Chapter 9 Asset Pricing

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Chapter 9 Outline9.1 The Efficient Frontier

•The role of risk•The efficient frontier with risk-free borrowing and lending

•Risk-free investing

•Risk-free borrowing

9.2 The Capital Asset

Pricing Model

•The market portfolio and the capital market line

•Risk-adjusted performance and the Sharpe ratio

9.3 The CAPM and Market

Risk•The security market line

•Estimating long-term discount rates

9.4 Alternative

Asset Pricing Models

•Fama and French Model

•Arbitrage Pricing Theory

9.5 Market Efficiency

•Efficient market hypothesis

•Forms of market efficiency

•Efficient market assumptions

•Anomalies and wacky indicators

9.6 Behavioral Finance and

Financial Management

•Behavioral finance

•Cognitive bias

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The efficient frontier, which is the set of assets that offer the best opportunities in terms of risk and return.

If someone turns down a fair gamble, he or she is likely risk averse. A risk-averse person prefers the risk-free situation—that is, not

gambling on a risky situation where there is an equal probability of winning or losing the same amount of money.

We assume that investors are risk averse when choosing among different investments: Investors are not willing to undertake fair gambles.

9.1 The Efficient Frontier

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The Role of Risk The corollary to turning down a fair gamble

is that the risk-averse person requires a risk premium to enter into a risky situation.

Generally, investor behavior is consistent with risk aversion and the existence of risk premiums to induce individuals to bear risk.

We represent this risk aversion by the required risk premium, with higher risk premiums indicating greater risk aversion.

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We can also reverse the situation and put the individual in a risky situation and ask how much he or she is willing to pay to get out of the risky situation.

The payment to get out of a risky situation as an insurance premium.

The Role of Risk

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The Efficient Frontier with Risk-Free Borrowing and Lending The existence of insurance markets indicates how risk aversion

creates a demand to remove risk, whereas the existence of capital markets indicates how risk aversion generates risk premiums required to induce people to bear risk.

We assume the following:1. Investors are risk averse, that is, they require a risk premium to

bear risk 2. The more risk averse an investor, the higher the risk premium

the investor requires3. We can represent risk by the standard deviation of the return

on the portfolio

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The Efficient Frontier with Risk-Free Borrowing and LendingThese assumptions lead to the result that investors choose only portfolios on the efficient frontier. The efficient frontier represents all portfolios of risky assets that have better risk-return situations than other portfolios of risky assets, which we represent below:

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A very risk-averse individual may choose portfolio A, and a less risk-averse individual may choose portfolio B. Why do we know that the investor choosing A is more risk averse than the investor choosing B?

Risk aversion, cont.

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The expected return on a portfolio is always a weighted average of the expected returns on the individual assets, so we can estimate the expected return on this portfolio as:

where E(rP) is the expected return on the portfolio, rf is the expected return on the risk free asset, E(rX) is the expected return on the risky portfolio X, and w is the proportion of the portfolio invested in the risk portfolio.

Risk-Free Investing

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Example: Suppose an investor begins with a 100% investment in the risk-free asset; that is, w = 0.

w = proportion of the portfolio in the risky asset

As the investor shifts the portfolio, adding more of the risky asset, w increases, so the investor increases the expected return on the portfolio, E(rP), at the cost of reducing the investment in the risk-free asset.

Risk-Free Investing

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Risk-free & risk investing

C

D

rf

Standard deviation

Expe

cted

retu

rn

lending

borrowing

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If the return on the risk-free asset is 3% and the return on the risky portfolio X is 11%, a portfolio consisting only of the risk-free asset has an expected return of 3%, whereas a portfolio consisting of only the risky portfolio X has an expected return of 11%.

However, a portfolio consisting of equal parts of the risk-free and risky portfolio X has an expected return of:

Risk-free + risky investing

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We estimate the standard deviation on the portfolio in as:

where σ2X and σ2

Y are the standard deviations of the returns on assets X and Y, respectively, wX and wY are the proportions invested in X and Y, respectively, and ρXY is the correlation of returns between X and Y.

If we replace portfolio Y with the risk-free security and (1 – w) for wY, we arrive at the following:

Calculating Risk

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We know exactly what we are going to get from the risk-free asset, so the standard deviation of its return is zero; that is, σrf = 0 = 0. Because the return does not vary, the correlation between the return on the risk-free asset and that on the risky portfolio X is also zero.

Therefore, the standard deviation with σrf = 0 and ρXrf = 0 reduces to:

Taking the square root of the final terms leaves us with:

Calculating Risk

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In other words, portfolio risk increases in direct proportion to the amount invested in the risky asset, w.

Therefore, the higher the proportion of the portfolio allocated to the risky asset, the higher the portfolio risk.

If the standard deviation of the returns on risky portfolio X is 20%, a portfolio comprising equal parts of the risk-free asset and the risky portfolio X will have a standard deviation of:

Calculating Risk

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Consider an investor who invests $10,000 in a portfolio of risky assets, portfolio V.Portfolio V has an expected return of 12% and a standard deviation of returns of 25%. If, in addition, the investor borrows $5,000 at the risk-free rate of 3% and invests it in portfolio V, the total invested in portfolio V increases to $15,000. What is the expected return on this investment?

The expected return calculation requires us to first calculate the weights: w = $15,000 ÷ $10,000 = 1.5

Risk-Free Borrowing

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In other words, there is a short position, borrowing at the risk-free rate. The expected return on the portfolio is therefore:

The standard deviation of the investor’s investment, considering both the investment in the risky portfolio V and the borrowing, is the weight invested in the risky portfolio (150%, in this example), multiplied by the standard deviation of the returns on the risky asset (the 25%):

Risk-Free Borrowing

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The Capital Asset Pricing Model (CAPM) is a model that describes expected returns in terms of the risk-free rate of interest and a premium for bearing market risk.

9.2 The Capital Asset Pricing Model

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All investors have identical expectations about expected returns, standard deviations, and correlation coefficients for all securities.

All investors have the same one-period time horizon. All investors can borrow or lend money at the risk-free

rate of return (rf). There are no transaction costs.

CAPM assumptions

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There are no personal income taxes so that investors are indifferent between capital gains and dividends.

There are many investors, and no single investor can affect the price of a stock through his or her buying and selling decisions. Therefore, investors are price takers.

Capital markets are in equilibrium.

Assumptions, cont.

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1. The “optimal” risky portfolio is the one that is tangent to the efficient frontier on a line that is drawn from the risk-free rate. This portfolio is the same for all investors.

2. This optimal risky portfolio is the market portfolio that contains all risky securities.

The value of this portfolio is the aggregate of the market values of all the individual assets in the portfolio. Therefore, the weights of these assets in the market portfolio are their proportionate weight in its total value.

Implications of the CAPM

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The relationship between expected return and standard deviation implied by the capital asset pricing model:

The Capital Market Line (CML)

The Capital Market Line

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The slope of the capital market line is the incremental expected return divided by the incremental risk.

The Capital Market Line

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This trade-off of risk and return is the market price of risk for efficient portfolios.

This indicates the additional expected return that the market demands for an increase in a portfolio’s risk. Adding the risk-free rate, rf , the CML is:

Market price of risk

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When the ideas underlying the CML are used in this way, it leads to the Sharpe ratio, the ratio of ex post excess returns to risks:

We also refer to the Sharpe ratio as the reward-to-variability ratio or the reward-to-risk ratio because this ratio is a comparison of the return, in excess of the risk-free rate, to risk.

Risk-Adjusted Performance and the Sharpe Ratio

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Suppose a portfolio has an expected return of 5% and a standard deviation of returns of 8%. If the risk-free rate of interest is 3%, the Sharpe ratio is:

We have derived the Sharpe ratio in terms of “expected” (i.e., ex ante) returns in our discussion. However, in practice, it is a commonly used measure of ex post (i.e., realized) returns, where we replace expected returns with historic realized returns.

Example: Sharpe Ratio

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The CML provides a method of estimating the required return on equity assets relative to their risk, but it applies only to efficient portfolios and not to individual securities.

In financial management, we are usually concerned with the risk associated with individual firms and the required return for investing in them.

9.3 The CAPM and Market Risk

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The figure shows the average risk of an individual security, where the curve gets very close to the vertical axis, that is, a one-stock portfolio.

The CAPM and Market Risk

Portfolio Risk and Diversification

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A result of the capital asset pricing model is that investors should be compensated for market risk, as measured by beta. The security market line (SML):

where E(ri) is the expected return on the asset (or portfolio) i. The term E(rM) - rf is the expected market risk premium and is also a function of market conditions.

The Security Market Line

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Company Ticker Beta

IBM IBM 0.72

Microsoft MSFT 0.80

Apple Computer AAPL 0.58

FB 2.10

Johnson & Johnson JNJ 0.49

General Electric GE 1.22

Examples of Beta

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Suppose the risk-free rate is 3% and the expected return on the market is 9%.

If an asset’s beta is 1.2, the expected return on this asset is 9.2%:

E(ri) = 0.03 + [(0.09-0.03) × 1.2] = 9.2% The market risk premium in this case is 6%, and the risk

premium for this asset is 7.2%.

The SML is the most widely used contribution of the CAPM.

Example: SML

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Generally, we use a risk-free security with a maturity that mimics investors’ typical holding period.

Example: 10-year U.S. Treasury As of 10/29/13, 2.5%

What risk-free rate?

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Using the SML to Estimate Long-Term Discount Rates

Estimated Market Risk Premiums

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Fama and French Model – a pricing model that uses three factors to relate expected returns to risk:

A market factor The market value of a firm’s common equity The ratio of a firm’s book equity value to its

market value of equity)

9.4 Alternative Pricing Models

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The Arbitrage Pricing Theory (APT) is a pricing model that uses multiple factors to relate expected returns to risk by assuming that asset returns are linearly related to a set of indexes, which are proxy risk factors that influence asset returns.

Arbitrage is the process of taking advantage of different pricing for the same item in different markets.

Alternative Asset Pricing Models

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An efficient market is one in which the prices of all assets accurately reflect all relevant and available information about the assets. This definition implies that asset prices in efficient capital markets are “correct.”

The efficient market hypothesis (EMH) formalized the concept of an efficient market into a theory that markets are efficient, and therefore, in its strictest sense, it implies that prices accurately reflect all information at any point in time.

9.5 Market Efficiency

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Weak form of market efficiency Asset prices fully reflect all market data, which refers

to all past price and volume trading information. Historical trading data is already reflected in current

prices and should be of no value in predicting future price changes. It is not possible to generate an abnormal profit, a

profit in excess of that expected for the asset’s level of risk on a consistent basis.

Efficient Market Hypothesis

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Semi-strong form of market efficiency Asset prices reflect all publicly known and

available information. For stock markets this includes information about

earnings, dividends, corporate investments, management changes, and so on. This also includes market data, which are publicly available.

This version of the EMH encompasses the weak form. In other words, if a market is semi-strong efficient, then it must also be weak form efficient.

Efficient Market Hypothesis

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Strong form of market efficiency Asset prices fully reflect all information, which

includes both public and private information. Public information is any information that is

readily available to investors, whereas private information is any information not available, which we generally think of as inside information.

Efficient Market Hypothesis

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A large number of rational, profit-maximizing investors exist, who actively participate in the market by analyzing, valuing, and trading assets. The markets are assumed to be competitive, which means that no one investor can significantly affect the price of an asset.

Information is costless and widely available to market participants at the same time.

Efficient Market Assumptions

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Information arrives randomly, and therefore announcements are not related to one another.

Investors react quickly and fully to the new information, which is reflected in stock prices.

Assumptions, cont.

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Researchers have documented numerous anomalies in security pricing over the past few decades.

An anomaly is a mispricing of an asset such that the pricing of the asset is not consistent with efficient markets. In other words, the pricing of an asset is not associated with relevant information about the asset that is known to all market participants.

Anomalies

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The size effect - small capitalization stocks outperform large capitalization stocks, which indicates that the market is not weak-form efficient.

The January effect - returns in January are higher than the rest of the year. This indicates that the market is not weak-form efficient.

The earnings surprise anomaly - when earnings are released by a company and there is an unexpected portion, the market not only reacts immediately, but also continues to react for some days following the earnings announcement. This is evidence against semi-strong market efficiency.

Examples of anomalies

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The value anomaly - value stocks (that is, low market/book ratios) outperform other securities, which would indicate that the market is not semi-strong-form efficient.

Anomalies, cont.

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Super Bowl winners - if a former member of the American Football League wins, the stock market will experience a down year.

Butter production in Bangladesh - the return on the S&P Index will be twice that of the change in butter production in Bangladesh.

Hemlines of women’s dresses - there is a positive relationship between movement in the market and hemlines: if hemlines get shorter, prices go up, and vice versa.

Sports Illustrated Swimsuit Model - the market does well when the swimsuit model chosen for the cover is of U.S. nationality.

Wacky Indicators

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Behavioral finance - the study of how human behavior affects economic decision making. While we devise theories that depend on rational, risk-averse

investors, there is always the possibility that financial managers do not act this way and, in fact, make decisions that are biased.

Cognitive bias - a mistake in decision making that results from one’s own preferences and beliefs. These biases result in decisions that are not consistent with pure

scientific or statistical reasoning. The many sources of bias include overconfidence, loss aversion, the

disposition effect, representativeness, anchoring, framing, and mental accounting.

9.6 Behavioral Finance and Financial Management

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Overconfidence - there are many ways that this becomes apparent when we observe behavior. For example, a decision maker who has a good understanding of the role

of diversification may nevertheless choose an investment portfolio without adequate diversification because of overconfidence in his or her ability to select investments.

Loss aversion - the willingness to avoid losses that is disproportionate to the willingness to seek similar-sized gains.

Disposition effect - the behavior of individuals in which they avoid realizing any paper losses, but tend to realize paper gains. In other words, people tend to hang onto losers longer than they should. Closely related to loss aversion.

Examples of Cognitive Bias

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Representativeness - the tendency for people to judge something, such as an investment, by the degree to which this something resembles something else in the person’s experience. In other words, an individual will overweight most recent events, giving

less weight to long-term averages. Anchoring - the tendency to use inappropriate or irrelevant factors in

decision making. Closely related to representativeness

Framing - the manner in which something is presented, which may influence how someone feels about this something.

Mental accounting - the process of investors separating money into different accounts in their thought process, which affects decision making.

Examples of Cognitive Bias

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The efficient frontier is the set of portfolios that provides the best return for a given level of risk (or, similarly, the lowest risk for a given level of return). If we combine borrowing or lending at the risk-free rate of interest, the set of investment opportunities improves.

The capital market line depicts the required return for efficient portfolios based on their standard deviations, whereas the security market line depicts the relation between the required return and market risk, as measured by beta.

An efficient market is a market in which asset prices reflect available information quickly and accurately. The EMH is commonly broken down into three forms that are based on the extent to which prices reflect different types of available information.

The weak-form EMH states that security prices reflect all market data; the semi-strong form states that prices reflect all publicly known and available information; and the strong form states that prices reflect all public and private information.

Chapter Summary

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An implication of the capital asset pricing model for financial decision making is that return on an investment is consistent with the amount of market risk of the investment, not on the investment’s total or stand-alone risk. The implications of market efficiency for financial decision making

include: The inability to time security offerings The expectation of returns commensurate with market risk

Behavioral finance is an area of study that examines how human behaviors can enter into decision making. Cognitive biases can enter into decision making, resulting in decisions

that are not in the best interest of owners.

Chapter Summary