Intro to Finance: Risk and Return - Arbitrage Pricing Theory, APT and MultiFactor Models

What This Is

Arbitrage Pricing Theory (APT) and Multi-Factor Models are used to estimate the expected return of an asset based on its exposure to various risk factors. This theory is essential in finance as it helps investors and analysts understand the relationship between asset returns and macroeconomic variables. For example, consider Apple Inc. (AAPL), a technology giant with a market capitalization of $2 trillion. Using APT, we can estimate the expected return of AAPL based on its exposure to risk factors such as the market return, inflation rate, and interest rate.

Key Formulas & Symbols

• ? = Cov(Ri, Rm) / ?m^2 where-= beta, Ri = return on asset i, Rm = market return, Cov = covariance, ?m = standard deviation of market return.
• Ri = Rf + ?i(Rm - Rf) + ?(wi × Ri) where Ri = return on asset i, Rf = risk-free rate, ?i = beta of asset i, Rm = market return, wi = weight of factor i, Ri = return on factor i.
• APT Model: Ri = Rf + ?i(Rm - Rf) + ?(wi × Ri) + ?i where Ri = return on asset i, Rf = risk-free rate, ?i = beta of asset i, Rm = market return, wi = weight of factor i, Ri = return on factor i, ?i = error term.
• Multi-Factor Model: Ri = Rf + ?i(Rm - Rf) + ?i2(Rm2 - Rf) + ?(wi × Ri) where Ri = return on asset i, Rf = risk-free rate, ?i = beta of asset i, Rm = market return, ?i2 = beta of asset i with respect to factor 2, Rm2 = return on factor 2, wi = weight of factor i, Ri = return on factor i.
• Rf = (1 + r)^(1/n) - 1 where Rf = risk-free rate, r = periodic interest rate, n = number of periods.
• ?m = ?(?(Ri - Rm)^2 / (n - 1)) where ?m = standard deviation of market return, Ri = return on asset i, Rm = market return, n = number of periods.
• Cov(Ri, Rm) = ?((Ri - Rm) × (Rm - Rf)) / (n - 1) where Cov = covariance, Ri = return on asset i, Rm = market return, Rf = risk-free rate, n = number of periods.

Step-by-Step Calculation

1. Estimate the market return (Rm) using historical data or a market index (e.g., S&P 500).
2. Estimate the risk-free rate (Rf) using a short-term government bond yield (e.g., 1-year Treasury bill).
3. Estimate the beta (?) of the asset using historical data or a regression analysis.
4. Estimate the weights (wi) of the risk factors using historical data or a factor model.
5. Plug in the values into the APT model to estimate the expected return of the asset.

Common Mistakes

• Mistake: Using the wrong risk-free rate (e.g., using a long-term bond yield instead of a short-term bond yield).
• Correction: Use a short-term government bond yield as the risk-free rate, as it is a better representation of the risk-free rate.
• Mistake: Failing to account for the error term (?i) in the APT model.
• Correction: Include the error term in the APT model to account for any omitted variables or measurement errors.
• Mistake: Using the wrong beta (?) estimate (e.g., using a beta estimate from a different time period).
• Correction: Use a beta estimate that is relevant to the time period of interest.

Exam / CFA Tips

• Tip: Be careful with the wording of the question, as it may ask for the expected return of an asset based on a specific risk factor (e.g., "What is the expected return of AAPL based on its exposure to the market return?").
• Tip: Make sure to include the error term in the APT model, as it is an essential component of the model.
• Tip: Be aware of the assumptions of the APT model, such as the assumption of a linear relationship between asset returns and risk factors.

Quick Practice Problem

Scenario: Estimate the expected return of Tesla Inc. (TSLA) based on its exposure to the market return and the inflation rate.

Question: What is the expected return of TSLA based on its exposure to the market return and the inflation rate?

Answer: 12.5% (using the APT model with a beta estimate of 1.2, a market return of 10%, and an inflation rate of 2%).

Explanation: The expected return of TSLA is estimated to be 12.5% based on its exposure to the market return and the inflation rate.

Last-Minute Cram Sheet

• The APT model assumes a linear relationship between asset returns and risk factors.
• The beta estimate should be relevant to the time period of interest.
• The risk-free rate should be a short-term government bond yield.
• The error term should be included in the APT model.
• The APT model is a linear model, not a non-linear model.
• The expected return of an asset is estimated based on its exposure to risk factors.
• The APT model is used to estimate the expected return of an asset based on its exposure to risk factors.
• The beta estimate is a measure of the asset's sensitivity to the market return.
• The risk-free rate is a measure of the return on a risk-free asset.
• The error term is a measure of the omitted variables or measurement errors in the APT model.