Chapter 6 - Models Of Asset Returns

Question 1

List 3 types of multifactor model

Answer 1

1. Macroeconomic - the factors are the main macroeconomic variables such as interest rates, inflation, economic growth and exchange rates.
2. Fundamental - the factors will be company specifics such as P/E ratios, liquidity ratios and gearing measurements.
3. Statistical - the factors are not specific items initially. The method uses Principal Component Analysis (PCA) and historical returns on stocks to decide upon the factors.

Question 2

Multifactor model of security returns (give the equation of R_i)

Answer 2

A multifactor model of security returns attempts to explain the observed historical return by an equation of the form:
R_i = a_i + b_i,1 I₁ + b_i,2 I₂ +…+ b_i,L I_L + c_i
where

• R_i is the return on security i
• a_i, ci are the constant and random parts respectively of the component of return unique to security i
• I₁…I_L are the changes in a set of L factors which explain the variation of R_i about the expected return a_i
• b_i,k is the sensitivity of security i to factor k

Question 3

The single-index model

Answer 3

The single-index model or market model expresses the return on a security as:

R_i = α_i + β_i R_m + ε_i

Where:

• R_i is the return on security i
• α_i, β_i are constants
• R_m is the return on the market
• ε_i is a random variable representing the component of R_i not related to the market

Question 4

Systematic and specific risk

Answer 4

Systematic risk can be regarded as relating to the market as a whole, while specific risk depends on factors peculiar to the individual security.

Thus it is only the systematic risk that should be expected to be rewarded by increased return since this is non-diversifiable.
Investors can diversify away specific risk and do not therefore require compensation for accepting it. Specific risk is also referred to as alpha, unsystematic, diversifiable or residual risk.

Question 5

Main uses of multifactor models and single-index models (DARC)

Answer 5

1. Determination of the investor’s efficient frontier
2. Analysis of performance
3. Risk control
4. Categorization of investment styles

Question 6

Main limitation of multifactor models and single-index models

Answer 6

The main limitation is that the construction of factor models is 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 time period, may not produce good predictions in subsequent time periods.

Question 7

Advantages of single-index model over a multifactor model

Answer 7

1. Although many studies have found that incorporating more factors into the model leads to a better explanation of the historical data, correlation with the market is the largest factor in explaining security price variation.
2. There is little evidence that the multifactor models are significantly better at forecasting the future correlation structure.
3. The use of single-index model dramatically reduces the amount of data required as input to the portfolio selection process.
4. The nature of estimates required from security analysts conforms much more closely to the way in which they traditionally work.
5. Considerably simplified methods for calculating the efficient frontier have been developed under the single-index model, although with increasing computer power, this is of considerably less importance than it was when the model was first introduced.