When investing in a security or a portfolio of assets, you are subject to two types of risk: systematic risk and unsystematic risk:

*Total risk of investment* = *systematic risk* + *unsystematic risk*

Systematic risk is the risk inherent in all investments to one degree or another. Unsystematic risk is company or industry-specific. We can reduce, and even eliminate, unsystematic risk by investing in a well-diversified portfolio of securities. In contrast, systematic risk is undiversifiable.

**What is Systematic Risk?**

Systematic risk is often referred to as “market risk.” It measures the degree to which a security’s return is affected by external economic forces, such as inflation, changes in interest rates, world politics, and economic growth. When an investor holds a well-diversified portfolio, it is the only relevant risk since the unsystematic risk has been diversified away. Individual securities and portfolios of securities differ in their amount of systematic risk.

**What is Unsystematic Risk?**

Unsystematic risk is the uncertainty that arises due to the unique business and financial aspects of the firm. These uncertainties can be diversified away by holding other securities. For example, a pharmaceutical firm’s profits may be affected by a lawsuit filed against it if one of the medications it produces has an adverse effect on the users. But this incident may lead to higher profits for other pharmaceutical companies that produce a medication to treat the same symptoms that doesn’t have these adverse effects.

**How Systematic Risk Is Calculated**

The systematic risk of a security or a portfolio of securities is measured by its Beta (β). Beta measures the comovement of the security’s (or portfolio’s) return with the market. The mathematical formula for beta is as follows:

where Covariance(i, M) is the covariance of the security’s returns with the market returns and Variance_{M } is a measure of the volatility of the market in general.

But what do we use as the “market” in this equation? Any broad market indicator can be used as a proxy for the market portfolio, but the S&P 500 Index is the one that is commonly used. So, to apply this formula, we would calculate the covariance of the returns of our security (or portfolio of securities) with the returns on the S&P 500 Index and divide that number by the variance of the returns on the S&P 500 Index.

As an example, if we have determined that the covariance of the returns on our stock and the S&P 500 Index is 0.6 and that the variance of the returns on the index is 12, then the β of our stock is 0.5:

This means our stock is half as volatile as the market.

Excel makes it relatively easy to calculate the beta of a security, given that you know the returns. The calculation of the beta of the returns on the stock of Amazon.com is shown below. Actual data for the years 2000 through 2016 has been used.

Beta Calculation | ||

Year | S&P 500 Index | AMZN |

2000 | -9.03% |
-79.56% |

2001 | -11.85% | -30.47% |

2002 | -21.97% | 74.58% |

2003 | 28.36% | 178.56% |

2004 | 10.74% | -15.83% |

2005 | 4.83% | 6.46% |

2006 | 15.61% | -16.31% |

2007 | 5.48% | 134.77% |

2008 | -36.55% | -44.65% |

2009 | 25.94% | 162.32% |

2010 | 14.82% | 33.81% |

2011 | 2.10% | -3.83% |

2012 | 15.89% | 44.93% |

2013 | 32.15% | 58.96% |

2014 | 13.52% | -22.18% |

2015 | 1.36% | 117.78% |

2016 | 9.84% | 10.95% |

Covariance(AMZN, S&P) | 0.059581 | |

Variance S&P | 0.031993 | |

Beta of Amazon | 1.862331 |

This indicates that the returns on Amazon.com are far more volatile than the returns on the S&P 500 Index, which you can observe by looking at the data. When the market was down, Amazon’s returns were down significantly more in most cases; likewise, when the market was up, Amazon’s returns were significantly higher in most years.

Another method of calculating beta is to regress the returns of the security (or portfolio) on the returns on the market portfolio. More commonly, the returns on the security in excess of the risk-free rate are regressed against the market returns in excess of the risk-free rate. The slope of the resulting regression line is the beta.

Regardless of which method is used, there is an issue inherent in these calculations because historical data is being used, and the past is not a good predictor of the future. For one thing, the composition of the firm may have changed. For this reason, many companies that report betas report what is called an “adjusted beta” to better reflect the movement of the future returns of the stock relative to the market. Because they sell their services, these companies don’t provide the formula that they use to adjust the beta, however.

The beta of a portfolio of securities can be easily calculated as a weighted average of the betas of the individual securities in the portfolio. For example, if you invest 1/3 of your funds equally in each of three securities with betas of 1.0, 0.3, and 2.0 respectively, the beta of your portfolio can be calculated as follows:

Β_{p} = 1/3(1.0) + 1/3(0.3) + 1/3(2) = 1.1

**The Fundamentals of Beta**

The beta of the market portfolio, regardless of the proxy used, is equal to 1.0 since the covariance of the market with itself is equal to the variance of the market.

Thus, if a stock has a beta of 1.0, it is considered to have the same risk as the market. If the market increases 10%, that stock can also be expected to increase by 10%. However, if the market decreases 10%, that stock can be expected to decrease by 10%. A beta of zero indicates a security that does not move with the market. Its covariance with the market portfolio is zero, and it is risk-free. U.S. Treasury bills, for example, would have a beta of zero.

A negative beta is indicative of a security for which the returns move opposite the market. Its returns have a negative covariance with the market portfolio. If the S&P 500 Index increases, we would expect the returns on that security to decrease. On the other hand, if the returns on the S&P 500 Index fall, the returns on that security would be expected to increase. Adding stocks with negative betas to a portfolio serves to decrease the risk of the portfolio. There are some time frames in which the covariance of a particular foreign stock index and the S&P 500 index has been negative; thus the beta of the portfolio comprising that foreign stock index would be negative This suggests that international diversification can be helpful in reducing the overall risk of a portfolio.

The betas of most securities fall between 0.0 and 2.0 when measured over the long-run. Betas are not stable over time, however, particularly for individual securities. The beta of a portfolio of securities tends to be more stable since a downward movement in the beta of one of the securities in the portfolio may be offset by an upward movement in the beta of another of the securities in the portfolio.

Because unsystematic, or company-specific, risk can be diversified away, researchers have concluded that the only risk investors are rewarded for taking is systematic risk. In other words, the expected return on a security or portfolio of securities is based on its level of systematic risk, i.e., its beta. An investor can, therefore, use this information to decide on his or her personal risk/return trade-off. A very risk-averse investor might decide to invest his investment monies in Treasury securities and low beta stocks. An average risk investor would target an investment portfolio that has a beta of 1.0, which would offer him the expected return on the market in return for the risk level he is assuming. A risk-lover would choose high beta stocks for his portfolio, taking on a lot more risk to earn a return significantly higher than what the market is expected to return.