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

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 Ove

Trang 1

Arbitrage Pricing Theory

Tô Thị Phương Thảo Nguyễn Hoàng Minh Huy

Chapter 10

Trang 2

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

Trang 3

10.1 Multifactor Models: An Overview

 SECURITY RISK INDEX MODEL:

Total risk = Systematic + firm-specific risk

 SINGLE – INDEX MODEL

Trang 4

Ri = E(Ri) + βiF + ei (1)

If the macro factor (F) = 0 in any particular period (i.e., no macro surprises), then

The nonsystematic components of returns, the ei, are assumed to

be uncorrelated across stocks and with the factor F

the effect of specific events

firm-ei

Trang 5

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

SINGLE-FACTOR MODEL

Trang 6

Ex:

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

Trang 7

• Systematic risk is not confined to a single factor.

• Systematic risk is representated explicitly =>

different stocks to exhibit different sensitivities

to its various components.

SINGLE-FACTOR MODEL

multifactor models can provide better descriptions of security returns

Trang 8

Suppose:

• 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) + iGDPGDP + iIRIR +ei (2)

MULTIFACTOR MODELS

two-factor model

Trang 9

• Both macro factors have zero expectation

• iGDP and iIR measure the sensitivity of share returns

to that factor factor loadings or factor betas

• An increase in interest rates is bad news for most

firms iIR < 0

• ei reflects firmspecific influences

two-factor model

MULTIFACTOR MODELS

two-factor model

Trang 10

• electric-power utility firm’s stock: eGDP low and eIR

high.

• airline firm’s stock: eGDP high and eIR 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

MULTIFACTOR MODELS

two-factor model

Trang 11

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%

MULTIFACTOR MODELS

two-factor model

Trang 12

• 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)

MULTIFACTOR MODELS

two-factor model

.

Trang 13

• Developed by Stephen Ross (1976)

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)

Trang 14

Ross’s APT relies on three key propositions:

(1) security returns can be described by a factor

model;

(2) There are sufficient securities to diversify away

idiosyncratic risk;

(3) Well-functioning security markets do not allow for

the persistence of arbitrage opportunities

Arbitrage Pricing Theory (APT)

Trang 15

• We begin with a simple version of Ross’s model, which assumes that only one systematic factor affects security returns.

(10.4) (10.5)

•

Single- Factor APT Model

Trang 16

• 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

Trang 18

• 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.

Trang 19

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

Trang 20

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.

Trang 21

Consider the risk of a portfolio of stocks in a 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

single-The excess return on an n -stock portfolio with weights ,

(10.3)

; (is uncorrelated with F)

•

Trang 22

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,

•

Trang 23

If the portfolio were equally weighted, =1/ n, then the

nonsystematic variance would be

•

Trang 24

• Because the expected value of for any diversified portfolio is zero, and its variance also is effectively zero, we can conclude that any realized value of will be virtually zero

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

•

Trang 25

The excess return on the portfolio A is therefore

•

Trang 26

• In a single-factor world, all pairs of well-diversified portfolios are perfectly correlated

• Perfect correlation means that in a plot of expected return versus standard deviation, any two well-

diversified portfolios lie on a straight line We will see later that this common line is the CML

• Their risk is fully determined by the same systematic factor

Trang 27

• Consider a second welldiversified portfolio, Portfolio

Q, with

We can compute the standard deviations of P and Q,

as well as the covariance and correlation between them:

•

Trang 28

Diversification and Residual Risk in Practice

Trang 29

• Since neither M nor portfolio P have residual risk,

the only risk to the returns of the two portfolios is

systematic, derived from their betas on the

common factor (the beta of the index is 1.0)

• Construct a zero-beta portfolio, called Z, from P and

M by appropriately selecting weights và on each

Trang 30

(10.6)

; Its alpha is:

The risk premium on Z must be zero because the risk

of Z is zero If its risk premium were not zero, you

•

Executing Arbitrage

Trang 31

• Since the beta of Z is zero, Equation 10.5 implies that its

risk premium is just its alpha Using Equation 10.7, its alpha is , so (10.8)

• If and the risk premium of Z is positive, borrow and invest the proceeds in Z, you get a net return:

• Similarly if and the risk premium is negative; therefore,

sell Z short and invest the proceeds at the risk-free rate

•

Executing Arbitrage

Trang 32

• We’ve seen that arbitrage activity will quickly pin the risk premium of any zero-beta well-diversified portfolio to zero From Equation 10.5, this

means that for any well-diversified P,

(10.9)

• Equation 10.9 thus establishes that the SML of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage” requirement of the APT

•

The No-Arbitrage Equation of the APT

Trang 33

• Another demonstration that the APT results in the same SML as the CAPM is more graphical in nature.

• First we show why all well-diversified portfolios with the same beta must have the same expected return

• Figure 10.2 plots the returns on two such portfolios, A and B, both with betas of 1, but with differing expected

returns:

and

•

Trang 34

Trang 35

• If you sell short $1 million of B and buy $1 million of A,

a zero-net-investment strategy, you would have a riskless payoff of $20,000, as follows:

Your profit is risk-free because the factor risk cancels out across the long and short positions

(.10 + 1.0 X F) X $1 million From long position in A

-(.08 + 1.0 X F) X $1 million From short position in B

.02 X $1 million = $20,000 Net proceeds

Trang 36

• What about portfolios with different betas? Their risk premiums must be proportional to beta.

Trang 37

• We have a new portfolio, D, composed of half of portfolio

A and half of the risk-free asset.

• Portfolio D’s beta will be (0.5 X 0 + 0.5 X 1.0) = 0.5, and its

expected return will be (0.5 X 4 + 0.5 X 10) = 7%

• Now portfolio D has an equal beta but a greater expected return than portfolio C.

• To preclude arbitrage opportunities, the expected return

on all well-diversified portfolios must lie on the straight line from the risk-free asset in Figure 10.3

Trang 38

• Risk premiums are indeed proportional to portfolio betas

in Figure 10.3

• As in the simple CAPM, the risk premium is zero for and rises in direct proportion to

•

Trang 39

The APT, CAPM

It should be noted, however, that when we replace the unobserved market portfolio of the CAPM with an observed, broad index portfolio that may not be efficient, we no longer can be sure that the CAPM predicts risk premiums of all assets with no bias Neither model therefore is free of limitations Compar-ing the APT arbitrage strategy to maximization of the Sharpe ratio in the context of an index model may well be the more useful framework for analysis

10.3 The APT, CAPM, Index Model

Trang 40

APT, Index Model

In effect, the APT shows how to take advantage of security mispricing when diversi-fication opportunities are abundant When you lock in and scale up an arbitrage opportunity, you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio, but only if the arbitrage portfolio is truly risk-free!

10.3 The APT, CAPM, Index Model

10.3 The APT, CAPM, Index Model

Trang 41

 too simplistic !!!!!!!!!

 single-factor  multifactor

E() = E() = 0 Cov(,) = Cov(, ) = 0

•

𝑹𝒊= 𝑬 (𝑹 ¿ ¿ 𝒊)+𝜷𝒊 𝟏 𝑭𝟏+ 𝜷𝒊 𝟐 𝑭 𝟐+ 𝒆𝒊 ¿

10.4 A Multifactor APT

Trang 42

10.4 A Multifactor APT

Portfolio A: a well-diversified portfolio : β A1 = 0,5 ; β A2 = 0,75

Suppose: 2 factor portfolio, and risk-free rate: 4%

Portfolio 1 Porfolio 2

risk premium = 6% risk premium = 8%

Portfolio 1 Porfolio 2

risk premium = 6% risk premium = 8%

The risk premium attributable to risk factor 1: βA1 [E(r1) – r f] = 3% The risk premium attributable to risk factor 2: βA2 [E(r2) – r f] = 6% The total risk premium: 3% + 6% = 9% The total return on the portfolio: 4% + 9% = 13%

10.4 A Multifactor APT

Trang 43

Portfolio Q: βP1 in the first factor portfolio,

βP2 in the second factor portfolio,

Trang 45

+ +

- SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the

return on a portfolio of large stocks.

- HML = High Minus Low, i.e., the return of

aportfolio of stock with a high book-to-market ratio in excess of the return on a portfolio of

stocks with a low high book-to-market ratio.

10.5 The Fama-French (FF) three-factor model

Trang 46

Capture sensitive to risk factor in the macroeconomy

Small stocks

High ratio

of market value

book-to-THE FAMA-FRENCH (FF) THREE-FACTOR MODEL

Trang 47

• The Fama-French model use proxies for

extramarket sources of risk, is that none

of the factors in the proposed models can

be clearly identified as hedging a significant source of uncertainty.

• Fama and French have shown that size

predicted average returns in various time periods and in markets all over the world.

THE FAMA-FRENCH (FF) THREE-FACTOR MODEL

Trang 48

• Whether the FF model reflect a multi-index ICAPM based on extra-market hedging demands

or just represent yet-unexplained anomalies?

values

associated with average returns are correlated

•

THE FAMA-FRENCH (FF) THREE-FACTOR MODEL

Định dạng
Số trang	49
Dung lượng	1,49 MB