*Last Update: December 15, 2020*

**Asset pricing models** consist of estimating asset expected return through its expected risk premium linear relationship with factors portfolios expected risk premiums and macroeconomic factors.

This topic is part of **Investment Portfolio Analysis with Excel** course. Feel free to take a look at Course Curriculum.

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An example of asset pricing models is **arbitrage pricing theory APT** [1] which consists of estimating asset expected return through its expected risk premium linear relationship with market portfolio expected risk premium and macroeconomic factors.

**1. Formula notation.**

1.1. Arbitrage pricing theory APT asset risk premium formula notation.

- Note: macroeconomic factor not fixed and only included for educational purposes.

Where = asset realized risk premiums, = asset realized returns, = realized risk free returns, = asset average realized excess return, = asset market beta coefficient, = market realized risk premiums, = market realized returns, = asset macroeconomic factor beta coefficient, = realized consumer price index percentage changes macroeconomic factor, = linear regression residuals or forecasting errors.

1.2. Arbitrage pricing theory APT model formula notation.

- Note: APT formula recasting using expected returns (ex-ante) instead of average realized returns (ex-post). For full reference, please read The Arbitrage Theory of Capital Asset Pricing [1].

Where = asset expected return, = expected risk-free return, = asset market beta coefficient, = market expected risk premium, = market expected return, = asset macroeconomic factor beta coefficient, = expected percentage change in consumer price index macroeconomic factor.

**2. Excel example.**

2.1. APT multiple factors model data.

- Data: S&P 500® index replicating ETF (ticker symbol: SPY) adjusted close prices and market portfolio [2] monthly arithmetic returns risk premiums, U.S. consumer price index monthly arithmetic change (2007-2016).

2.2. APT multiple factors model multiple linear regression calculation.

- Multiple linear regression calculation done using Microsoft Excel Data Analysis® Add-In Regression Analysis Tool.

##### 3. References

[1] Stephen Ross. “The Arbitrage Theory of Capital Asset Pricing”. *Journal of Economic Theory*. 1976.

[2] Eugene F. Fama and Kenneth F. French. “Common Risk Factors in the Returns on Stocks and Bonds,” *Journal of Financial Economics*. 1993.