## The Capital Asset Pricing Model

The capital asset pricing model is a set of predictions concerning equilibrium expected returns on risky assets. Harry Markowitz laid down the foundation of modern portfolio management in 1952. The CAPM was developed 12 years later in articles by William Sharpe,1 John Lintner,2 and Jan Mossin.3 The time for this gestation indicates that the leap from Markowitz's portfolio selection model to the CAPM is not trivial.

1 William Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium," Journal of Finance, September 1964.

2 John Lintner, "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics, February 1965.

3 Jan Mossin, "Equilibrium in a Capital Asset Market," Econometrica, October 1966.

### 264 PART III Equilibrium in Capital Markets

We will approach the CAPM by posing the question "what if," where the "if' part refers to a simplified world. Positing an admittedly unrealistic world allows a relatively easy leap to the "then" part. Once we accomplish this, we can add complexity to the hypothesized environment one step at a time and see how the conclusions must be amended. This process allows us to derive a reasonably realistic and comprehensible model.

We summarize the simplifying assumptions that lead to the basic version of the CAPM in the following list. The thrust of these assumptions is that we try to ensure that individuals are as alike as possible, with the notable exceptions of initial wealth and risk aversion. We will see that conformity of investor behavior vastly simplifies our analysis.

1. There are many investors, each with an endowment (wealth) that is small compared to the total endowment of all investors. Investors are price-takers, in that they act as though security prices are unaffected by their own trades. This is the usual perfect competition assumption of microeconomics.

2. All investors plan for one identical holding period. This behavior is myopic (shortsighted) in that it ignores everything that might happen after the end of the single-period horizon. Myopic behavior is, in general, suboptimal.

3. Investments are limited to a universe of publicly traded financial assets, such as stocks and bonds, and to risk-free borrowing or lending arrangements. This assumption rules out investment in nontraded assets such as education (human capital), private enterprises, and governmentally funded assets such as town halls and international airports. It is assumed also that investors may borrow or lend any amount at a fixed, risk-free rate.

4. Investors pay no taxes on returns and no transaction costs (commissions and service charges) on trades in securities. In reality, of course, we know that investors are in different tax brackets and that this may govern the type of assets in which they invest. For example, tax implications may differ depending on whether the income is from interest, dividends, or capital gains. Furthermore, actual trading is costly, and commissions and fees depend on the size of the trade and the good standing of the individual investor.

5. All investors are rational mean-variance optimizers, meaning that they all use the Markowitz portfolio selection model.

6. All investors analyze securities in the same way and share the same economic view of the world. The result is identical estimates of the probability distribution of future cash flows from investing in the available securities; that is, for any set of security prices, they all derive the same input list to feed into the Markowitz model. Given a set of security prices and the risk-free interest rate, all investors use the same expected returns and covariance matrix of security returns to generate the efficient frontier and the unique optimal risky portfolio. This assumption is often referred to as homogeneous expectations or beliefs.

These assumptions represent the "if" of our "what if" analysis. Obviously, they ignore many real-world complexities. With these assumptions, however, we can gain some powerful insights into the nature of equilibrium in security markets.

We can summarize the equilibrium that will prevail in this hypothetical world of securities and investors briefly. The rest of the chapter explains and elaborates on these implications.

1. All investors will choose to hold a portfolio of risky assets in proportions that duplicate representation of the assets in the market portfolio (M), which includes all traded assets. For simplicity, we generally refer to all risky assets as stocks. The proportion of each stock in the market portfolio equals the market value of the

CHAPTER 9 The Capital Asset Pricing Model 265

stock (price per share multiplied by the number of shares outstanding) divided by the total market value of all stocks.

2. Not only will the market portfolio be on the efficient frontier, but it also will be the tangency portfolio to the optimal capital allocation line (CAL) derived by each and every investor. As a result, the capital market line (CML), the line from the risk-free rate through the market portfolio, M, is also the best attainable capital allocation line. All investors hold M as their optimal risky portfolio, differing only in the amount invested in it versus in the risk-free asset.

3. The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the representative investor. Mathematically,

where vM is the variance of the market portfolio and A is the average degree of risk aversion across investors.4 Note that because M is the optimal portfolio, which is efficiently diversified across all stocks, vM is the systematic risk of this universe.

4. The risk premium on individual assets will be proportional to the risk premium on the market portfolio, M, and the beta coefficient of the security relative to the market portfolio. Beta measures the extent to which returns on the stock and the market move together. Formally, beta is defined as

„ = Cov(ri, M vu and the risk premium on individual securities is

We will elaborate on these results and their implications shortly.

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