**Quick Summary**

1. Calculate the geometric average return on 10-year treasury bonds

2. Calculate the geometric average return on the S&P 500 Total Return index

3. Estimate the historical equity risk premium as the excess returns of stocks over Treasury bonds. Calculate the standard error to check for reasonability.

**1. Geometric average return on 10-year treasury bonds**

One method for estimating the equity risk premium involves looking at excess historical returns on stocks above risk-free securities. This requires making assumptions about the equity index to represent equity market returns, the time period for computing the estimate, the type of mean calculated, and the proxy for the risk-free return. Based on the parameters used in the calculation, the premium estimate can vary greatly. Best practice suggests using a long-term government bond for the risk-free security, a very long time period to minimize the standard error of the estimate, and the geometric average to compute returns.

Historical data on US Treasury bonds can be found on the website for the Federal Reserve Bank of St. Louis (here). On the *Research & Data* tab, select *FRED Economic Data*. Choose the *10-Yr. Treas. Rate* link in the *At A Glance* section. Once on the page for the historical 10-year Treasury rates, select *Max* for the time horizon. Scroll down to the *Edit Data Series* section and change the return *Frequency* from *Daily* to *Annual*. The *Aggregation Method* should then be changed from *Average* to *End of Period*. Navigate back up to the tabs immediately below the maturity rate graph and select the *Export* tab and then *Export to Excel*.

The rate on a Treasury is only one component of its return. Returns are also affected by price changes due to interest rate fluctuations. Use the following equation to compute the annual return on a 10-year Treasury bond. The first component (in brackets) captures the gain or loss due to interest rate changes. The second component is the promised coupon rate on the bond. After the annual returns are determined, calculate the geometric average return from 1988 to the present.

**2. Geometric average return on the S&P 500 Total Return index**

To accurately estimate the historical returns on stocks we must assume that all dividends are reinvested over the period. This requires the use of a total return index. Four of these exist, each with a different base period (1936, 1970, 1988, and year-to-date). Unfortunately, a subscription to a service provider like Compustat is required to access these, except for the index with the start date of 1988.

The total annual return for the S&P 500 index can be found on the S&P 500 Dow Jones Indices website (here). Click on the drop-down button for *Additional Info* and select *Monthly and Annual Returns*. The downloaded Excel file will have annual total returns in column E of the second tab. Since the annual changes are already calculated for you, all that is left to do is determine the geometric average returns over the period.

**3. Historical Equity Risk Premium and Standard Error**

After the geometric average returns have been calculated for stocks and Treasury bonds, the historical equity risk premium is estimated as the difference between the two. It is also worth calculating the standard error of the mean to ensure there is not too much noise in the estimate. In order to make this calculation, it is first necessary to calculate the difference in returns for each year of the period. The standard error is then calculated as the sample standard deviation of these spreads divided by the square root of the number of observations.

Using the method and data sources delineated above, the geometric average return for stocks and treasury bonds is 10.27% and 6.74%, respectively. This means that the historical equity risk premium for the period from 1988 to 2015 is 3.53% (=10.27% – 6.74%). The sample standard deviation of excess returns is 21.50%, therefore the standard error of the mean is 4.06% (=0.215 / √28). A 95 percent confidence level would imply a range of ±4.06% * 1.96 = ±7.96%, which is arguably too large to rely upon. Unfortunately, lowering the standard error would require access to a total return index with an earlier start period.

**Concluding Thoughts**

The analysis outlined above attempts to mirror the same methodology employed by Aswath Damodaran to calculate the historical equity risk premium (here). Using Damodaran’s complied dataset to determine the risk premium from 1988 to 2015 yields an estimate of 3.40% with a standard deviation of 4.05%. The variance in estimates is attributable to the different data sources used.

The advantage of Damodaran’s dataset is that it includes returns as far back as 1928. (The data likely comes from Ibbotson Associates’ Stocks, Bonds, Bills, and Inflation (or SBBI), which is currently available through licensing with Morningstar). The equity risk premium calculation using the 88 years of data yields an estimate of 4.54% with a standard error of 2.29%.

According to the CFA Level II curriculum, the equity premium relative to bonds historically is 3.2% globally. The premium is higher for the US, though it is left unspecified. The authors go on to note that some analysts adjust historical US equity premium estimates downward by 1.25% due to favorable circumstances in the past that are unlikely to continue.

**Suggested Reading**

Mercer. *A Review of the Equity Risk Premium*.

J.P. Morgan Chase. *The Most Important Number in Finance*.