** FORMULA AND DERIVATION**

Different models have different ways of calculating cost of equity. The most widely used models are Capital Asset Pricing Model (CAPM) and Gordon model.

Cost of equity under CAPM:

The CAPM is a model that considers the relation between the risk and expected return for a security. The risk that is considered here is systematic risk or market risk, which is the risk that affects the returns of the entire market. It is also called un-diversifiable risk (risk that cannot be mitigated by diversifying our portfolio of securities). A measure that indicates a security’s exposure to systematic risk is the beta for that security. Beta(\(\left(\beta \right)\) ) is a measure of a security’s volatility of returns (compared to market returns). A beta of 1 indicates that the security’s return is similar to the market returns. A beta of value higher than 1 indicates a security with returns that are more volatile than the market returns. Beta value less than 1 means that the returns are less volatile than market returns. A negative value of beta indicates that the return on the security is inversely related with market returns.

The CAPM approach towards cost of equity is based on the theory that the expected return on equity would be higher than the risk-free rate of return. This extra margin of return, above the risk-free rate, is called the equity risk premium. It represents the premium (additional reward) to be provided to shareholders for assuming a greater risk (by investing into a company’s shares) than the risk from other securities like debentures and other debt.

So, the conventional formula for cost of equity is

\[cost\ of\ equity=risk\ free\ rate+equity\ risk\ premium\ \]

If \(r_i\) is the cost of equity, \(r_f\) is the risk-free rate and \(r_m\) is the market return, then the above equation can be written as follows:-

\[r_i=\ r_f+\left(r_m-r_f\right)\]

where,

\(\left(r_m-r_f\right)\) is the equity risk premium, i.e. the excess of market returns over the risk-free rate.

But this formula does not exactly determine the cost of equity for a company, as it does not take into consideration the beta of the company’s stock. A security with a high beta is more volatile than the market (more risky) and must therefore provide a higher risk premium, whereas a low beta security is less volatile than the market and must therefore provide a relatively low risk premium. To incorporate this factor into the formula, the risk premium is multiplied by beta of the stock.

So, the actual formula that determines the return on equity or cost of equity is

\[cost\ of\ equity=risk\ free\ rate+\ \left(beta\ \times equity\ risk\ premium\right)\]

\[r_i=\ r_f+\ \beta \left(r_m-r_f\right)\]

Cost of equity under Gordon model:

\[P=\ \frac{D}{r-g}\]

where,

P is the price of the share

D is the amount of the next dividend

r is the return on equity or cost of capital

g is the expected growth rate of the dividend yield.

The above formula can be rearranged to give

\[r=\ \frac{D}{P}+g\]

This is the formula to calculate the cost of equity under Gordon model. Here, the cost of equity is the sum of the current dividend yield (amount of dividend/price of share or \(\left({D}/{P}\right)\)) and the growth in dividend yield.

So, the formula can be written as

\[r=d+g\]

Where,

\(d\) is the dividend yield \(\left({D}/{P}\right)\)

\(g\) is the growth rate of the dividend yield.