A return-generating model is a model that provides estimates of expected return given certain parameters, such as the level of systematic risk.

There are different variants of such models including the market model, multi-factor model, etc.

## Multi-factor model

A multi-factor model is a model that links an asset’s expected return to different macroeconomic, fundamental and/or statistical factors. Macroeconomic factors include factors such as economic growth, interest rate, inflation, etc. Fundamental factors include company-specific factors such as earnings, sales, cash flow generation, etc. Statistical factors analyze historical and cross-sectional return data based on factors that might not have an economic or fundamental connection with returns. Since statistical models may generate spurious relationships, analysts often prefer macroeconomic and fundamental models.

A general return-generating model is expressed as follows:

$$E(R_i)-R_f=β_1×E(F_1)+β_2×E(F_2)+…+ β_n×E(F_n)$$

Where E(Ri) is the expected return on the asset, β represents the factor weight (called factor loading) and E(Fn) represents return resulting from each factor.

We can separate the market return from other factors as follows:

$$E(R_i)-R_f=β_{i1}×[E(R_m) -R_f] +β_1×E(F_1) + β_2×E(F_2)+…+ β_n×E(F_n)$$

## The single-index model

From the relationship between expected return, relative weights and standard deviation of a risky portfolio and the risk-free asset, we derived an equation for the CML. The equation has the y-intercept equal to the risk-free rate rf, and slope [E(Rm)-Rf]/σm. This can be used to formulate the following equation rearranged as follows:

$$E(R_i)-R_f=\frac{σ_i}{σ_m} ×[E(R_m)-R_f]$$

Where σim is the ratio of total security risk to total market risk.

Based on the single-factor model, we can write an expression for realized excess return as follows:

$$R_i-R_f=β_i×(R_m-R_f)+e_i$$

Where βi is the factor weight and ei is an error term that represents the non-market realized return. Using the formula for the variance of a portfolio containing market and non-market factors and using the fact that the correlation between the market and non-market factors should be zero, we can derive the following equation for the standard deviation of total security risk (σi):

$$σ_i=β_i σ_m$$

Substituting this in the equation for the single-factor model, we get the following express:

$$E(R_i)-R_f=β_i×[E(R_m)-R_f]$$

This is the equation for a single-index model, the simplest return-generating model which considers only one factor, the market factor. However, this holds only for diversified portfolios.

## The market model

The most common implementation of the single-index model is the market model.

$$R_i=α_i+β_i R_m+e_i=R_f×(1-β_i)+β_i R_m+e_i$$

Where the first expression (αi) is the y-intercept, and βiRm is the slope.

## Three-factor and four-factor models

The capital asset pricing model has a single factor, the beta coefficient. The three-factor model (developed by Eugene Fama and Kenneth French) includes the relative size of a company and relative book value. The four-factor model also adds momentum as the fourth factor.

## Test

Which of the following factors is common in all return-generating multi-factor models?

1. Market return
2. Momentum.
3. Size 