# Single-Index Model for Security Returns

To minimize firm-specific risk, a portfolio should consist of securities with no, or preferably, negative covariances. But to calculate covariances for large portfolios requires large amounts of computing power. Moreover, since returns and variances have to be estimated, these estimations sometimes lead to nonsensical results when applied to the portfolio as a whole. Index models greatly reduce the computations needed to calculate the optimum portfolio while also eliminating nonsensical results.

## Markowitz Portfolio Selection Model

In 1952, Harry Markowitz published a portfolio selection model that maximized a portfolio's return for a given level of risk. A graph of these portfolios constitutes the **efficient frontier of risky assets**. This model required the estimation of expected returns and variances for each security and a covariance matrix that calculated the covariance between each possible pair of securities within the portfolio based on historical data or through scenario analysis. For n securities, that would require n estimates of expected returns, n estimates of their variances, and a covariance matrix that consisted of (n^{2} – n) / 2 estimates of covariances. The calculations increase rapidly as n increases.

## Single-Index Model

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 *i* is:

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* **microeconomic 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.

## Security Characteristic Line

The comparison of a stock's excess return can be plotted against the market's excess return on a scatter diagram using linear regression to construct a line that best represents the data points. This regression line, called the **security characteristic line** (**SCL**), is a graph of both the systematic and the unsystematic risk of a security. The **intercept** of the regression line is the alpha of the security while the **slope** of the line is equal to its beta.

## Single-Index Model and the Capital Asset Pricing Model

The alpha of a portfolio is the average of the alphas of the individual securities. For a large portfolio the average will be zero, since some stocks will have positive alpha whereas others will have negative alpha. Hence, the alpha for a market index will be zero.

Likewise, the average of firm-specific risk (aka **residual risk**) diminishes toward zero as the number of securities in the portfolio is increased. This, of course, is the result of diversification, which can reduce firm-specific risk, but not market risk, to zero.

Hence, the alpha component and the residual risk tends toward zero as the number of securities are increased, which reduces the single-index model equation to the market return multiplied by the risky portfolio's beta, which is what the Capital Asset Pricing Model predicts.

## Profiting from Alphas with Tracking Portfolios

The decomposition of a stock's return into alpha and beta components allows an investor to profit from stocks with positive alpha while neutralizing the risk of the beta component. Suppose that a portfolio manager has identified, through research, one or more securities with positive alpha but the manager also forecasts that the market may decline in the near future. The positive alphas indicate that the stocks are mispriced, and, therefore, can be expected to correct to their proper price in time.

To isolate the potential profits of the mispriced stocks, a portfolio manager can construct a **tracking portfolio**, which consists of a portfolio that has a beta equal to the portfolio with the positive alpha stocks, and sell it short. This eliminates market risk, since if the market declines, then the shorted tracking portfolio will increase in value by the same amount that the long position will decline due to its systematic risk as measured by its beta. Hence, the manager can profit from the positive alphas without worrying about what the market will do. Many hedge funds use this **long-short strategy** to profit in any market.