# Methods of Capital Budgeting Under Risk and Uncertainty

A capital expenditure decision may not be sound, if taken on the basis of only one set of assumptions as regards the profitability, without perceiving the risk and uncertainty connected with the assumptions. Well then, how the firm perceives uncertainty. There are different techniques developed for the purpose, both simple and highly complicated and mathematical. Common and non-mathematical Methods of Capital Budgeting Under Risk and Uncertainty are discussed below:

**(1) Risk Adjusted Rate of Return** – One way of adjusting for uncertainty is to simply vary the expected return keeping in view the degree of risk. For instance, if the cost of capital to the firm is 15%, on the assumption that the proposed project has the same degree of risk as the existing projects, the cash-flows may be discounted at 15% to ascertain the NPV.

However, if the proposal were to be more or less risky than the typical existing ones the discount rate would be more or less than 15%. The greater the risk and uncertainty, the higher the discount rate and vice versa, the magnitude being dependent on the degree of risk involved. If the net present value is zero or positive at the adjusted discount rate the project will be considered. If DRR is used, the DRR will be compared with risk adjusted rate of return.

The main disadvantage of this technique is the arbitrariness associated with the adjustment of rate of return. Though it may be possible to divide the investments into risk classes and assign different rates of return, such a categorization is based on only hunch and intuition. Moreover, it does not use all the information which is available from a probability distribution of cash-inflows (see probability approach).

**(2) Certainty Equivalents (CE):** This is an alternative method to risk adjusted rate of return in which the adjustment is done for risk in the expected future cash flows before arriving at the present value. The expected uncertain cash-flows before arriving at the present value. The expected uncertain cash-flows of each year are modified by multiplying them with what is known as **‘certainty equivalent coefficient**‘ (CEC) to remote the element of uncertainty. This co-efficient is determined by management’s preference with respect to risk.

For example, assume that the expected cash-flow from an investment at the end of the first year is $10,000 and that the management ranked this investment on par with another alternative investment with a certain cash-flows of $7,000. The CEC in this case is 0.7 that is equal to Certain cash-flows divided by Uncertain cash-flows.

**CEC=Certain cash-flows/ Uncertain cash-flows**

Similarly, a CEC can be assigned to each year’s expected uncertain cash-flows and convert them into certainty equivalents and then proceed to ascertain the rate of return or NPV. CE approach also presents practical problem of implementation as it is very difficult to assign the exact CEC’s for a given stream to expected future cash flows. However, it is superior to risk adjusted rate of return because distant discount rate implies that risk increases at a constant rate with time which is not realistic. But under CE approach CEC is determined period by period recognizing increasing uncertainty as the future advances.

The CEC varies invertly with the risk. Again CE approach is upheld on the ground that it is the future cash-flow a project which is subject to risk and not the rate of return.

**(3) Sensitivity Analysis** – Sensitivity analysis is a technique designed to measure the response or change in the profitability of a project caused by changes in the factors that affect a project’s cash-inflows. Sensitivity analysis is used in association with the method of evaluation chosen to obtain such information as :

- what is the effect of a 10% decrease in selling price on the NPV (or other measure of profitability) ?
- what happens to the NPV if production effect of a change in demand for the product ?
- what would be the profitability if the economic life of the asset is one year less than the predicted life ?
- what is the effect of change in the price level ?, and so on.

If a small change in one factor results in a major change in the profitability of a proposed investment the project is considered sensitive to that factor and more risky. Other things being equal, less sensitive project is preferred to more sensitive ones, since a small change in any factor would have a substantial effect on the sensitive project and might change the estimated profit from the investment (project) to loss.

Sensitivity analysis is also not a precise and flawless technique. It does not usually consider the effect of a combination of changes in various factors on the profitability of the prospective project and does not normally provide the information on the probability that these changes may occur, thus, making the changes in factors arbitrary. However, sensitivity analysis provides the management with information about the sensitivity of a project to changes in different factors and warm them of the risk. To say the least, this analysis helps in identifying areas where additional information is needed to improve the estimates by pointing out the effect of errors in estimates of certain factors.

**(4) The Probability Approach** – It has already been pointed out that the capital budgeting decision based on estimation of only one set of cash-flows and profitability of a prospective project may prove to be erroneous for the simple reason that we are dealing with uncertain future. Let us assume that two alternative projects ‘A’ and ‘B’ offer the same profitability, say, DRR of 20%. If we do not analyze further both are equally desirable. Let us further assume that in case of project ‘A’ there is a possibility of DRR going down to 10% if benefits are not fully realized and up to 30% if benefits are better than expected and that the corresponding figures for project ‘B’ are 5% and 35%. The lowest return expected (pessimistic return) from ‘A’ is 10% and the maximum (optimistic) return is 30% giving an average (most likely) return of 20%. The corresponding estimates for project B’ are 5%, 35% and 20th. Both have the same most likely DRR. Which would be preferred ? The answer is project ‘A’ for, its profitability is less variable (or dispersed) compared to that of project `B’ investment. ‘A’ has a profitability range of 20% (30-10) and project ‘B’ has 30% (35-5). Project ‘B’ is considered more risky as it has a wider profitability range. For a more accurate result standard deviation of Profitability may be used as measure of dispersion instead of range.

The above analysis could be made more meaningful and realistic through probability approach. Instead of estimating the pessimistic, most likely and optimistic values arbitrarily is possible to assign definite probabilities to each of the expected stream of cash-flows and construct a probability distribution. Probability of something is an expression of the chances or the likelihood of its occurrence.

For example, the probability of getting a head when a coin is tossed is .5. which is the ratio of number of ways can occur divided by all possible outcomes. However, this theoretical approach to probability is not applicable to business situations. What is applicable is the subjective approach whereby probability is determined on the basis of strength of belief, experience and educated guess. ‘Reasonable men base the probabilities which they assign to events in the real world on their experience with events in the real world, and where two reasonable men have roughly the same experience with a certain kind of event they assign it roughly the same probability. Two methods are widely used under probability approach to incorporate risk and uncertainty in capital budgeting decision.

**Dispersion on of Probability Distribution:**The first step is to construct a probability distribution of cash flows by assigning probabilities (which vary from 0 total and the sum of which is always 1) to each stream of expected cash-flows. For instance, the probability of cash-flows being $10,000 in the first year is, say, .5, being $15,000, is .3 and being $7,500 is .2, the probabilities adding upto 1. The most probable cash-flow would be the sum of the expected cash-flow multiplied by its probability, which, in this case, is (10,000 x .5) – 4 – (15,000 x .3) + (7,500 x .2)= $11,000.

Secondly, the probability distribution of cash-flows, obtained as above, of two or more proposals may be represented on a graph for the purpose of comparison of probability distribution. Probability of occurrence is plotted on the ‘Y’ axis and cash- flows on the ‘X’ axis. Alternatively, a measure of dispersion (usually standard deviation) may be computed for each of the alternative’s probability distribution for comparison. Finally, decision is made in favor of the investment that has the lowest dispersion or variability of cash-flows from the expected cash-flow.

(**d) Decision-Tree Analysis:** The decision-tree approach to the evaluation of risk and uncertainty rests on the impact of all probabilistic estimate of potential outcomes. in other words, when using decision-tree analysis every potential event is weighted in probabilistic terms and that is the basis for evaluation. The decision-tree is an analytical technique used especially in sequential decisions, where various decision points are studied in relation to subsequent events. A decision-tree is a pictorial presentation in a ‘tree form’ indicating the magnitude, probability and interrelationship of all possible outcomes.

These results may be compared with those of the alternative projects and on that basis a more objective and consistent decision can be taken. It may be noted that the decision-tree analysis is used in various other fields of managerial decision making and it is not exclusively a technique of uncertainty analysis in capital expenditure evaluation.

**Conclusion**: Uncertainty is an umbrella under which all firms operate and the pretention of its non-existence when making a capital expenditure decision is a serious and cowardly error. The series techniques described above to incorporate the element of uncertainty into the Capital Budgeting Decision are not independent methods of evaluation, but are supplements to the basic methods. They attempt to measure risk and uncertainty quantitatively to help the management to assess the expected impact of an investment decision on the firm’s profitability. These techniques are not fool-proof as uncertainty is caused by a host of problems.

Nevertheless, they do provide insight to the important dimensions of uncertainty and these dimensions should not be ignored in capital budgeting simple because evaluating them is difficult. There are other mathematical techniques like stimulation models, linear programming, etc., used for analyzing and incorporating risk and uncertainty which have for not yet become very popular among businessmen. The entire analysis is applicable only to those investments the benefits from which can be quantified. There are many capital projects the profitability of which cannot be evaluated with any reasonable degree of accuracy (labour welfare investments, strategic investments, etc.), hence they have not been considered here.