Winning the game of investing is largely about understanding what the market (NYSEARCA:SPY) perceives about the future and determining if that perception is right or wrong. In other words, taking a different view from the market and being right about it. But if you don’t know what the market is thinking in the first place then how can you even begin to beat the market? For this reason we need to understand or at the very least estimate the risk the market is assessing about the future. This is analogous to determining the *expected* risk investors are willing to pay for equities. More specifically, we need a measure of the excess risk investors are willing to pay for equities over a risk-free asset such as a government bond. This is known as the expected equity risk premium and is likely the most important variable in finance as it helps us estimate the opportunity cost investors are willing to take on a particular risky asset.

There are two common approaches to estimating the equity risk premium, a historical and a forward-looking approach. In this paper we will apply both approaches to help us assess the expected risk investors have given to U.S. equities over time and the expected risk investors perceive of equities into the future.

**Historical Equity Risk Premium**

The historical approach is calculated by taking the average differences between the return of a broad equity market index, such as the S&P 500, and the return of a government bond or bill, such as U.S. 10-year treasuries, over a specified time frame. It is sometimes called the “future equals the past” approach because it assumes reversion to a mean value and it also assumes that factors which have described returns in the past will continue to describe returns into the future. As many of us well know this will not always be true. If you believe returns will revert to a historical mean then it may be okay to use the historical record, but remember we are trying to estimate an *expected* risk premium which depends on *expectations of the future* not the past. Consequently, adjustments may be needed when looking forward.

A related issue is that we assume an equal set of positive events have balanced out an equal set of negative events over the past time frame. If this is not the case we may have bias in our estimate of the mean equity risk premium. There are a plethora of events researchers have suggested that may have biased the equity risk premium upward (Dimson, Marsh, Staunton). For example, the change in regulations to allow pension funds to allocate a percentage of their portfolio to venture capital firms in the late 1970s opened the door for new investments into smaller riskier companies. The massive credit expansion from the 1980s to the 2000s allowed for a greater amount of people to purchase new homes and products. The transformation of closed world economies to free world economies allowed new companies to be created and along with globalization allowed for free trade of products, services and ideas. The investment into information technology infrastructure allowing for greater opportunities for new companies to flourish in the 1990s. We may never have such events of this magnitude contribute to equity returns again. If you believe this to be true, we may need to adjust the equity risk premium downward to reduce bias in the historical data set due to these past phenomena.

Let’s first take a look at historical returns of the S&P 500, U.S. Treasury Bonds and Bills.

*Source: **Damodaran*

Perhaps not surprising but always amazing, investing $100 in U.S. equities with dividend reinvestment in 1928 would have returned you over $382,850 by 2018. In comparison, investing $100 in U.S. bonds returned only $7,308 and in U.S. bills only $2,063 by 2018.

Now let’s take a look at the annual differences between stock returns to bond returns and stock returns to bill returns. As mentioned before, if we take the average of these differences we will have calculated the historical equity risk premium. It’s clear from the chart below that the annual differences in returns have and will vary over time. Consequently, this will vary the mean equity risk premium.

*Source: Damodaran*

Finally let us plot the distribution and calculate the mean and standard deviations of the annual risk premiums.

*Source: Damodaran*

Stocks-Bills | Stocks-Bonds | |

Arithmetic Mean | 7.93% | 6.26% |

Geometric Mean | 5.96% | 3.95% |

Standard Deviation | 19.90% | 21.19% |

Standard Error | 2.09% | 2.22% |

*Source: Calculations By Author*

We can see the large deviations from the mean as represented by the standard deviation and standard error. Assuming normality in the distribution of the historical equity risk premium, we can be 95% confident that the true population equity risk premium may lie within two standard deviations from the sample mean. In other words, the true population mean may lie between 3.75% and 12.11% for the equity risk premium relative to bills and 1.80% and 10.70% for the equity risk premium relative to bonds.

**Forward-Looking Equity Risk Premium**

The second approach is the forward-looking estimate of the equity risk premium. In many respects, this approach is more robust versus the historical approach because it is forward looking in nature and accounts for projected risks in earnings, cash flows and interest rates. Finance is a game about expectations of the future, more so than about descriptions of the past. We can utilize the H-Model to help us estimate a forward-looking equity risk premium. This simple discounting model forecasts current dividends *D _{0}* into the future and discounts those future dividends by a required return

*r*to find the current value

*P*today. The only complexity of this model is that it accounts for two growth rates, a short-term rate

_{0}*g*and a long-term growth rate

_{S}*g*and assumes a linear change in growth over

_{L}*H*half-life years. The equation below has two terms. The first term models the present value of the asset in the long term and the second term models the present value of the asset in the short term. Summed together we have the total present value of the asset in question.

If we rearrange the H-model and do some algebra to solve for *r* we find the required return or implied required return, given estimates of the other variables.

We can then subtract the yield on a government bond *rf* to find the expected equity risk premium *E(ERP)*.

Why are we using this model? As investors for any asset (or equity market in this case), we have a minimum return that we require in order for us to invest in that asset over a certain time period given the risk of that asset. This required return reflects the relationship between future cash flows and the value today. The H-model is doing just that, it is equating future cash flows to a price today using a required rate of return. What we are attempting to do is solve for the market's required return, given the market's forecast of future cash flows and the current price the market is paying today for those future cash flows.

The dividend component in our equation will comprise of dividends and buybacks of the S&P 500. In estimating the short-term growth rate, we will use consensus analyst S&P 500 earnings growth estimates from Zacks Investment Research beginning in 1985. We will assume that earnings growth will track dividend growth over time and maintain a constant payout ratio of dividends from earnings. To help avoid any subjective bias, we will assume the long-term growth rate in perpetuity will be the 10-year government bond yield rather than use forecasts from other financial analysts and economists. Lastly, we will assume the short-term rate will gradually shift over a 10-year period to the long-term growth rate (half-life years equal to 5). Already you can see the many assumptions going into this model. Garbage in, garbage out as they say. Fortunately, if we are consistent in our input variables, we may be able to reliably estimate changes in the expected equity risk premium over time. Let’s calculate the expected equity risk premium in 2018 using the H-Model and input variables we specified.

Solving for *r* will return a value of 8.67%. Subtracting the T. Bond Rate of 2.68% will equate to a forward-looking equity risk premium of 5.99%. We can also chart the past expected risk premiums using the forward-looking approach.

*Source: Calculations By Author*

*Source: Damodaran*

A significant difference between the two approaches can be seen in 2008. If you look at the earlier chart where we plotted annual equity returns minus bond returns, this value was negative in 2008. Therefore, when calculating an average equity risk premium over the past time frame, a lower value would be added into the computed average, which would lower the equity risk premium. However, this does not make intuitive economic sense. As equity markets fell in 2008, investors should demand a higher risk premium over bonds as risk has increased.

In contrast, when using the forward equity risk premium model, the equity risk premium increases, which does make intuitive economic sense. As equity markets fall, an investor would require a greater return in order to compensate for the greater risk perceived in equities. This is just a mathematical necessity. If we again look at the equation of the H-Model and keep the numerators *D _{0}( 1 + g_{L })* and

*D*constant (which is analogous to keeping analyst predicted growth unchanged), but decrease

_{0}H( g_{S}- g_{L })*P*(which is analogous to a drop in the market value of the S&P 500), then

_{0}*r*(the required return) will have to increase in order to keep the equation balanced.

The difference between the historical and forward-looking approach can also be seen in the years 1999-2000. When using the historical approach, the equity risk premium reached new highs, while the forward-looking equity risk premium reached new lows in 1999-2000. This leads us to some interesting insights. While a string of positive market returns increases optimism for equities and increases the historical equity risk premium, the forward-looking risk premium may have actually decreased. On the other hand, while a string of negative market returns decreases optimism for equities and decreases the historical equity risk premium, the forward-looking risk premium may have actually increased.

*Source: Calculations By Author*

Forward-Looking Equity Risk Premium | |

Average | 4.36% |

Median | 3.93% |

Max | 8.31% |

Min | 1.60% |

Standard Deviation | 1.79% |

*Source: Calculations By Author*

Let’s look at the distribution of forward-looking equity risk premiums. The distribution is rightly skewed, which implies a greater frequency of small risk premiums and a lesser frequency of large risk premiums have occurred in the past. If history is any guide, then one would expect a greater likelihood of risk premiums to be less than 4.36% (the average) going into the future. The expected equity risk premium today is equal to 5.69%. The market is demanding a greater risk premium as compared to the average and median forward equity risk premiums of the past. If you believe the market is correctly forecasting cash flow growth, then the market may have already fallen to a price that compensates for future uncertainty; thus today may be a good time to allocate towards equities.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.