The CAPM has implications for firm decision making. The cost of equity for a firm is given directly by the CAPM. This is because the company’s beta is measured by calculating the covariance between the return on its common stock and the market index. As a result, beta measures the market risk of a common stock, and if we know the market risk we can use the CAPM to determine the required rate of return on equity. The equation is given below:
E(r) = rf +[E(rm) – rf]β
If we can estimate the market risk of a firm’s equity and the market rate of return, then E(r) is the required return on equity, i.e. the cost of equity for the firm. If we assign the cost of equity as ke, then
E(r) = ke
This is one advantage of the CAPM. Firms can calculate the required rate of return on different projects and therefore maximise shareholder wealth. Fig 3. below shows a firm’s cost of equity and two various projects it is considering to take on. We can analyse the effect these projects will have on shareholder wealth by using the CAPM.
As long as all projects have the same risk as the firm, then ke may also be interpreted as the minimum required rate of return on new projects. However, the project may have a different risk level from the whole firm.
In this situation it is necessary to estimate the specific risk of the project and use the CAPM to determine the required level of return. For example, in the diagram below the projected rate of return on project A, rA, is higher than the cost of equity for the firm E(r). However, the project is also riskier than the firm because it has greater specific risk. If the managers of the firm were to demand that it earn the same rate as the firm [kA = E(r)], the project would be accepted since its projected rate of return rA is greater than the firm’s cost of equity. However this would be incorrect. The market requires a rate of return E(rA) for a project with systematic risk of βA but the project will earn less. Therefore since rA < E(rA) the project is clearly unacceptable.
Fig. 3: Evaluating the Impact of a New Project
Because the CAPM allows decision makers to estimate the required rate of return for projects of different risk, it is a very useful concept.
We must remember that capital asset pricing model is just another theory. The really important question is: does it work?
Many institutional investors have embraced the beta concept. Beta is, however, an academic creation. What could be more staid? Simply created as a number that describes a stock’s risk, it appears almost sterile in nature. Is beta a useful measure of risk? Is it true that high beta projects will provide larger long run returns than low beta projects as the CAPM suggests? Does beta alone summarise a security’s total systematic risk, or do we need to take into account other factors as well? Does beta really deserve alpha? In short, every assumption in which the CAPM is derived is violated in reality.
Firstly, lenders may not be able to lend at a risk free rate. This problem was solved by Black [1972]. He suggests that it’s not a very good assumtion for many investors and one feels that the model would be changed substantially if this assumption were dropped. He changed the CAPM formula by replacing the element of the risk free asset with the expected rate of return on a zero beta portfolio, E(rz).
E(r) = E(rz) +[E(rm) – E(rz)]β
As a result, he proved that the major results of the CAPM do not require the existence of a pure risk free asset. Beta is still the appropriate measure of systematic risk of an asset, and the linearity of the model still obtains. The model given above is called the two-factor model. One limitation of the two-factor model is that it relies heavily on the assumption that there are no short sales constraints. Empirically, most asset returns have positive correlations. This makes it virtually impossible to construct a zero beta portfolio composed of only long positions in securities. Therefore the use of short sales is a practical necessity to obtain zero beta portfolios. In general, zero beta portfolios would have to be composed of both long and short positions of risky assets. Ross [1977] has shown that in a world with short sales restrictions and no risk free asset the linear CAPM is invalid. Therefore to obtain the CAPM in a linear form one needs either a risk free asset that can be freely short sold or no constraints on short sales.
The second disadvantage of the CAPM is that it doesn’t take into account investor preference for skewness. The CAPM assumes returns are normally distributed. Over long periods of time, returns are best measured by a lognormal distribution.
CAPM assumes that investors are price takers and have homogenous expectations about the distribution of long run returns. This is a weak assumption. Investors do not have the same information and as a consequence will value future returns differently.
Another assumption of CAPM is that all of the firm’s assets are perfectly marketable. In reality this is false. A good example of this is human capital. You can rent your skills in return for wages but you cannot sell yourself or buy anyone else. Human capital is therefore a non-diversifiable asset. What affect does this have on the CAPM? The CAPM assumes no transaction costs and if all assets are perfectly divisible, investors hold either the risk free asset or the market portfolio. Empirical testing shows us this is not what happens. People do hold different positions of risky assets. The reason for this could be due to the existence of non-marketable assets.
Furthermore, the model does not take into account personal or corporation taxes. Brennan [1970] suggested that higher rates of returns would be required on assets with higher dividend yields. This is due to investors disliking dividends because they must pay income tax on them.
In a study published in 1992, Eugene Fama and Kenneth French divided all traded stocks on the New York, American and NASDAQ exchanges into deciles according to their beta measurements over the 1963-90 period. The findings showed that there was essentially no relationship between the returns of the decile portfolios and their beta measures.
Despite its critics, the CAPM can still be a useful management tool. Pioneers of the risk management field sought to improve the capital asset pricing model. The work of Ross [1976] gave birth to the Arbitrage Pricing Theory (APT). The APT aims to analyse the equilibrium relationship between the risk and expected return of assets, just as the CAPM does. Various key assumptions are also maintained; perfectly competitive and efficient markets, homogeneous expectations, and that investors should not be compensated for taking on extra specific risk.
According to Sharpe [1985] there are two main differences between the APT and CAPM. Firstly, in the APT, explicit modelling of several factors affects an asset’s return as opposed to the CAPM’s focus on the market portfolio. Secondly, the equilibrium relationship is only approximation, as market equilibrium in the CAPM rests on the observability and efficiency of the market portfolio.
Bodie et al. [1999] and Brigham et al. [1999] suggest that the CAPM was derived from a single factor model whereby the sensitivity of the asset’s returns is defined by the asset’s beta. But systematic elements of risk in particular stocks and portfolios may be too complicated to be captured by a measure of beta. Hence the APT was born.
The APT is a multi factor model:
E(r) = E(rf) +[E(r1) – E(rf)]β 1 + [E(r2) – E(rf)]β 2 +…[E(rk) – E(rf)]β k
Where E(r) is the Expected Return (i.e. The Cost of Equity), rf is the risk free rate, βk is the sensitivity of the asset to the kth factor and rk is the Expected Return of a on a portfolio with unit sensitivity to the kth factor and zero sensitivity to all other factors.
Multi factor means that the returns on an asset are a function of several factors generally excluding the market portfolio. Such factors maybe changes in national income, interest rates, inflation, and particular sector-specific influences. The covariance of each factor with the asset then leads to an extension of measuring risk by the beta attached to each factor. The fact that identification of the market portfolio is not required in the APT is a big advantage over the CAPM.
However, the APT does not suggest which factors should be considered. Its therefore easy to price assets incorrectly by including the wrong or irrelevant risk factors.
Bodie et al [1999] suggest another important difference between the CAPM and the APT. The CAPM assumes there is an efficient market portfolio that every investor desires to hold. In contrast the APT relies on the absence of free arbitrage opportunities. For example, two portfolios with the same risk cannot offer different expected returns because that would present an arbitrage opportunity with a net investment of zero. An investor could then guarantee a risk free positive expected return by short-selling one portfolio and holding an equal and opposite long position in another. As such free arbitrage cannot persist; equilibrium in the APT specifies a linear relationship between expected returns and the betas of the corresponding risk factors.
Financial practitioners have embraced the APT as it allows a more detailed and customised approach to risk management then the CAPM. The rise of the derivatives market has increased the use of the APT even more as it proves particularly useful in analysing the types of risk associated with them.
The final method of measuring the cost of equity I will analyse in his study is the Dividend Valuation Model (DVM)
The model examines the relationship between the market value of equity, the cost of equity and the dividends.
Where Ve = Market Value of Equity
Ke = Cost of Equity
Divn = Dividend at time n
This equation is a variation of the NPV equation. The dividends represent the cash flows and the cost of equity is equivalent to the discount rate (which for an unleveraged firm would be same thing). When the dividends are expected to grow at a constant rate, g, this equation reduces down to:
or
The DVM implies that the expected return on equity equals the prospective dividends plus expected capital gains. It has the advantage of using information which doesn’t need to be estimated; the current market price of the firm’s shares and its dividends are readily available. However the DVM does have its limitations.
Firstly, the DVM allows one to determine a relationship between the constant growth rate and the expected return on equity, but it is necessary to know one in order to determine the other.
Secondly, the model assumes a constant growth. This is a limiting assumption as most companies are not stable. Hurley and Johnson [1994] argue that dividend payout a firm makes will remain fairly constant – the company will not increase it in every period, but only when it is confident that the higher rates can be maintained in the future.
However, one can argue against this limitation. In general, even a fully specified discounted cash flow analysis will proceed in two steps: firstly, one estimates the PV of net cash flows for five to ten years into the future; and secondly, one estimates a ‘horizon value’ or ‘terminal value’ by using some variant of a constant growth scenario. Typically, more than 75% of a company’s value comes from the horizon value, and the assumption about future growth. Therefore, even though constant growth assumption is concerning, it will be part of most valuation procedures anyway.
Measuring the cost of equity is beneficial for the firm. However, the gain can be diluted if the cost of equity is measured inaccurately.
When executives evaluate a potential investment, whether it’s to build a new factory or acquire a company, they weigh the cost against potential returns. If the company were to apply too high a cost of capital in its valuations, then it would reject valuable opportunities that its competitors would take on. Applying too low a cost of capital would guarantee that the company would commit resources to projects that will erode profitability and destroy shareholder value. I’d like to show the significance of this through an example.
“An all equity firm is evaluating the potential gains of a new one-period project. It will require an initial investment of £1m and provide an expected income at the end of the period of £1.15m. The firm has a beta of 1 and the risk free rate is 5%. Should the firm take the project if the expected return on the market is a) 10% b) 20%?”
For this question we can incorporate NPV and CAPM.
NPV = -I + x/(r+1)
NPV = -1 +1.15/(1+r)
To solve r, we can use CAPM
E(r) = rf +[E(rm) – rf]β
For a),
E(r) = 0.05 + [0.1 - 0.05]1
E(r) = 10%
Using NPV,
NPV = -1 + 1.15/(1+.1)
NPV = 0.045
For b)
E(r) = 0.05 +[0.2 - 0.05]1
E(r) = 20%
Using NPV
NPV = -1 + 1.15/(1 + 0.2)
NPV = -0.042.
By assuming different market returns the firm can get different NPV results. For a) the NPV is positive and will increase shareholder wealth and therefore should be taken. Part b) on the other hand has a negative NPV and should be avoided. Using the wrong estimation can have a detrimental effect on the firm.
Throughout this study we have seen the various methods of valuing the cost of equity. Unfortunately the various methods have their downfalls. There is no perfect way of measuring the cost of equity. Undoubtedly, there will be many improvements in the techniques of risk and equity cost analysis. My own guess is that future risk measures will be even more sophisticated – not less so. For firms, the development of these theories is extremely important. Measuring the cost of equity inaccurately can have a detrimental affect on shareholder wealth.
Bibliography
Ross, S., Westerfield, R., Jaffe, J., 2002. Corporate Finance. 6th edition. New York: McGraw-Hill Irwin.
Brealey, R., Myers, S., 2000. Principles of Corporate Finance. 6th edition. New York: McGraw-Hill Irwin.
Copeland and Weston, 1988. Financial Theory and Corporate Policy. 3rd edition.
McNulty, J., 2002. What’s Your Real Cost of Capital? 114-121
