Chapter 5 - Models of asset returns

Question 1

Give the equation for the multifactor model

Answer 1

Ri = ai + bi,1* l1 + bi,2* l2 +…. +ci
_
where
Ri= return on security i
ai, ci -> constant & random parts respectively pf the component of return unique to security i
l1, …. -> changes in a set of L factors which explain the variation of Ri about the expected return ai
bi,k is the sensitivity of a security i to factor k
___
the goal of building such models is to find a set of factors which explain as much of the possible historical variation, w/o introducing too much noise into prediction of future returns
___
systematic return => b * L
specific return => ai & ci

Question 2

Types of multifactor models

Answer 2

MFS
_
1) MACROECONOMIC - factors are macroeconomic variables such as interest rates, inflation, economic growth & exchange rates
2) FUNDAMENTAL - company specific variables such as P/E ratio, level of gearing, levels of R&D spending, industry group to which the company belongs (models are constructed via regression techniques)
3) STATISTICAL - factors are not specific items initially (use PCA* to determine a set of indices which explain as much as possible of the observed variance)
_
*PCA = principal component analysis
__
useful if the factors are uncorrelated/orthogonal (check pg 7-9)

Question 3

Single-index model (or market model)

Answer 3

def: single factor - normally return on the investment as a whole
__
Ri = αi + βi*Rm + εi. where
__
Ri = return on security i
αi, βi = constants
Rm = return on the market
εi = random variable representing the component of Ri, not related to the market

Question 4

In the single-index model, how can we interpret αi & βi?

Answer 4

αi -expected value of the component of security i’s return that is independent of the markets performance & specific to that particular security
____
βi- quantifies the component of the security return that is directly related to movements in the market - so that if βi = x, then security i’s return is expected to increase by x% when the market return increases by 1%.

Question 5

What are the expected return & variance of return on security i & the covariance of the returns on securities i & j?

Answer 5

Ei=αi + βiEm
__
Vi = βi^2Vm + Vei. &
Cij = βiβjVm. where Vei is the variance of ei
__
It is only the systematic risk (i.e βi which should be expected to be rewarded by increased return since this is non-diversifiable)

Question 6

Data requirements of single-index model

Answer 6

Although incorporating more factors into the model may lead to a better explanation of historical data, correlation with the market is the largest factor explaining security price variation.
_
using a single-index model dramatically reduces the amount of data required. It reduces from N(N+3)/2 to 3N+2.
_
N values for αi’s
N values for βi’s
N values for Vei’s
Em & Vm

___
Main use for multifactor & single-index is to find an efficient frontier

Question 7

Main uses of multifactor & single index models?

Answer 7

1)determination of the investors EFFICIENT FRONTIER as part of the derivation of the investor’s optimal portfolio
2) RISK CONTROL - by enabling the investor to forecast the variability of portfolio returns both absolutely & relatively to some benchmark
3) PERFORMANCE ANALYSIS - by comparing the actual to that of predicted models
4) CATEGORISATION OF INVESTMENT STYLES - according to the extent of the exposure to particular factors

Question 8

What is the main limitation of multifactor & single index models?

Answer 8

Construction of these models are based on historical data that reflect conditions that may not be replicated in the future. Moreover, a model that does produce good predictions in one period, may not produce good predictions in subsequent time periods.