Which one is correct?

If the markets are informationally efficient, the price in the market, HK$89.40, is an unbiased estimate of all future discounted cash flows (recall the formula for the price of an asset). In other words, the price aggregates and reflects all information that is publicly available, and investors cannot expect to earn a return that is greater than the required rate of return for that asset. If, however, the price reflects all publicly available information and there is no way to outperform the market, then there is little point in investing time and money in evaluating HSBC to arrive at your price using your own estimates of cash flows and rates of return.

In that case, a simple and convenient approach to investing is to rely on the prices set by the market. Portfolios that are based on the assumption of unbiased market prices are referred to as passive portfolios. Passive portfolios most commonly replicate and track market indices, which are passively constructed on the basis of market prices and market capitalizations. Examples of market indices are the S&P 500 Index, the Nikkei 300, and the CAC 40. Passive portfolios based on market indices are called index funds and generally have low costs because no significant effort is expended in valuing securities that are included in an index.

In contrast to passive investors’ reliance on market prices and index funds, active investors may not rely on market valuations. They have more confidence in their own ability to estimate cash flows, growth rates, and discount rates. Based on these estimates, they value assets and determine whether an asset is fairly valued. In an actively managed portfolio, assets that are undervalued, or have a chance of offering above-normal returns, will have a positive weight (i.e., overweight compared to the market weight in the benchmark index), whereas other assets will have a zero weight, or even a negative weight if short selling is permitted (i.e., some assets will be underweighted compared with the market weight in the benchmark index). This style of investing is called active investment management, and the portfolios are referred to as active portfolios. Most open-end mutual funds and hedge funds practice active investment management, and most analysts believe that active investing adds value. Whether these analysts are right or wrong is the subject of continuing debate.

2.2.2. What Is the “Market”?

In the previous discussion, we referred to the “market” on numerous occasions without actually defining the market. The optimal risky portfolio and the capital market line depend on the definition of the market. So what is the market?

Theoretically, the market includes all risky assets or anything that has value, which includes stocks, bonds, real estate, and even human capital. Not all assets are tradable, however, and not all tradable assets are investable. For example, the Taj Mahal in India is an asset but is not a tradable asset. Similarly, human capital is an asset that is not tradable. Moreover, assets may be tradable but not investable because of restrictions placed on certain kinds of investors. For example, all stocks listed on the Shanghai Stock Exchange are tradable. Class A shares, however, are available only to domestic investors, whereas Class B shares are available to both domestic and foreign investors. For investors not domiciled in China, Class A shares are not investable—that is, they are not available for investment.

If we consider all stocks, bonds, real estate assets, commodities, and so forth, probably hundreds of thousands of assets are tradable and investable. The “market” should contain as many assets as possible; we emphasize the word “possible” because it is not practical to include all assets in a single risky portfolio. Even though advancements in technology and interconnected markets have made it much easier to span the major equity markets, we are still not able to easily invest in other kinds of assets like bonds and real estate except in the most developed countries.

For the rest of this chapter, we will define the “market” quite narrowly because it is practical and convenient to do so. Typically, a local or regional stock market index is used as a proxy for the market because of active trading in stocks and because a local or regional market is most visible to the local investors. For our purposes, we will use the S&P 500 Index as the market’s proxy. The S&P 500 is commonly used by analysts as a benchmark for market performance throughout the United States. It contains 500 of the largest stocks that are domiciled in the United States, and these stocks are weighted by their market capitalization (price times the number of outstanding shares).

The stocks in the S&P 500 account for approximately 80 percent of the total equity market capitalization in the United States, and because the U.S. stock markets represent about 32 percent of the world markets, the S&P 500 represents roughly 25 percent of worldwide publicly traded equity. Our definition of the market does not include non-U.S. stock markets, bond markets, real estate, and many other asset classes, and therefore, “market” return and the “market” risk premium refer to U.S. equity return and the U.S. equity risk premium, respectively. The use of this proxy, however, is sufficient for our discussion, and is relatively easy to expand to include other tradable assets.

2.2.3. The Capital Market Line (CML)

A capital allocation line includes all possible combinations of the risk-free asset and any risky portfolio. The capital market line is a special case of the capital allocation line, where the risky portfolio is the market portfolio. The risk-free asset is a debt security with no default risk, no inflation risk, no liquidity risk, no interest rate risk, and no risk of any other kind. U.S. Treasury bills are usually used as a proxy of the risk-free return, Rf.

The S&P 500 is a proxy of the market portfolio, which is the optimal risky portfolio. Therefore, the expected return on the risky portfolio is the expected market return, expressed as E(Rm). The capital market line is shown in Exhibit 6-3, where the standard deviation (σp), or total risk, is on the x-axis and expected portfolio return, E(Rp), is on the y-axis. Graphically, the market portfolio is the point on the Markowitz efficient frontier where a line from the risk-free asset is tangent to the Markowitz efficient frontier. All points on the interior of the Markowitz efficient frontier are inefficient portfolios in that they provide the same level of return with a higher level of risk or a lower level of return with the same amount of risk. When plotted together, the point at which the CML is tangent to the Markowitz efficient frontier is the optimal combination of risky and risk-free assets, on the basis of market prices and market capitalizations. The optimal risky portfolio is the market portfolio.

The CML’s intercept on the y-axis is the risk-free return (Rf) because that is the return associated with zero risk. The CML passes through the point represented by the market return, E(Rm). With respect to capital market theory, any point above the CML is not achievable and any point below the CML is dominated by and inferior to any point on the CML.

Note that we identify the CML and CAL as lines even though they are a combination of two assets. Unlike a combination of two risky assets, which is usually not a straight line, a combination of the risk-free asset and a risky portfolio is a straight line, as illustrated in the following by computing the combination’s risk and return.

Risk and return characteristics of the portfolio represented by the CML can be computed by using the return and risk expressions for a two-asset portfolio:

and

The proportion invested in the risk-free asset is given by w1, and the balance is invested in the market portfolio, (1 – w1). The risk of the risk-free asset is given by σf, the risk of the market is given by σm, the risk of the portfolio is given by σp, and the covariance between the risk-free asset and the market portfolio is represented by Cov(Rf, Rm).

By definition, the standard deviation of the risk-free asset is zero. Because its risk is zero, the risk-free asset does not co-vary or move with any other asset. Therefore, its covariance with all other assets, including the market portfolio, is zero, making the first and third terms under the square root sign zero. As a result, the portfolio return and portfolio standard deviation can be simplified and rewritten as:

and

By substitution, we can express E(Rp) in terms of σp. Substituting for w1, we get:

Note that the expression is in the form of a line, y = a + bx. The y-intercept is the risk-free rate, and the slope of the line referred to as the market price of risk is [E(Rm) − Rf]/σm. The CML has a positive slope because the market’s risky return is larger than the risk-free return. As the amount of the total investment devoted to the market increases—that is, as we move up the line—both standard deviation (risk) and expected return increase.

2.2.4. Leveraged Portfolios

In the previous example, Mr. Miles evaluated an investment of between 0 percent and 100 percent in the market and the balance in T-bills. The line connecting Rf and M (market portfolio) in Exhibit 6-4 illustrates these portfolios with their respective levels of investment. At Rf, an investor is investing all of his or her wealth into risk-free securities, which is equivalent to lending 100 percent at the risk-free rate. At point M he or she is holding the market portfolio and not lending any money at the risk-free rate. The combinations of the risk-free asset and the market portfolio, which may be achieved by the points between these two limits, are termed “lending” portfolios. In effect, the investor is lending part of his or her wealth at the risk-free rate.

If Mr. Miles is willing to take more risk, he may be able to move to the right of the market portfolio (point M in Exhibit 6-4) by borrowing money and purchasing more of Portfolio M. Assume that he is able to borrow money at the same risk-free rate of interest, Rf, at which he can invest. He can then supplement his available wealth with borrowed money and construct a borrowing portfolio. If the straight line joining Rf and M is extended to the right of point M, this extended section of the line represents borrowing portfolios. As one moves further to the right of point M, an increasing amount of borrowed money is being invested in the market. This means that there is negative investment in the risk-free asset, which is referred to as a leveraged position in the risky portfolio. The particular point chosen on the CML will depend on the individual’s utility function, which, in turn, will be determined by his risk and return preferences.

2.2.4.1. Leveraged Portfolios with Different Lending and Borrowing Rates

Although we assumed that Mr. Miles can borrow at the same rate as the U.S. government, it is more likely that he will have to pay a higher interest rate than the government because his ability to repay is not as certain as that of the government. Now consider that although Mr. Miles can invest (lend) at Rf, he can borrow at only Rb, a rate that is higher than the risk-free rate.

With different lending and borrowing rates, the CML will no longer be a single straight line. The line will have a slope of [E(Rm) – Rf]/σm between points Rf and M, where the lending rate is Rf, but will have a smaller slope of [E(Rm) – Rb]/σm at points to the right of M, where the borrowing rate is Rb. Exhibit 6-5 illustrates the CML with different lending and borrowing rates.

The equations for the two lines are given next.

and

The first equation is for the line where the investment in the risk-free asset is zero or positive—that is, at M or to the left of M in Exhibit 6-5. The second equation is for the line where borrowing, or negative investment in the risk-free asset, occurs. Note that the only difference between the two equations is in the interest rates used for borrowing and lending.

All passive portfolios will lie on the kinked CML, although the investment in the risk-free asset may be positive (lending), zero (no lending or borrowing), or negative (borrowing). Leverage allows less risk-averse investors to increase the amount of risk they take by borrowing money and investing more than 100 percent in the passive portfolio.

3. PRICING OF RISK AND COMPUTATION OF EXPECTED RETURN

In constructing a portfolio, it is important to understand the concept of correlation and how less than perfect correlation can diversify the risk of a portfolio. As a consequence, the risk of an asset held alone may be greater than the risk of that same asset when it is part of a portfolio. Because the risk of an asset varies from one environment to another, which kind of risk should an investor consider and how should that risk be priced? This section addresses the question of pricing of risk by decomposing the total risk of a security or a portfolio into systematic and nonsystematic risk. The meaning of these risks, how they are computed, and their relevance to the pricing of assets are also discussed.

3.1. Systematic Risk and Nonsystematic Risk

Systematic risk, also known as nondiversifiable or market risk, is the risk that affects the entire market or economy. In contrast, nonsystematic risk is the risk that pertains to a single company or industry and is also known as company-specific, industry-specific, diversifiable, or idiosyncratic risk.

Systematic risk is risk that cannot be avoided and is inherent in the overall market. It is nondiversifiable because it includes risk factors that are innate within the market and affect the market as a whole. Examples of factors that constitute systematic risk include interest rates, inflation, economic cycles, political uncertainty, and widespread natural disasters. These events affect the entire market, and there is no way to avoid their effect. Systematic risk can be magnified through selection or by using leverage, or diminished by including securities that have a low correlation with the portfolio, assuming they are not already part of the portfolio.

Nonsystematic risk is risk that is local or limited to a particular asset or industry that need not affect assets outside of that asset class. Examples of nonsystematic risk could include the failure of a drug trial, major oil discoveries, or an airliner crash. All these events will directly affect their respective companies and possibly industries, but have no effect on assets that are far removed from these industries. Investors are capable of avoiding nonsystematic risk through diversification by forming a portfolio of assets that are not highly correlated with one another.

We will derive expressions for each kind of risk later in this chapter. You will see that the sum of systematic variance and nonsystematic variance equals the total variance of the security or portfolio:

Total variance = Systematic variance + Nonsystematic variance

Although the equality relationship is between variances, you will find frequent references to total risk as the sum of systematic risk and nonsystematic risk. In those cases, the statements refer to variance, not standard deviation.

3.1.1. Pricing of Risk

Pricing or valuing an asset is equivalent to estimating its expected rate of return. If an asset has a known terminal value, such as the face value of a bond, then a lower current price implies a higher future return and a higher current price implies a lower future return. The relationship between price and return can also be observed in the valuation expression shown in Section 2.1.2. Therefore, we will occasionally use price and return interchangeably when discussing the price of risk.

Consider an asset with both systematic and nonsystematic risk. Assume that both kinds of risk are priced—that is, you receive a return for both systematic risk and nonsystematic risk. What will you do? Realizing that nonsystematic risk can be diversified away, you would buy assets that have a large amount of nonsystematic risk. Once you have bought those assets with nonsystematic risk, you would diversify, or reduce that risk, by including other assets that are not highly correlated. In the process, you will minimize nonsystematic risk and eventually eliminate it altogether from your portfolio. Now, you would have a diversified portfolio with only systematic risk, yet you would be compensated for nonsystematic risk that you no longer have. Just like everyone else, you would have an incentive to take on more and more diversifiable risk because you are compensated for it even though you can get rid of it. The demand for diversifiable risk will keep increasing until its price becomes infinite and its expected return falls to zero. This means that our initial assumption of a nonzero return for diversifiable risk was incorrect and that the correct assumption is zero return for diversifiable risk. Therefore, we can assume that in an efficient market, no incremental reward can be earned for taking on diversifiable risk.

In the previous exercise we illustrated why investors should not be compensated for taking on nonsystematic risk. Therefore, investors who have nonsystematic risk must diversify it away by investing in many industries, many countries, and many asset classes. Because future returns are unknown and it is not possible to pick only winners, diversification helps in offsetting poor returns in one asset class by garnering good returns in another asset class, thereby reducing the overall risk of the portfolio. In contrast, investors must be compensated for accepting systematic risk because that risk cannot be diversified away. If investors do not receive a return commensurate with the amount of systematic risk they are taking, they will refuse to accept systematic risk.

In summary, systematic or nondiversifiable risk is priced and investors are compensated for holding assets or portfolios based only on that investment’s systematic risk. Investors do not receive any return for accepting nonsystematic or diversifiable risk. Therefore, it is in the interest of risk-averse investors to hold only well-diversified portfolios.

3.2. Calculation and Interpretation of Beta

As previously mentioned, in order to form the market portfolio, you should combine all available assets. Knowledge of the correlations among those assets allows us to estimate portfolio risk. You also learned that a fully diversified portfolio will include all asset classes and essentially all assets in those asset classes. The work required for construction of the market portfolio is formidable. For example, for a portfolio of 1,000 assets, we will need 1,000 return estimates, 1,000 standard deviation estimates, and 499,500 (1,000 × 999 ÷ 2) correlations. Other related questions that arise with this analysis are whether we really need all 1,000 assets and what happens if there are errors in these estimates.

An alternate method of constructing an optimal portfolio is simpler and easier to implement. An investor begins with a known portfolio, such as the S&P 500, and then adds other assets one at a time on the basis of the asset’s standard deviation, expected return, and impact on the portfolio’s risk and return. This process continues until the addition of another asset does not have a significant impact on the performance of the portfolio. The process requires only estimates of systematic risk for each asset because investors will not be compensated for nonsystematic risk. Expected returns can be calculated by using return-generating models, as we will discuss in this section. In addition to using return-generating models, we will also decompose total variance into systematic variance and nonsystematic variance and establish a formal relationship between systematic risk and return. In the next section, we will expand on this discussion and introduce the CAPM as the preferred return-generating model.

3.2.1. Return-Generating Models

A return-generating model is a model that can provide an estimate of the expected return of a security given certain parameters. If systematic risk is the only relevant parameter for return, then the return-generating model will estimate the expected return for any asset given the level of systematic risk.

As with any model, the quality of estimates of expected return will depend on the quality of input estimates and the accuracy of the model. Because it is difficult to decide which factors are appropriate for generating returns, the most general form of a return-generating model is a multifactor model. A multifactor model allows more than one variable to be considered in estimating returns and can be built using different kinds of factors, such as macroeconomic, fundamental, and statistical factors.

Macroeconomic factor models use economic factors that are correlated with security returns. These factors may include economic growth, the interest rate, the inflation rate, productivity, employment, and consumer confidence. Past relationships with returns are estimated to obtain parameter estimates, which are, in turn, used for computing expected returns. Fundamental factor models analyze and use relationships between security returns and the company’s underlying fundamentals, such as earnings, earnings growth, cash flow generation, investment in research, advertising, and number of patents. Finally, in a statistical factor model, historical and cross-sectional return data are analyzed to identify factors that explain variance or covariance in observed returns. These statistical factors, however, may or may not have an economic or fundamental connection to returns. For example, the conference to which the American football Super Bowl winner belongs, whether the American Football Conference or the National Football Conference, may be a factor in U.S. stock returns, but no obvious economic connection seems to exist between the winner’s conference and U.S. stock returns. Moreover, data mining may generate many spurious factors that are devoid of any economic meaning. Because of this limitation, analysts prefer the macroeconomic and fundamental factor models for specifying and estimating return-generating models.

A general return-generating model is expressed in the following manner:

The model has k factors, E(F1), E(F2), . . . E(Fk). The coefficients, βij, are called factor weights or factor loadings associated with each factor. The left-hand side of the model has excess return, or return over the risk-free rate. The right-hand side provides the risk factors that would generate the return or premium required to assume that risk. We have separated out one factor, E(Rm), which represents the market return. All models contain return on the market portfolio as a key factor.

3.2.1.2. Three-Factor and Four-Factor Models

Eugene Fama and Kenneth French2 suggested that a return-generating model for stock returns should include relative size of the company and relative book-to-market value of the company in addition to beta. Fama and French found that past returns could be explained better with their model than with other models available at that time, most notably, the capital asset pricing model. Mark Carhart (1997) extended the Fama and French model by adding another factor: momentum, defined as relative past stock returns. We will discuss these models further in Section 5.3.2.

3.2.1.3. The Single-Index Model

The simplest form of a return-generating model is a single-factor linear model, in which only one factor is considered. The most common implementation is a single-index model, which uses the market factor in the following form: E(Ri) – Rf = βi[E(Rm) – Rf].

Although the single-index model is simple, it fits nicely with the capital market line. Recall that the CML is linear, with an intercept of Rf and a slope of [E(Rm) – Rf]/σm. We can rewrite the CML by moving the intercept to the left-hand side of the equation, rearranging the terms, and generalizing the subscript from p to i, for any security:

The factor loading or factor weight, σi/σm, refers to the ratio of total security risk to total market risk. To obtain a better understanding of factor loading and to illustrate that the CML reduces to a single-index model, we decompose total risk into its components.

3.2.2. Decomposition of Total Risk for a Single-Index Model

With the introduction of return-generating models, particularly the single-index model, we are able to decompose total variance into systematic and nonsystematic variances. Instead of using expected returns in the single index, let us use realized returns. The difference between expected returns and realized returns is attributable to nonmarket changes, as an error term, ei, in the second equation shown here:

and

The variance of realized returns can be expressed in the next equation (note that Rf is a constant). We can further drop the covariance term in this equation because, by definition, any nonmarket return is uncorrelated with the market. Thus, we are able to decompose total variance into systematic and nonsystematic variances in the second equation shown here:

Total variance = Systematic variance + Nonsystematic variance, which can be written as

Total risk can be expressed as

Because nonsystematic risk is zero for well-diversified portfolios, such as the market portfolio, the total risk of a market portfolio and other similar portfolios is only systematic risk, which is βiσm. We can now return to the CML discussed in the previous subsection and replace σi with βiσm because the CML assumes that the market is a diversified portfolio. By making this substitution for the previous equation, we get the following single-index model:

Thus, the CML, which is only for well-diversified portfolios, is fully consistent with a single-index model.

In this section, you have learned how to decompose total variance into systematic and nonsystematic variances and how the CML is the same as a single-index model for diversified portfolios.

3.2.3. Return-Generating Models: The Market Model

The most common implementation of a single-index model is the market model, in which the market return is the single factor or single index. In principle, the market model and the single-index model are similar. The difference is that the market model is easier to work with and is normally used for estimating beta risk and computing abnormal returns. The market model is

To be consistent with the previous section, αi = Rf(1 – β). The intercept, αi, and slope coefficient, βi, can be estimated by using historical security and market returns. These parameter estimates are then used to predict company-specific returns that a security may earn in a future period. Assume that a regression of Wal-Mart’s historical daily returns on S&P 500 daily returns gives an αi of 0.0001 and a βi of 0.9. Thus, Wal-Mart’s expected daily return = 0.0001 + 0.90 × Rm. If on a given day the market rises by 1 percent and Wal-Mart’s stock rises by 2 percent, then Wal-Mart’s company-specific return (ei) for that day = Ri −E(Ri) = Ri – (αi + βiRm) = 0.02 – (0.0001 + 0.90 × 0.01) = 0.0109, or 1.09%. In other words, Wal-Mart earned an abnormal return of 1.09 percent on that day.

3.2.4. Calculation and Interpretation of Beta

We begin with the single-index model introduced in Section 3.2.2 using realized returns and rewrite it as

Because systematic risk depends on the correlation between the asset and the market, we can arrive at a measure of systematic risk from the covariance between Ri and Rm, where Ri is defined using the preceding equation. Note that the risk-free rate is a constant, so the first term in Ri drops out.

The first term is beta multiplied by the variance of Rm. Because the error term is uncorrelated with the market, the second term drops out. Then, we can rewrite the equation in terms of beta as follows:

The preceding formula shows the expression for beta, βi, which is similar to the factor loading in the single-index model presented in Section 3.2.1. For example, if the correlation between an asset and the market is 0.70 and the asset and market have standard deviations of return of 0.25 and 0.15, respectively, the asset’s beta would be (0.70)(0.25)/0.15 = 1.17. If the asset’s covariance with the market and market variance were given as 0.026250 and 0.02250, respectively, the calculation would be 0.026250/0.02250 = 1.17. The beta in the market model includes an adjustment for the correlation between asset i and the market because the market model covers all assets whereas the CML works only for fully diversified portfolios.

As shown in the previous equation, beta is a measure of how sensitive an asset’s return is to the market as a whole and is calculated as the covariance of the return on i and the return on the market divided by the variance of the market return; that expression is equivalent to the product of the asset’s correlation with the market with a ratio of standard deviations of return (i.e., the ratio of the asset’s standard deviation to the market’s). As we have shown, beta captures an asset’s systematic risk, or the portion of an asset’s risk that cannot be eliminated by diversification. The variances and correlations required for the calculation of beta are usually based on historical returns.

A positive beta indicates that the return of an asset follows the general market trend, whereas a negative beta shows that the return of an asset generally follows a trend that is opposite to that of the market. In other words, a positive beta indicates that the return of an asset moves in the same direction of the market, whereas a negative beta indicates that the return of an asset moves in the opposite direction of the market. A risk-free asset’s beta is zero because its covariance with other assets is zero. In other words, a beta of zero indicates that the asset’s return has no correlation with movements in the market. The market’s beta can be calculated by substituting σm for σi in the numerator. Also, any asset’s correlation with itself is 1, so the beta of the market is 1:

Because the market’s beta is 1, the average beta of stocks in the market, by definition, is 1. In terms of correlation, most stocks, especially in developed markets, tend to be highly correlated with the market, with correlations in excess of 0.70. Some U.S. broad market indices, such as the S&P 500, the Dow Jones 30, and the Nasdaq 100, have even higher correlations that are in excess of 0.90. The correlations among different sectors are also high, which shows that companies have similar reactions to the same economic and market changes. As a consequence and as a practical matter, finding assets that have a consistently negative beta because of the market’s broad effects on all assets is unusual.

3.2.5. Estimation of Beta

An alternative and more practical approach is to estimate beta directly by using the market model described previously. The market model, Ri = αi + βiRm + ei, is estimated by using regression analysis, which is a statistical process that evaluates the relationship between a given variable (the dependent variable) and one or more other (independent) variables. Historical security returns (Ri) and historical market returns (Rm) are inputs used for estimating the two parameters αi and βi.

Regression analysis is similar to plotting all combinations of the asset’s return and the market return (Ri, Rm) and then drawing a line through all points such that it minimizes the sum of squared linear deviations from the line. Exhibit 6-6 illustrates the market model and the estimated parameters. The intercept, αi (sometimes referred to as the constant), and the slope term, βi, are all that is needed to define the security characteristic line and obtain beta estimates.

Although beta estimates are important for forecasting future levels of risk, there is much concern about their accuracy. In general, shorter periods of estimation (e.g., 12 months) represent betas that are closer to the asset’s current level of systematic risk. Shorter period beta estimates, however, are also less accurate than beta estimates measured over three to five years because they may be affected by special events in that short period. Although longer period beta estimates are more accurate, they may be a poor representation of future expectations, especially if major changes in the asset have occurred. Therefore, it is necessary to recognize that estimates of beta, whether obtained through calculation or regression analysis, may or may not represent current or future levels of an asset’s systematic risk.

3.2.6. Beta and Expected Return

Although the single-index model, also called the capital asset pricing model (CAPM), will be discussed in greater detail in the next section, we will use the CAPM in this section to estimate returns, given asset betas. The CAPM is usually written with the risk-free rate on the right-hand side:

The model shows that the primary determinant of expected return for a security is its beta, or how well the security correlates with the market. The higher the beta of an asset, the higher its expected return will be. Assets with a beta greater than 1 have an expected return that is higher than the market return, whereas assets with a beta of less than 1 have an expected return that is less than the market return.

In certain cases, assets may require a return that is less than the risk-free return. For example, if an asset’s beta is negative, the required return will be less than the risk-free rate. When combined with the market, the asset reduces the risk of the overall portfolio, which makes the asset very valuable. Insurance is one such asset. Insurance gives a positive return when the insured’s wealth is reduced because of a catastrophic loss. In the absence of such a loss or when the insured’s wealth is growing, the insured is required to pay an insurance premium. Thus, insurance has a negative beta and a negative expected return, but helps in reducing overall risk.

4. THE CAPITAL ASSET PRICING MODEL

The capital asset pricing model is one of the most significant innovations in portfolio theory. The model is simple, yet powerful; is intuitive, yet profound; and uses only one factor, yet is broadly applicable. The CAPM was introduced independently by William Sharpe, John Lintner, Jack Treynor, and Jan Mossin and builds on Harry Markowitz’s earlier work on diversification and modern portfolio theory.3 The model provides a linear expected return–beta relationship that precisely determines the expected return given the beta of an asset. In doing so, it makes the transition from total risk to systematic risk, the primary determinant of expected return. Recall the following equation:

The CAPM asserts that the expected returns of assets vary only by their systematic risk as measured by beta. Two assets with the same beta will have the same expected return irrespective of the nature of those assets. Given the relationship between risk and return, all assets are defined only by their beta risk, which we will explain as the assumptions are described.

In the remainder of this section, we will examine the assumptions made in arriving at the CAPM and the limitations those assumptions entail. Second, we will implement the CAPM through the security market line to price any portfolio or asset, both efficient and inefficient. Finally, we will discuss ways in which the CAPM can be applied to investments, valuation, and capital budgeting.

4.1. Assumptions of the CAPM

Similar to all other models, the CAPM ignores many of the complexities of financial markets by making simplifying assumptions. These assumptions allow us to gain important insights into how assets are priced without complicating the analysis. Once the basic relationships are established, we can relax the assumptions and examine how our insights need to be altered. Some of these assumptions are constraining, whereas others are benign. And other assumptions affect only a particular set of assets or only marginally affect the hypothesized relationships.

The main objective of these assumptions is to create a marginal investor who rationally chooses a mean–variance-efficient portfolio in a predictable fashion. We assume away any inefficiency in the market from both operational and informational perspectives. Although some of these assumptions may seem unrealistic, relaxing most of them will have only a minor influence on the model and its results. Moreover, the CAPM, with all its limitations and weaknesses, provides a benchmark for comparison and for generating initial return estimates.

4.2. The Security Market Line

In this subsection, we apply the CAPM to the pricing of securities. The security market line (SML) is a graphical representation of the capital asset pricing model with beta on the x-axis and expected return on the y-axis. Using the same concept as the capital market line, the SML intersects the y-axis at the risk-free rate of return, and the slope of this line is the market risk premium, Rm – Rf. Recall that the capital allocation line (CAL) and the capital market line (CML) do not apply to all securities or assets but only to efficient portfolios. In contrast, the security market line applies to any security, efficient or not. The difference occurs because the CAL and the CML use the total risk of the asset rather than its systematic risk. Because only systematic risk is priced and the CAL and the CML are based on total risk, the CAL and the CML can only be applied to those assets whose total risk is equal to systematic risk. Total risk and systematic risk are equal only for efficient portfolios because those portfolios have no diversifiable risk remaining. We are able to relax the requirement of efficient portfolios for the SML because the CAPM, which forms the basis for the SML, prices a security based only on its systematic risk, not its total risk.

Exhibit 6-7 is a graphical representation of the CAPM, the security market line. As shown earlier in this chapter, the beta of the market is 1 (x-axis) and the market earns an expected return of Rm (y-axis). Using this line, it is possible to calculate the expected return of an asset. The next example illustrates the beta and return calculations.

4.2.1. Portfolio Beta

As we stated previously, the security market line applies to all securities. But what about a combination of securities, such as a portfolio? Consider two securities, 1 and 2, with a weight of wi in Security 1 and the balance in Security 2. The return for the two securities and return of the portfolio can be written as:

The last equation gives the expression for the portfolio’s expected return. From this equation, we can conclude that the portfolio’s beta = w1β1 + w2β2. In general, the portfolio beta is a weighted sum of the betas of the component securities and is given by:

The portfolio’s return given by the CAPM is

This equation shows that a linear relationship exists between the expected return of a portfolio and the systematic risk of the portfolio as measured by βp.

4.3. Applications of the CAPM

The CAPM offers powerful and intuitively appealing predictions about risk and the relationship between risk and return. The CAPM is not only important from a theoretical perspective but is also used extensively in practice. In this section, we will discuss some common applications of the model. When applying these tools to different scenarios, it is important to understand that the CAPM and the SML are functions that give an indication of what the return in the market should be, given a certain level of risk. The actual return may be quite different from the expected return.

Applications of the CAPM include estimates of the expected return for capital budgeting, comparison of the actual return of a portfolio or portfolio manager with the CAPM return for performance appraisal, and the analysis of alternate return estimates and the CAPM returns as the basis for security selection. The applications are discussed in more detail in this section.

4.3.1. Estimate of Expected Return

Given an asset’s systematic risk, the expected return can be calculated using the CAPM. Recall that the price of an asset is the sum of all future cash flows discounted at the required rate of return, where the discount rate or the required rate of return is commensurate with the asset’s risk. The expected rate of return obtained from the CAPM is normally the first estimate that investors use for valuing assets, such as stocks, bonds, real estate, and other similar assets. The required rate of return from the CAPM is also used for capital budgeting and determining the economic feasibility of projects. Again, recall that when computing the net present value of a project, investments and net revenues are considered cash flows and are discounted at the required rate of return. The required rate of return, based on the project’s risk, is calculated using the CAPM.

Because risk and return underlie almost all aspects of investment decision making, it is not surprising that the CAPM is used for estimating expected return in many scenarios. Other examples include calculating the cost of capital for regulated companies by regulatory commissions and setting fair insurance premiums. The next example shows an application of the CAPM to capital budgeting.

4.3.2. Portfolio Performance Evaluation

Institutional money managers, pension fund managers, and mutual fund managers manage large amounts of money for other people. Are they doing a good job? How does their performance compare with a passively managed portfolio—that is, one in which the investor holds just the market portfolio? Evaluating the performance of a portfolio is of interest to all investors and money managers. Because active management costs significantly more than passive management, we expect active managers to perform better than passive managers or at least to cover the difference in expenses. For example, Fidelity’s passively managed Spartan 500 Index fund has an expense ratio of only 0.10 percent whereas Fidelity’s actively managed Contrafund has an expense ratio of 0.94 percent. Investors need a method for determining whether the manager of the Contrafund is worth the extra 0.84 percent in expenses.

In this chapter, performance evaluation is based only on the CAPM. However, it is easy to extend this analysis to multifactor models that may include industry or other special factors. Four ratios are commonly used in performance evaluation.

4.3.2.1. Sharpe and Treynor Ratios

Performance has two components, risk and return. Although return maximization is a laudable objective, comparing just the return of a portfolio with that of the market is not sufficient. Because investors are risk averse, they will require compensation for higher risk in the form of higher returns. A commonly used measure of performance is the Sharpe ratio, which is defined as the portfolio’s risk premium divided by its risk:

Recalling the CAL from earlier in the chapter, one can see that the Sharpe ratio, also called the reward-to-variability ratio, is simply the slope of the capital allocation line; the greater the slope, the better the asset. Note, however, that the ratio uses the total risk of the portfolio, not its systematic risk. The use of total risk is appropriate if the portfolio is an investor’s total portfolio—that is, the investor does not own any other assets. Sharpe ratios of the market and other portfolios can also be calculated in a similar manner. The portfolio with the highest Sharpe ratio has the best performance, and the one with the lowest Sharpe ratio has the worst performance, provided that the numerator is positive for all comparison portfolios. If the numerator is negative, the ratio will be less negative for riskier portfolios, resulting in incorrect rankings.

The Sharpe ratio, however, suffers from two limitations. First, it uses total risk as a measure of risk when only systematic risk is priced. Second, the ratio itself (e.g., 0.2 or 0.3) is not informative. To rank portfolios, the Sharpe ratio of one portfolio must be compared with the Sharpe ratio of another portfolio. Nonetheless, the ease of computation makes the Sharpe ratio a popular tool.

The Treynor ratio is a simple extension of the Sharpe ratio and resolves the Sharpe ratio’s first limitation by substituting beta risk for total risk. The Treynor ratio is

Just like the Sharpe ratio, the numerators must be positive for the Treynor ratio to give meaningful results. In addition, the Treynor ratio does not work for negative-beta assets—that is, the denominator must also be positive for obtaining correct estimates and rankings. Although both the Sharpe and Treynor ratios allow for ranking of portfolios, neither ratio gives any information about the economic significance of differences in performance. For example, assume the Sharpe ratio of one portfolio is 0.75 and the Sharpe ratio for another portfolio is 0.80. The second portfolio is superior, but is that difference meaningful? In addition, we do not know whether either of the portfolios is better than the passive market portfolio. The remaining two measures, M2 and Jensen’s alpha, attempt to address that problem by comparing portfolios while also providing information about the extent of the overperformance or underperformance.

4.3.2.2. M-Squared (M2)

M2 was created by Franco Modigliani and his granddaughter, Leah Modigliani—hence the name M-squared. M2 is an extension of the Sharpe ratio in that it is based on total risk, not beta risk. The idea behind the measure is to create a portfolio (P′) that mimics the risk of a market portfolio—that is, the mimicking portfolio (P′) alters the weights in Portfolio P and the risk-free asset until Portfolio P′ has the same total risk as the market (i.e., σp′ = σm). Because the risks of the mimicking portfolio and the market portfolio are the same, we can obtain the return on the mimicking portfolio and directly compare it with the market return. The weight in Portfolio P, wp, that makes the risks equal can be calculated as follows:

Because the correlation between the market and the risk-free asset is zero, we get wp as the weight invested in Portfolio P and the balance invested in the risk-free asset. The risk-adjusted return for the mimicking portfolio is:

The return of the mimicking portfolio based on excess returns is .5 The difference in the return of the mimicking portfolio and the market return is M2, which can be expressed as a formula:

M2 gives us rankings that are identical to those of the Sharpe ratio. They are easier to interpret, however, because they are in percentage terms. A portfolio that matches the performance of the market will have an M2 of zero, whereas a portfolio that outperforms the market will have an M2 that is positive. By using M2, we are not only able to determine the rank of a portfolio but also which, if any, of our portfolios beat the market on a risk-adjusted basis.

4.3.2.3. Jensen’s Alpha

Like the Treynor ratio, Jensen’s alpha is based on systematic risk. We can measure a portfolio’s systematic risk by estimating the market model, which is done by regressing the portfolio’s daily return on the market’s daily return. The coefficient on the market return is an estimate of the beta risk of the portfolio (see Section 3.2.5 for more details). We can calculate the risk-adjusted return of the portfolio using the beta of the portfolio and the CAPM. The difference between the actual portfolio return and the calculated risk-adjusted return is a measure of the portfolio’s performance relative to the market portfolio and is called Jensen’s alpha. By definition, αm of the market is zero. Jensen’s alpha is also the vertical distance from the SML measuring the excess return for the same risk as that of the market and is given by

If the period is long, it may contain different risk-free rates, in which case Rf represents the average risk-free rate. Furthermore, the returns in the equation are all realized actual returns. The sign of αp indicates whether the portfolio has outperformed the market. If αp is positive, then the portfolio has outperformed the market; if αp is negative, the portfolio has underperformed the market. Jensen’s alpha is commonly used for evaluating most institutional managers, pension funds, and mutual funds. Values of alpha can be used to rank different managers and the performance of their portfolios, as well as the magnitude of underperformance or overperformance. For example, if a portfolio’s alpha is 2 percent and another portfolio’s alpha is 5 percent, the second portfolio has outperformed the first portfolio by 3 percentage points and the market by 5 percentage points. Jensen’s alpha is the maximum amount that you should be willing to pay the manager to manage your money.

The measures of performance evaluation assume that the benchmark market portfolio is the correct portfolio. As a result, an error in the benchmark may cause the results to be misleading. For example, evaluating a real estate fund against the S&P 500 is incorrect because real estate has different characteristics than equity. In addition to errors in benchmarking, errors could occur in the measurement of risk and return of the market portfolio and the portfolios being evaluated. Finally, many estimates are based on historical data. Any projections based on such estimates assume that this level of performance will continue in the future.

4.3.3. Security Characteristic Line

Similar to the SML, we can draw a security characteristic line (SCL) for a security. The SCL is a plot of the excess return of the security on the excess return of the market. In Exhibit 6-11, Jensen’s alpha is the intercept and the beta is the slope. The equation of the line can be obtained by rearranging the terms in the expression for Jensen’s alpha and replacing the subscript p with i:

As an example, the SCL is drawn in Exhibit 6-11 using Manager X’s portfolio from Exhibit 6-8. The security characteristic line can also be estimated by regressing the excess security return, Ri – Rf, on the excess market return, Rm – Rf.

4.3.4. Security Selection

When discussing the CAPM, we assumed that investors have homogeneous expectations and are rational, risk-averse, utility-maximizing investors. With these assumptions, we were able to state that all investors assign the same value to all assets and, therefore, have the same optimal risky portfolio, which is the market portfolio. In other words, we assumed that there is commonality among beliefs about an asset’s future cash flows and the required rate of return. Given the required rate of return, we can discount the future cash flows of the asset to arrive at its current value, or price, which is agreed upon by all or most investors.

In this section, we introduce heterogeneity in beliefs of investors. Because investors are price takers, it is assumed that such heterogeneity does not significantly affect the market price of an asset. The difference in beliefs can relate to future cash flows, the systematic risk of the asset, or both. Because the current price of an asset is the discounted value of the future cash flows, the difference in beliefs could result in an investor-estimated price that is different from the CAPM-calculated price. The CAPM-calculated price is the current market price because it reflects the beliefs of all other investors in the market. If the investor-estimated current price is higher (lower) than the market price, the asset is considered undervalued (overvalued). Therefore, the CAPM is an effective tool for determining whether an asset is undervalued or overvalued and whether an investor should buy or sell the asset.

Although portfolio performance evaluation is backward looking and security selection is forward looking, we can apply the concepts of portfolio evaluation to security selection. The best measure to apply is Jensen’s alpha because it uses systematic risk and is meaningful even on an absolute basis. A positive Jensen’s alpha indicates a superior security, whereas a negative Jensen’s alpha indicates a security that is likely to underperform the market when adjusted for risk.

Another way of presenting the same information is with the security market line. Potential investors can plot a security’s expected return and beta against the SML and use this relationship to decide whether the security is overvalued or undervalued in the market.6 Exhibit 6-12 shows a number of securities along with the SML. All securities that reflect the consensus market view are points directly on the SML (i.e., properly valued). If a point representing the estimated return of an asset is above the SML (Points A and C), the asset has a low level of risk relative to the amount of expected return and would be a good choice for investment. In contrast, if the point representing a particular asset is below the SML (Point B), the stock is considered overvalued. Its return does not compensate for the level of risk and should not be considered for investment. Of course, a short position in Asset B can be taken if short selling is permitted.

4.3.5. Constructing a Portfolio

Based on the CAPM, investors should hold a combination of the risk-free asset and the market portfolio. The true market portfolio consists of a large number of securities, and an investor would have to own all of them in order to be completely diversified. Because owning all existing securities is not practical, in this section we will consider an alternate method of constructing a portfolio that may not require a large number of securities and will still be sufficiently diversified. Exhibit 6-13 shows the reduction in risk as we add more and more securities to a portfolio. As can be seen from the exhibit, much of the nonsystematic risk can be diversified away in as few as 30 securities. These securities, however, should be randomly selected and represent different asset classes for the portfolio to effectively diversify risk. Otherwise, one may be better off using an index (e.g., the S&P 500 for a diversified large-cap equity portfolio and other indices for other asset classes).

Let’s begin constructing the optimal portfolio with a portfolio of securities like the S&P 500. Although the S&P 500 is a portfolio of 500 securities, it is a good starting point because it is readily available as a single security for trading. In contrast, it represents only the large corporations that are traded on the U.S. stock markets and, therefore, does not encompass the global market entirely. Because the S&P 500 is the base portfolio, however, we treat it as the market for the CAPM.

Any security not included in the S&P 500 can be evaluated to determine whether it should be integrated into the portfolio. That decision is based on the αi of the security, which is calculated using the CAPM with the S&P 500 as the market portfolio. Note that security i may not necessarily be priced incorrectly for it to have a nonzero αi; αi can be positive merely because it is not well correlated with the S&P 500 and its return is sufficient for the amount of systematic risk it contains. For example, assume a new stock market, ABC, opens to foreign investors only and is being considered for inclusion in the portfolio. We estimate ABC’s model parameters relative to the S&P 500 and find an αi of approximately 3 percent, with a βi of 0.60. Because αi is positive, ABC should be added to the portfolio. Securities with a significantly negative αi may be short sold to maximize risk-adjusted return. For convenience, however, we will assume that negative positions are not permitted in the portfolio.

In addition to the securities that are correctly priced but enter the portfolio because of their risk–return superiority, securities already in the portfolio (S&P 500) may be undervalued or overvalued based on investor expectations that are incongruent with the market. Securities in the S&P 500 that are overvalued (negative αi) should be dropped from the S&P 500 portfolio, if it is possible to exclude individual securities, and positions in securities in the S&P 500 that are undervalued (positive αi) should be increased.

This brings us to the next question: What should the relative weight of securities in the portfolio be? Because we are concerned with maximizing risk-adjusted return, securities with a higher αi should have a higher weight, and securities with greater nonsystematic risk should be given less weight in the portfolio. A complete analysis of portfolio optimization is beyond the scope of this chapter, but the following principles are helpful. The weight in each nonmarket security should be proportional to , where the denominator is the nonsystematic variance of security i. The total weight of nonmarket securities in the portfolio is proportional to

The weight in the market portfolio is a function of

The information ratio, (i.e., alpha divided by nonsystematic risk), measures the abnormal return per unit of risk added by the security to a well-diversified portfolio. The larger the information ratio is, the more valuable the security.

5. BEYOND THE CAPITAL ASSET PRICING MODEL

In general, return-generating models allow us to estimate an asset’s return given its characteristics, where the asset characteristics required for estimating the return are specified in the model. Estimating an asset’s return is important for investment decision making. These models are also important as a benchmark for evaluating portfolio, security, or manager performance. The return-generating models were briefly introduced in Section 3.2.1, and one of those models, the capital asset pricing model, was discussed in detail in Section 4.

The purpose of this section is to make readers aware that, although the CAPM is an important concept and model, the CAPM is not the only return-generating model. In this section, we revisit and highlight the limitations of the CAPM and preview return-generating models that address some of those limitations.

5.1. The CAPM

The CAPM is a model that simplifies a complex investment environment and allows investors to understand the relationship between risk and return. Although the CAPM affords us this insight, its assumptions can be constraining and unrealistic, as mentioned in Section 5.2. In Section 5.3, we discuss other models that have been developed along with their own limitations.

5.2. Limitations of the CAPM

The CAPM is subject to theoretical and practical limitations. Theoretical limitations are inherent in the structure of the model, whereas practical limitations are those that arise in implementing the model.

5.2.1. Theoretical Limitations of the CAPM

• Single-factor model: Only systematic risk or beta risk is priced in the CAPM. Thus, the CAPM states that no other investment characteristics should be considered in estimating returns. As a consequence, it is prescriptive and easy to understand and apply, although it is very restrictive and inflexible.
• Single-period model: The CAPM is a single-period model that does not consider multiperiod implications or investment objectives of future periods, which can lead to myopic and suboptimal investment decisions. For example, it may be optimal to default on interest payments in the current period to maximize current returns, but the consequences may be negative in the next period. A single-period model like the CAPM is unable to capture factors that vary over time and span several periods.

5.2.2. Practical Limitations of the CAPM

In addition to the theoretical limitations, implementation of the CAPM raises several practical concerns, some of which are listed next.

• Market portfolio: The true market portfolio according to the CAPM includes all assets, financial and nonfinancial, which means that it also includes many assets that are not investable, such as human capital and assets in closed economies. Richard Roll7 noted that one reason the CAPM is not testable is that the true market portfolio is unobservable.
• Proxy for a market portfolio: In the absence of a true market portfolio, market participants generally use proxies. These proxies, however, vary among analysts, the country of the investor, and so on. and generate different return estimates for the same asset, which is impermissible in the CAPM.
• Estimation of beta risk: A long history of returns (three to five years) is required to estimate beta risk. The historical state of the company, however, may not be an accurate representation of the current or future state of the company. More generally, the CAPM is an ex ante model, yet it is usually applied using ex post data. In addition, using different periods for estimation results in different estimates of beta. For example, a three-year beta is unlikely to be the same as a five-year beta, and a beta estimated with daily returns is unlikely to be the same as the beta estimated with monthly returns. Thus, we are likely to estimate different returns for the same asset depending on the estimate of beta risk used in the model.
• The CAPM is a poor predictor of returns: If the CAPM is a good model, its estimate of asset returns should be closely associated with realized returns. However, empirical support for the CAPM is weak.8 In other words, tests of the CAPM show that asset returns are not determined only by systematic risk. Poor predictability of returns when using the CAPM is a serious limitation because return-generating models are used to estimate future returns.
• Homogeneity in investor expectations: The CAPM assumes that homogeneity exists in investor expectations for the model to generate a single optimal risky portfolio (the market) and a single security market line. Without this assumption, there will be numerous optimal risky portfolios and numerous security market lines. Clearly, investors can process the same information in a rational manner and arrive at different optimal risky portfolios.

5.3. Extensions to the CAPM

Given the limitations of the CAPM, it is not surprising that other models have been proposed to address some of these limitations. These new models are not without limitations of their own, which we will mention while discussing the models. We divide the models into two categories and provide one example of each type.

5.3.1. Theoretical Models

Theoretical models are based on the same principle as the CAPM but expand the number of risk factors. The best example of a theoretical model is the arbitrage pricing theory (APT), which was developed by Stephen Ross.9 Like the CAPM, APT proposes a linear relationship between expected return and risk:

where

Unlike the CAPM, however, APT allows numerous risk factors—as many as are relevant to a particular asset. Moreover, other than the risk-free rate, the risk factors need not be common and may vary from one asset to another. A no-arbitrage condition in asset markets is used to determine the risk factors and estimate betas for the risk factors.

Although it is theoretically elegant, flexible, and superior to the CAPM, APT is not commonly used in practice because it does not specify any of the risk factors and it becomes difficult to identify risk factors and estimate betas for each asset in a portfolio. So from a practical standpoint, the CAPM is preferred to APT.

5.3.2. Practical Models

If beta risk in the CAPM does not explain returns, which factors do? Practical models seek to answer this question through extensive research. As mentioned in Section 3.2.1, the best example of such a model is the four-factor model proposed by Fama and French (1992) and Carhart (1997).

Based on an analysis of the relationship between past returns and a variety of different factors, Fama and French (1992) proposed that three factors seem to explain asset returns better than just systematic risk. Those three factors are relative size, relative book-to-market value, and beta of the asset. With Carhart’s (1997) addition of relative past stock returns, the model can be written as follows:

where

Historical analysis shows that the coefficient on MKT is not significantly different from zero, which implies that stock return is unrelated to the market. The factors that explain stock returns are size (smaller companies outperform larger companies), book-to-market ratio (value companies outperform glamour companies), and momentum (past winners outperform past losers).

The four-factor model has been found to predict asset returns much better than the CAPM and is extensively used in estimating returns for U.S. stocks. Note the emphasis on U.S. stocks; because these factors were estimated for U.S. stocks, they have worked well for U.S. stocks over the past several years.

Three observations are in order. First, no strong economic arguments exist for the three additional risk factors. Second, the four-factor model does not necessarily apply to other assets or assets in other countries, and third, there is no expectation that the model will continue to work well in the future.

5.4. The CAPM and Beyond

The CAPM has limitations and, more importantly, is ineffective in modeling asset returns. However, it is a simple model that allows us to estimate returns and evaluate performance. The newer models provide alternatives to the CAPM, although they are not necessarily better in all situations or practical in their application in the real world.

6. SUMMARY

In this chapter, we discussed the capital asset pricing model in detail and covered related topics such as the capital market line. The chapter began with an interpretation of the CML, uses of the market portfolio as a passive management strategy, and leveraging of the market portfolio to obtain a higher expected return. Next, we discussed systematic and nonsystematic risk and why one should not expect to be compensated for taking on nonsystematic risk. The discussion of systematic and nonsystematic risk was followed by an introduction to beta and return-generating models. This broad topic was then broken down into a discussion of the CAPM and, more specifically, the relationship between beta and expected return. The final section included applications of the CAPM to capital budgeting, portfolio performance evaluation, and security selection. The highlights of the chapter are as follows.

• The capital market line is a special case of the capital allocation line, where the efficient portfolio is the market portfolio.
• Obtaining a unique optimal risky portfolio is not possible if investors are permitted to have heterogeneous beliefs because such beliefs will result in heterogeneous asset prices.
• Investors can leverage their portfolios by borrowing money and investing in the market.
• Systematic risk is the risk that affects the entire market or economy and is not diversifiable.
• Nonsystematic risk is local and can be diversified away by combining assets with low correlations.
• Beta risk, or systematic risk, is priced and earns a return, whereas nonsystematic risk is not priced.
• The expected return of an asset depends on its beta risk and can be computed using the CAPM, which is given by E(Ri) = Rf + βi[E(Rm) – Rf].
• The security market line is an implementation of the CAPM and applies to all securities, whether they are efficient or not.
• Expected return from the CAPM can be used for making capital budgeting decisions.
• Portfolios can be evaluated by several CAPM-based measures, such as the Sharpe ratio, the Treynor ratio, M2, and Jensen’s alpha.
• The SML can assist in security selection and optimal portfolio construction.

By successfully understanding the content of this chapter, you should feel comfortable decomposing total variance into systematic and nonsystematic variance, analyzing beta risk, using the CAPM, and evaluating portfolios and individual securities.

PROBLEMS10

1. The line depicting the risk and return of portfolio combinations of a risk-free asset and any risky asset is the:

A. Security market line.

B. Capital allocation line.

C. Security characteristic line.

2. The portfolio of a risk-free asset and a risky asset has a better risk–return tradeoff than investing in only one asset type because the correlation between the risk-free asset and the risky asset is equal to:

A. −1.0.

B. 0.0.

C. 1.0.

3. With respect to capital market theory, an investor’s optimal portfolio is the combination of a risk-free asset and a risky asset with the highest:

A. Expected return.

B. Indifference curve.

C. Capital allocation line slope.

4. Highly risk-averse investors will most likely invest the majority of their wealth in:

A. Risky assets.

B. Risk-free assets.

C. The optimal risky portfolio.

5. The capital market line, CML, is the graph of the risk and return of portfolio combinations consisting of the risk-free asset and:

A. Any risky portfolio.

B. The market portfolio.

C. The leveraged portfolio.

6. Which of the following statements most accurately defines the market portfolio in capital market theory? The market portfolio consists of all:

A. Risky assets.

B. Tradable assets.

C. Investable assets.

7. With respect to capital market theory, the optimal risky portfolio:

A. Is the market portfolio.

B. Has the highest expected return.

C. Has the lowest expected variance.

8. Relative to portfolios on the CML, any portfolio that plots above the CML is considered:

A. Inferior.

B. Inefficient.

C. Unachievable.

9. A portfolio on the capital market line with returns greater than the returns on the market portfolio represents a(n):

A. Lending portfolio.

B. Borrowing portfolio.

C. Unachievable portfolio.

10. With respect to the capital market line, a portfolio on the CML with returns less than the returns on the market portfolio represents a(n):

A. Lending portfolio.

B. Borrowing portfolio.

C. Unachievable portfolio.

11. Which of the following types of risk is most likely avoided by forming a diversified portfolio?

A. Total risk.

B. Systematic risk.

C. Nonsystematic risk.

12. Which of the following events is most likely an example of nonsystematic risk?

A. A decline in interest rates.

B. The resignation of chief executive officer.

C. An increase in the value of the U.S. dollar.

13. With respect to the pricing of risk in capital market theory, which of the following statements is most accurate?

A. All risk is priced.

B. Systematic risk is priced.

C. Nonsystematic risk is priced.

14. The sum of an asset’s systematic variance and its nonsystematic variance of returns is equal to the asset’s:

A. Beta.

B. Total risk.

C. Total variance.

15. With respect to return-generating models, the intercept term of the market model is the asset’s estimated:

A. Beta.

B. Alpha.

C. Variance.

16. With respect to return-generating models, the slope term of the market model is an estimate of the asset’s:

A. Total risk.

B. Systematic risk.

C. Nonsystematic risk.

17. With respect to return-generating models, which of the following statements is most accurate? Return-generating models are used to directly estimate the:

A. Expected return of a security.

B. Weights of securities in a portfolio.

C. Parameters of the capital market line.

Use the following data to answer questions 18 through 20:

An analyst gathers the following information:

18. Which security has the highest total risk?

A. Security 1.

B. Security 2.

C. Security 3.

19. Which security has the highest beta measure?

A. Security 1.

B. Security 2.

C. Security 3.

20. Which security has the least amount of market risk?

A. Security 1.

B. Security 2.

C. Security 3.

21. With respect to capital market theory, the average beta of all assets in the market is:

A. Less than 1.0.

B. Equal to 1.0.

C. Greater than 1.0.

22. The slope of the security characteristic line is an asset’s:

A. Beta.

B. Excess return.

C. Risk premium.

23. The graph of the capital asset pricing model is the:

A. Capital market line.

B. Security market line.

C. Security characteristic line.

24. With respect to capital market theory, correctly priced individual assets can be plotted on the:

A. Capital market line.

B. Security market line.

C. Capital allocation line.

25. With respect to the capital asset pricing model, the primary determinant of expected return of an individual asset is the:

A. Asset’s beta.

B. Market risk premium.

C. Asset’s standard deviation.

26. With respect to the capital asset pricing model, which of the following values of beta for an asset is most likely to have an expected return for the asset that is less than the risk-free rate?

A. −0.5.

B. 0.0.

C. 0.5.

27. With respect to the capital asset pricing model, the market risk premium is:

A. Less than the excess market return.

B. Equal to the excess market return.

C. Greater than the excess market return.

Use the following data to answer questions 28 through 31:

An analyst gathers the following information:

Security	Expected Standard Deviation	Beta
Security 1	25%	1.50
Security 2	15%	1.40
Security 3	20%	1.60

28. With respect to the capital asset pricing model, if the expected market risk premium is 6% and the risk-free rate is 3%, the expected return for Security 1 is closest to:

A. 9.0%.

B. 12.0%.

C. 13.5%.

29. With respect to the capital asset pricing model, if expected return for Security 2 is equal to 11.4% and the risk-free rate is 3%, the expected return for the market is closest to:

A. 8.4%.

B. 9.0%.

C. 10.3%.

30. With respect to the capital asset pricing model, if the expected market risk premium is 6% the security with the highest expected return is:

A. Security 1.

B. Security 2.

C. Security 3.

31. With respect to the capital asset pricing model, a decline in the expected market return will have the greatest impact on the expected return of:

A. Security 1.

B. Security 2.

C. Security 3.

32. Which of the following performance measures is consistent with the CAPM?

A. M-squared.

B. Sharpe ratio.

C. Jensen’s alpha.

33. Which of the following performance measures does not require the measure to be compared to another value?

A. Sharpe ratio.

B. Treynor ratio.

C. Jensen’s alpha.

34. Which of the following performance measures is most appropriate for an investor who is not fully diversified?

A. M-squared.

B. Treynor ratio.

C. Jensen’s alpha.

35. Analysts who have estimated returns of an asset to be greater than the expected returns generated by the capital asset pricing model should consider the asset to be:

A. Overvalued.

B. Undervalued.

C. Properly valued.

36. With respect to capital market theory, which of the following statements best describes the effect of the homogeneity assumption? Because all investors have the same economic expectations of future cash flows for all assets, investors will invest in:

A. The same optimal risky portfolio.

B. The S&P 500 Index.

C. Assets with the same amount of risk.

37. With respect to capital market theory, which of the following assumptions allows for the existence of the market portfolio? All investors:

A. Are price takers.

B. Have homogeneous expectations.

C. Plan for the same, single holding period.

38. The intercept of the best fit line formed by plotting the excess returns of a manager’s portfolio on the excess returns of the market is best described as Jensen’s:

A. Beta.

B. Ratio.

C. Alpha.

39. Portfolio managers who are maximizing risk-adjusted returns will seek to invest more in securities with:

A. Lower values of Jensen’s alpha.

B. Values of Jensen’s alpha equal to 0.

C. Higher values of Jensen’s alpha.

40. Portfolio managers, who are maximizing risk-adjusted returns, will seek to invest less in securities with:

A. Lower values for nonsystematic variance.

B. Values of nonsystematic variance equal to 0.

C. Higher values for nonsystematic variance.

1.N risky assets.

2.Fama and French (1992).

3.See, for example, Markowitz (1952), Sharpe (1964), Lintner (1965a, 1965b), Treynor (1961, 1962), and Mossin (1966).

4.Short selling shares involves selling shares that you do not own. Because you do not own the shares, you (or your broker) must borrow the shares before you can short sell. You sell the borrowed shares in the market hoping that you will be able to return the borrowed shares by buying them later in the market at a lower price. Brokerage houses and securities lenders lend shares to you to sell in return for a portion (or all) of the interest earned on the cash you receive for the shares that are short sold.

5.Note that the last term within parentheses on the right-hand side of the previous equation is the Sharpe ratio.

6.In this chapter, we do not consider transaction costs, which are important whenever deviations from a passive portfolio are considered. Thus, the magnitude of undervaluation or overvaluation should be considered in relation to transaction costs prior to making an investment decision.

7.Roll (1977).

8.See, for example, Fama and French (1992).

9.Ross (1976).

10.Practice questions were developed by Stephen P. Huffman, CFA (University of Wisconsin, Oshkosh).