Comm 324 --- W. Suo Slide 1. Comm 324 --- W. Suo Slide 2 Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio

Comm 324 --- W. SuoSlide 1Slide 1

Arbitrage Arbitrage Pricing ModelsPricing Models


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


Example: Returns with Equal Probability

Stocks State 1 State 2 State 3 State 4

A -20% 20% 40% 60%

B 0% 70% 30% -20%

C 90% -20% -10% 70%

D 15% 23% 15% 36%


Arbitrage Example

Stock Current Price

($)

Expected Return

(%)

Standard Deviation (%)

A 10 25.0 29.58

B 10 20.0 33.91

C 10 32.5 48.15

D 10 22.5 8.58


Arbitrage Action and Returns

Action: Short 3 shares of D and buy 1 of A, B & C to form portfolio PReturns: You earn a higher rate on the investment than you pay on the short sale

E(r)

s

P D

25.8322.25

6.40 8.58


Payoffs (short 300 shares of D. buy 100 shares of A. B & C each)

Stock $ Inv State 1 State 2 State 3 State 4

A 1,000 -200 200 400 600

B 1,000 0 700 300 -200

C 1,000 900 -200 -100 700

D -3,000 -450 -690 -450 -1,080

Portfolio $0 $250 $10 $150 $20


Example - continued

What will happen in the market?


APT & Well-Diversified Portfolios

F is some macroeconomic factor For a well-diversified portfolio eP approaches zero

The result is similar to CAPM

( )P P P Pr E r F e


Arbitrage Pricing Theory Line

E(r

i) =

Exp

ecte

d R

etur

n

Slope = = risk premium

Risk class of assets O and U

Overpriced assetO

U Underpriced asset

0

RFR

Factor beta


F

E(r)(%)

Portfolio

F

E(r)(%)

Individual Security

Portfolio & Individual Security Comparison


E(r)%

Beta for F

10

7

6

Risk Free = 4

AD

C

.5 1.0

Disequilibrium Example


Disequilibrium Example

Short Portfolio C Use funds to construct an equivalent risk higher

return Portfolio D D is comprised of A & Risk-Free Asset

Arbitrage profit of 1%


M

Beta (Market Index)

Risk Free

1.0

[E(rM) - rf]

Market Risk Premium

E(r)

APT with Market Index Portfolio


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 Compared


A Two-Factor APT Model

The single factor APT can be extended to include more independent risk factors that work together to determine market prices

APT is more flexible than SML However, APT offers no clues as to what factors

are relevant• Research must be done to determine best explanatory

factor

, 1, 2,1 2 .i t t ti i i i ta b b er F F


Three Highly Diversified Portfolios

Three risk-averse investors form portfolios B, C and D (each contains N assets) with two risk factors

These are arbitrage portfolios, requiring no cash investment When N is large, unsystematic residual risk is diversified away

PortfolioExpected

ReturnRisk Factor

bp1

Risk Factor bp2

B 16% 1.0 0.7

C 14% 0.6 1.0

D 11% 0.5 0.4


Three Highly Diversified Portfolios

The general form of the APT model with two factors is:

The specific APT model for the three portfolios on the previous slide is:

0 1 1 2 2( )iE r b b

1 2( ) 5.629 7.777 3.703i i iE r b b


The Arbitrage Portfolio

Consider a mispriced asset

Portfolio E(rp) Bi1 Bi2 Definition

S 13.66% 0.7 0.7 3B + 3C + 3D

U 15.66% 0.7 0.7 Underpriced

Portfolios S and U have the same risk but different expected returns Portfolio U is underpriced

• Smart investors would buy portfolio U


The Arbitrage Portfolio

It is possible to set up a perfect hedge with portfolios S and U to create a riskless profit opportunity

PortfolioInitial Cash

flow

Ending

Cash flow Bi1 Bi2

S = short position +$100 -$113.66 -0.7 -0.7

U = underpriced (long position) -$100 $115.66 0.7 0.7

A = arbitrage (perfectly hedged) 0 +$2.00 0 0


The k-Dimensional APT Hyperplane

A more elaborate model with k risk factors is:

Salomon Smith Barney uses a multi-factor arbitrage pricing model including factors such as: The market’s trend or drift Economic growth Credit quality Interest rates Inflation shock Small-cap premiums

, 1, 2, ,1 2 .i t t t k ti i i ik i ta b b b er F F F

Documents

Comm 324 --- W. Suo Slide 1. Comm 324 --- W. Suo Slide 2 Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio