May 2013

Equity Risk Preiums

By Victor Modugno


This is a summary of my paper, “Estimating Equity Risk Premiums,” published online by the Society of Actuaries1. The purpose was to help actuaries develop forward thinking long-term estimates of future equity risk premiums. Equity risk premium is the amount by which the total return of a stock market index exceeds that of government bonds. Equity risk premiums, calculated from historical data, have been used to project long term values of equity portfolios in retirement plans. The validity of using historical data to project future equity returns was examined along with other forward looking methods.

My paper was primarily a literature review. The Bibliography contains summaries of 25 papers reviewed. References for the data in this summary can also be found in there. The best papers include Damodaran (2012), which contains a description of methods and the CFA Institute (2011), which contains an update of 11 papers published before the financial crisis. Most of the economic literature focuses on individuals who are concerned with short term losses and inflation. U.S. private sector defined benefit plans typically pay fixed dollar benefits with less concern about market fluctuations and inflation. There was one actuarial paper, Derrig and Orr (2004 – pre-financial crisis) on this topic.

An excel model was constructed to back test equity forecasts based on various methods of calculating equity risk premiums. One conclusion reached was that arithmetic mean of historical returns produced estimates that were consistently too high and geometric mean was a better estimator.

Historical Equity Risk Premiums

The historical equity risk premium (ERP), also referred to as the realized ERP, ex post ERP or the excess return, can be defined as the return of a stock market index minus the risk free return calculated as an annual percent over some historical period. The historical ERP can be expressed as an arithmetic average of the annual rates or a geometric average, which is the total return over the period. In the U.S., the S&P 500 index or its predecessors are frequently used to measure stock market returns while the 6 month Treasury (T) bill or 10 year T-bond are used for the risk free rate. The 10 year T-bond is used here due to its long history and the long term nature of pension benefits.

From the founding of the New York Stock Exchange, which commenced trading five bank stocks and government bonds in 1792, to the present, the geometric excess return of stocks over 10 year U.S. T-bonds is 3 percent. There have been long periods where bonds have outperformed stocks, including a 40 year period from 1969 to 2009. Table 1 below breaks this return down by century while Table 2 shows international comparison.



The following Table decomposes equity returns by source:

Issues in Using Historical Data to Determine the Ex Ante ERP

Using historical stock and bond market performance data to estimate future performance raises a number of issues beyond simply what time period and data series to use. These include the validity of using historical data to project future returns and whether arithmetic or geometric mean or some other measurement should be used.

Standard Error

The standard error of a sample is standard deviation divided by the square root of the number in a sample. Plus or minus two standard errors should cover 95 percent of the outcomes. Table 4 shows that even with 50 years of U.S. data, we cannot be confident that the equity premium is greater than 0. The standard error would be larger if the returns are correlated. 



A data series is stationary if the mean and standard deviation do not change over time. The U.S. went from a developing agricultural economy to an industrial economy in the mid-19th to 20th centuries to a service and technology based economy today. The stocks comprising the market that are being measured have changed significantly. The earlier market performance may not be predictive of the future. Significant stock market changes in the 20th century bring in to question the stationarity of the return series:

Geometric Versus Arithmetic Mean

The arithmetic mean of a sample of n entries is the sum of the entries divided by n while the geometric mean is the nth root of the product of the entries. In much of the statistical work, the historical returns each year are assumed to be independent and identically distributed random variables. However, investment returns are serially negatively correlated. As an extreme example, an investor with $100 has returns of 50 percent and minus 50 percent over two years. The arithmetic mean of these two returns is 0, but the investor ends up with $75.

Jacquier developed an unbiased estimate of the mean (U) from historical data by weighing the geometric (G) and arithmetic (A) means by the ratio of number of years in the projection (P) to the number of years in the sample (S):

U = A*(1-P/S) + G*(P/S)

As the projection time gets longer, the geometric mean becomes more important. When the projection time equals the sample time the geometric mean is the unbiased estimate of the mean. Since most pension work involves long projection periods, the geometric mean is a more appropriate measure for future projections.

Using Market Based Factors to Estimate ERP

Another method is using current market value measures to determine the ERP. The most commonly used of these implicit methods is the Dividend Discount Method. Under this method the value of equity is the present value of all future dividends. Assuming a constant growth rate in dividends (the Gordon Model):

ERP = Dividend Yield + (Dividend Growth Rate – Risk Free Rate)

The unknown quantity is the dividend growth rate. Payout ratios have fallen recently as firms use stock buybacks instead of dividends, so share buybacks less new issuance could be added. Retained earnings, whether used for share buybacks or reinvested should yield higher future dividend growth. If retained earnings are reinvested at the expected equity return rate (ERP + Risk Free Rate) with a constant payout ratio, then:

ERP = (Earnings/Price) – Risk Free Rate

Here the expected return on stocks is simply the earnings yield, 1/ (Price Earnings Ratio). Since earnings are an accounting construct that can change drastically each year, Shiller uses a rolling 10 year historical average. Both the Dividend growth and earnings yield models are consistent with long term historical equity returns.

Damodaran’s model uses cash dividends plus an estimate of share buybacks (averaging 4.7 percent over the past 10 years) with projected growth using consensus analysts’ earnings estimates for the next five years and then the risk free rate thereafter to obtain the total expected return on stocks.

Issues in Using Market Based Factors to Determine the ERP

In addition to determining what should be included in dividends and the growth rate for dividends, the ERP under implicit methods will be changing significantly over business cycles unlike historical ERPs, which change only gradually as new years are added to the historical data.


Dividend yields have been declining, partially due to a decline in payout ratios. Rather than using dividends, share buybacks have become a common way to return capital to shareholders. Prior to 2001, capital gains had lower tax rates than dividends. Also it’s easier to change or stop buybacks. By increasing share prices, buybacks increase the value of stock options, which have become a major component of executive compensation. Dividends could be adjusted to add buybacks less new issuance at the firm level. Looking at the market as a whole, all share purchases for cash (buybacks, LBOs) less share issuance (stock options, IPOs) could be added to dividends. These quantities vary significantly by year, but on average 2.2 percent of shares are bought back compared to 2 percent new issuance, leaving a net addition to dividends of .2 percent. Free cash flow (funds available to pay dividends) could also be used instead of dividends in these formulas.

Dividend Growth Rate

One possible assumption is continuation of the 1.34 percent per year real long term historical growth in dividends. Another is that dividends will increase with earnings (constant payout ratio), which is proportional to the increase with GDP. Since a portion of increased earnings will be captured by executives and entrepreneurs (stock options and IPOs), a lower amount such as per capita GDP could be used for existing shareholders.

Model of Long-Term Forecasting Accuracy

An excel model was developed to test the forecasting accuracy of four different methods for 50, 40, 30, and 20 year time periods starting in January 1, 1962. The most common method is to use geometric total return of historical stock data going back to 1926 minus the total return of 10 year T-bonds during that period (HERP-G). The next most common is arithmetic mean of the same historical data minus the arithmetic return of T-bills for during that period (HERP-A). The next two are implied methods DAMDARAN and EARNINGS (reflecting the annual earnings yield). For Historical ERPs, data is used from 1927 to end of the year prior to the date of projection. For implied methods using T-bills or bonds for the risk free rate produces the same forecast. The following chart shows the accuracy of 30 year forecasts made in years between 1962–1982


This and other Charts in the paper show that long-term forecasts based upon the historical ERP using the arithmetic mean are far too high, with a few exceptions. The geometric mean is the better historical ERP forecast. The earnings yield forecasts are usually below the actual returns but tend to follow them.

ERP and Long-Term Stock Market Forecast 2012

As of 12/30/2011, the 6 month T-bill yielded .06 percent, while the 10 year bond was at 1.94 percent. The 10 year inflation adjusted yield was -.11 percent. The 30 year rates were 2.98 percent nominal and .78 percent for inflation indexed. The ERP and stock market return forecast using the methods in the last section and a few other ones are shown below. The ERP is based upon the 10 year T-bond except for the historical ERP based upon arithmetic mean, which uses T-bills. The dividend yield on the S&P 500 was 2.06 percent.

Basis for Estimate ERP Stock Market Forecast

Earnings Yield 5.78% 7.72%
Shiller 10 Year Earnings Yield 2.67% 4.61%
Damodaran 6.04% 7.98%
Historical Geometric from 1927 4.10% 6.04%
Historical Arithmetic from 1927 2 7.62% 7.68%
Historical Geometric from 1792 3.00% 4.94%
Hassatt 2.87% 4.81%
Fernandez 4.00% 5.94%
1.96 x Baa Credit Spread 5.64% 7.58%
Gordon DDM3 3.60% 5.54%



Stationarity and standard error would indicate that there is significant uncertainty in using any historical ERP estimate to forecast returns. On both theoretical and empirical grounds, the geometric mean is preferred to the arithmetic mean for pension plans. Using the arithmetic mean would have led to forecast returns substantially higher than those actually realized.

Implicit or market based ERP methods have the advantage of reflecting current market conditions. When pension plan stocks are valued at market as of the date of valuation, it would be consistent to have an ERP calculated as of the same day. Implied ERPs fall in bull markets and rise in bear markets, while historical ERPs do the opposite. Prior to 2000, the historical ERPs produced higher forecasts than implicit methods, but after 12 years of poor performance, the historical and implied ERPs are much closer.

Victor Modugno, FSA, MAAA, FCA, is a consulting actuary in Redono Beach, Calif. He can be reached at

1 /Research/Research-Projects/Pension/research-est-equity-risk-premiums.aspx

2 The Arithmetic Mean uses T-bills as the risk free rate, while all others use 10-year T-bonds.

3 Dividend Discount Model using 2.06 percent dividend yield plus .2 percent net buybacks and real growth based upon 50 year historic average of 1.34 percent.