Portfolio Management #53 - Introduction to Multi-factor Models

Question 1

arbitrage pricing theory (APT)

Answer 1

LOS 53.a

APT - describes the equilibrium relationship between expected returns for well-diversified portfolios and their multiple sources of systematic risk.

E(R_p) = R_F + ß_P1(A₁) + … + ß_Pk(A_k​), where

A (lamda) = risk premiums

ß = portfolio factor betas (or “loadings”)

APT assumes there are no market imperfections preventing investors from exploiting arbitrage i.e. extreme long/short positions are permitted

Question 2

types of multifactor models

Answer 2

LOS 53.d

1. Macroeconomic Factor Models

• surprises in macro variables that explain differences in stock returns
• e.g. interest rates, inflation, business cycle, credit spreads, etc.

2. Fundamental Factor Models

• attributes of stocks that are important in explaining cross-sectional differences in returns
• e.g. B/M, market cap, P/E, leverage, etc.

3. Statistical Factor Models

• Principal components models - factors are portfolios of securities that best reproduce historic return variances
• Factor analysis models - factors are portfolios of securities that best reproduce historic return covariances
• Issue: attaching economic meaning to statistical factors is difficult

Question 3

macroeconomic factor model

Answer 3

LOS 53.d

Two-factor example:

R_i = E(R_i) + b_i1F_GDP + b_i2F_INT + e_i

E(R_i) – ex-ante (forecasted) return (no surprises)

b_i1 – sensitivity to GDP surprises
F_GDP – GDP surprise actual - consensus predicted

b_i2 – sensitivity to INT surprises
F_INT – interest rate surprise actual - consensus expected

e_i – the part of the return that cannot be explained by the model; it represents unsystematic risk related to firm-specific events e.g. a strike, warehouse fire, etc.

Question 4

fundamental factor model

Answer 4

LOS 53.d

fundamental factor model:

R_i = a_i + b_i1F_P/E + b_i2F_SIZE + e_i, where

R<sub>i</sub> = return for stock i
F<sub>P/E</sub> = return assoc. with the P/E factor
F<sub>SIZE</sub> = return assoc with the SIZE (mkt cap) factor
a<sub>i</sub> = intercept
b<sub>i</sub><sub>1</sub> = standardized sensitivity of stock to P/E factor
b<sub>i2</sub> = standardized sensitivity of stock to SIZE factor

• factors are returns, not surprises
• intercept != expected return
• factor sensitivities are standardized e.g.:

b_i1 = [(P/E)_i - avg(P/E)] / σ_P/E, where

(P/E)<sub>i</sub> = P/E for _stock i_
avg(P/E) = average P/E calculated across _all stocks_
σ<sub>P/E</sub> = std deviation of P/E ratios across _all stocks_

Question 5

macro vs. fundamental models

Answer 5

LOS 53.d

Macro Factor Fundamental
Model Factor Model

regression time series of cross-sectional
surprises asset returns

factor sens. (ß) regression based standardized from
attribute data

factor rtns. (F) surprises in computed from
macro variables multiple regression

intercept expected return undefined

Question 6

active return, active risk, information ratio

Answer 6

LOS 53.e

• active return - differences in returns between a managed portfolio and its benchmark:

Active Return = R_P - R_B

• active risk (aka tracking error, tracking risk) - the standard deviation of the active return:

Active Risk = TE = s(R_P - R_B)

• information ratio - the active portfolio’s average active return per unit risk:

IR = [avg(R_P) - avg(R_B)] / s(R_P - R_B)

Question 7

multifactor model use for return attribution

Answer 7

LOS 53.f

Fundamental Factor Model
(macro & stat factor models not commonly used for return attribution)

decompose active return into factor return and security selection:

active return = factor return + security selection return

active return = sum_(i=1,k)[(ß_pk - ß_bk) * A_k] + sec. select. rtn.

• use: decompose sources of an asset manager’s return relative to a benchmark
• model uses easily understood factors
• can express investment style choices and security characteristics in detail

Question 8

risk attribution

Answer 8

LOS 53.f

decompose active risk into two compoments:

• active factor risk - attributable to factor tilts
• active specific risk - attributable to stock selection

σ²(R_P - R_B) = active risk + active specific risk

active specific risk = sum_(i=1,n)[(W_Pi - W_Bi)²σ_<em>e</em>i²

active factor risk = residual

Question 9

strategic portfolio decisions

Answer 9

LOS 53.g

Two questions for investors:

1. what kind of risk do I have a comparitive advantage in bearing?
2. what kind of risk do I have a comparitive disadvantage in bearing?

Examples:

• pension funds have long investment horizons, so less exposed to liquidity risk
• unemployed worker reliant on income is exposed to business cycle risk