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Week 4

Capital Asset Pricing and Arbitrage Pricing Theory

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Capital Asset Pricing Model (CAPM)

CAPM: A model that relates the required rate of return for a security to its risk measured by beta.

CAPM predicts the relationship between the risk and equilibrium expected returns on risky assets.

CAPM is derived using principles of diversification with simplified assumptions.

CAPM is first proposed by Sharpe (1964), and Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

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CAPM: Assumptions Individual investors cannot affect prices by their

individual traders There are many investors, each with an endowment of

wealth that is small compared with total endowment of all investors.

Single-period investment horizon All investors plan for one identical holding period.

Investors from portfolios from a universe of publicly traded financial assets, and have access to unlimited risk-free borrowing or lending.

Investors pay neither taxes on returns nor transaction costs on trades in securities.

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CAPM: Assumptions (continued) All investors attempt to construct efficient frontier

portfolio. Investors are rational mean-variance optimizers

Information is costless and available to all investors Homogeneous expectations

All investors analyze securities in the same way and share same economic view of the world.

They all end with identical estimates of the probability distribution of future cash flows from investing in the available securities.

All investors use the same expected return, standard deviations, and correlations to generate efficient frontier and the unique optimal risky portfolio.

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Resulting Equilibrium Conditions All investors will choose to hold the same optimal

risky portfolio – Market portfolio. Market portfolio contains all securities in the market and the

proportion of each security is its market value as the percentage of total market value.

The market portfolio will be on the efficient frontier, and it will be the optimal risky portfolio. The CML, the line from risk-free rate through the market

portfolio, is the best attainable CAL. All investors will hold the market portfolio as their optimal

risk portfolio, differing only in the amount invested in it compared to investment in risk-free asset.

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The risk premium on the market portfolio will be proportional to the variance of the market portfolio and investor’s typical degree of risk aversion. E(rM) – rf = A*M

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The risk premium on individual securities will be proportional to the risk premium on market portfolio and to the beta coefficient of the security on the market portfolio. Rate of return on the market portfolio is the single factor of

the security market. The beta measures the extent to which returns on the stock

respond to the returns of the market portfolio.

Resulting Equilibrium Conditions (cont.)

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CAPM: Implications Why all investors would hold the market portfolio.

The assumed necessary conformity of investors make all investors hold the market portfolio.

Each investors uses the market portfolio for the optimal risky portfolio, the CAL in this case is called CML.

The price adjustment process guarantees that all stocks will be included in the optimal portfolio.

Suppose Stock A is not included in optimal portfolio. The demand is zero for Stock A, so Stock A will get cheaper,

and will look relatively more attractive. Ultimately, Stock A’s price will reach a price at which it is

desirable to be included in the optimal portfolio.

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E(r)E(r)

E(rE(rMM))

rrff

M (MarketM (MarketPortfolio)Portfolio)

CMLCML

mm

Capital Market Line

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CAPM: Implications The Passive Strategy is Efficient

The market portfolio proportions are optimal, and in the simple world of CAPM, all investors use precious resources in security analysis.

A passive investor who simply holds market portfolio benefits from the efficiency of that portfolio (CML).

Mutual Fund Theorem: All investors desire the same portfolio of risky assets and can be satisfied by a single mutual fund composed of that portfolio.

Logical inconsistency of the CAPM: If a passive strategy is costless and efficient, why would anyone follow an active strategy? But if no one does any security analysis, what brings about the efficiency of market portfolio.

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The Risk Premium on Market Portfolio In equilibrium, the risk premium on the market

portfolio [E(rM) – rf], must be just high enough to induce investors to hold the available supply of stocks. If the risk premium is too high compared to the average

degree of risk aversion, there will be excess demand for securities, and prices will rise.

If it is too low, investors will not hold enough stock to absorb the supply, and prices will fall.

In equilibrium risk premium of the market portfolio is proportional to both the risk of the market, as measured by the variance of its return, and to the degree of risk aversion of average investor.

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Expected Return on Individual Securities As unsystematic risk can be diversified away,

investors do not require a risk premium for bearing unsystematic risk.

Investors only need to be compensated for bearing systematic risk, which cannot be diversified.

The contribution of a single security to the risk of a large diversified portfolio depends only on the systematic risk of the security measured by its beta.

Individual security’s risk premium is proportional to its beta, that is, a function of the contribution of individual security to the risk of market portfolio.

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Expected Return on Individual Securities The ratio of risk premium to beta should be the same

for all securities and portfolios.

[E(rM) – rf] / M = [E(ri) – rf] / i, where M = 1.0 The CAPM’s expected return-beta relationship

E(ri) = rf + i [E(rM) – rf], i = [COV(ri,rM)] / M2

Even if one does not hold the market portfolio, a well-diversified portfolio will be so highly corre-lated with the market that a stock’s beta relative to the market still will be a useful risk measure.

If the CAPM holds for any individual asset, it must hold for any combination of assets (portfolios).

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Security Market Line (SML) SML: Graphical representation of the expected

return-beta relationship of the CAPM – Slope is the market risk premium and intercept is risk-free rate.

SML vs. CML CML: Risk premiums of efficient portfolio as a function of

portfolio standard deviation. SML: Risk premium of individual asset as a function of the

contribution of the asset to the portfolio standard deviation measured by beta.

Whenever CAPM holds, all securities must lie on the SML in market equilibrium; Fairly priced assets plot exactly on the SML.

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E(r)E(r)

E(rE(rMM))

rrff

SMLSML

MM = 1.0 = 1.0

Security Market Line

Over-pricedOver-priced

Under-pricedUnder-priced

MM

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Sample Calculations for SML

E(rM) - rf = .08 rf = .03

x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6

E(ry) = .03 + .6(.08) = .078 or 7.8%

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E(r)E(r)

E(rE(rMM)=11%)=11%

rrff=3%=3%

SMLSML

MM = =

1.01.0

Graph of Sample Calculations

xx = =

1.251.25yy = =

0.60.6

E(rE(ryy)=7.8%)=7.8%

E(rE(rxx)=13%)=13%

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Application of the CAPM The CAPM provides a required rate of return on an

investment for a given risk. The SML provides a benchmark for evaluation of

investment performance. Use in the investment management industry. Suppose the SML is taken as a benchmark to assess the fair

expected return on a risky asset: An analyst calculate own expected return; If a stock is perceived as a good buy (underpriced), it will provide a positive alpha ().

The CAPM is useful in capital budgeting decisions. Managers can use the CAPM to obtain the cutoff IRR or

hurdle rate for the project.

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The CAPM and Index Models The CAPM has two limitations.

It relies on the theoretical market portfolio. It deals with expected as opposed to actual returns.

To implement the CAPM, we cast it in the form of an index model using actual market index, and use realized returns.

True market portfolio cannot be observable and would not be easily accessed by investors.

The CAPM’s reliance of the market portfolio should not faze us if we can verify that predictions of CAPM are sufficiently accurate when the index portfolio is substituted for the market.

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The CAPM and Index Models Single-index regression equation with realized

excess returns

ri – rf = i + i (rM – rf) + ei

Expressed in terms of expectation;

E(ri) – rf = i + i [E(rM)– rf] Comparing this relationship with the CAPM reveals

that the CAPM predicts that i = 0. Convert the CAPM predictions about unobserved

expectations of security returns relative to an unobserved market portfolio into a prediction about the intercept in a regression of observed variables: realized excess returns of a security relative to those of a specified index.

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The CAPM and Index Models If intercepts of regressions of returns on an index

differ significantly from zero, you cannot tell whether it is because you chose a bad index to proxy for the market or because the theory is not useful.

Index models are widely used to operationalize CAPM (Security Characteristic Line). In actuality, few instances of persistent, positive alpha values

have been identified, and future alphas are practically impossible to predict from past values.

Even if a single-index model representation is not fully consistent with the CAPM, the concept of systematic vs. diversifiable risk is still useful.

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Using Text Example Table 5:

Jan.Jan.Feb.Feb.....Dec.Dec.

MeanMeanStd. Dev.Std. Dev.

5.415.41-3.44-3.44

..

..2.432.43

-.60-.604.974.97

7.247.240.930.93

..

..3.903.90

1.751.753.323.32

ExcessExcessMarket ReturnMarket Return

ExcessExcessGM ReturnGM Return

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Estimated coefficientEstimated coefficientStd error of estimateStd error of estimate

Variance of residuals = 12.601Variance of residuals = 12.601Std dev of residuals = 3.550Std dev of residuals = 3.550R-square = 0.575R-square = 0.575

-2.590-2.590(1.547)(1.547)

1.13571.1357(0.309)(0.309)

(r(rGMGM - r - rff) =) = ii + + GMGM(r(rmm - r - rff))

Regression Results:

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Predicting Beta Estimate betas to forecast the rate of return based on

the past history. Betas exhibit a statistical property called “regression

toward the mean.” High (1) securities in one period tend to exhibit a lower

beta in the future, while low (1) securities exhibit a higher in future periods.

A simple way to account for this tendency of future betas is to use as your forecast of beta a weighted average of sample estimate with the value 1.0.

Suppose past data yield a beta estimate of 0.65. Adjusted beta = 2/3 0.65 + 1/3 1.0 = 0.77.

Some complicated techniques have been used, but not very successful to provide accurate beta estimate.

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The CAPM and The Real World In only limited ways, portfolio theory and CAPM

have become accepted tools in the practitioner community.

Many empirical studies argue that beta does not tell the whole story of risk.

Nevertheless, beta is not dead yet. Other study shows that when we use more inclusive (even

including human capital) proxy than S&P500 for market portfolio and allows for the fact that beta changes over time, the performance of beta is considerably improved.

The logic of the model still compelling and more sophisti-cated pricing models all rely on the key distinction between systematic vs. diversifiable risk.

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Arbitrage Pricing Theory APT is introduced by Ross (1976). Like the CAPM, APT predicts the relationship

between the risk and equilibrium expected returns on risky assets.

However, the APT relies on no-arbitrage condition rather than the market portfolio.

To explain the APT, we begin with the concept of Arbitrage, which is the exploitation of relative mispricing among two or more securities to earn risk-free profits

A riskless arbitrage opportunity arises if an investor can construct a zero investment portfolio with a sure profit.

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Arbitrage Pricing Theory Since no investment is required, an investor can

create large positions to secure large levels of profit. One has to be able to sell short at least one asset and use

the proceeds to purchase one or more assets.

An obvious case of an arbitrage opportunity arises in the violation of the law of one price: When an asset is trading at different prices in two markets, sell short in the high priced market and buys it in the low priced market.

In efficient markets, profitable arbitrage opportunity will quickly disappear – Program trading and index arbitrage.

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Arbitrage Example from Text Current Expected Standard

Stock Price$ Return% Dev. %

A 10 25.0 29.58

B 10 20.0 33.91

C 10 32.5 48.15

D 10 22.5 8.58

Mean Stan.Dev. Correlation

Portfolio P(A,B,C) 25.83 6.40 0.94

D 22.25 8.58

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Arbitrage Action and Returns

Expected Expected ReturnReturn

Standard Standard DeviationDeviation

* P* P

* D* D

Short 3 shares of D and Buy 1 of A, B & C to form P Short 3 shares of D and Buy 1 of A, B & C to form P (Arbitrage Portfolio: Zero-investment Portfolio).(Arbitrage Portfolio: Zero-investment Portfolio).

You earn a higher rate on the investment than you You earn a higher rate on the investment than you pay on the short sale.pay on the short sale.

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Arbitrage Pricing Theory The critical property of an arbitrage portfolio is than

any investor, regardless of risk aversion or wealth, will want to take an infinite position in it so that profits will be driven to an infinite level.

Because those large positions will force some prices up and/or some down until the opportunity is vanished, we can derive restrictions on security prices that satisfy that no arbitrage opportunities are left in the market place.

There is an important distinction between arbitrage and CAPM risk-versus-return dominance arguments in support of equilibrium price relationships.

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Arbitrage Pricing Theory Risk-vs-Return Dominance argument in CAPM

It holds that when an equilibrium relationship is violated, many investors will make limited portfolio changes, depending on wealth and risk-aversion.

Aggregation of limited portfolio changes over many investors will restore the equilibrium price.

Arbitrage argument in APT When arbitrage opportunities exist, each investor wants to

take as large a position as possible. It will not take many investors to restore equilibrium. Implications derived from no-arbitrage argument is

stronger, because they do not depend on a large, well-educated investors.

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Well-diversified Portfolio and APT APT: A theory of risk-return relationship derived

from no arbitrage conditions in large capital market. APT posits a single-factor security market.

Ri = i + iRM + e, where Ri = (ri – rf)

Suppose we construct a well-diversified portfolio with a given beta – No firm-specific risk.

Rp = p + pRM

If the portfolio beta is zero, Rp = p, implying a riskless excess return over risk-free rate.

This implies that p should be zero, or else an immediate arbitrage opportunity opens up (borrow at risk free rate and buy zero-beta portfolio).

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Well-diversified Portfolio and APT Portfolio V (v and v) and Portfolio U (u and u).

To form zero-beta portfolio (V+U); buy Portfolio V and sell Portfolio U with proportions of wv = [-u/(v-u)], and wu = [v/(v-u)]

Riskless portfolio, but non-zero excess return unless v and u equal zero.

Beta(V+U) = v[-u/(v-u)] + u[v/(v-u)] = 0

R(V+U) = v[-u/(v-u)] + u[v/(v-u)] 0 The alpha of any well-diversified portfolio must be

zero, even if the beta is not zero. (rp – rf) = p(rM– rf); E(rp) = rf + p[E(rM) – rf]: Same

as CAPM without any assumption about either investor preferences or access to market portfolio.

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APT and CAPM Compared APT applies only to well diversified portfolios and

not necessarily to individual stocks in equilibrium. However, APT relationship must almost surely hold

true for individual securities. If APT relationship is violated by many individual assets, it

would be virtually impossible for all well-diversified portfolios to satisfy the relationship.

APT serves many of same functions as the CAPM. 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.

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Multifactor Generalization of APT

Use a multifactor version of APT to accommodate multiple sources of risk.

Generalize the single-factor model to a two-factor model: Ri = i + i1RM1 + i2RM2 + e.

Two-Factor APT

E(rp) =

rf + p1 [E(rM1) – rf] + p2 [E(rM2) – rf]