Chap012 Edited

An Alternative View of Risk and Return: The Arbitrage Pricing Theory

Chapter 12

Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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Key Concepts and Skills Discuss the relative importance of systematic

and unsystematic risk in determining a portfolio’s return

Compare and contrast the CAPM and Arbitrage Pricing Theory

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Chapter Outline12.1 Introduction12.2 Systematic Risk and Betas12.3 Portfolios and Factor Models12.4 Betas and Expected Returns12.5 The Capital Asset Pricing Model and the Arbitrage

Pricing Theory12.6 Empirical Approaches to Asset Pricing

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APT Arbitrage pricing theory (APT) is a well-known method of estimating the price

of an asset. The theory assumes an asset's return is dependent on various macroeconomic, market and security-specific factors.

The general idea behind APT is that two things can explain the expected return on a financial asset: 1) macroeconomic/security-specific influences and 2) the asset's sensitivity to those influences

There are an infinite number of security-specific influences for any given security including inflation, production measures, investor confidence, exchange rates, market indices or changes in interest rates, RND, GNP, Rival Product e.t.c. It is up to the analyst to decide which influences are relevant to the asset being analyzed.

APT may be more customizable than CAPM, but it is also more difficult to apply because determining which factors influence a stock or portfolio takes a considerable amount of research. It can be virtually impossible to detect every influential factor much less determine how sensitive the security is to a particular factor. But getting "close enough" is often good enough

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Bahria Enterprises Any Stock traded in Financial Markets has two parts. A) Normal or Expected return(info available to SH & mkt undstding influence in

next month), B) Uncertain or Risky Return (info that is to be reveled with in a month, list of

such info is endless) News abt Bahria Reraserch Sudden drop in interest Discovery about Rivals product News that sales figure are higher then expected Un expected retirement of Founder and president

R= R+U where R is actual total return, R is expected, U is unexpected part of return

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Expected Part of info and unexpected part of info e.g. GNP diff b/w actual and forecast

Announcement = Expected part + surprise (innovation)

So the expected part is there part of info the mkt uses to form the expectation R and the surprise is the news that influences the unanticipated return on stock U.

Un anticipated part is the true risk of any investment. Because if we got what is expected then there'll be no uncertainty

U is further categorized as Systematic Risk and Unsystematic risk.

S.Risk = is any risk affect large no. of assets, each to a greater or lesser degree.

For e.g GNP, Inflation, interest rates

U.Srisk or idiosyncratic risk = Risk that specifically affects a single asset or small group of assets. For e.g Rivals info, Retirement of CEO, RnD

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U = m+ϵ where m is systematic or market risk, ϵ company unique risk

ϵ is specific to company , is unrelated to specific risk of most other companiesBahria Entp specific risk is unrelated to Xerox stock. If Bahria's stock go up or down because of its new

discovery by RnD probably unrelated to Xerox so it mean both the stocks are uncorrelated with each other i.e.

Corr (ϵb, ϵx)

But companies are influenced by same systematic risk, individual companies systematic risks and therefore total return as well.

Lecture Tip: It is easy to see the effect of unexpected news on stock prices and returns. Consider the following two cases: (1) On November 17, 2004 it was announced that K-Mart would acquire Sears in an $11 billion deal. Sears’ stock price jumped from a closing price of $45.20 on November 16 to a closing price of $52.99 (a 7.79% increase) and K-Mart’s stock price jumped from $101.22 on November 16 to a closing price of $109.00 on November 17 (a 7.69% increase). Both stocks traded even higher during the day. Why the jump in price? Unexpected news, of course. (2) On November 18, 2004, Williams-Sonoma cut its sales and earnings estimates for the fourth quarter of 2004 and its share price dropped by 6%. There are plenty of other examples where unexpected news causes a change in price and expected returns.

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Risk: Systematic and Unsystematic

Systematic Risk: m

Nonsystematic Risk:

n

2

Total risk

We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk:

risk icunsystemat theis

risk systematic theis

where

becomes

ε

m

εmRR

URR

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Systematic Risk and Betas For example, suppose we have identified three

systematic risks: inflation, GNP growth, and the interest rate change

Our model is:

risk icunsystemat theis

change rateinterest isr

beta GNP theis

betainflation theis

ε

β

β

β

εβrFrFβFβRR

εmRR

GNP

I

GNPGNPII

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Systematic Risk and Betas: Example

Suppose we have made the following estimates:1. I = 2

2. GNP = 1

3. S = -1.8 Finally, the firm was able to attract a “superstar” CEO, and this unanticipated

development contributes 1% to the return.

εFrβFβFβRR rGNPGNPII

%5ε

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Arbitrage Pricing Theory

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.

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Total Risk Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure

of total risk. For well-diversified portfolios, unsystematic

risk is very small. Consequently, the total risk for a diversified

portfolio is essentially equivalent to the systematic risk.

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12.2 Systematic Risk and Betas The beta coefficient, , tells us the response of the stock’s

return to a systematic risk. In the CAPM, measures the responsiveness of a security’s

return to a specific risk factor, the return on the market portfolio.

)(

)(2

,

M

Mii R

RRCov

• We shall now consider other types of systematic risk.

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We must decide what surprises took place in the systematic factors.

If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:

FI = Surprise in the inflation rate = actual – expected

= 8% – 3% = 5%

%150.050.130.2 SGNPI FFFRR

%150.050.1%530.2 SGNP FFRR

ϵ

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If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:

FGNP = Surprise in the rate of GNP growth

= actual – expected = 1% – 4% = – 3%

%150.050.1%530.2 SGNP FFRR

%150.0%)3(50.1%530.2 SFRR

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If it were the case that the dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then:

FS = Surprise in the exchange rate

= actual – expected = 0% – 10% = – 10%

%150.0%)3(50.1%530.2 SFRR

%1%)10(50.0%)3(50.1%530.2 RR

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Finally, if it were the case that the expected return on the stock was 8%, then:

%1%)10(50.0%)3(50.1%530.2 RR

%12

%1%)10(50.0%)3(50.1%530.2%8

R

R

%8R

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12.3 Portfolios and Factor Models Now let us consider what happens to portfolios of stocks when each

of the stocks follows a one-factor model. We will create portfolios from a list of N stocks and will capture the

systematic risk with a 1-factor model. The ith stock in the list has return:

iiii εFβRR

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Relationship Between the Return on the Common Factor & Excess Return

Excess return

The return on the factor F

i

iiii εFβRR

If we assume that there is no unsystematic risk, then i = 0.

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Excess return


If we assume that there is no unsystematic risk, then i = 0.

FβRR iii

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Excess return


Different securities will have different betas.

0.1Bβ

50.0Cβ

5.1Aβ

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Portfolios and Diversification We know that the portfolio return is the weighted

average of the returns on the individual assets in the portfolio:

NNiiP RXRXRXRXR 2211

)(

)()( 22221111

NNNN

P

εFβRX

εFβRXεFβRXR

NNNNNN

P

εXFβXRX

εXFβXRXεXFβXRXR

222222111111

iiii εFβRR

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Portfolios and DiversificationThe return on any portfolio is determined by three sets of parameters:

In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

NNP RXRXRXR 2211

1. The weighted average of expected returns.

FβXβXβX NN )( 2211

2. The weighted average of the betas times the factor.

NN εXεXεX 2211

3. The weighted average of the unsystematic risks.

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Portfolios and DiversificationSo the return on a diversified portfolio is determined by two sets of parameters:

1. The weighted average of expected returns.

2. The weighted average of the betas times the factor F.

FβXβXβX

RXRXRXR

NN

NNP

)( 2211

2211

In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.

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12.4 Betas and Expected Returns

The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.

FβXβXRXRXR NNNNP )( 1111

FβRR PPP

NNP RXRXR 11

that Recall

NNP βXβXβ 11

and

PR Pβ

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Relationship Between & Expected Return

If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.

FβRR PPP

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Relationship Between & Expected Return

Exp

ecte

d re

turn

FR

A B

C

D

SML

)( FPF RRβRR

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12.5 The Capital Asset Pricing Model and the Arbitrage Pricing Theory 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.

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12.6 Empirical Approaches to Asset Pricing Both the CAPM and APT are risk-based models. Empirical methods are based less on theory and

more on looking for some regularities in the historical record.

Be aware that correlation does not imply causality. Related to empirical methods is the practice of

classifying portfolios by style, e.g., Value portfolio Growth portfolio

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Quick Quiz Differentiate systematic risk from unsystematic risk. Which type is essentially eliminated with well diversified portfolios? Define arbitrage. Explain how the CAPM be considered a special case of Arbitrage Pricing Theory?

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