Introduction

In this chapter, we delve into the methodologies for calculating the expected return of an asset using both single-factor and multifactor models. Understanding these calculations is pivotal for assessing the potential risk and return of investments and making informed financial decisions.

Calculating Expected Return with a Single-Factor Model

A single-factor model simplifies the relationship between an asset’s expected return and its exposure to a single systematic risk factor. The Capital Asset Pricing Model (CAPM) is a classic example of a single-factor model. The formula for calculating expected return using a single-factor model is as follows:

$\text{Expected Return} = \text{Risk-Free Rate} + \text{Beta} \times (\text{Market Return - Risk-Free Rate})$

Where:

• Risk-Free Rate: The rate of return on a risk-free investment (e.g., government bonds).
• Beta: The asset’s sensitivity to market fluctuations.
• Market Return: The expected return of the overall market.

Example: let’s consider an asset with a beta of 1.2, a risk-free rate of 3%, and an expected market return of 8%. The expected return of the asset using the single-factor model would be:

$$\text{Expected Return} = 3\% + 1.2 × (8\% - 3\%) = 9\%$$

Calculating Expected Return with a Multifactor Model

Multifactor models capture the influence of multiple risk factors on an asset’s expected return. They offer a more nuanced approach to risk assessment and return estimation. The formula for calculating expected return using a multifactor model is:

$\text{Expected Return = } \text{Risk-Free Rate} + (\text{Factor 1 Beta} \times \text{Factor 1 Risk Premium})$ + $(\text{Factor 2 Beta} \times \text{Factor 2 Risk Premium})$ + \ldots + $(\text{Factor n Beta} \times \text{Factor n Risk Premium})$

Here, each factor beta represents the asset’s sensitivity to a specific risk factor, and the corresponding factor risk premium quantifies the compensation for that risk.

Example: Consider an asset exposed to two risk factors: market return and interest rate changes. If the asset has factor betas of 1.5 for the market and 0.8 for interest rates, and the risk premiums for these factors are 6% and 2%, respectively, the expected return using the multifactor model would be:

$$\text{Expected Return = } 3\% + (1.5 × 6\%) + (0.8 × 2\%) = 12\%$$