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# Return Generating Models

A return generating model is one that can provide investors with an estimate of the return of a particular security given certain input parameters. The most general form of a return generating model is a multi-factor model. The multi-factor model, in its simplest form, is the single index model, a common implementation which gives the market model.

## Multi-factor Models

A multi-factor model is a financial model that employs multiple factors in its calculations to explain asset prices. These models introduce uncertainty stemming from multiple sources. CAPM, on the other hand, limits risk to one source – covariance with the market portfolio. Multi-factor models can be used to calculate the required rate of return for portfolios as well as individual stocks.

CAPM uses one factor to determine the required return – the market factor. However, the market factor can be split up even further into different macroeconomic factors. These may include inflation, interest rates, business cycle uncertainty, etc.

A factor can be defined as a variable which explains the expected return of an asset.

A factor beta is a measure of the sensitivity of a given asset to a specific factor. The bigger the factor, the more sensitive the asset is to that factor.

A multi-factor appears as follows:

$${ R }_{ i }=E\left( { R }_{ i } \right) +{ \beta }_{ i1 }{ F }_{ 1 }+{ \beta }_{ i2 }{ F }_{ 2 }+\cdots +{ \beta }_{ ik }{ F }_{ k }+{ e }_{ i }$$

Where:

$${ R }_{ i }$$= rate of return on stock $$i$$;

$$E\left( { R }_{ i } \right)$$= expected return on stock $$i$$;

$${ \beta }_{ ik }$$= sensitivity of the stock’s return to a one unit change in factor $$k$$;

$${ F }_{ k }$$= macroeconomic factor $$k$$; and

$${ e }_{ i }$$= the firm-specific return or portion of the stock’s return unexplained by macro factors.

The expected value of the firm-specific return is always zero.

### Calculating the Expected Return of an Asset Using a Multi-factor Model

Assume that the common stock of BRL is examined using a multi-factor model, based on two factors: unexpected percentage change in GDP and unexpected percentage change in interest rates. Further, assume that the following data is provided:

• expected return for BRL = 10%;
• GDP factor beta = 1.50;
• interest rate factor beta = 2.0;
• expected growth in GDP = 2%; and
• expected growth in interest rates = 1%.

Compute the required rate of return on BRL stock, assuming there’s no new information regarding firm-specific events.

$${ R }_{ i }=E\left( { R }_{ i } \right) +{ \beta }_{ i1 }{ { F }_{ 1 } }+{ \beta }_{ i2 }{ { F }_{ 2 } }$$

$$=10\%+1.5\times 2\%+2.0\times 1\%$$

$$=15\%$$

## Three and Four Factor Models

One widely used multi-factor model that has been developed in recent times is the Fama and French three-factor model. A major weakness of the multi-factor model is that it is silent on the issue of the appropriate risk factors for use. The FF three-factor model puts three factors forward:

• size of firms;
• book-to-market values; and
• excess return on the market.

The firm size factor, also known as SMB (small minus big) is equal to the difference in returns between portfolios of small and big firms $$\left( { R }_{ s }-{ R }_{ b } \right)$$.

The book-to-market value factor, also known as HML (high minus low) is equal to the difference in returns between portfolios of high and low book-to-market firms $$\left( { R }_{ H }-{ R }_{ L } \right)$$.

Note: book-to-market value is book value per share divided by the stock price.

### This begs the question: Why SMB and HML?

Fama and French put forth the argument that returns are higher on small versus big firms as well as on high versus low book-to-market firms. This argument has indeed been validated through historical analysis. Fama and French contend that small firms are inherently riskier than big ones, and high book-to-market firms are inherently riskier than low book-to-market firms.

The equation for the Fama-French three-factor model is:

$${ R }_{ i }-{ R }_{ F }={ \alpha }_{ i }+{ \beta }_{ i,M }\left( { R }_{ M }-{ R }_{ F } \right) +{ \beta }_{ i,SMB }SMB+{ \beta }_{ i,HML }HML+{ e }_{ i }$$

The intercept term, $${ \alpha }_{ i }$$, equals the abnormal performance of the asset after controlling for its exposure to the market, firm size, and book-to-market factors. As long as the market is in equilibrium, the intercept should be equal to zero, assuming the three factors adequately capture all systematic risks.

Exam tip: SMB is a hedging strategy – long small firms, short big firms. HML is also a hedging strategy – long high book-to-market firms, short low book-to-market firms.

## The Single Index Model

The simplest return-generating model contains a single factor – the market factor. It looks much like the Capital Market Line.

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

Where the factor weight $$\frac{ \sigma_i }{ \sigma_m }$$ reflects the ratio of the security risk to the market risk.

## The Market Model

The market model is a common implementation of the single index model. The market or index return is used as the single factor. The market model is constructed as follows:

$$R_i = \alpha_i + \beta_iR_m + e_i$$

Where $$\alpha_i = R_f(1 – \beta)$$

The historic relationship between security returns and market returns is used to estimate the beta or slope coefficient.

## Decomposition of Risk

Using a single index model, the systematic and non-systematic risk can be decomposed.

$$E(R_i) – R_f = \beta_i[E(R_m)-R_f]$$

Instead of using the expected returns of the market, E(Rm), we can use realized returns. The difference between the expected return and the realized return is attributable to non-market changes and is represented as an error term e.

$$R_i -R_f = \beta_i[R_m-R_f] + e_i$$

The variance of realized returns is expressed as follows:

$$\sigma_i^2 = β_i^2 σ_m^2 + σ_e^2 + 2Cov(R_m, e_i)$$

We can drop the term 2Cov(Rmei) because any non-market return is by definition uncorrelated with the market. In fact, this leaves:

$$σ_i^2 = β_i^2 σ_m^2 + σ_e^2$$

Which states that total variance $$(σ_i^2)$$ is equal to systematic variance $$(β_i^2 σ_m^2)$$ and non-systematic variance $$(σ_e^2)$$.

## Question

If the beta of a security is 1.3, the risk-free rate is 2% and the market expected return is 8%, use the market model to calculate the expected return for the security. (Ignore error terms.)

A. 8.4%

B. 12.4%

C. 9.8%

Solution

The correct answer is C.

The market model is given as:

Ri = αi + βiR + e

Where: αi = Rf(1 – β)

Ri = 2%(1 – 1.3)  + 1.3(8%) = -0.6% + 10.4%

Ri = 9.8%

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