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1

Modified single-factor model

(BKM - 10)

R = E[R] + beta * F + e

replaces the market factor from the single-factor model with F

F = macroeconomic surprise = deviation of common factor from expected value of 0

*e's = residual, are uncorrelated with F and b/w stocks

2

Use for the modified single-factor model

(BKM - 10)

risk management (example: measure exposure and hedge specific risks)

3

Main idea of multifactor models

(BKM - 10)

model systematic risk as a combination of factors instead of a single factor (F)

4

Example multifactor model (GDP & IR)

(BKM - 10)

R = E[R] + beta(GDP) * GDP + beta(IR) * IR + e

measures 2 macroeconomic forces:
1. unanticipated GDP growth
2. interest rate changes

5

Factor betas in multifactor models

(BKM - 10)

beta coefficients that measure the sensitivity of the firm's excess returns to each macroeconomic factor

6

Advantage of multifactor models

(BKM - 10)

captures differences in sensitivity to each macroeconomic factor

7

Arbitrage

(BKM - 10)

ability to earn riskless profits with zero net investment

8

Assumptions of the Arbitrage Pricing Theory (APT) model (3)

(BKM - 10)

1. security returns can be described with a factor model

2. there are enough securities to diversify away non-systematic risk

3. arbitrage opportunities DNE in well-functioning markets

9

Law of one price

(BKM - 10)

two economically equivalent assets should have identical prices

10

General arbitrage strategy

(BKM - 10)

long (buy) the cheaper asset and short (sell) the more expensive one

11

Fundamental concept of capital market theory

(BKM - 10)

idea that market prices will move to rule out arbitrage opportunities

(e.g. prices of cheaper assets are bid up and prices of more expensive assets are forced down until equilibrium position is reached)

12

Difference between market equilibrium under CAPM and APT

(BKM - 10)

CAPM: many investors making small trades (aka risk-return dominance argument)

APT: few investors making extremely large trades

13

Excess returns (R(P)) under a well-diversified portfolio (APT)

(BKM - 10)

R = E[R] + beta * F

(b/c e goes to 0 with diversification)

14

Variance (sigma^2(P)) under a well-diversified portfolio (APT)

(BKM - 10)

sigma^2(P) = beta(P)^2 * sigma^2(F) + sigma^2(e(P))

first term = systematic risk, second term = firm-specific risk

15

Expected return for well-diversified portfolios with the same beta (APT)

(BKM - 10)

well-diversified portfolios with the same beta must produce the same expected return (o/w arbitrage)

16

SML, arbitrage, and well-diversified portfolios (APT)

(BKM - 10)

all well-diversified portfolios must live on the SML with slope F (o/w arbitrage!)

exploiting arbitrage opportunities forces all portfolios to the SML

17

SML for APT vs. CAPM (& rationale)

(BKM - 10)

SML of CAPM also applies to well-diversified portfolios b/c of the no arbitrage assumption of APT

18

Advantages of APT over CAPM (2)

(BKM - 10)

1. APT does not rely on an "impossible to observe" market portfolio - instead provides a mean-beta relationship that works for a well-diversified portfolio

2. APT does not require all investors to be mean-variance optimizers

19

Disadvantages of APT vs. CAPM (2)

(BKM - 10)

1. only assumes the mean-beta relationship holds for nearly all securities (vs. all securities)

2. relies on holding a well-diversified portfolio that eliminates firm-specific risk, but this is practically difficult

20

Reason it is difficult to completely eliminate all firm-specific risk

(BKM - 10)

empirical evidence that it is hard to hold a large enough number of securities to have a negligible amount of firm-specific risk

21

Similarities between APT and CAPM (2)

(BKM - 10)

Both:

1. produce the same SML

2. make the distinction between systematic and firm-specific risk

22

Implication of imperfectly diversified portfolios for APT

(BKM - 10)

means there is some amount of residual, or firm-specific risk, which means that arbitrage cannot exist (b/c of riskless profit condition)

23

Single-index model (aka Treynor-Black or TB model) vs. APT

(BKM - 10)

T-B model is more flexible because it responds to the practical limitations of diversification

24

Additional factors in multi-factor APT model (aka factor or tracking portfolios)

(BKM - 10)

each factor is a different source of systematic risk

each factor portfolio has a beta = 1 for one of the risk factors (perfect correlation) and beta = 0 (uncorrelated) for the other risk factor

25

Fama-French (FF) three factor model excess returns (R(i))

(BKM - 10)

R = alpha + beta(M) * R(M) + beta(SMB) * SMB + beta(HML) * HML + e

SMB = small minus big = difference in XS returns based on firm-size

HML = high minus low = difference in XS returns based on book-to-market ratio

26

Reason high book-to-market ratios and firm size are intuitive for predicting average stock returns

(BKM - 10)

high book-to-market ratio can signal distress

small firms are more sensitive to changes in business conditions

27

Ways to interpret the Fama-French (FF) model (2)

(BKM - 10)

1. model may signify a departure from rational equilibrium (b/c there is no theoretical reason for preferences due to firm size or book-to-market ratio)

2. firm characteristics such as firm size and book-to-market ratio are correlated with other risk factors (proxies for unidentified risk factors)

28

Risk-return dominance argument of CAPM

(BKM - 10)

if there is a mis-priced security, many investors will shift their portfolios to capitalize on the mis-pricing & restore the equilibrium price

29

Expected return, E[r], for the multifactor APT model

(BKM - 10)

E[r] = risk-free rate + sum(beta(j) * risk premium for factor j)

30

Arbitrage portfolio for multifactor APT model

(BKM - 10)

weight in each asset = factor beta for the portfolio
weight in risk-free asset = 1 - sum of weights

employ buy/sell strategy with the portfolios and the arbitrage profit = difference in expected returns