A.4 The CAPM and Multifactor Models

Keith Cuthbertson and Dirk Nitzsche 1

The capital asset pricing model (CAPM) is central in determining the average return required by investors for holding a particular stock (or portfolio of stocks). It therefore determines the cost of equity capital for a firm raising funds when issuing more shares, and this ‘cost’ is used to discount the future cash flows expected from a risky physical investment project, in order to decide whether the project is viable. Hence the CAPM is useful for issues in corporate finance. What use is the CAPM for an investor? In the CAPM, the riskiness of a stock when held as part of a well-diversified portfolio is determined by how much the stock adds to overall portfolio risk, and this is measured by the stock’s ‘beta’. When the market as a whole goes up or down by 10% and the return on a particular stock then moves by plus or minus 20%, then the beta of the stock equals 2. A stock with a beta of 0.5 is therefore less risky than one with a beta of 2. You can reduce the overall riskiness of a portfolio of stocks by selling ‘high beta’ stocks and using the funds to purchase ‘low beta’ stocks. However, you might incur high transaction costs (e.g. bid– ask spreads, brokerage fees) especially if you just wanted to change your ‘risk exposure’ for a relatively short period (e.g. 3–6 months). Once you know the beta of your stock portfolio, we see in later chapters how this ‘risk reduction’ can be achieved more easily and cheaply by using stock index futures. Every financial institution in developed economies has to hold capital against the riskiness of its assets (and liabilities) and, as we see in later chapters, the beta of a stock can be used in calculating the financial intermediaries’ ‘dollar’ exposure to risk (called its value at risk) on its stock portfolio (in domestic and foreign assets). The CAPM is therefore a key ‘tool’ in analysing and solving a wide range of important practical problems.

In this chapter we analyse the (basic one-period) CAPM, the single-index model and multifactor models such as the arbitrage pricing theory (APT). The CAPM is widely used in the finance literature to determine the average return required by shareholders on a particular asset based on its contribution to overall portfolio risk. We also present a brief account of the APT, which relates the expected return on a security to a set of variables called ‘factors’, which could include market-wide effects due to interest rates, exchange rates, etc. Throughout this chapter we consider that the only risky securities are equities (stocks), although strictly the model applies to choices among all risky assets (e.g. stocks, bonds, real estate).

1_{Keith Cuthbertson is Professor of Finance and Dirk Nitzsche is Senior Lecturer at the Cass Business} School, City University, London.

I.A.4.1

Overview

The CAPM provides an elegant model of the determinants of the equilibrium expected or required return on any individual risky asset. It predicts that the expected return on a risky asset ERi consists of the risk-free rate r plus a risk premium (rp_i ):

) ( Ƣ ER r r rp r ERi  i  i m  (I.A.4.1)

where the risk premium, rpi = Ƣi (ERm– r) and Ƣi = cov(Ri, Rm)/var(Rm). The risk premium is proportional to the excess market return (ERm– r) with the constant of proportionality given by the beta(Ƣi) of the individual risky asset. The excess market return (ERm r) is also known as the market risk premium since it is the additional average return on the market portfolio over and above the risk-free rate. It is a ‘payment’ for holding the risky market portfolio. The definition of security i’s beta, Ƣi, indicates that it:

x depends positively on Ƴim, the covariance between the return on security i and the market portfolio, cov(Ri,Rm);

x is inversely related to the variance Ƴ2mof the market portfolio, var(Rm).

Loosely speaking, if ex post (or actual average returns) approximate the ex ante expected return ERi, then we can think of the CAPM as explaining the average monthly return (over, say, a 36- month period) on security i.

What does the CAPM tell us about equilibrium-required returns on individual securities in the stock market? First, note that (ERm – r) > 0, otherwise no risk-averse investor would hold the market portfolio of risky assets when she could earn more, for certain, by investing all her wealth in the risk-free asset. The CAPM predicts that for those stocks that have a zero covariance with the market portfolio, they will be willingly held as long as they have an expected return equal to the risk-free rate (putƢi = 0 in equation (I.A.4.1)).

Second, returns on individual stocks tend to move in the same direction and hence, in general, cov(Ri,Rm) > 0 andƢi > 0. Securities that have a large positive covariance with the market return (Ƣi > 0) will have to earn a relatively high average return. As we have seen in the previous chapter, this is because the addition of such a security to the portfolio does little to reduce overall portfolio variance, and to offset the latter you require a high average return on this security.

The CAPM also allows one to assess the relative volatility of the expected returns on individual stocks on the basis of theirƢi values. Stocks for which Ƣi = 1 have a return that is expected to move one-for-one with the market portfolio (i.e. ERi = ERm) and are termed ‘neutral stocks’. IfƢi > 1 the stock is said to be an ‘aggressive stock’, since, on average, it moves more than changes in

the expected market return (either up or down). Conversely, ‘defensive stocks’ have Ƣi < 1. Therefore investors can use betas to rank the relative ‘safety’ of various securities and can combine different shares to give a desired beta for the portfolio.