Finance

Introduction
The cost of equity is the opportunity cost of raising funds through equity. In other words, it is the minimum rate of return that a firm must offer to its shareholders to compensate for bearing risk and waiting for their returns. A firms shareholders always have an opportunity to sell their existing shares and invest in another company. The return they require from investments in a firm is, at the same time, the firms cost of equity, that is, the opportunity cost of raising funds through equity. There are three basic models of estimating the firms cost of equity Dividend Growth Model, Capital Asset Pricing Model (CAPM), and, Arbitrage Pricing Theory (APT) model. Each of these models has different assumptions, inputs, ease of use and accuracy. On the basis of these criteria the Capital Asset Pricing Model can be considered as the best model to estimate the require rate of return for the shareholders.

Three Models
Dividend Growth Model This model is based on the expected growth of the future dividend issued by the company to its shareholders. In dividend growth model

Required Rate of Return  (Next expected dividend)  (Current Share Price)  Growth Rate
Capital Asset Pricing Model This model is based on the theory that the shareholders require compensation for the market risk they bear.

Required Rate of Return  Risk free rate of return  Beta  (Market Risk Premium)
Arbitrage Pricing Theory APT assumes that the return on an asset is linearly related to a set of risk factors as shown below

Required Rate of Return  Risk Free Return on Asset   Sensitivity of asset  (Risk Factor)

Ease of use
CAPM is very simple and easy in use. It requires only three inputs for the estimation of the required rate of return risk free rate of return, beta and the market premium. If the firm is already listed in a market then its beta can be calculated using the historical price movement of the share with respect to the market price. In case the firm is not listed in any market then the beta can be estimated using the other similar firms in the industry which are already listed in the market. The risk free rate can be estimated by the looking at the yields of bonds issued by the government. And, finally market premium is the premium over risk free return, which market pays for the additional risk taken. It can also be estimated using the historical market prices (Ross, Wetserfield,  Jaffe, 2005).
Dividend growth model is also a simplistic model but there is a significant challenge in estimating the reasonable growth rate. Commonly three approaches are used to estimate the growth rate using analysts forecasts, the historical time series approach, and the sustainable growth model. The analysts forecasts method can be quite unreliable at times due to the biasness of the analysts towards a particular industry or company (Arnold, 2008). The other two approaches require analysis of past data of the firm with detailed analysis of financial figures of the firm.

The APT model is quite cumbersome in nature. Since the model does not specify the factors which can affect the riskiness of the firms, the first task in using APT is to identify various risk factors which can add to the risk profile of the firm. This may require the use of multivariate techniques like factor analysis. Then it requires estimating the sensitivity of the firms share price with respect to the risk factors identified, obviously most of the times this will not be easy with the availability of the data regarding the risk factors (Fabozzi, Focardi,  Kolm, 2006). The model becomes complex as the number of factors increases so their interpretation also become abstruse. Empirical studies of this kind so far suggest that there is hardly any consistency in terms of the number of basic factors, the interpretation that may be put on these factors (typically, the factors identified are artificial construct representing several economic variables), and the stability of these factors from test to test.

Accuracy of Results
Several studies have been done to check the accuracy of above models with different-2 findings. In case of CAPM, several empirical studies have cross verified the linear relationship between the expected return and the market risk. Their finding reflects that the relation is linear the y-intercept is greater than risk free rate while the slope is less than market premium, so, the actual relationship may be flatter than what CAPM says in addition to beta, some other factors, such as standard deviation of returns and company size, too have a bearing on return (Chandra, 2007). While reviewing these empirical evidences, two important problems need to bear in mind. First, the studies use historical returns as proxies for expectations. This assumes that expected returns will be the same as realized returns. Second, the studies use a market index as a proxy for the market portfolio (Chandra, 2007). However, certain studies have shown that broad based indexes (such as SP 500 Index) mirror stock market movements quite well (Ross, Wetserfield,  Jaffe, 2005). So, the CAPM may not offer a surgical accuracy but its results are certainly conclusive enough to estimate the required rate of return.
Because many factors appear on the right hand side of the APT equation, the APT formulation has the potential to measure expected returns more accurately than does CAPM. However as mentioned earlier, one cannot easily determine which are the appropriate factors. These factors cannot be derived theoretically.

In case of dividend growth model, the accuracy of results depends entirely on the accuracy of the forecasts about future dividends (Brigham  Ehrhardt, Financial Management, 2008). Accurate forecasting is not an easy task. Despite the increased sophistication of forecasting technology and more powerful computers, we are, after all, living in a world of uncertainty, and any forecast about the future should reflect this uncertainty. The PE ratio may be used just to check the cost of equity calculations quickly, but under very limited conditions.

Assumptions
The dividend growth model assumes, only dividends as the returns from an asset one gets out of it. However, in reality the returns are not only the dividend returns but also the capital gains. The model is also based on assumption that firms always issue dividends to its shareholders. In reality, there are number of firms in the market which are performing outstandingly but do not issue dividends to their shareholders (such as Infosys). The model is also inapplicable to the firms which are not traded on the market, because it requires the current market price to estimate the required return.

The CAPM assumes that an investor always has a portfolio of shares and hence can diversify some of the risks, and these risks are called unique risks. There is still some part of the risk that cannot be diversified which is called market risk. So CAPM assumes that firms compensate the shareholders for only the market risk taken by them. Some more assumptions used by CAPM are as following
Individuals are risk averse.

Individuals seek to maximize the expected utility of their portfolio over a single period of planning horizon.

Individuals have homogeneous expectations  they have identical subjective estimates of the means, variances, and co-variances among returns.

Individuals can borrow and lend freely at a riskless rate of interest (Arnold, 2008).
The market is perfect there are no taxes there are no transaction costs securities are completely divisible market is competitive.

Looking at these assumptions, one may feel that the CAPM is unrealistic. However, the value of model depends not on the realism of its assumption, but on the validity of its conclusions.

Although, the APT seems an extension of CAPM, but it does not require above assumptions as it is in case of CAPM. The APT only assumes that the capital markets are perfectly competitive and that investors always prefer more wealth to less wealth with certainty (Ross, Wetserfield,  Jaffe, 2005).

Conclusions
Notwithstanding the problems mentioned above, the CAPM is the most widely used risk return model. It has advantages over other models in terms of its simple and intuitive nature. Its basic message that diversifiable risk does not matter is accepted by nearly everyone. It offers some objective estimate of risk premium which is better than a completely subjective estimate or no estimate.