# Arbitrage Pricing Theory

Arbitrage pricing theory (APT) is a model of asset pricing that holds that the expected return on an asset is a linear function of various market factors. The APT was developed by Stephen Ross in the 1970s and has become one of the most popular models in finance.

The main idea behind APT is that assets are priced according to their riskiness, or beta. Beta measures how much an asset's price changes when there is a change in the overall market. For example, if the market goes up by 10% and an asset's price also increases by 10%, then that asset has a beta of 1.0.

APT is often used to explain the one-equation model of investing, which states that the expected return on investment is equal to its beta times the market risk premium. The market risk premium is the difference between the expected return on a portfolio of risky assets and the risk-free rate (the rate you would expect to earn on

## What is the Arbitrage Pricing Theory (APT)?

The Arbitrage Pricing Theory (APT) is an asset pricing theory which seeks to calculate the fair market price of a security. It states that an asset’s expected return can be forecasted with the linear relationship between its risk and macroeconomic factors. APT is used by investors and analysts to assess and anticipate returns of assets, portfolios and other securities in order to gain an advantage by buying assets at a lower price than they are sold for (arbitrage). The APT model shows how changes in macroeconomic factors can affect the expected return of an asset or portfolio. In comparison to the Capital Asset Pricing Model (CAPM), APT is more flexible and complex, making it easier to gauge and implement than traditional models of risk-return.

What are the components of APT's one-equation model?

1. Asset returns

Asset return is an important component of the Arbitrage Pricing Theory (APT) one-equation model because it accounts for the systematic risk associated with investing in assets. Since investors are not able to find arbitrage opportunities in well-diversified portfolios, they must rely on buying low and selling high to make money. The systematic risk associated with investing can be eliminated through diversification, making returns predictable and reliable. The APT's one-equation model takes this into account by using asset return to determine the value of an asset over time.

2. Systematic risk factors

Systematic risk factors are risks that cannot be diversified away and affect the entire market or market segment. They are important in the Arbitrage Pricing Theory's one-equation model because they help to calculate the return on assets by affecting things like beta (which measures how sensitive an asset’s price is to changes in the index), as well as a risk premium (the expected return of an investor over and above the risk-free rate). By taking these systematic risks into account, investors can better identify potential opportunities for arbitrage when portfolios are well diversified.

3. Factor sensitivities

The significance of factor sensitivities in Arbitrage Pricing Theory's (APT) one-equation model is that they can be used to determine the likelihood of success for an attacker. Factor sensitivities allow attackers to adjust their attack strategy, which can improve their chances of success. The one-equation model is also beneficial as it breaks down securities into factors and residuals, helping investors identify how different securities contribute to the portfolio return. In addition, the one-equation model provides a more detailed analysis than simpler statistical models and allows investors to approach portfolios from multiple angles.

4. Expected returns

Expected returns significantly impact the one-equation model as they are used to calculate the expected return of any security product. The expected return for a security will be impacted by factors such as the current risk factor, price to earnings ratio (P/E), and dividend yield. As market conditions change, so do the expected returns, which in turn can affect how companies make their investment decisions.

5. Risk-free rate of return

The risk-free rate of return is the theoretical return an investment can earn without any risk of loss. The T-bill is often used as a benchmark for these rates, since it reflects the government's ability to repay its debt. This rate serves as a standard against which other investments with higher risks are evaluated and compensated for with their own risk premiums. Risk premiums act as hazard pay for investors who are willing to take on higher levels of risk, allowing them to potentially earn higher returns. Therefore, the concept of a risk-free rate of return is important in asset pricing and investing decisions because it helps set expectations regarding what level of compensation investors should demand when taking on additional levels of risk.

6. Systematic risk

Systematic risk is a type of risk that affects the entire market or market segment. It is unsystematic and thus cannot be diversified away. Systematic risk affects APT's one-equation model by creating an opportunity for arbitrageurs to exploit any differences in expected returns and by reducing the accuracy of targeting decisions due to its inherent uncertainty. Beta measures how sensitive an asset's price is to changes in the market index, which can then be used to calculate the equity risk premium as part of a return on assets formula.

7. Unsystematic risk

Unsystematic risk is a type of risk that is not diversified and can lead to financial losses. The one-equation model of asset pricing used in Arbitrage Pricing Theory (APT) takes into account systematic risks such as inflation and premiums, but does not allow for the exploitation of arbitrage opportunities due to its concrete nature. Unsystematic risks include five different portfolios which have their own limitations: risk-free, factor, mean, cost, and mean-variance efficient. These are implementable using the hyperfinite model by allowing distributions over the random variables in each ensemble to vary from one another. As a result of this universality property, unsystematic risks can be accounted for when evaluating assets with APT's one-equation model.

8. Risk-neutrality assumption

The risk-neutrality assumption states that investors should not be affected by the specific risks associated with any particular asset. This is important in APT's one-equation model because it allows for the replication and pricing of securities without taking into account the risk preferences of investors. By assuming an efficient market and risk aversion among investors, the equation governing asset prices must hold true across all assets. The resulting equation incorporates factors such as expected return, volatility, and other market conditions to ensure that all assets are priced fairly relative to each other regardless of their individual risks.

9. Equity market median

The equity market median is used in the Arbitrage Pricing Theory (APT)'s one-equation model to compare the expected return of an asset to its benchmark. The median is used as a reference point for comparison and helps establish whether or not an asset's expected return is worth investing in when considering risk. It also assists investors with determining if there may be arbitrage opportunities between two assets, helping determine if a good/profitable investment opportunity exists.

10. Arbitrage pricing theory

Arbitrage pricing theory is a model used to estimate the fair market value of a financial asset by exploiting price differences between markets. The model assumes that markets are efficient and contains no risk when trading. By buying assets which are currently overvalued and selling those which are currently undervalued, investors can make money on the difference in prices. This theory has been tested using empirical data, and has been found to be an accurate predictor of market prices. Arbitrage pricing theory is used to explain how stock market prices stocks based on expected future profits, with stock prices being reflective of those expected future profits.

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