1. Arbitrage Pricing Theory
Lecture 12
Arbitrage Pricing Theory

2. 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.

3. 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.

4. 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.

5. 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.

6. Single-Factor APT The return on security i is ri = E(ri) + biF + ei.
E(ri) is the expected return.
F is the factor.
bi measures the sensitivity of ri to F.
ei is the firm specific return.
E(ei) = 0 and E(F) = 0.

7. Well Diversified Portfolios
rP = E(rP) + bPF + eP.
bP = Swibi
eP = Swiei ’ 0
s 2(eP) = Swi2 s 2(ei) ’ 0
sP2 = bP2sF2 + s 2(eP) ’ bP2sF2
sP ’ bPsF

8. Single-Factor APT r r F F Diversified Portfolio Security i i i i i i i

9. Single-Factor APT r
Two well diversified portfolios with the same beta must have the same expected return. p A B
Factor Realization

10. Single-Factor APT
The expected return on a well diversified portfolio is a linear function of the portfolio’s beta.
E(rP ) = rf + [RP]bP
RP is the risk premium.
rf is the risk-free rate.

11. Single-Factor APT C B A D Expected Return 20% i i 15% 10% i i 5% 0.5
1.0
1.5
Beta

12. Single-Factor APT Let P be a well diversified portfolio.
E(rP ) = rf + [RP]bP
RP is the risk premium = E*- rf
E* is the expected return on any well diversified portfolio with b* = 1.0.
rf is the risk-free rate or return on a zero beta portfolio.

13. Single-Factor APT
E[r ] P * E *
RP = E - r * f r f
1.0 b P

14. 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]bi.

15. 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% 0.6 B $40,000 10% C $40,000 13% 1.6
E(rP) = .2x8+.4x10+.4x13 = 10.8%
P = .2x0.6+.4x1.2+.4x1.6 = 1.24

16. 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% 0.6 C $64,000 13% 1.6
E(rP) = .36x x13 = 11.2%
P = .36x x1.6 = 1.24

17. Multi-Factor APT
The return on security i is ri = E(ri) + b1iF bkiFk+ ei.
E(ri) is the expected return.
Fj is factor j, (j = 1,...,k).
bji measures the sensitivity of ri to factor j, (j = 1,...,k).
ei is the firm specific return.

18. Multi-Factor APT
The return on a well diversified portfolio is rP = E(rP) + b1PF bkPFk.
E(rP) is the expected return.
Fj is factor j, (j = 1,...,k).
bjP measures the sensitivity of rP to factor j, (j = 1,...,k).
eP = Swiei g 0.

19. Multi-Factor APT
Diversified Portfolio
The relationship between the return on a well diversified portfolio and factor j, holding other factors equal to zero. r P i i i i i i F j

20. Multi-Factor APT
Arbitrage causes the expected return on a well diversified portfolio to be
E[rP] = rf + [RP1]b1P [RPk]bkP
bjP is the sensitivity of portfolio P to unexpected changes in factor j.
RPj is the risk premium on factor j.

21. Multi-Factor APT E[r ] E r b RP = E - rj
RP = E - r j j f r f
1.0 b j
Relationship when all other betas are zero.

22. 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]b1i [RPk]bki

23. 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.

24. 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.

25. Portfolio Sensitivities
Productivity Beta
Portfolios
S - Stocks
B – Bonds
U – Unit Beta
Z – Zero Beta h U
1.0 S h B h Z h
1.0
Inflation Beta

26. 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.

27. 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.

28. 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.

29. Salomon Brothers Growth rate in GNP. Rate of inflation.
Rate of interest.
Rate of change in oil prices.
Rate of growth in defense spending.