What is Arbitrage Pricing Theory? A Beginner’s Guide to APT Explained

The biggest challenge in dealing with stocks and other financial assets is determining the fair value and expected return. A wide range of trading algorithms considers various pricing models and theories to estimate the values of underlying assets accurately. This article discusses arbitrage pricing theory (APT) as a potent and powerful model, offering an alternative to market risk-based pricing models.

What is arbitrage pricing theory?

The theory of arbitrage pricing of financial assets is an asset pricing model based on the simple yet powerful no-arbitrage principle. According to this principle, no risk-free return (profit) can exist in an efficient market. A risk-free return implies there is zero chance of financial loss in earning the profit. If such an opportunity exists, investors (or traders) will exploit it instantaneously, closing the price gap and eliminating the window to earn the risk-free return.

Example of arbitrage:

Assume equity stocks of a company are selling at a price of ₹100 on the National Stock Exchange (NSE) and ₹105 on the Bombay Stock Exchange (BSE). Considering negligible to zero transaction costs, a trader can buy on NSE and sell the stocks on BSE, making a ₹5 profit on each stock. This trading practice is known as arbitrage. It raises prices on the NSE and decreases prices on the BSE, balancing prices across both exchanges.

APT can also serve as a multi-factor asset pricing tool. It helps construct asset prices as a linear function of macro and microeconomic variables and parameters. A few of the common factors that APT models can accommodate are:

• Consumer price index
  
• Wholesale price index
  
• GDP
  
• Industry growth rate
  
• Bank rate
  
• Foreign exchange rates
  
• Corporate net worth
  
• Promoter stakes
  
• Corporate bond rate
  
• Company rating
  

The ability to accommodate a wide variety of factors makes APT stand out against conventional asset valuation theories like CAPM (Capital Asset Pricing Model).

How does arbitrage pricing theory work?

Before understanding how APT works, let us look at a simple mathematical representation of the theory.

• E(Ri) = Rf + β1F1+ β2F2 +...+ βkFk

Here, E(R) = expected return from the asset F = risk-free rate of return (e.g., sovereign bond rate) β = sensitivity of the underlying asset to a risk factor (macro and microeconomic factors) F = premium associated with a risk factor

Application of the formula:

Assume we are determining the price of an asset XYZ. It is sensitive to three broad factors: GDP, gold price, and the performance of Nifty50.

• F(GDP) = 4%
• Β (GDP) = 0.6
• F(gold) = 6%
• Β (gold) = 0.7
• F(Nifty50) = 8%
• Β (Nifty50) = 1.2
• Sovereign rate = 2%

Expected return from XYZ = 2% + (4%*0.6) + (6%*0.7) + (8%*1.2) = 2% + 2.4% + 4.2% + 9.6% = 18.2%

Inserting the expected return value into a discounted cash flow or dividend discount model will help determine the present value of the asset XYZ.

Applications of arbitrage pricing theory

The major applications of APT-based models and tools are seen in asset valuation for private equity funds, portfolio management, algorithm-based trading applications, and asset performance analysis.

Advantages of APT

The major advantages of arbitrage pricing-based models and tools for asset valuation are:

• Scope for multi-factor assessment: Unlike single-factor CAPM-based tools, APT tools allow factoring in multiple risk factors, including markets and non-markets.

• Accommodate a wide variety of risk factors: APT-based models can include economy, industry, and asset-specific risks. This makes the model highly versatile.

• Compensation for non-market risks: Conventional asset pricing theories usually account for market-related risks efficiently. However, they are not effective in adjusting valuation based on external factors. Arbitrage pricing theory has no such limitations.

Limitations of APT

Arbitrage pricing models are versatile and can accommodate very complex pricing functions. However, such models are not free of limitations.

• Identification of risk factors: A prerequisite for plotting a linear function using APT is accurately identifying risk factors for a specific asset or portfolio. Otherwise, the expected return may contain too much noise.

• Measurement of risk premiums and sensitivity: It is also important to estimate the risk premium and the sensitivity of the underlying asset or portfolio to individual risk factors.

• Dynamism of sensitivity to risk factors: Risk premium (F) and sensitivity (β) to individual risk factors may vary over time.

• Absence of true risk-free rate of return: APT depends heavily on the risk-free return as an independent parameter. However, the risk-free rate of return can be difficult to estimate in many markets.

• Availability of reliable historical data: It is not always possible to access or collect reliable historical data for every risk factor. Plotting an APT-based function can be difficult without robust and reliable data.

Conclusion

It is essential to estimate the fair value of an asset for any fundamental analysis in portfolio management and value investing. CAPM-based tools are simple to use but archaic and limited in scope. Comparatively, the zero-arbitrage principle makes arbitrage pricing theory more versatile for a wide variety of assets.

Sources

Certificate in Quantitative Finance
Investopedia