IAI Actuarial Core Principles · Economic Modelling
Single and Multifactor Models for Investment Returns Explained
A factor model says a security's return is driven by a small number of common factors plus a security-specific error. The single-factor market model uses one factor, the market return. Multifactor models and APT add more factors. You solve questions by writing the model, finding the coefficients and computing expected return and risk.
What this chapter covers
This chapter in CM2 (Economic Modelling) explains how to describe the return on a security or portfolio using a few common sources of risk. In a single factor model, the return on security i is written as Rᵢ = aᵢ + bᵢ·I + cᵢ, where I is the factor, bᵢ is the sensitivity to it and cᵢ is a random error with zero mean that is uncorrelated with I and with the errors of other securities. The market model is the case where the factor is the return on a market index.
Multifactor models extend this to several factors: Rᵢ = aᵢ + bᵢ₁·I₁ + bᵢ₂·I₂ + ... + bᵢₖ·Iₖ + cᵢ. Arbitrage pricing theory (APT) links expected returns to factor sensitivities, using a no-arbitrage argument rather than assumptions about investor preferences. You also meet the three usual types: macroeconomic factor models, fundamental factor models and statistical factor models.
The chapter connects to the rest of the paper in several ways. It builds on the portfolio theory and mean-variance ideas in the measures of investment risk and rational economic theory areas. It also feeds into asset valuation, because a factor model gives a way to estimate expected returns and covariances without estimating every pair of securities directly. Check the current IAI syllabus for exactly which models and results are examinable.
Factor models turn up in both the multiple-choice section and the written questions, and in computer-based work where you may estimate a regression for the market model. The chapter is mostly definitions, a few formulas and clear reasoning, so it is a good place to secure marks with modest effort. The same reasoning on risk, diversification and no-arbitrage pricing also helps you in other parts of CM2, so the time you spend here is not wasted.
Single and multifactor models for investment returns: topics in the order to study them
- 1Introduction to Investment Return ModelsStart here to see why we model returns with common factors at all, and to fix the idea of systematic versus specific risk before any formula.
- 2Single Factor Models and the Market ModelThe single factor form is the base case. Learn the equation, its assumptions, and how to get expected return, variance and covariance from it.
- 3Multifactor Models and Arbitrage Pricing TheoryThis extends the single factor algebra to several factors and adds the APT pricing idea, so you need the single factor results first.
- 4Types of Multifactor ModelsOnce you know the general form, you can compare macroeconomic, fundamental and statistical models by how their factors and sensitivities are obtained.
- 5Applications and Limitations of Factor ModelsFinish with uses and weaknesses, which are easier to judge once you know what each model assumes and how it is built.
How to prepare Single and multifactor models for investment returns
Treat this as a short chapter with one core equation that grows. Master the algebra of the single factor model and the rest follows.
- Read the introduction and write in your own words the difference between systematic risk and specific risk. Keep this as a one-line note.
- Write out the single factor model and list its assumptions: the error has zero mean, is uncorrelated with the factor, and errors of different securities are uncorrelated.
- Derive from those assumptions the expected return, the variance and the covariance of securities. Do it by hand until you can do it without notes.
- Practise portfolio questions: the portfolio sensitivity is the weighted average of the individual sensitivities, and the portfolio specific risk shrinks as the number of securities grows under the stated assumptions.
- Learn APT as a logical argument: state the assumptions, say what no-arbitrage implies for expected returns, and write the result in terms of factor risk premiums.
- Make a comparison table of the three multifactor model types on your own, covering what the factors are and how each is estimated.
- Finish with past-style questions that mix calculation and discussion, and rehearse a short list of applications and limitations to use in written answers.
Common mistakes in Single and multifactor models for investment returns
Forgetting the assumptions on the error term.
Fix: Write the assumptions at the start of every answer. The covariance formula holds only because the errors are uncorrelated with each other and with the factor.
Adding specific variance into the covariance between two securities.
Fix: Under the model assumptions, the covariance between different securities comes only from the common factor, so it is bᵢ·bⱼ·Var(I).
Mixing up variance and standard deviation when working with weights.
Fix: Square the sensitivity and the weight correctly in variance, and take the square root only at the final step if the question asks for a standard deviation.
Describing APT as needing a specific utility function or a market portfolio.
Fix: State that APT relies on a factor structure for returns and on no-arbitrage. Compare it with CAPM only if the question asks for it.
Writing vague limitations such as 'models are not perfect'.
Fix: Tie each limitation to a model feature, for example that factors in a statistical model may have no clear economic meaning.
Confusing the three types of multifactor model.
Fix: Remember the source of the inputs: macroeconomic uses observed economic variables, fundamental uses company characteristics, statistical extracts factors from return data.
Last-day revision: Single and multifactor models for investment returns
- Single factor model: Rᵢ = aᵢ + bᵢ·I + cᵢ, with cᵢ having zero mean.
- The market model uses a market index return as the factor.
- bᵢ measures sensitivity to the factor; it is the slope in a regression on the factor.
- Total variance of a security = bᵢ²·Var(I) + Var(cᵢ), as the error is uncorrelated with the factor.
- Covariance between two securities i and j = bᵢ·bⱼ·Var(I) when errors are uncorrelated.
- Systematic risk comes from the factors and cannot be diversified away; specific risk can be.
- A portfolio's sensitivity is the weighted average of its securities' sensitivities.
- Multifactor model adds further factors, each with its own sensitivity.
- APT is based on no-arbitrage, not on investor preferences.
- Three types: macroeconomic, fundamental and statistical factor models.
- Limitations: factors may be wrongly chosen, sensitivities may change over time, and past data may not predict the future.
- Always state your assumptions before computing.
Single and multifactor models for investment returns practice questions
- An equally weighted portfolio holds 25 shares, each with beta 1.0 and residual variance 0.0400. The market variance is 0.0100. Residuals are…
- Two stocks, X and Y, follow a single-index model with market variance 0.04. Beta of X is 0.8 and beta of Y is 1.5. Residuals are uncorrelate…
- A two-factor model for a fund's return is R = 2% + 0.9 F1 + 0.5 F2 + e, where F1 is the surprise in market return and F2 is the surprise in …
- A two-factor model for a portfolio's excess return has factor loadings of 0.9 on the market factor and 0.5 on an interest-rate factor. The e…
- An analyst uses a single-index model, R_i = a_i + b_i R_M + e_i, to split the risk of a portfolio of Indian equities. Which statement about …
- A single-index model gives security returns R_i = a_i + b_i R_M + e_i. Security A has b = 1.2 and security B has b = 0.8. The market return …
- A two-factor APT model for an Indian equity portfolio has a risk-free rate of 6%, factor risk premiums of 4% for factor 1 (say, market) and …
- In a multifactor model of security returns, R_i = a_i + b_i1 F1 + b_i2 F2 + e_i, which statement about the term e_i is correct under the sta…
Single and multifactor models for investment returns in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Single and multifactor models for investment returns: frequently asked questions
What is the difference between the market model and a general single factor model?
The market model is a single factor model in which the factor is the return on a market index. A general single factor model could use another factor, such as an economic variable. The algebra for expected return and risk is the same.
Do I need to memorise derivations for this chapter?
You should be able to derive the basic variance and covariance results from the model assumptions, because written questions can ask you to show them. For multiple-choice questions you mainly need to apply the results quickly.
How is APT different from CAPM?
APT starts from a factor structure for returns and a no-arbitrage argument, and allows several factors. CAPM is built on investor behaviour and the market portfolio. Learn each model's assumptions separately so you can contrast them clearly.
Can this chapter appear in the computer-based paper?
It can, because estimating factor sensitivities from return data is a regression task. Practise setting out the model, running the regression and interpreting the slope and residual variance in your answer.