## The Capital Asset Pricing Model: Highlights

### Optimal Portfolio Allocation

• A risky investment portfolio (referred to here as the risky asset) can be characterized by its reward-to-variability ratio. This ratio is the slope of the capital allocation line (CAL), the line that goes from the risk-free asset through the risky asset. All combinations of the risky and risk-free assets lie on this line. Investors would prefer a steeper sloping CAL, because that means higher expected returns for any level of risk.
• An investor's preferred choice among the portfolios on the capital allocation line will depend on risk aversion. Risk-averse investors will weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their complete portfolios in the risky asset.
• The efficient frontier of risky assets is the graphical representation of the set of portfolios that maximizes portfolio expected return for a given level of portfolio standard deviation. Rational investors will choose a portfolio on the efficient frontier.
• If a risk-free asset is available and input data are identical, all investors will choose the same portfolio on the efficient frontier, the one that is tangent to the CAL. All investors with identical input data will hold the identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio selection.

### The Capital Asset Pricing Model

• Some, but not all, of the risk associated with a risky investment can be eliminated by diversification. The reason is that unsystematic risks, which are unique to individual assets, tend to wash out in a large portfolio, but systematic risks, which affect all of the assets in a portfolio to some extent, do not.
• Because unsystematic risk can be freely eliminated by diversification, the systematic risk principle states that the reward for bearing risk depends only on the level of systematic risk. The level of systematic risk in a particular asset, relative to average, is given by the beta of that asset.
• The reward-to-risk ratio for Asset i is the ratio of its risk premium, E(Ri) - Rf, to its beta, Bi:
[E(Ri) - Rf]/Bi
• In a well-functioning market, this ratio is the same for every asset. As a result, when asset expected returns are plotted against asset betas, all assets plot on the same straight line, called the security market line (SML).
• From the SML, the expected return on Asset i can be written:
E(Ri) = Rf +Bi[E(Rm) - Rf]
• This is the capital asset pricing model (CAPM). The expected return on a risky asset thus has three components. The first is the pure time value of money (Rf), the second is the market risk premium, [E(Rm) - Rf], and the third is the beta for that asset, Bi.
• The CAPM implies that the risk premium on any individual asset or portfolio is the product of the risk premium of the market portfolio and the asset's beta.
• The CAPM assumes investors are rational single-period planners who agree on a common input list from security analysis and seek mean-variance optimal portfolios.
• The CAPM assumes ideal security markets in the sense that: (a) markets are large, and investors are price takers, (b) there are no taxes or transaction costs, (c) all risky assets are publicly traded, and (d) any amount can be borrowed and lent at a fixed, risk-free rate. These assumptions mean that all investors will hold identical risky portfolios.
• The CAPM implies that, in equilibrium, the market portfolio is the unique mean-variance efficient tangency portfolio, which indicates that a passive strategy is efficient.
• The market portfolio is a value-weighted portfolio. Each security is held in a proportion equal to its market value divided by the total market value of all securities. The risk premium on the market portfolio is. proportional to its variance and to the risk aversion of the average investor.
• In a single-index security market, once an index is specified, any security beta can be estimated from a regression of the security's excess return on the index's excess return.
• This regression line is called the security characteristic line (SCL). The intercept of the SCL, called alpha, represents the average excess return on the security when the index excess return is zero. The CAPM implies that alphas should be zero.
• Estimates of beta from past data often are adjusted when used to assess required future returns.

### The Arbitrage Pricing Theory

• An arbitrage opportunity arises when the disparity between two or more security prices enables investors to construct a zero net investment portfolio that will yield a sure profit. Rational investors will want to take infinitely large positions in arbitrage portfolios regardless of their degree of risk aversion.
• The presence of arbitrage opportunities and the resulting volume of trades will create pressure on security prices that will persist until prices reach levels that preclude arbitrage. Only a few investors need to become aware of arbitrage opportunities to trigger this process because of the large volume of trades in which they will engage.
• When securities are priced so that there are no arbitrage opportunities, the market satisfies the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are important because we expect them to hold in real-world markets.
• Portfolios are called well diversified if they include a large number of securities in such proportions that the residual risk or diversifiable of the portfolio is negligible.
• In a single-factor security market, all well-diversified portfolios must satisfy the expected retum-beta relationship of the SML in order to satisfy the no-arbitrage condition.
• If all well-diversified portfolios satisfy the expected retum-beta relationship, then all but a small number of securities also must satisfy this relationship.
• The APT implies the same expected return-beta relationship as the CAPM yet does not require that all investors be mean-variance optimizers. The price of this generality is that the APT does not guarantee this relationship for all securities at all times.
• A multifactor APT generalizes the single-factor model to accommodate several sources of influence on a stock's expected return.
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