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Equity risk premium (ERP) is the difference between the benchmark risk-free rate and expected equity return. Analysts use ERP to calculate a company’s cost of equity capital.

$$ \text{Company } ir_e={Er}_{(f)}+(ERP+IRP) $$

We can estimate the ERP of a company using two broad approaches:

- The historical approach (ex-post).
- The forward-looking approach (ex-ante).

This approach is used when reliable long equity return data is available. A historical ERP is the mean value of the difference between a government debt return and a broad-based equity market index return. Before using this approach, an analyst will need to determine if historical market returns provide useful information about the future. The four key decisions an analyst makes when coming up with a historical ERP are:

Analysts select an equity index that represents the returns equity investors earned in a market. They usually choose market-value-weighted indexes and broad-based equity indexes.

Analysts use tradeoffs to determine the best estimation time period. A longer data period may be problematic because the past may not represent the current market environment in a market. Fluctuating market volatility has an insignificant effect on estimates from long data series.

A shorter data period uses less representative contained in longer data series. This means that the ERP estimate closely represents the current market environment and current portion(s) of the business cycle. The tradeoff is that a shorter period has a greater likelihood of noise in the ERP estimate.

An analyst attempting to estimate ERP will either choose an arithmetic mean or a geometric to calculate the average difference between the benchmark risk-free rate and the returns from the equity market.

The * geometric mean return *best represents the compound growth rate equal to the first value to the last value of one money unit invested in an asset. It is the most preferred due to its ability to be consistent with expected terminal wealth estimates and low sensitivity to outliers.

The * arithmetic mean return*, or the average single-period return, best represents the mean return in one period. It is popularly used for single-period models, e.g., multifactor and capital asset pricing models.

The choices include YTM on long-term government bonds and short-term government debt. Government bonds are preferred to the highest-rated corporate bonds because of their near-zero default risk.

Several analysts prefer using a short-term bond rate as a proxy because a short-term government bond is a zero-coupon bond with a known return independent of coupon reinvestment. A limitation of using short-term government bonds is that they do not match the duration of most investments.

The industry tends to favor the use of long-term bonds as the risk-free rate proxy. The disadvantage of using the YTM on a long-term government bond is that it is not a known risk-free return at the time of purchase.

- ERPs can vary over time. ERPs that have shifted permanently to a different level in recent years will not produce estimates representing the future ERP.
- Survivorship bias inflates historical ERP estimates. This bias is present in data from equity markets when companies are removed from the index due to poor performance.

A forward-looking approach suggests that future expectations affect the ERP. Current information and expectations concerning such variables are used to estimate ERP. These estimates are often referred to as ex-ante or forward-looking estimates. The following methods are used in forward-looking estimation.

- Survey-based estimates.
- Dividend discount models.
- Macroeconomic modeling.

This approach involves gauging expectations by asking people questions. Such surveys reveal that the ERP is lower in developed markets as compared to developing markets. The limitation of using surveys is that estimates are sensitive to recent market returns.

The dividend discount model expresses the value of a share as the present value of future expected dividends. Gordon’s growth model is a simplified DDM used to estimate ERP. Gordon growth model equation:

$$ V_0=\frac{D_1}{r_e-g} $$

Where:

\(V_0\) = Value of a stock.

\(D_1\) = Year ahead dividend.

\(r_e\) = Required return on Equity.

\(g\) = Expected earnings growth rate.

The forward-looking ERP estimate is:

$$ ERP=E\left(\frac{D_1}{V_0}\right)+E\left(g\right)-r_f $$

We assume that P/E will remain constant since the prices, dividends, and earnings will grow at the same rate. If an analyst believes it is unlikely that this will happen in the future, they will make an adjustment to reflect the decrease or increase in P/E. This is because the return from capital appreciation cannot be less than the earnings growth rate unless P/E decreases. An analyst can assume multiple earnings growth stages for rapidly growing economies as follows:

- Fast growth stage.
- Transition growth stage.
- Mature growth stage.

The required rate of return, in this case, will be:

$$ \text{Equity Index Price}={PV}_{0,\text{stage } 1}+{PV}_{0,\text{stage } 2}+{PV}_{0,\text{stage } 3} $$

ERP estimates calculated using macroeconomic models depend on forecasted economic variables. These models are more reliable in developed markets where publicly traded equities represent a large proportion of the economy.

Grinold-Kroner’s decomposition of equity return is one such model.

$$ ERP=\left[\text{Dividend yield}+\text{Expected capital gains}\right]-E_{(rf)} $$

Or

$$ ERP=\left[DY+\text{Expected repricing}+\text{Earnings growth per share}\right]-E_{(rf)} $$

Where earnings growth per share is expressed as:

$$ \text{Earnings growth per share}=i+g+ \Delta S $$

Where:

\(i\) = Expected inflation.

\(g\) = Real economic growth.

\(\Delta S\) = Change in shares outstanding.

From empirical studies, the dilution effect or change in outstanding shares will vary from one country to another. For this reading, we assume \(\Delta S=0\).

$$ ERP=\left[DY+\Delta \left(\frac {P}{E} \right)+i+g+\Delta S \right]-E_{(rf)} $$

An analyst can estimate the expected inflation rate by comparing real and nominal yields for government benchmark bonds with a similar maturity.

- Survey-based estimates are often subject to response and sampling and behavioral biases such as confirmation and recency biases.
- DDM assumes the P/E is constant when dividends, earnings growth, and prices differ.
- Macroeconomic models could have behavioral biases or errors in forecasting.

## Question

Sarah is looking to estimate an ERP using the historical approach. Which of the following is the

least likelydecision that she will make in her estimate of an ERP:

- The equity index to use to represent market returns.
- The mean measure to use.
- The expected growth in P/E.
## Solution

The correct answer is C.The expected growth in P/E is used in the forward-looking approach to estimating ERP.

A and B are incorrect. The equity index and the mean measures are two of the four decisions needed to estimate ERP using the historical approach.

Reading 20: Cost of Capital: Advanced Topics

*LOS 20 (c) Explain historical and forward-looking approaches to estimating an equity risk premium.*