Lecture 12

Arbitrage Pricing Theory

Pure Arbitrage A pure (or risk-free) arbitrage

opportunity exists when an investor can construct a zero-investment portfolio that yields a sure profit.

Zero-investment means that the investor does not have to use any of his or her own money.

Pure Arbitrage

One obvious case is when a violation of the law of one price occurs.

Example: The exchange rate is $1.50/£ in New York and $1.48/£ in London.

Arbitrage Pricing Theory

The APT is based on the premise that equilibrium market prices ought to be rational in the sense that they rule out risk-free arbitrage opportunities.

Arbitrage Pricing Theory

The APT assumes that:

1. Security returns are a function of one or more macroeconomic factors.

2. All securities can be sold short and the proceeds can be used to purchase other securities.

Single-Factor APT

The return on security i is

ri = E(ri) + iF + ei.

E(ri) is the expected return. F is the factor. i measures the sensitivity of ri to F.

ei is the firm specific return.

E(ei) = 0 and E(F) = 0.

Well Diversified Portfolios

rP = E(rP) + PF + eP.

P = wii

eP = wiei ’ 0

2(eP) = wi2 2(ei) ’ 0

P2 = P

2F2 + 2(eP) ’ P

2F2

P ’ PF

Single-Factor APT

Diversified Portfolio

F F

Security i

r rP i

Single-Factor APT

Two well diversified portfolios with the same beta must have the same expected return.

rp

Factor Realization

A

B

Single-Factor APT

The expected return on a well diversified portfolio is a linear function of the portfolio’s beta.

E(rP ) = rf + [RP]P

RP is the risk premium.

rf is the risk-free rate.

Single-Factor APTExpected Return

5%

10%

15%

20%

0.5 1.0 1.5 Beta

A

B

C

D

Single-Factor APT Let P be a well diversified portfolio.

E(rP ) = rf + [RP]P

RP is the risk premium = E*- rf

E* is the expected return on any well diversified portfolio with * = 1.0.

rf is the risk-free rate or return on a zero beta portfolio.

Single-Factor APT

1.0 P

E[r ]P

E*

rf

RP = E - r* f

*

Single-Factor APT

Risk-free arbitrage applies only to well diversified portfolios.

However, an investor can increase the expected return on her portfolio without increasing systematic risk if individual securities violate the relationship

ri = E(ri) + [RP]i.

Single-Factor APT Consider the following portfolio which

is part of a well diversified portfolio.

Amount Security Invested E(ri) i

A $20,000 8% 0.6B $40,000 10% 1.2 C $40,000 13% 1.6

E(rP) = .2x8+.4x10+.4x13 = 10.8%

P = .2x0.6+.4x1.2+.4x1.6 = 1.24

Single-Factor APT

Sell B and purchase $16,000 of A and $24,000 of C.

Amount Security Invested E(ri) i

A $36,000 8% 0.6C $64,000 13% 1.6

E(rP) = .36x8 + .64x13 = 11.2%

P = .36x0.6 + .64x1.6 = 1.24

Multi-Factor APT

The return on security i is ri = E(ri) + 1iF1+ ... + kiFk+ ei.

E(ri) is the expected return.

Fj is factor j, (j = 1,...,k).

ji measures the sensitivity of ri to factor j, (j = 1,...,k).

ei is the firm specific return.

Multi-Factor APT The return on a well diversified

portfolio is rP = E(rP) + 1PF1+ ... + kPFk.

E(rP) is the expected return. Fj is factor j, (j = 1,...,k). jP measures the sensitivity of rP to

factor j, (j = 1,...,k). eP = wiei 0.

Multi-Factor APT

Diversified Portfolio

F

rP

j

The relationship between the return on a well diversified portfolio and factor j, holding other factors equal to zero.

Multi-Factor APT

Arbitrage causes the expected return on a well diversified portfolio to be

E[rP] = rf + [RP1]1P +...+ [RPk]kP

jP is the sensitivity of portfolio P to

unexpected changes in factor j.

RPj is the risk premium on factor j.

Multi-Factor APT

1.0 j

E[r ]P

E j

rf

RP = E - rj j f

Relationship when all other betas are zero.

Multi-Factor APT

Risk-free arbitrage applies only to well diversified portfolios.

However, an investor can increase the expected return on her portfolio without increasing systematic risk if individual securities violate the relationship

E[ri] = rf + [RP1]1i +...+ [RPk]ki

Portfolio Strategy

Portfolio strategy involves choosing the optimal risk-return tradeoff.

The APT can be used to estimate

> security expected returns,

> security variances, and

> covariances between security returns.

Portfolio Strategy

The APT can also be used to refine the measure of risk.

Factor risks can affect investors differently.

The appropriate pattern of factor sensitivities depends upon a variety of considerations unique to the investor.

Portfolio Sensitivities

1.0

1.0 U

B

S

Z

Portfolios

S - Stocks

B – Bonds

U – Unit Beta

Z – Zero Beta

Inflation Beta

Productivity Beta

Identifying Factors

The biggest problem is identifying the factors that systematically affect security returns.

Theory is silent regarding the factors.

A variety of macroeconomic factors have been used.

Chen, Roll & Ross

Growth rate in industrial production.

Rate of inflation.

Expected rate of inflation.

Spread between long-term and short-term interest rates.

Spread between low-grade and high-grade bonds.

Berry, Burmeister & McElroy

Growth rate in aggregate sales.

Rate of return on the S&P500.

Rate of inflation.

Spread between long-term and short-term interest rates.

Spread between low-grade and high-grade bonds.

Salomon Brothers

Growth rate in GNP.

Rate of inflation.

Rate of interest.

Rate of change in oil prices.

Rate of growth in defense spending.