LOS E requires us to:

##### calculate and interpret the cost of equity capital using the capital asset pricing model approach and the bond yield plus risk premium approach

There are many methods of calculating and estimating the cost of common equity, the most prominent of them being: the Capital Asset Pricing Model (CAPM), Dividend Discount Model (DDM), and Bond Yield plus Risk Premium Approach.

# 1. Capital Asset Pricing Model (CAPM)

a. The CAPM developed by William F Sharpe, John Lintener, and Jan Mossin is one of the major developments in financial theory. The CAPM establishes a linear relationship between the required rate of return of a security and its systematic or undiversifiable risk or beta.

CAPM can be mathematically represented as:

**R _{e }= R_{f }+ β_{j }(k_{m }– R_{f})**

Where,

R_{e }= expected or required rate of return on security j.

R_{f }= risk – free rate of return.

β_{j }= beta coefficient of security j

k_{m}= return on market portfolio.

b. The selection of a ** risk-free rate** of return is usually guided by the duration of the project’s cash flow. So, if the duration of the project’s cash flow is expected to be around 10 years, then the risk-free rate on the government bonds, etc. of a similar duration should be considered.

c. ** Beta** is a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. It should be estimated relative to the market index.

d. The term ‘k_{m }– R_{f}’ is the expected ** market risk premium**, also referred to as Equity Risk Premium (ERP).

e. Here, as per the CAPM, we use β_{j }to estimate the risk associated with the project. This beta may not be able to capture all the risks. In such cases, we could use the **multi-factor models** with multiple betas.

The equation for such a model is:

**R _{e }= R_{f }+ β_{j1 }(Factor Risk Premium) + β_{j2 }(Factor Risk Premium) + …… + β_{jn }(Factor Risk Premium)**

## 1.1. Estimating Equity Risk Premium (ERP)

ERP can be estimated using one of the following methods:

**Historical Equity Risk Premium Approach. **

a. The two main components of ERP are market return and the risk-free rate. The historical approach makes use of historical data to find out an average market return and the risk-free rate of return.

b. While using this approach, analysts must consider different time periods to average the rates (i.e. 1-year period, 5-year period, 10-year period, etc.). They may also consider overlapping vs. non-overlapping time periods.

c. For averaging the returns, the analyst may use arithmetic or geometric means. A geometric mean may be used to measure the profitability of an investment over a multi-period horizon.

d. Some of the limitations of using such an approach are:

i. The levels of risks of an index can change over time,

ii. The levels of risk aversions can also change,

iii. Estimates are sensitive to the methodologies of choosing that we use, etc.

**Dividend Discount Model or Implied Risk Premium Approach. **

a. This approach is implemented using the Gordon Growth Model. According to this approach, the market returns are reflected in the long-run trends in growth rates.

b. Assuming a constant rate of growth of dividends, the following equation shows how the price is calculated using this approach:

**P _{o }= D_{1} / ( r_{e} – g)**

Where,

D_{1} = Dividends expected in the next period,

P_{o} = Current Market Value of Equity Market Index,

r_{e} = Required rate of return on the market, and

g = Expected growth rates of dividends

Solving this equation for the value of r_{e,} we get the value of the required rate of return on the market index:

**r _{e} = ( D_{1} / P_{o} ) + g**

Where,

D_{1} = Dividends expected in the next period,

P_{o} = Current Market Value of Equity Market Index,

r_{e} = Required rate of return on market, and

g = Expected growth rates of dividends

This value of r_{e} is considered as a market return and is reduced from the risk-free rate to get the value of ERP.

**Survey Approach.**

According to this approach, a panel of experts is asked for their opinion as to what should be the market rate of return. The average of all the responses is considered as the market rate of return on equity. This is reduced from the risk-free rate (which may also be suggested by the experts), to arrive at the figure of ERP.

# 2. Dividend Discount Model Approach

Instead of using this approach to find the value of equity risk premium (as done above), we can also use this approach to find the value or cost of equity.

According to this method:

**V _{o }= D_{1} / ( r_{e} – g )**

Where,

D_{1} = Dividends expected in the next period,

V_{o} = Current Market Value of Equity,

r_{e} = Required rate of return on the market, and

g = Expected growth rates of dividends

If we let the value of equity be the current market price of equity, we can find the value of r_{e} i.e. required return on equity, as follows:

**r _{e} = ( D_{1} / P_{o} ) + g**

Where,

D_{1} = Dividends expected in the next period,

P_{o} = Current Market Value of Equity Market Index,

r_{e} = Required rate of return on market, and

g = Expected growth rates of dividends

**NOTE:**

i. The term in the above equation is nothing but the forward annual dividend yield.

ii. Re-arranging the above equation, we get:

g=(1-D/EPS)×ROE

Where (D/EPS) represents the assumed stable dividend payout ratio and ROE is the historical return on equity.

iii. The term 1 – (D/EPS) represents the company’s earnings retention rate. Thus, the growth rate of a company equals its earnings retention rate times its return on equity.

# 3. Bond Yield Plus Risk Premium Approach

a. As per this approach, the cost of equity to a company equals the corporate bond yield plus some premium i.e. a spread representing premium against risk on investing in equity over the bonds.

**R _{e }= R_{d} + Risk Premium**

b. The risk premium is usually estimated from historical spreads between stocks and bonds.

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