Tiểu luận môn định giá doanh arbitrage pricing theory and multifactor models of risk and return

Chapter 10

Arbitrage Pricing Theory
and
Multifactor Models of Risk and Return
GVHD: TS. Trần Thị Hải Lý

Nhóm 5
Lâm Bá Du
Huỳnh Thái Huy
Phan Tuyết Trinh
Trần Thị Ngọc Hạnh
Tô Thị Phương Thảo
Nguyễn Hoàng Minh Huy


OUTLINE

Multifactor Models: An Overview

Arbitrage Pricing Theory

The APT, the CAPM, and the Index Model

A Multifactor APT

The Fama-French (FF) Three-Factor Model


10.1 Multifactor Models: An Overview

 SECURITY RISK INDEX MODEL:
Total risk = Systematic + firm-specific risk

 SINGLE – INDEX MODEL


10.1 Multifactor Models: An Overview

SINGLE-FACTOR MODEL

Ri = E(Ri) + βiF + ei (1)
If the macro factor (F) = 0 in any particular period (i.e., no macro surprises), then

the

excess

security

return on

the

its previously expected value

the effect of firm-specific events

E(Ri)

ei

Ri nonsystematic components of returns, the e , are assumed to be uncorrelated across
The
i
stocks and with the factor F.


10.1 Multifactor Models: An Overview

SINGLE-FACTOR MODEL

Ri = E(Ri) + βiF + ei (1)
F: the deviation of the common factor from its expected value.
βi: the sensitivity of firm i to that factor.
ei: the firm-specific disturbance.
The actual excess return on firm i will equal its initially expected value plus a (zero
expected value) random amount attributable to unanticipated economywide events, plus
another (zero expected value) random amount attributable to firm-specific events


10.1 Multifactor Models: An Overview
SINGLE-FACTOR MODEL

Ex:
Ri = E(Ri) + βiF + ei (1)
Suppose F is taken to be news about the state of the business cycle.

•
•

Consensus is GDP will increase by 4% this year. stock’s β value is 1.2.
If GDP increases by only 3% => F = -1%, then (1):
Ri = E(Ri) -1.2% + ei

F and ei determines the total departure of the stock’s return from its E(Ri).


10.1 Multifactor Models: An Overview

•
•

Systematic risk is not confined to a single factor.

SINGLE-FACTOR MODEL

Systematic risk is representated explicitly => different stocks to exhibit different
sensitivities to its various components.

multifactor models can provide better descriptions of security
returns


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS

Suppose:

•
•

two-factor model

macroeconomic sources of risk are measured by unanticipated growth in GDP and
unexpected changes in interest rates IR
The return on any stock will respond both to sources of macro risk and to its own
firm-specific influences. Then:
Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model
two-factor model

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)

•
•

Both macro factors have zero expectation

•
•

An increase in interest rates is bad news for most firms

βiGDP and βiIR measure the sensitivity of share returns to that factor
loadings or factor betas.
ei reflects firmspecific influences.

βiIR < 0 .

factor


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)

•
•
•

electric-power utility firm’s stock: βeGDP low and βeIR high.
airline firm’s stock: βeGDP high and βeIR low.

Economy will expand suggestion
both GDP and Interest
rates are expected increase.
“macro news” are the bad news for the utility but good ones for
the airline


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)
Suppose the result of Northeast Airlines estimation by using multifactor
models is

•
•
•

R = .133 + 1.2(GDP) - .3(IR) + e
E(R) for Northeast is 13.3%
With every percentage point increase in GDP beyond current
expectations, the return on Northeast shares increases on average by
1.2%,
With every unanticipated percentage point that interest rates
increases, Northeast’s shares fall on average by .3%.


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model

•
•

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)
where E ( R) comes from? What determines a security’s expected
excess rate of return.
This is where we need a theoretical model of equilibrium security
returns
arbitrage pricing theory can help determine the

expected value, E (R), in (1) and (2)
.


10.2 Arbitrage Pricing Theory (APT)
•
•

Developed by Stephen Ross (1976)
Basic

idea:

Like

the

CAPM,

the APT predicts a security market line linking expected returns to
risk.

•

Arbitrage: Creation of riskless profits made possible by relative
mispricing among securities.


10.2 Arbitrage Pricing Theory
(APT)

Arbitrage
Pricing Theory (APT)

Ross’s APT relies on three key propositions:

(1)
(2)
(3)

security returns can be described by a factor model;
There are sufficient securities to diversify away idiosyncratic risk;
Well-functioning security markets do not allow for the persistence of arbitrage
opportunities.


10.2 Arbitrage Pricing Theory (APT)
Single- Factor APT Model

••

We begin with a simple version of Ross’s model, which assumes that only one systematic

 

factor affects security returns.
(10.4)
(10.5)


10.2 Arbitrage Pricing TheoryArbitrage

(APT)Pricing Theory

Arbitrage:

•

An arbitrage opportunity arises when an investor can earn riskless profits without making
a net investment.

•

A trivial example of an arbitrage opportunity would arise if shares of a stock sold for
different prices on two different exchanges.


10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory

Arbitrage:

195$

193$

gain 2$

NASDAQ

NYSE

10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory
•
•

The Law of One Price states that
if two assets are equivalent in all economically relevant respects, then they should have
the same market price.

•
•

The Law of One Price is enforced by arbitrageurs:
If they observe a violation of the law, they will engage in arbitrage activity simultaneously
buying the asset where it is cheap and selling where it is expensive. In the process, they
will bid up the price where it is low and force it down where it is high until the arbitrage
opportunity is eliminated.


10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory
They will engage in arbitrage activity
simultaneously buying the asset
where it is cheap and selling where
it is expensive. In the process, they
will bid up the price where it is low
and force it down where it is high
until the arbitrage opportunity is
eliminated

10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory

A dominance argument holds that when an equilibrium price relationship is violated, many
investors will make limited portfolio changes, depending on their degree of risk aversion.
Aggregation of these limited portfolio changes is required to create a large volume of buying
and selling, which in turn restores equilibrium price.


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model
Consider the risk of a portfolio of stocks in a single-factor market. We first show that if a
portfolio
  is well diversified, its firm-specific or nonfactor risk becomes negligible, so that only
factor (or systematic) risk remains.

•

The excess return on an n -stock portfolio with weights ,
(10.3)
;
(is uncorrelated with F)


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model

•

We can divide the variance of this portfolio into systematic and nonsystematic sources:

 

Where:

•
•

is the variance of the factor F
is the nonsystematic risk of the portfolio, with,


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model

•

If the portfolio were equally weighted, =1/ n, then the nonsystematic variance would be

 


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model
••

Because the expected value of for any well-diversified portfolio is zero, and its variance

 

also is effectively zero, we can conclude that any realized value of will be virtually zero.

•

Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio, for all practical
purposes:


10.2 Arbitrage Pricing Theory (APT)
Single- Factor APT Model

•

The excess return on the portfolio A is therefore