Level 1 CFA® Exam:
Beta & Return-Generating Models

Last updated: October 07, 2022

CFA Exam: Return-Generating Models

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Return-generating models can provide an estimate of the expected return of risky assets given certain parameters. Therefore, it is critically important to determine the parameters considered in a given model. In principle, those factors can be divided into three main categories:

  • macroeconomic factors,
  • fundamental factors, and
  • statistical factors.

Macroeconomic factors include GDP growth, the rate of inflation, the level of consumption, and the like. Fundamental factors include, among other things, sales growth, size of the company, or earnings.

And let's not forget about the statistical factors that are meant to identify statistical correlations between certain variables. Statistical factors do not necessarily present macroeconomic or fundamental correlations. What's more, you often cannot find a logical correlation between the factor and the rate of return on the assets.

Generally speaking, a multi-factor model, i.e. a model in which the rate of return depends on a few factors, may be expressed in the following manner:

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Return-Generating Model
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\(E(R_{i}) - R_{f} = \Sigma^{k}_{j=1} \beta_{i,j}\times E(F_{j}) = \\=\beta_{i1}\times [E(R_{m}) - R_{f}] + \Sigma^{k}_{j=2} \beta_{i,j}\times E(F_{j})\)

  • \(E(R_{i})\) - return on an asset
  • \(R_{f}\) - return on the risk-free asset
  • \(\beta_{i,j}\) - factor weights
  • \(E(R_{m})\) - market return
  • \(E(F)\) - factor
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The expected rate of return on the portfolio minus the risk-free rate is equal to the sum of factors multiplied by the weights of given factors.

Such a model allows you to calculate the rate of return on a given asset in excess of the risk-free rate.

Very often, one of the factors is the rate of return on the market portfolio over the risk-free rate, the so-called market risk premium.

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Fama, French Three-Factor Model

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The most well-known and commonly used return-generating model is the three-factor model created by Fama and French. In this model the factors are:

  • the size of the company,
  • the book-to-market value, and
  • the market risk premium.

In its time, this model explained the rates of return on the stocks of American corporations relatively well and was often presented as an alternative to the CAPM model, which we will discuss in detail in the next lesson.

You must know that three- and four-factor models are commonly used in finance. Such number of factors seems optimal – it's not too high, nor is it too low. Three- and four-factor models are also very popular for example in bankruptcy valuation.

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CFA Exam: Single-Index Model

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The simplest model that allows return estimation is a single-index model. It is a single-factor linear model. Most often the single factor is the rate of return on the market portfolio over the risk-free rate. The model may be expressed this way:

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Single-Index Model
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\(E(R_{i}) - R_{f} = \beta_{i}\times [E(R_{m}) - R_{f}]\)

  • \(E(R_{i})\) - return on the asset "i"
  • \(R_{f}\) - return on the risk-free asset
  • \(\beta_{i}\) - beta coefficient
  • \(E(R_{m})\) - return on the market
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Also, it is consistent with the Capital Market Line (CML) for well-diversified portfolios:

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Capital Market Line (CML)
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\(E(R_{p}) - R_{f} = (\frac{\sigma_{p}}{\sigma_{m}})\times [E(R_{m}) - R_{f}]\)

  • \(E(R_{p})\) - expected return on portfolio
  • \(R_{f}\) - expected return on the risk-free asset 
  • \(\sigma_{p}\) - risk of portfolio
  • \(\sigma_{m}\) - market risk
  • \(E(R_{m})\) - market return
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Example 1 (single-index model)

Let's assume that the return on the risk-free assets is 3%, and the return on the market portfolio is 10%. The standard deviation of the market portfolio is 16% and that of KG's well-diversified portfolio is 22%. Let's estimate the expected rate of return on the KG portfolio.

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Market Model

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The simplest implementation of the single-index model is the market model. It is expressed like this:

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Market Model
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\(R_{i} = \alpha_{i} + \beta_{i}\times R_{m} + e_{i}\)

  • \(R_{i}\) - return on asset "i"
  • \(\alpha_{i}\) - intercept
  • \(\beta_{i}\) - slope coefficient
  • \(R_{m}\) - market return
  • \(e_{i}\) - difference between expected returns and realized returns

\(\alpha_{i} = R_{f}\times (1 - \beta)\)

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Example 2 (market model)

The rate of return on ACP's securities for the last week was 8%. The beta of ACP shares is 0.79 and the alpha is 0.3%. At this time, the rate of return on the stock index was 3.1%. Calculate the abnormal return in the previous week.

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Beta in CFA Exam – Formula, Interpretation, Estimation

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Beta in the market model is the measure of systematic risk, i.e. the market risk, because it shows how sensitive the return on the stock is to the return on the market portfolio. From a mathematical point of view, beta is a standardized covariance and may be expressed as follows:

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Beta Coefficient
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\(\beta_{i} = \frac{\rho_{i,m}\times \sigma_{i}}{\sigma_{m}}\)

  • \(\beta_{i}\) - beta coefficient
  • \(\rho\) - correlation between asset "i" and market
  • \(\sigma_{i}\) - risk of asset "i"
  • \(\sigma_{m}\) - market risk
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Example 3 (beta estimation)

We know that the correlation coefficient between the stock index and the stocks of PU Company is 0.87. The standard deviation of the stock index is 12% and the stock of PU Company is 8%. Calculate the beta coefficient of PU Company stocks.

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Lesson Video

Level 1 CFA Exam Takeaways: Beta & Return-Generating Models

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  1. Return-generating models can provide an estimate of the expected return of risky assets given certain parameters.
  2. Single-index model is a single-factor linear model.
  3. The simplest implementation of the single-index model is the market model.
  4. Beta in the market model is the measure of systematic risk, i.e. the market risk, because it shows how sensitive the return on the stock is to the return on the market portfolio.
  5. When the beta of a stock is 1, it means that the rate of return on that stock moves in the same direction as the rate of return on the market and the stock has the same systematic risk as the market.
  6. Beta is usually estimated by using regression analysis of the market model.
  7. The beta of the market is always 1.
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