Arbitrage Pricing
Theorem
Chapter 7
1
Learning Objectives
 Develop
an understanding of multi-factor
pricing models
 Use
the APT to identify mispriced
securities
 Compare
and Contrast the APT and
CAPM
2
Single Index v Multi Index Models

Both break total risk into systematic and
unique
 Single
Index models assumes that systematic risk
comes from a single source
 Multi Index models allow for systematic risk to
come from several sources
3
Two Index Model

Risk comes from
1) Business Cycle (S&P 500)
2) Interest Rate Changes (Treasury bond portfolio)
 Two
Index Realized Returns
rit = αi + βiMrMt + βiTBrTBt + eit
 Two Factor SML
E(ri) = rf +βiM[E(rM)–rf ]+βiTB[E(rTB)–rf ]
4
Interpretation

The expected return on a security is the sum
of:
1.
The risk-free rate
2.
The sensitivity to the business cycle times the business
cycle risk premium (S&P 500)
3.
The sensitivity to interest rate risk times the interest
rate risk premium (Treasury Portfolio)
5
Two Factor Example

What is the expected return of a portfolio with
a beta of 1.5 on the market and a beta of 0.75
on oil? The market risk premium is 8%, the oil
risk premium is 7%, and the risk free rate is
3%.
Two Factor Example: Fun with Algebra
Risk Comes from: Market and Oil
 Risk-free rate = 6%
 The following are well diversified portfolios:

Portfolio

Market Beta
Oil Beta
Expected Return
A
1.5
2.0
31%
B
2.2
-0.2
27%
What are the expected returns for Market and
Oil?
Fama-French Three-Factor Model
Rit  r f   iM RMt   iSMB SMBt   iHML HMLt  eit


Market Factor
 Same as CAPM
SML: Small minus Big (Market Cap)
 Return
on the averages small firm minus the average large
firm

HML: High minus Low (Book to Market)
 Return
on the average value firm minus the average growth
firm

Argues these firm characteristics are correlated with
actual (but currently unknown) systematic risk factors
10-8
Example
What is a stock’s expected return if its betas
are: SML: 0.5; HML: 3.0; Mkt: 2.0
 The expected returns are

 16%
small firms, 8% large firms
 14% value firms, 9% growth firms
 7% market, and the risk free rate is 3%
9
Measuring Model Success

A model is successful if it can accurately
explain a stock’s return
 The
return actually earned equals what we would
expect given the actual movements in the market

Measure success with:
 Higher
Adjusted R-Square
 Lower residual standard deviation
 Smaller α’s
10
What Drives These Models?

Arbitrage ≡ Exploiting the mispricing of two (or
more) securities to earn risk free profit
 EX:
Imagine that Google stock is selling for $10 on
the NYSE and $12 on NASDAQ, how would you
make money?
 In truest form no investment is required so they can
be scaled up easily
How do risk averse investors feel about these?
 Technological advancements has made it
extremely difficult to find simple arbitrage
opportunities

11
Arbitrage Pricing Theory

Starts with the idea that arbitrage opportunities
cannot exist in an efficient market
 In
a well functioning economy no one should be
able to earn money for nothing

Uses the absence of arbitrage to derive the
risk-return relation
 Avoid
the CAPM assumptions
12
APT and CAPM
APT
CAPM
Assumes
a well-diversified

portfolio.
Residual still important
Arbitrage Opportunities
Equilibrium
is quickly
restored
Only
Uses

Model is based on an
inherently unobservable
“market” portfolio.
Rests on mean-variance
efficiency. The actions of
many small investors
restore CAPM equilibrium.
takes a few arbitragers
an observable, market
index
13
Arbitrage in a 1 Factor Economy

Portfolio (P): has a positive alpha
 We



need to buy P to earn alpha
However, P has systematic risk
How do we remove the systematic risk?
What if we had an investment that moved in the exact
opposite way of the market?
 We
short the market (M) (βM of 1), but how much?
 We short so that our total portfolio βT = 0



βT = wP * βp + wM * (-βM)
0 = 1 * βp + wM * (-1)
Use the risk free asset to ensure weights balance
 Just
used to make up cash difference, borrow/lend
14
Explanation Math
Steps to convert a well-diversified portfolio into an
arbitrage portfolio
*When alpha is negative, you would reverse the signs of each portfolio weight to
achieve a portfolio A with positive alpha and no net investment.
Arbitrage Portfolio: Generates risk free profits with 0 net
investment
-This is a money making machine
15
APT: 2 Factors

Now need two benchmark portfolios
 P1:
benchmark portfolio with a beta of 1 on factor 1
and a beta of 0 on factor 2
 P2: benchmark portfolio with a beta of 0 on factor 1
and a beta of 1 on factor 2

Then follow the same basic methodology as in
the single factor example
APT: 2 Factors Math
Constructing an arbitrage portfolio with two
systemic factors
Question




Consider a one-factor economy. All portfolios are
well diversified.
Portfolio
Expected Return
Beta
A
12%
1.2
F
6%
0.0
Suppose that another portfolio, E, is well
diversified with a beta of .6 and an expected
return of 8%.
Would an arbitrage opportunity exist?
If so, what would be the arbitrage strategy?
18
Actual v Expected Returns
E(rP)
rP
𝑟𝑝 = 𝐸 𝑟𝑝 + 𝛽𝑝 𝐹 + 𝑒𝑝






Actual return = Expected return + the effect of surprises
rp = Actual return earn on the portfolio
E(rp )= Expected return on the portfolio
βp= Portfolio Factor Betas
F = Surprise in macro-economic factors (+/-)
ep = Firm specific events
 For
a well diversified portfolio ep should be 0
10-19
Two Factor Example



Risk Comes from: Market and Oil
Risk-free rate = 6%
The follow is a well diversified portfolio:
Portfolios
A


Mkt Beta
1.5
E(Mkt)
10%
Oil Beta
0.75
E(Oil)
8%
What is this portfolio’s expected return?
If the Mkt earned 8% and Oil earned 13% what
did our portfolio actually return?