## Discount Rate in Project Appraisal

Discount rate is a measure of risk and the cost of capital used to finance a project. There are various models, which have been designed, to determine the discount rate to use in appraising projects of multi-activity companies. The models include capital asset pricing model, arbitrage pricing theory, weighted average cost of capital and marginal cost of capital. These alternative ways of deciding on discount rates are discussed in this paper.

One of the approaches is the interaction between equity; asset and debt through information available can be utilized. In this case, a sector that has identical characteristics as the new project is identified and determined with its note of beta. The analysts know the returns on the portfolio on the market and the rate of return that is free of risk; the only unknown is the beta for the new project. A new formula is designed to help make the sector beta to give a new beta for equity in the new project. A second formula helps convert these two sectors’ assets beta to be similar.

## First formula

e u = e g This is when using the sector’s information

## Second formula

e u calculated above = + d this is when using the information for the new project.

## Exact project discount rate

Ko= Ke+ Ki

Capital asset pricing model can only be used in the generation of discount rates if the company in question is fully financed by equity. The formula used in this model equates equity costs to the sum of the riskless rate and a product of beta and the difference of market portfolio average return and the risk free rate. Risk free rate is an implied interest rate, for example that of treasury bills that depends on the maturity time. The market rate of return and beta usually vary depending on the size of the company in question.

This model of discounting emphasizes on the several investments in terms of the market. In this model, there is the assumption that risks can be avoided by diversification. Therefore, the expected return will only have provision for risks that are not possible to eliminate using diversification. The risk free return when taxes have been deducted equals the expected return in this model. Risks are rated according to how the securities behave in relation to the conditions in the market. It is assumed that investors do not take risks and they attain their maximum utility when the period of wealth ends. Further assumptions are that there are assets that have no risk, capital markets are perfect and efficient, assets are perfectly divisible and marketable and that investors can lend and borrow at that rate.

## The CAPM is represented as below;

Re=Rfr+[E(MR-Rfr)]B

Where, Re is the return of security e required

Rfr is the rate of return that is free

E(MR) is the market rate of return that is expected

B is Beta.

However, the CAPM has some disadvantages. This approach is not empirically practical since investors’ expectations cannot be tested. The model is only applicable in a single period and focuses only at the returns at the end of the year. There is also an assumption that market beta is the only determinant of the rate of return required.

The second approach of determining the rate of return is the Weighted Average Cost of Capital. This approach is only applicable if a company is financed by both debt and equity. However, it is assumed that when using this approach, the present debt equity ratio will not be altered during the implementation of the new project. A comprehensive investigation is done to determine a similar company with identical attributes in that particular industry and risk. These attributes will be used in the new project’s cash flows. Beta sectors are better than a single company beta because of the effect of diversification.

This approach is sometimes referred to as the composite cost of capital. Different components of capital have different cost percentages. It is advantageous to find a single capital average cost that is attributable to different capital costs. The cost percentage cost of every capital attribute is used to determine this. The market weight value for every component of capital is given by the formula;

W.A.C.C=ke(E/V)+kp(P/V)+kd(1-T)(D/V)

All the functions of k are percentage equity costs, debt capital and preference share capital.

E,P and D are values of market equity, preference share and debt capital.

The market value is found by multiplying the number of securities by the market security prices.

WACC has some disadvantages when used to as a discounting rate. This approach assumes that the structure of capital is optimal but it is not attainable in the real world. The other assumption is that the risk of the entire firm and that of the project is equal, which is not true. This approach is also built on the foundations of capital market values which change very often hence; changing the W.A.C.C.

The Arbitrage Pricing Theory is another model used to determine a discount rate in the appraisal of a project. This model can be used in a multiple activity company and it can be tested using the CAPM approach. The difference between the APT and CAPM approaches is that the APT has the assumption that rates of return of securities are linearly related to single common factors like the market portfolio rate of return. At equilibrium, a portfolio that has zero investment and risk has a zero rate of return. In cases where the rate of return is positive, the positive effect will be done away with instantly through arbitrage trading activities so as to increase the expected returns.

The expected rate of return in this particular model is:

E(Re)=Rfr+B1(R1-Rfr)+B2(R2-Rf)+Bn(Rn-Rf)+Ei

Where; E(Re) is a security’s expected rate of return

Rfr is a rate that is risk free

B1 is elasticity factor

Ei is a random error value

The advantage of the APT is that there are no assumptions concerning the returns of assets’ distribution empirically. Functions of individual utility are not assumed and the model can be extended into a multiple period framework.

The certainty-equivalent approach (CEA) and the risk adjusted rate discount are approaches of adjusting and incorporating risk into decisions of capital budgeting. The risk is included in the discount rate used in calculating present values.

In conclusion, the Arbitrage pricing theory is the most ideal of all the discount rate appraisal approaches. This is because this approach is very practical and covers many aspects of the market such as investments, risks and market equilibrium. This approach does not also include assumptions like in the other approaches. The approach can also be used over a multiple period frameworks unlike in other approaches. This makes this approach very effective and friendly to use.

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