The Arbitrage Pricing Theory (APT) formula explains the expected return of an asset or portfolio based on its sensitivity to various macroeconomic factors and their associated risk premiums.
The APT Formula
The formula is:
E(ri) = rf + βi1 * RP1 + βi2 * RP2 + ... + βkn * RPn
Where:
- E(ri): Expected return of the asset or portfolio
- rf: Risk-free rate of return
- βi1, βi2, ..., βkn: Sensitivity of the asset or portfolio to factor 1, factor 2, ..., factor n. This is also known as the asset's beta with respect to each factor.
- RP1, RP2, ..., RPn: Risk premium associated with factor 1, factor 2, ..., factor n.
Explanation of Components
-
Risk-Free Rate (rf): This is the theoretical rate of return of an investment with zero risk. Treasury bills are often used as a proxy for the risk-free rate.
-
Sensitivity (β): Beta measures the asset's responsiveness to changes in each factor. A beta of 1 indicates that the asset's return will move in line with the factor. A beta greater than 1 suggests the asset is more volatile than the factor, and a beta less than 1 suggests it's less volatile.
-
Risk Premium (RP): This represents the additional return investors expect for taking on the risk associated with a particular factor. It's the difference between the expected return on a factor and the risk-free rate.
Example
Let's say we want to calculate the expected return of a stock using a two-factor APT model:
- Risk-free rate (rf) = 2%
- Factor 1: Inflation rate, with a risk premium (RP1) of 3% and the stock's sensitivity (βi1) is 0.8.
- Factor 2: GDP growth, with a risk premium (RP2) of 5% and the stock's sensitivity (βi2) is 1.2.
Using the APT formula:
E(ri) = 2% + (0.8 * 3%) + (1.2 * 5%) = 2% + 2.4% + 6% = 10.4%
Therefore, the expected return of the stock is 10.4%.
Key Differences between APT and CAPM
While both APT and the Capital Asset Pricing Model (CAPM) aim to explain asset returns, they differ in several ways:
| Feature | APT | CAPM |
|---|---|---|
| Number of Factors | Multiple macroeconomic factors | Single factor: market risk (beta) |
| Assumptions | Fewer restrictive assumptions | More restrictive assumptions |
| Implementation | Requires identifying relevant factors and estimating their risk premiums | Simpler to implement, relies on market data |
Practical Considerations
- Identifying Factors: Choosing the right factors is crucial. Common factors include inflation, interest rates, GDP growth, and commodity prices.
- Estimating Betas and Risk Premiums: Accurately estimating the sensitivities (betas) and risk premiums of the chosen factors can be challenging. Statistical methods and historical data are often used.
- Model Complexity: Adding more factors increases the complexity of the model. It's important to strike a balance between model accuracy and simplicity.
