R.47 Intro to Multi-factor Models

Question 1

General Multifactor Equation

Carhart 4-factor model

Arbitrage Pricing Theory

• assumptions
• formula

Answer 1

Equation (1): General Multifactor Equation

R_{<span>i</span>} = a ​_{<span>i</span>}+ b_{<span>i1</span>}I₁ + b_i2I₂ + … + b_iKI_K + ε_i,

where

Ri = the return to asset i

ai = an intercept term

Ik = the return to factor k, k = 1, 2, …, K

bik = the sensitivity of the return on asset i to the return to factor k, k = 1, 2, …, K

εi = an error term with a zero mean that represents the portion of the return to asset i not explained by the factor model

The intercept term ai is the expected return of asset i, given that all the factors take on a value of zero.

Equation (3a) - Carhart 4-factor model

Rp − RF = ap + bp1RMRF + bp2SMB + bp3HML + bp4WML + εp

where

Rp and RF = the return on the portfolio and the risk-free rate of return, respectively

ap = “alpha” or return in excess of that expected given the portfolio’s level of systematic risk (assuming the four factors capture all systematic risk)

bp = the sensitivity of the portfolio to the given factor

RMRF = the return on a value-weighted equity index in excess of the one-month T-bill rate

SMB = small minus big, a size (market capitalization) factor; SMB is the average return on three small-cap portfolios minus the average return on three large-cap portfolios

HML = high minus low, the average return on two high book-to-market portfolios minus the average return on two low book-to-market portfolios

WML = winners minus losers, a momentum factor; WML is the return on a portfolio of the past year’s winners minus the return on a portfolio of the past year’s losers6

εp = an error term that represents the portion of the return to the portfolio, p, not explained by the model

Following Equation 2, the Carhart model can be stated as giving equilibrium expected return as

Equation (3b)

E(Rp) = RF + βp,1RMRF + βp,2SMB + βp,3HML + βp,4WML

because the expected value of alpha is zero.

APT Assumptions

1. A factor model describes the asset returns.
2. Well-diversified portfolios can be created to eliminate asset-specific risk. Factor risk is
   still present.
3. No arbitrage opportunities exist in well-diversified portfolios.

Equation (2): APT

E(Rp) = RF + λ₁β_p,1 + … + λ_Kβ_p,K

where

E(Rp) = the expected return to portfolio p

RF = the risk-free rate

λj = the expected reward for bearing the risk of factor j

βp,j = the sensitivity of the portfolio to factor j

K = the number of factors

Question 2

1. Information Ratio
2. Sharpe Ratio
3. Active risk squared
4. Active specific risk

Answer 2

Question 3

Type of Multifactor models (broadly)

Types of fundamental factor models

Answer 3

Type of Multifactor models (broadly) - see attached

Types of fundamental factor models - most fundamental factor models for equities into three broad groups:

Company fundamental factors. These are factors related to the company’s internal performance.

• earnings growth, earnings variability, earnings momentum, and financial leverage.

Company share-related factors. These factors include valuation measures and other factors related to share price or the trading characteristics of the shares. In contrast to the previous category, these factors directly incorporate investors’ expectations concerning the company. Examples include:

• price multiples like earnings yield, dividend yield, and book to market.
• Market capitalization falls under this heading. Various models incorporate variables relating to share price momentum, share price volatility, and trading activity that fall in this category.

Macroeconomic factors. Sector or industry membership factors fall under this heading. Includes:

• CAPM beta, other similar measures of systematic risk, and yield curve level sensitivity, all of which can be placed in this category.

For global factor models in particular, a classification of country, industry, and style factors is often used.

Question 4

Inflation and growth combos and Asset classes that benefit

Answer 4