To simplify analysis, the **single-index model** assumes that there is only 1 macroeconomic factor that causes the **systematic risk** affecting all stock returns and this factor can be represented by the rate of return on a market index, such as the S&P 500. According to this model, the return of any stock can be decomposed into the **expected excess return** of the individual stock due to firm-specific factors, commonly denoted by its alpha coefficient (α), which is the return that exceeds the risk-free rate, the return due to macroeconomic events that affect the market, and the ** unexpected** microeconomic events that affect only the firm. Specifically, the return of stock

**is:**

*i***r _{i} = α_{i} + β_{i}r_{m} + e_{i }**

The term β_{i}r_{m} represents the stock’s return due to the movement of the market modified by the stock’s **beta** (β_{i}), while e_{i} represents the **unsystematic risk** of the security due to firm-specific factors.

**Macroeconomic events**, such as interest rates or the cost of labor, causes the systematic risk that affects the returns of all stocks, and the **firm-specific events** are the *unexpected ***microeconomics**** events** that affect the returns of specific firms, such as the death of key people or the lowering of the firm’s credit rating, that would affect the firm, but would have a negligible effect on the economy. The unsystematic risk due to firm-specific factors of a portfolio can be reduced to zero by diversification.

The index model is based on the following:

Most stocks have a positive covariance because they all respond similarly to macroeconomic factors.

However, some firms are more sensitive to these factors than others, and this firm-specific variance is typically denoted by its beta (β), which measures its variance compared to the market for one or more economic factors.

Covariances among securities result from differing responses to macroeconomic factors. Hence, the covariance (σ^{2}) of each stock can be found by multiplying their betas by the market variance:

**Cov(R _{i}, R_{k}) = β_{i}β_{k}σ^{2}**

This last equation greatly reduces the computations, since it eliminates the need to calculate the covariance of the securities within a portfolio using historical returns and the covariance of each possible pair of securities in the portfolio. With this equation, only the betas of the individual securities and the market variance need to be estimated to calculate covariance. Hence, the index model greatly reduces the number of calculations that would otherwise have to be made for a large portfolio of thousands of securities.

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