The cost of equity is the rate of return expected by shareholders for holding a risky share in the business so it is the price of the risk that they take. The rate of return is defined as the total rate of return:

• Rate of return over the period = (Dividend + Increase in share price) / (Opening share price)

Companies’ risk can be broken down into specific risk (also called non-systematic) and unspecific risk (also called systematic). As investors can diversify away the specific risk but can’t get rid of the unspecific risk, it is only the non-diversifiable risk that they should expect to be compensated for. This is the cost of equity.

A common approach for estimating the cost of equity of a quoted company is to take a historical view, calculate the rate of return of the company share and average over time. There is controversy in the academia and among practitioners whether it is better to use an arithmetic or a geometrical average. A geometric average is more appropriate to model the behaviour of buy-and-hold investors, whereas arithmetic averages better portray a one-year invest horizon. The geometric average also has the advantage that, unlike the arithmetic average, it is unbiased by the measurement units (days, weeks, months). Note that the arithmetic average is typically 2% points higher than the geometric average. As there is no consensus between practitioners, a pragmatic approach is to take the average between the arithmetic and geometric averages.

Although historical returns provide a convenient approach, they have two major drawbacks:

• the future cost of equity is not necessarily the same as the past cost as the risk of a company’s businesses might be changing over time and the price of risk expected by shareholders is also believed to change over (long periods of) time
• the cost of risk of an individual project is not the same as the cost of risk for the company as a whole, which represents an average across all projects and businesses the company is engaged in.

For a non-quoted company or an individual project, a possible work-around is to find a quoted company with a similar risk structure and use it as proxy. Both companies should have a similar un-levered cost of equity. Finding a proxy is not always possible though. In the extreme case, it is necessary to derive the cost of equity from a Monte Carlo simulation applied after having identified the key uncertainties in the input parameters and their volatility.

Estimating the cost of equity from a simple model

The challenge here is to find a model for the forward-looking cost of equity. One idea is to express the cost of equity as a function of macroeconomic parameters that are changing over time and can be forecast, in particular the risk-free interest rate and the market risk premium, to derive a measure of the intrinsic risk of the business. Performing a historical regression analysis against changing parameters allows us to derive estimates for parameters that characterise the business and are broadly constant over time.

The most common model used in practice is the Capital Asset Pricing Model (CAPM). It is important to be clear that this is a model only. There are other models available for instance the Arbitrage Pricing Model (APM), which regresses the return on equity not on a single factor, the market risk premium, but on a larger number of macroeconomic indicators. These models are more complex to apply and go beyond the scope of this book.

The CAPM states that the expected rate of return of a given asset can be expressed as the sum of a market risk-free rate and an asset-risk factor, itself a function of the market-risk premium. E( ) designates the expected value of a variable.

•  E(rasset) = E(rf) + ßasset x [ E(rmarket)-E(rf) ].

E(rmarket)-E(rf) is the expected value of the market premium. Assuming that ßasset changes slowly over time, the value of the asset beta can be obtained by regressing the historical rate of return of the asset on the risk-free market rate and the market-risk premium. By definition, for the market portfolio, the expected rate of return E(rasset) is equal to E(rmarket) so that the market beta is equal to one.

The beauty of the CAPM model is that for an asset of a certain risk, we only have to estimate one parameter, beta, and then forecast the other market parameters, which are hopefully easier to forecast, to derive a forward-looking cost of equity for the asset. Note that the market parameters vary from one country to another though.

When beta is higher than one, then the asset is said to be more volatile that the market, and conversely, when beta is lower than one, it is less volatile. Volatile industries are: telecom equipment, air transportation, automotive manufacturers, investment banking. Less volatile ones are: food business, utility companies, telecom service providers, private banking.