The equity risk premium (ERP) is the additional return (premium) required by investors for holding equities rather than risk-free assets. It is the difference between the required return on equities and the expected risk-free rate of return.

$$\text{Required return on equity}=\text{Current expected risk-free return}+\text{Equity risk premium}$$

Analysts estimate the required return in two ways:

$$\text{Required return on share }i=\text{Current expected risk-free return}+\beta_{i}(\text{Equity risk premium})$$

Or

$$\text{Required return on share }i=\text{Current expected risk-free return}+\\ \text{Equity risk premium}± \text{Other risk premia or discount appropriate for }i$$

There are two approaches for estimating the equity risk premium.

1. Historical estimates
2. Forward-looking estimates

## Historical Estimates

A historical equity risk premium is calculated as the mean value of the differences between broad-based equity market index returns and government debt returns over a specific period.

Historical equity risk premium estimation involves the selection of the following:

• The equity index to represent equity market returns.
• The period for computing the estimate.
• The type of mean calculated.
• The proxy for the risk-free return.

Analysts attempt to select an equity index, usually broad-based and market-value weighted indexes, that accurately represent the average returns earned by equity investors.

Extending the length of the sample period increases its precision. Therefore, the best option is to use the longest reliable return series available.

There are two approaches for computing the mean and two choices for the proxy for the risk-free return. The mean return of the difference between equity market index returns and government debt returns can be calculated using a geometric mean or an arithmetic mean.

1. A geometric mean equity risk premium estimate is the compound annual excess return of equities over the risk-free return or
2. An arithmetic mean equity risk premium estimate is the sum of the annual return differences divided by the number of observations in the sample

The risk-free rate can be computed in two ways:

1. As a long-term government bond return.
2. As a short-term government debt instrument (Treasury bill) return.

### Arithmetic Mean or Geometric Mean

The arithmetic mean return as the average one-period return best represents the mean return in a single period.

The geometric mean return of a sample represents the compound rate of growth that equates the beginning value to the ending value of one unit of money initially invested in an asset.

The geometric mean reflects future value more accurately relative to the arithmetic mean.

### Long-Term Government Bonds or Short-Term Government Bills

A short-term government debt rate such as a 30-day T-Bill rate or a long-term government bond yield to maturity can be used as the risk-free rate. Government bonds are preferred because they have near-zero default and equity market risk.

A risk premium based on a bill rate may produce a better estimate of the required rate of return for discounting a cash flow one year from now, but a premium relative to bonds should produce a more likely required return rate in a multiperiod scenario. Analysts should match the duration of the risk-free-rate measure to the duration of the asset being valued.

A historical risk premium may need to be adjusted to reduce the effect of biases in the data series being used and to take account of an independent estimate of the equity risk premium.

A common bias in the calculations is the survivorship bias. This arises when poorly performing companies are removed from an index and only good performing companies remain. Survivorship bias inflates the historical estimate of the equity risk premium. Analysts should make a downward adjustment to the estimate derived from such an index.

A series of positive or negative surprises may result in a series of high or low returns that increase or decrease the historical mean estimate of the equity risk premium.

## Forward-Looking Estimates

A forward-looking estimate is an estimation of the equity risk premium using current information and expectations concerning economic and financial variables. These estimates are called forward-looking or ex-ante estimates and they are subject to other potential errors like modeling errors and potential behavioral bias in forecasting.

### Gordon Growth Model Estimates

The Gordon Growth Model (GGM) estimate is more appropriately applied in developed countries where broad-based equity indexes are associated with a dividend yield and year ahead dividends are fairly predictable.

This model assumes a stable rate of earnings growth. An assumption of multiple earnings growth stages is more appropriate for rapidly growing economies, i.e., a fast-growth stage, a transition stage in which growth rates decline, and a mature growth stage characterized by moderate growth.

The Gordon Growth Model (GGM) equity risk premium estimate is:

\begin{align} \text{GGM equity risk premium estimate}&=\text{Dividend yield on the index based on year-ahead aggregate forecasted dividends and aggregate market value}\\&+ \text{Consensus long-term earnings growth rate}-\text{Current long-term government bond yield} \end{align}

i.e., $$\text{GGM equity risk premium estimate}=\frac{\text{D}_{1}}{\text{P}_{o}}+\text{g}-\text{r}_{\text{LTGD}}$$

#### Example: Gordon Growth Model Estimates

Consider the following information

• Dividend yield = 2%.
• Earnings growth = 8%.
• 15-year US Government bond yield = 6%.

The estimated market’s equity risk premium using Gordon Growth model is closest to:

#### Solution

$$\text{GGM equity risk premium}= 2\%+8\%-6\%=4\%$$

### Macroeconomic Model Estimates

Macroeconomic models use relationships between macroeconomic variables and financial variables used in equity valuation models to estimate the equity risk premium. The use of macroeconomic models is appropriate in developed countries where public equities represent a large share of the economy.

The equity risk premium can be estimated as:

$$\text{Equity risk premium}={[(1+\text{EINFL})(1+\text{EGREPS})(1+\text{EGPE})-1.0]+\text{EINC}}\\-\text{Expected risk}-\text{free return}$$

Where:

• $$EINFL$$ (Expected inflation) is calculated as the difference between the yields for long-term Treasury bonds and Treasury Inflation-Protected Securities (TIPS) of similar maturity.

$$\text{Implicit inflation forecast}=\frac{1+\text{YTM of 20 year maturity T}-\text{bonds}}{1+\text{YTM of 20 year maturity TIPS}}-1$$

• $$EGREPS$$ (Expected growth rate in real earnings per share) is equal to the sum of labor productivity growth and the labor supply growth rate which can be estimated as the sum of the population growth rate and the increase in the labor force participation rate.
• $$EGPE$$ (Expected growth rate in the P/E ratio) has a baseline value of zero reflecting an efficient market’s view. However, it may be positive (negative) if the analyst views the market as currently undervalued (overvalued).
• $$EINC$$ (Expected income component) can be based on the historical rate, but a forward-looking estimate based on the forward expected dividend yield can be used.

#### Example: Macroeconomic Model Estimates

Given the following data:

$$\small{\begin{array}{l|r}\text{Expected inflation} & 3.00\% \\ \hline\text{Expected real EPS growth} & 4\% \\ \hline\text{Expected growth in P/E ratio} & 0\% \\ \hline\text{Expected income component} & 3.80\% \\ \hline\text{20-year US government bond yield} & 6\%\\ \end{array}}$$

The equity risk premium is closest to:

#### Solution

\begin{align*}\text{ERP}&={[(1+\text{EINFL})(1+\text{EGREPS})(1+\text{EGPE})-1.0]+\text{EINC}}-\text{R}_{\text{F}}\\&=(1.03)(1.04)(1)-1+0.0380-0.06\\&=4.9\%\end{align*}

### Survey Estimates

Survey estimates of the equity risk premium involve asking a sample of experts about their expectations regarding the equity risk premium or their capital markets expectations from which the equity risk premium can be inferred.

## Question

Which of the following factors is least likely to be considered when using macroeconomic model estimates?

1. Expected inflation.
2. Expected growth rate in the P/E ratio.
3. Survivorship bias.

#### Solution

Survivorship bias is not considered in macroeconomic model estimates. It is a bias that arises when using the historical estimate approach of estimating equity risk premium.

A and B are incorrect. Expected inflation and expected growth rate in the P/E ratio are both factors that are considered when using macroeconomic model estimates.

LOS 21 (b) Calculate and interpret an equity risk premium using historical and forward-looking estimation approaches.

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