Arbitrage Pricing Theory
February 16, 2024
What is the Arbitrage Pricing Theory?
Arbitrage Pricing Theory (APT) is a financial model that investors use to determine the expected return of an asset based on various risk factors. It suggests that the future performance of an asset can be modeled by its sensitivity to multiple sources of risk.
Key Learning Points
- APT is used to determine the expected return of an asset by considering multiple sources of risk (factors) and its sensitivity to these factors. This differs from the Capital Asset Pricing Model (CAPM), which uses a single-factor assumption
- While CAPM assumes that markets are perfectly efficient, the APT (although still considering markets are somewhat efficient) is less stringent and admits that temporary mispricing opportunities may exist before the market eventually corrects and asset prices return to their fair value
- Due to the huge number of factors that can be used, the model is ambiguous, and most academics tend to focus on three to five factors to model returns
- As each stock has different sensitivities to various factors, one of the main limitations of APT is that it doesn’t suggest factors for specific assets or stocks
Elements of Risk / Arbitrage Pricing Theory Factors
The APT aims to establish the intrinsic value of a security that may be temporarily mispriced. Its core assumption is that markets could over- or under-react, and therefore occasionally create arbitrage opportunities for a short period. However, markets should eventually correct and move the price of that asset back to its fair value.
In APT, the elements of risk and uncertainty are represented by different factors that influence asset prices. These factors could encompass various economic or market variables such as interest rates, inflation, market volatility, industry-specific trends, or macroeconomic factors. Unlike the CAPM, which relies on a single factor , the APT considers a range of factors, each contributing to the overall risk associated with an asset. The relationship between these factors and the asset’s return helps to evaluate and forecast the risks involved in investing in that particular asset.
There are various sources or information that can feed factors data into the calculation of APT. Below is an example of the BlackRock’s Capital Market Assumptions.
Source: BlackRock’s Capital Market Assumptions.
Arbitrage Pricing Theory Formula
As discussed, the APT is based on a linear factor model that relates an asset’s expected return to various factors. Below is the formula:
Where:
- Ri – the expected return on asset i
- Rf – risk-free rate
- βij – represents the sensitivity of asset i to factor j
- Fj – denotes the various underlying factors influencing asset returns
- εi – the asset-specific random error term
This formula illustrates how an asset’s expected return is influenced by multiple factors (represented by F1, F2, … ,Fn) weighted by their respective sensitivities (βi1, βi2, …, βin). The model assumes that in an arbitrage-free market, any excess returns would be eliminated by investors exploiting mispricing, aligning asset prices with their expected returns.
Below is an example that takes GDP growth as the main factor.
Stock A has sensitivity to GDP growth of 1.5
Stock A has sensitivity to interest rate changes of 0.8
Stock B has sensitivity to GDP growth of 0.8
Stock A has sensitivity to interest rate changes of 1.2
Assuming the current risk-free rate is 3%, the expected GDP growth rate is 2% and the expected change in interest rates is 1%.
For Stock A:
Ra = Rf + βa,gdp × change in GDP growth + βa,interest × change in interest rates
Ra = 0.03 + 1.5×0.02 + 0.8×0.01 = 0.03 + 0.03 + 0.008 = 0.068 or 6.8%
For Stock B:
Rb = Rf + βb,gdp × change in GDP growth + βb,interest × change in interest rates
Rb = 0.03 + 0.8×0.02 + 1.2×0.01 = 0.03 + 0.016 + 0.012 = 0.058 or 5.8%
Considering a two-factor model, based on factor sensitivity to GDP growth and changes in interest rates, Stock A is expected to have a return of 6.8% compared to 5.8% for Stock B.
In this example, an investor with a time horizon of one year is considering the following two stocks and macro projections.
Calculate the expected return for each stock deriving from an APT model based on the unexpected changes in Inflation and GDP Growth.
There are four steps of the calculation:
- The Risk-free rate is already provided: 1.5%
- Calculating the inflation effect: (Actual Value of Inflation – Expected Value) x stock value
- Calculating the GDP Growth Effect: (Actual Value of GDP Growth – Expected Value) x stock value
- Calculating the Expected Return: Sum of the Risk-free, inflation and GDP growth effects
Expected Return-Rf+Inflation Beta*(Inflation surprise)+GDP Beta*(GDP Surprise)
Assumptions of APT
The APT relies on several key assumptions:
Factor Sensitivity
Asset returns are linearly related to multiple risk factors. These factors are not specified but are assumed to affect asset prices.
No Arbitrage
APT assumes that in an efficient market, no arbitrage opportunities exist. If an asset is mispriced, investors would immediately exploit the opportunity, eliminating any abnormal returns.
Factors Capture Risk
All relevant risk factors affecting asset prices are considered in the model. The factors included are assumed to capture all systematic risk, leaving no residual risk after considering these factors.
Homogeneous Expectations
Investors share homogeneous expectations regarding risk and return, meaning they evaluate assets based on similar beliefs about the future.
Arbitrage Pricing Theory vs. Capital Asset Pricing Model (CAPM)
Although both the APT and the CAPM are used to calculate the expected returns of assets, they differ in several ways:
| APT | CAPM | |
| Factors | Multiple such as interest rate changes, inflation, market indicators, that collectively influence asset prices | Single — the market risk factor, typically represented by the market return less the risk-free rate |
| Complexity | More complex by considering multiple factors and providing a more realistic representation of asset pricing, but however requiring more data and analysis.
|
Simpler in structure and calculation as it is based on a single factor, which makes it easier to apply in practice, but potentially less accurate in capturing the nuances of asset pricing.
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| Market Applicability | More applicable in diverse markets or when analysing assets with various sources of risk, as it accommodates multiple influencing factors | Often used as a baseline or starting point for estimating expected returns, especially in situations where market risk is the primary concern
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| Type | Considered a “supply-side” model, since its beta reflect the sensitivity of the underlying asset to economic and/or market factors. Factors shocks would result in structural changes to the expected return of an asset | Considered a “demand side” model as its results arise from a maximization problem of each investor’s utility function, and from the resulting market equilibrium |
Conclusion
The Arbitrage Pricing Theory is a multi-factor model explaining asset returns based on systematic risks, assuming no arbitrage opportunities in a perfectly efficient market and a linear return-factor link. More flexible than CAPM, APT allows customisation by adding multiple different factors, but has also faced criticism due to the difficulty of identifying and measuring factors. It is often assumed that APT works best when complemented by other models and tools. It finds application in areas like risk management, asset pricing and portfolio construction.