# Portfolio Risk and the Capital Asset Pricing Model - Financial Management

Portfolio Risk and the Capital Asset Pricing Model

The preceding analysis illustrates the possibilities for portfolio risk reduction when two or more securities are combined to form a portfolio. Unfortunately, when more than two securities are involved—as is usually the case—the number of calculations required to compute the portfolio risk increases geometrically. For example, whereas 45 correlation coefficients are needed for a portfolio containing 10 securities, 4,950 correlation coefficients must be computed for a portfolio containing 100 securities. In other words, a 10-fold increase in securities causes a greater than 100-fold increase in the required calculations.

In addition, a substantial computational undertaking is required to find the particular portfolio of securities that minimizes portfolio risk for a given level of return or maximizes return for a given level of risk, even for a portfolio that contains only a few securities. Obviously, a more workable method is needed to assess the effects of diversification on a portfolio of assets.

One method that has gained widespread use in analyzing the relationship between portfolio risk and return is the Capital Asset Pricing Model (CAPM). This model provides a strong analytical basis for evaluating risk–return relationships—both in the context of financial management and securities investment decisions. The remainder of this section discusses the development and application of the CAPM.

Systematic and Unsystematic Risk

As illustrated in the previous section, whenever the individual securities in a portfolio are less than perfectly positively correlated, diversification can reduce the portfolio’s risk below the weighted average of the total risk (measured by the standard deviation) of the individual securities. Because most securities are positively correlated with returns in the securities market in general, it is usually not possible to eliminate all risk in a portfolio of securities.

As the economic outlook improves, returns on most individual securities tend to increase; as the economic outlook deteriorates, individual security returns tend to decline. In spite of this positive “comovement” among the returns of individual securities, each security experiences some “unique” variation in its returns that is unrelated to the underlying economic factors that influence all securities. In other words, there are two types of risk inherent in each security:

• Systematic (market), or nondiversifiable, risk
• Unsystematic (unique), or diversifiable, risk

The sum of these two types of risk equals the total risk of the security:

Systematic risk refers to that portion of the variability of an individual security’s returns caused by factors affecting the market as a whole; as such, it can be thought of as being nondiversifiable.

Total Risk = Systematic Risk + Unsystematic Risk

Systematic risk accounts for 25 to 50 percent of the total risk of any security. Some of the sources of systematic risk, which cause the returns from all securities to tend to move in the same direction over time, include the following:

1. Interest rate changes
2. Changes in purchasing power (inflation)
3. Changes in investor expectations about the overall performance of the economy

Because diversification cannot eliminate systematic risk, this type of risk is the predominant determinant of individual security risk premiums.

Unsystematic risk is risk that is unique to the firm. It is the variability in a security’s returns caused by such factors as the following:

• Management capabilities and decisions
• Strikes
• The availability of raw materials
• The unique effects of government regulation, such as pollution control
• The effects of foreign competition
• The particular levels of financial and operating leverage the firm employs

Since unsystematic risk is unique to each firm, an efficiently diversified portfolio of securities can successfully eliminate most of the unsystematic risk inherent in individual securities, as is shown in Figure. To effectively eliminate the unsystematic risk inherent in a portfolio’s individual securities, it is not necessary for the portfolio to include a large number of securities. In fact, randomly constructed portfolios of as few as 20 to 25 securities on average can successfully diversify away a large portion of the unsystematic risk of the individual securities.

The risk remaining after diversification is market-related risk, or systematic risk, and it cannot be eliminated through diversification. Because unsystematic risk commonly accounts for 50 percent or more of the total risk of most individual securities, it should be obvious that the risk-reducing benefits of efficient diversification are well worth the effort.

Given the small number of securities required for efficient diversification by an individual investor, as well as the dominance of the securities markets by many large institutional investors who hold widely diversified portfolios, it is safe to conclude that the most relevant risk that must be considered for any widely traded individual security is its systematic risk. The unsystematic portion of total risk is relatively easy to diversify away.

Security Market Line (SML)

As discussed earlier in the chapter, the return required of any risky asset is determined by the prevailing level of risk-free interest rates plus a risk premium. The greater the level of risk an investor perceives about a security’s return, the greater the required risk premium will be. In other words, investors require returns that are commensurate with the risk level they perceive. In algebraic terms, the required return from any Security j, kj , is equal to the following:

kj= rf+ uj

where rf is the risk-free rate and uj is the risk premium required by investors.

The security market line (SML) indicates the “going” required rate of return (kj ) on a security in the market for a given amount of systematic risk and is illustrated in Figure . The SML intersects the vertical axis at the risk-free rate, rf, indicating that any security with an expected risk premium equal to zero should be required to earn a return equal to the risk-free rate. As systematic risk increases, so do the risk premium and the required rate of return. According to Figure, for example, a security having a risk level of a_ should be required to earn a 10 percent rate of return.

Unsystematic Risk and Portfolio Diversification

Beta: A Measure of Systematic Risk

Thus far, we have not addressed the question of the appropriate risk measure to use when considering the risk–return trade-offs illustrated by the SML. The previous discussion of risk in a portfolio context suggests that a measure of systematic risk is an appropriate starting point.

The Security Market Line (SML)

The systematic risk of a security is a function of the total risk of a security as measured by the standard deviation of the security’s returns, the standard deviation of the returns from the market portfolio, and the correlation of the security’s returns with those of all other securities in the market. A broad-based security market index, such as the Standard & Poor’s 500 Market Index or the New York Stock Exchange Index, is normally used as a measure of total market returns.

One useful measure of the systematic risk of a Security j is the value called beta. Beta is a measure of the volatility of a security’s returns relative to the returns of a broad-based Market Portfolio m. It is defined as the ratio of the covariance (or comovement) of returns on Security j and Market Portfolio m to the variance of returns on the Market Portfolio:

where bj is the measure of systematic risk for Security j ; sj is the standard deviation of returns for Security j ; sm is the standard deviation of returns for the Market Portfolio m; s2 m is the variance of returns for the Market Portfolio m; and rjm is the correlation coefficient between returns for Security j and Market Portfolio m.

In practice, beta may be computed as the slope of a regression line between periodic (usually yearly, quarterly, or monthly) rates of return on the Market Portfolio (as measured by a market index, such as the Standard & Poor’s 500 Market Index) and the periodic rates of return for Security j, as follows:

kj = aj + bjrm + ej

where kj is the periodic percentage holding period rate of return for Security j ; aj is a constant term determined by the regression; bj is the computed historical beta for Security j ; rm is the periodic percentage holding period rate of return for the market index; and ej is a random error term. This equation describes a line called Security j ’s characteristic line.

characteristic Line for General Motors

Figure shows the characteristic line for General Motors. The slope (and intercept) of this line can be estimated using the least-squares technique of regression analysis. The slope of this line, or beta, is 0.97, indicating that the systematic returns from General Motors common stock are slightly less variable than the returns for the market as a whole. A beta of 1.0 for any security indicates that the security is of average systematic risk; that is, a security with a beta of 1.0 has the same risk characteristics as the market as a whole when only systematic risk is considered.

When beta equals 1.0, a 1 percent increase (decline) in market returns indicates that the systematic returns for the individual security should increase (decline) by 1 percent. A beta greater than 1.0—for example, 2.0—indicates that the security has greater -than-average systematic risk. In this case, when market returns increase (decline) by 1 percent, the security’s systematic returns can be expected to increase (decline) by 2 percent. A beta of less than 1.0—for example, 0.5—is indicative of a security of less-than-average systematic risk. In this case, a 1 percent increase (decline) in market returns implies a 0.5 percent increase (decline) in the security’s systematic returns. Table summarizes the interpretation of selected betas.

Interpretation of Selected Beta Coefficients

The beta for the Market Portfolio as measured by a broad-based market index equals 1.0. This can be seen in Equation . Because the correlation of the market with itself is 1.0, the beta of the Market Portfolio must also be 1.0.

Finally, the beta of any portfolio of n securities or assets is simply the weighted average of the individual security betas:

This concept is useful particularly when evaluating the effects of capital investment projects or mergers on a firm’s systematic risk.

Fortunately for financial managers, it is not necessary to compute the beta for each security every time a security’s systematic risk measure is needed. Several investment advisory services, including the Value Line Investment Survey and Merrill Lynch, regularly compute and publish individual security beta estimates, and these are readily available. Table lists the Value Line computed betas for selected stocks.

Security Market Line and Beta

Given the information presented thus far, it is possible to compute risk premiums u that are applicable to individual securities. The SML may also be defined in terms of beta. The risk premium for any Security j is equal to the difference between the investor’s required return, kj, and the risk-free rate, rf :

θj= kjrj

If we let rˆm be the expected rate of return on the overall Market Portfolio and rˆf be the expected (short-term) risk -free rate (that is, the rate of return on Treasury bills), then the market risk premium is equal to

θˆm = rˆm– rˆf

Based on historic stock market data over the time period from 1926 through 2003, the average market risk premium has been about 8.6 percent. Other estimates based on expected returns using security analyst data lead to somewhat lower equity risk premium estimates. For a security with average risk (bj equal to 1.0), the risk premium should be equal to the market risk premium, or 8.6 percent. A security whose beta is 2.0, however, is twice as risky as the average security, so its risk premium should be twice the market risk premium:

θj = βj (rˆm– rˆf ) = 2.0 (8.6%) = 17.2%

The required return for any Security j may be defined in terms of its systematic risk, bj, the expected market return, rˆm, and the expected risk-free rate, rˆf, as follows:

kj = rˆf + bj (rˆm– r&circ;f )

For example, if the risk-free rate is 6 percent and (rˆm rˆf ) is 8.6 percent, then the required rate of return for any security j is given by:

kj = 6 + 8.6βj

The required return for Krispy Kreme, which has a beta of 1.20, can be computed using

kj = 6% + 8.6% (1.20) = 16.3%

Betas for Selected Stocks

Equation provides an explicit definition of the SML in terms of the systematic risk of individual securities. The slope of the SML is constant throughout. When measured between a beta of 0 and a beta of 1.0, it is equal to (rˆm rˆf )/(1 – 0), or simply rˆm rˆf. This slope represents the risk premium on an average risk security. Figure illustrates the SML for Equation 6.18.21 Given a risk-free rate of 6 percent and a market risk premium of 8.6 percent, the return required on a low-risk stock (for example, a security with a beta equal to 0.60, such as Hershey Foods) is 11.2 percent.

The return required on a high-risk stock (for example, a security with a beta equal to 1.50, such as MorganChase) is 18.9 percent, and the return required on a stock of average risk (such as DuPont, with a beta equal to 1.0) is 14.6 percent, the same as the market required return. Also, from Figure we can determine what securities (assets) are attractive investments by comparing the expected return from a security with the return required for that security, given its beta. For example, Security A with a beta of 1.0 and an expected return of 17 percent would be an attractive investment because the expected return exceeds the 14.6 percent required return. In contrast, Security B with a beta of 1.50 is not and acceptable investment because its expected return (18 percent) is less than its required return (18.9 percent).

The Security Market Line in Terms of Beta

Inflation and the Security Market Line

As discussed earlier in the chapter, the risk-free rate of return, rf, consists of the real rate of return and the expected inflation premium. Because the required return on any risky security, kj, is equal to the risk-free rate plus the risk premium, an increase in inflationary expectations effectively increases the required return on all securities. For example, if the risk-free rate of return increases by two percentage points, the required returns of all securities increase by two percentage points —the change in expected inflation.

Thus, the required rate of return on a security of average risk (that is, beta equal to 1.0) would increase from 14.6 to 16.6 percent. When investors increase their required returns, they become unwilling to purchase securities at existing prices, causing prices to decline. It should come as no surprise, then, that security analysts and investors take a dim view of increased inflation.

Uses of the CAPM and Portfolio Risk Concepts

The concepts of portfolio risk, and the Capital Asset Pricing Model (CAPM), which relates required returns to systematic risk (beta), are powerful pedagogical tools to explain the nature of risk and its relationship to required returns on securities and physical assets. In Chapter 12, the CAPM is discussed as one technique that can be used to estimate the cost of equity capital. Chapter also considers where the necessary data may be obtained to apply the model. Chapter considers the use of CAPM-determined required rates of return as a technique to adjust for risk in the capital budgeting process.

INTERNATIONAL ISSUES

Diversification and Multinational Corporations

As we discussed earlier in the chapter, the degree to which diversification can reduce risk depends on the correlation among security returns.The returns from domestic companies (DMCs)—companies that are based and operate within a given country —tend to be positively related to the overall level of economic activity within the given country. Hence, these companies tend to have a relatively high degree of systematic risk. Since overall economic activity in different countries is not perfectly correlated, the returns from multinational companies (MNCs) —companies that operate in a number of different countries —may tend to have less systematic risk than those of DMCs.This suggests that further risk reduction benefits may be achieved by either

1. Investing in MNCs, or
2. Investing directly in DMCs located in countries in which the MNC would otherwise operate.

If securities are traded in perfect financial markets, there should be no systematic advantage to holding shares in MNCs (Strategy 1) compared with owning shares directly in DMCs located in different countries (Strategy 2). However, if market imperfections exist, such as controls on capital flows, differential trading costs, and different tax structures, then MNCs may be able to provide diversification benefits to investors. The empirical evidence suggests that MNCs tend to have lower systematic risk (as measured by beta), as well as lower unsystematic risk, than DMCs.

Overall, MNCs tend to have a lower total risk (as measured by the standard deviation of rates of return on equity) than DMCs. Hence, MNCs appear to provide investors with substantial diversification benefits. a See Israel Shaked, “Are Multinational Corporations Safer?” Journal of International Business Studies (Spring 1986):

Assumptions and Limitations of the CAPM

The theoretical CAPM and its applications are based upon a number of crucial assumptions about the securities markets and investors’ attitudes, including the following:

• Investors hold well-diversified portfolios of securities. Hence, their return requirements are influenced primarily by the systematic (rather than total) risk of each security.
• Securities are traded actively in a competitive market, where information about a given firm and its future prospects is freely available.
• Investors can borrow and lend at the risk-free rate, which remains constant over time.
• There are no brokerage charges for buying and selling securities.
• There are no taxes.
• All investors prefer the security that provides the highest return for a given level of risk or the lowest amount of risk for a given level of return.

High-Risk Securities

The decade of the 1980s experienced a dramatic growth in high-risk, so-called “junk” bonds.These are bonds with credit ratings below investment grade(BBB from Standard & Poor’s and Baa3 from Moody’s).a The lure of these securities was their high returns relative to the returns available from investment- grade corporate and U.S. government-issued debt securities. Junk bonds appeared to offer an easy way to quickly increase the yield on the portfolios of assets held by many financial institutions.

Although there are many complex reasons for the financial collapse of the savings and loan industry, at least in some cases the failure of large institutions can be traced to their overinvestment in high-risk debt securities. In the early 1990s, similar problems came to light in parts of the insurance industry. For example, in April 1991 insurance regulators in California seized control of Executive Life Insurance Company.

The bond portfolio of Executive Life’s parent company, First Executive Corporation, had a market value of $3 billion less than its$9.85 billion book value at the end of 1990. During the first quarter of 1991, the insurance company lost \$465.9 million. The potential collapse of Executive Life placed in jeopardy the private pensions of thousands of workers whose employers had purchased retirement annuity contracts through the firm.

The problems caused by the savings and loan industry failure and the failure of large life insurance companies, such as First Executive, which had invested heavily in junk bonds, raise interesting issues of ethics and good business practice. Both savings and loan institutions and First Executive made a conscious decision to accept additional risk in the investment securities they purchased in exchange for additional expected return.

What standards of prudent business practice should be followed by firms when they attempt to improve their investment earnings performance? It is well known that, in the financial markets, higher returns can usually be achieved only by assuming greater risks. If this is true, should insurance companies completely avoid high-risk securities? How can these risks be managed effectively? Whatstandards of voluntary disclosure of information to depositors and policyholders about the risk and return characteristics of the assets held by the institution do you believe are appropriate?

• All investors have common (homogeneous) expectations regarding the expected returns, variances, and correlations of returns among all securities. While these assumptions may seem fairly limiting at first glance, extensions of the basic theory presented in this chapter, which relax the assumptions, have generally yielded results consistent with the fundamental theory.

Empirical studies of the CAPM have produced mixed results. Some researchers have found positive relationships between systematic (beta) risk and return. However, depending on the time period under examination, the results sometimes do not meet the standard tests of statistical significance. Other investigators, using different stock price data, have found that variables other than systematic risk are better predictors of the performance of common stocks.

Their results suggest that differences in company size and the ratio of book -to -market values explain the differences in stock returns. Still other researchers have argued that the results of statistical tests of the relationship between systematic risk and return may be flawed because of the difficulty of obtaining accurate estimates of beta.Their studies suggest that the use of market indexes, such as the Standard and Poor’s 500 Market Index, to measure returns on the true Market Portfolio can introduce significant errors into the process for estimating betas.

Despite the controversy concerning the validity of the CAPM, the model has been used extensively, both practically and conceptually, to consider the risk–return trade-off required by investors in the securities markets. For example, the CAPM (or a modification thereof) has been used in regulated public utility rate case testimony aimed at determining a reasonable allowed rate of return for the utility’s investors.

However, users of this approach should also be aware of some of the major problems encountered in practical applications, which include the following:

• Estimating expected future market returns
• Determining the most appropriate estimate of the risk -free rate
• Determining the best estimate of an asset’s future beta
• Reconciling the fact that some empirical tests have shown that investors do not totally ignore unsystematic risk, as the theory suggests
• Recognizing that measures of beta have been shown to be quite unstable over time, making it difficult to measure confidently the beta expected by investors
• Recognizing the growing body of evidence that suggests that required returns on most securities are determined by macroeconomic factors, such as interest rates and inflation, in addition to the risk-free rate of interest and the systematic risk of the security