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### CHAPTER 9

### CHAPTER 10 Returns

The Capital Asset Pricing Model

It is the equilibrium model that underlies all modern financial theory

Derived using principles of diversification with simplified assumptions

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Capital Asset Pricing Model (CAPM)Bahattin Buyuksahin, JHU, Investment

Individual investors are price takers financial theory

Single-period investment horizon

Investments are limited to traded financial assets

There are homogeneous expectations

Assumptions: InvestorsBahattin Buyuksahin, JHU, Investment

Information is costless and available to all financial theoryinvestors

No taxes and transaction costs

Risk-free rate available to all

Investors are rational mean-variance optimizers

Assumptions: AssetsBahattin Buyuksahin, JHU, Investment

All investors will hold the same portfolio for risky assets – market portfolio, which contains all securities and the proportion of each security is its market value as a percentage of total market value

held by all investors

includes all traded assets

suppose not: then price… -> included

is on the efficient frontier

asset weights: for each $ in risky assets, how much is in IBM?

for stock i: market cap of stock i / market cap of all stocks

Resulting Equilibrium ConditionsBahattin Buyuksahin, JHU, Investment

Risk premium on the market depends on the average risk aversion of all market participants

Risk premium on an individual security is a function of its covariance with the market

Resulting Equilibrium Conditions ContinuedBahattin Buyuksahin, JHU, Investment

Figure 9.1 The Efficient Frontier and the Capital Market Line

Bahattin Buyuksahin, JHU, Investment

Market Risk Premium Line

Bahattin Buyuksahin, JHU, Investment

- The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:

The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Return and Risk For Individual SecuritiesBahattin Buyuksahin, JHU, Investment

Using GE Text Example the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

Covariance of GE return with the market portfolio:

Therefore, the reward-to-risk ratio for investments in GE would be:

Using GE Text Example Continued the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

Reward-to-risk ratio for investment in market portfolio:

Reward-to-risk ratios of GE and the market portfolio:

And the risk premium for GE:

Expected Return-Beta Relationship the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

CAPM holds for the overall portfolio because:

This also holds for the market portfolio:

Figure 9.2 The Security Market Line the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

Figure 9.3 The SML and a Positive-Alpha Stock the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

The Index Model and Realized Returns the individual security’s contribution to the risk of the market portfolio

Bahattin Buyuksahin, JHU, Investment

To move from expected to realized returns—use the index model in excess return form:

The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship

Figure 9.4 Estimates of Individual Mutual Fund Alphas, 1972-1991

Bahattin Buyuksahin, JHU, Investment

The CAPM and Reality 1972-1991

Is the condition of zero alphas for all stocks as implied by the CAPM met

Not perfect but one of the best available

Is the CAPM testable

Proxies must be used for the market portfolio

CAPM is still considered the best available description of security pricing and is widely accepted

Bahattin Buyuksahin, JHU, Investment

Econometrics and the Expected Return-Beta Relationship 1972-1991

Bahattin Buyuksahin, JHU, Investment

- It is important to consider the econometric technique used for the model estimated
- Statistical bias is easily introduced
- Miller and Scholes paper demonstrated how econometric problems could lead one to reject the CAPM even if it were perfectly valid

Extensions of the CAPM 1972-1991

Zero-Beta Model

Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks

Consideration of labor income and non-traded assets

Merton’s Multiperiod Model and hedge portfolios

Incorporation of the effects of changes in the real rate of interest and inflation

Bahattin Buyuksahin, JHU, Investment

Extensions of the CAPM Continued 1972-1991

Bahattin Buyuksahin, JHU, Investment

- A consumption-based CAPM
- Models by Rubinstein, Lucas, and Breeden
- Investor must allocate current wealth between today’s consumption and investment for the future

- Models by Rubinstein, Lucas, and Breeden

Liquidity and the CAPM 1972-1991

Liquidity

Illiquidity Premium

Research supports a premium for illiquidity.

Amihud and Mendelson

Acharya and Pedersen

Bahattin Buyuksahin, JHU, Investment

Figure 9.5 The Relationship Between Illiquidity and Average Returns

Bahattin Buyuksahin, JHU, Investment

Three Elements of Liquidity Returns

Bahattin Buyuksahin, JHU, Investment

Sensitivity of security’s illiquidity to market illiquidity:

Sensitivity of stock’s return to market illiquidity:

Sensitivity of the security illiquidity to the market rate of return:

CAPM: Examples of Practical ReturnsProblems 1

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 2

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 3

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 4

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 5

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 6

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 7

Bahattin Buyuksahin, JHU Investments

CAPM: Examples of Practical ReturnsProblems 8

Bahattin Buyuksahin, JHU Investments

Index model Returnsvs. CAPM

Bahattin Buyuksahin, JHU Investments

- Risk
- CAPM (theoretical, unobservable portfolio)
- Index model (observable, “proxy” portfolio)

Index model Returnsvs. CAPM 2

Bahattin Buyuksahin, JHU Investments

- Beta Relationship
- CAPM (no expected excess return for any security)
- Index model (average realized alpha is 0)
- Fig 10.3

Market Model Returns Equivalence

Bahattin Buyuksahin, JHU Investments

- Idea
- use realized excess returns

- CAPM + Market model = Index model

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Single Factor Model Returns

Returns on a security come from two sources

Common macro-economic factor

Firm specific events

Possible common macro-economic factors

Gross Domestic Product Growth

Interest Rates

Bahattin Buyuksahin, JHU, Investment

Single Factor Model Equation Returns

ri= Return for security I

= Factor sensitivity or factor loading or factor beta

F = Surprise in macro-economic factor

(F could be positive, negative or zero)

ei = Firm specific events

Bahattin Buyuksahin, JHU, Investment

Multifactor ReturnsModels 1 Solution

Bahattin Buyuksahin, JHU Investments

- Necessity
- CAPM
- not practical

- CAPM
- Index model
- practical
- unique factor is unsatisfactory
- example: Table 10.2 (very small R2)

- multiple factors

Multi-factor Models 2 Returns

Bahattin Buyuksahin, JHU Investments

- Factors in practice
- business cycles factors
- examples (Chen Roll Ross)
- industrial production % change
- expected inflation % change
- unanticipated inflation % change
- LT corporate over LT gvt. bonds
- LT gvt. bonds over T-bills

- interpretation
- residual variance = firm specific risk

- examples (Chen Roll Ross)

- business cycles factors

Multi-factor Models 3 Returns

Bahattin Buyuksahin, JHU Investments

- Factors in practice
- firm characteristics (Fama and French)
- firm size
- difference in return
- between firms with low vs. high equity market value
- proxy for business cycle sensitivity?

- market to book
- difference in return
- between firms with low vs. high BTM ratio
- proxy for bankruptcy risk?

- firm size

- firm characteristics (Fama and French)

Multifactor ReturnsModels 4

Use more than one factor in addition to market return

Examples include gross domestic product, expected inflation, interest rates etc.

Estimate a beta or factor loading for each factor using multiple regression.

Bahattin Buyuksahin, JHU, Investment

Multifactor Model Equation Returns

ri= E(ri) + GDPGDP + IRIR + ei

ri= Return for security I

GDP= Factor sensitivity for GDP

IR= Factor sensitivity for Interest Rate

ei= Firm specific events

Bahattin Buyuksahin, JHU, Investment

Multifactor SML Models Returns

E(r) = rf + GDPRPGDP + IRRPIR

GDP = Factor sensitivity for GDP

RPGDP = Risk premium for GDP

IR = Factor sensitivity for Interest Rate

RPIR= Risk premium for Interest Rate

Bahattin Buyuksahin, JHU, Investment

Arbitrage Pricing ReturnsTheory (APT) APT vs. CAPM APT vs. Index models

Bahattin Buyuksahin, JHU Investments

- Nature of arbitrage
- APT
- well-diversified portfolios
- individual assets

- single factor
- multi-factor

Arbitrage Pricing Theory Returns

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit

Since no investment is required, an investor can create large positions to secure large levels of profit

In efficient markets, profitable arbitrage opportunities will quickly disappear

Bahattin Buyuksahin, JHU, Investment

APT & Well-Diversified Portfolios Returns

rP = E (rP) + bPF + eP

F = some factor

For a well-diversified portfolio:

eP approaches zero

Similar to CAPM,

Bahattin Buyuksahin, JHU, Investment

Figure 10.1 Returns as a Function of the Systematic Factor Returns

Bahattin Buyuksahin, JHU, Investment

Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

Bahattin Buyuksahin, JHU, Investment

Figure 10.3 An Arbitrage Opportunity An Arbitrage Opportunity

Bahattin Buyuksahin, JHU, Investment

Figure 10.4 The Security Market Line An Arbitrage Opportunity

Bahattin Buyuksahin, JHU, Investment

APT applies to well diversified portfolios and not necessarily to individual stocks

With APT it is possible for some individual stocks to be mispriced - not lie on the SML

APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio

APT can be extended to multifactor models

APT and CAPM ComparedBahattin Buyuksahin, JHU, Investment

Multifactor APT necessarily to individual stocks

Use of more than a single factor

Requires formation of factor portfolios

What factors?

Factors that are important to performance of the general economy

Fama-French Three Factor Model

Bahattin Buyuksahin, JHU, Investment

Two-Factor Model necessarily to individual stocks

Bahattin Buyuksahin, JHU, Investment

- The multifactor APR is similar to the one-factor case
- But need to think in terms of a factor portfolio
- Well-diversified
- Beta of 1 for one factor
- Beta of 0 for any other

- But need to think in terms of a factor portfolio

Example of the Multifactor Approach necessarily to individual stocks

Bahattin Buyuksahin, JHU, Investment

- Work of Chen, Roll, and Ross
- Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy

Another Example: necessarily to individual stocksFama-French Three-Factor Model

Bahattin Buyuksahin, JHU, Investment

- The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums
Where:

- SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks
- HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio

The Multifactor CAPM and the APM necessarily to individual stocks

Bahattin Buyuksahin, JHU, Investment

A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge

The APT is largely silent on where to look for priced sources of risk

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