CHAPTER 12 RISK ANALYSIS IN CAPITAL BUDGETING
Q. 1 Explain the concept of risk? How can risk be measured? A.1 Risk can be defined as variability of returns of an investment. Risk arises in
investment evaluation because we cannot make any correct prediction about the cash flow sequence since the future events on which they depend are uncertain. Risk can be measured by using statistical techniques to measure the variability.
Standard deviation, variance or coefficient variation can be used to measure risk.
Q.2 What are the advantages of risk-adjusted discount rate? What is the major
problem in using this approach to handle risk in capital budgeting? A.2 The
risk-adjusted discount rate is a composite rate, which allows for both time
preference and risk preference and will be the sum of the risk-free rate and risk premium.
Risk-adjusted rate = Risk-free rate + Risk premium = kf + kr
A higher rate will be used for riskier projects and a lower rate for less risky projects. The method is very simple, and can be easily understood. It has a great deal of intuitive appeal for risk-averse businessman. The constant risk-adjusted discount rate is not valid over the life span of project. It is also quite difficult to specify risk-adjusted discount rate properly to measure the degree of increasing risk.
Q.3 What are the advantages of using certainty-equivalent approach? Does it suffer
from any limitation? A. 3 The certainty-equivalent coefficient establishes a
under certainty equivalent approach may be expressed as:
where NCFt = the forecast of net cash flow without risk-adjustment; αt = the risk adjustment factor or certainty equivalent coefficient, and kf = risk-free rate. αt
assume a value between 0 and 1, and varies inversely with risk. Certainty-
equivalent approach explicitly recognizes the risk, but the procedure for reducing the forecasts of cash flows is implicit. In large enterprise, the forecaster, expecting that the reduction will be made in his forecast, as it pass through several layers of management, may inflate them in anticipation. By focusing only on the gloomy outcomes, chances are increased for passing by some good investments.
Q.4 "The certainty equivalent approach is theoretically superior to the risk-adjusted
discount rate". Do you agree? Give reasons.
A.4 Yes, because certainty equivalent approach measures risk more accurately by
adjusting estimated cash flows and employs risk-free rate to discount the adjusted cash flows. On the other hand, the risk-adjusted discount rate adjusts for risk by adjusting the discount rate. If the investor thinks that an investment may be more risky during the gestation period, and once established, risk may reduce. In such case, the use of a constant risk-adjusted discount rate is not valid. But the increased or decreased risks over a period of time can easily be accounted for by changing the certainty equivalent factors, when the certainty equivalent approach is used.
Q.5 What are the limitations of payback method as a risk handling technique? Can it
be used as a supplement to more sophisticated techniques?
A.5 The payback period is the number of years required to recover the initial outlay of
the investment. The payback period method ignores the time value of money, ignores the cash flows occurring after the payback period, and also does not measure the profitability of the project. The payback period method can be used, only in case of project having special type of risk like civil wars, introduction of new product by a competitor, and natural disasters such as flood or fire, which will suddenly cease the entire project altogether and be worth nothing. This method makes an allowance for risk by focusing attention on the short term project and thereby emphasizing the liquidity of the firm through early recovery of project.
Q.6 How can you conduct the DCF break even analysis? Why is the DCF analysis
important in risk analysis in capital budgeting?
A.6 DCF break-even means a situation where NPV is zero. The NPV of a project
depends on cash outlay, volume, price, variable costs, fixed costs, depreciation rate, tax rate, life of the project etc. For calculating DCF break-even point, one can take one variable and determine its minimum value at which NPV is zero. DCF analysis is important in risk analysis since it indicates the sensitivity of the project NPV to changes in variables. It helps to identify variables which are critical to the project NPV.
Q.7 What is sensitivity analysis? What are its advantages and limitations? A.7 Sensitivity analysis is a way of analyzing change in the project's NPV (or IRR)
for a given change in one of the variables, e.g., sales volume, unit selling price, unit variable costs, etc. It indicates how sensitive a project's NPV (or IRR) is to changes in particular variables. The more sensitive the NPV, the more critical is the variable. It has the following advantages: 1. Helps the decision maker to identify the variables which affect the cash flow
forecasts, and understand the project in totality.
2. Indicates the critical variables, and helps in strengthening the 'weak spots' of
3. Helps to expose inappropriate forecasts. It has the following limitations: 1. Does not provide clear cut results. 2. Fails to focus on interrelationship between variables.
Q.8 How is the probability theory utilized in analyzing risk of investment projects?
A.8 Probability may be defined as a measure of someone's opinion about the
likelihood that an event will occur. Risk is referred to a situation where the probability distribution of the cash flow of an investment proposal is known. The capital budgeting decision is a forecast of future cash flows. A forecaster should not give just one estimate, but a range of associated probability - a probability distribution. The probability of all events to occur lies between 0 and 1. Once the probability assignments have been made to the future cash flows, the next step is to find out the expected net present value. It can be found out by multiplying the monetary values of the possible events by their probabilities. The dispersion of cash flow indicates the degree of risk. The risk can be measured by standard deviation or variance. The standard deviation is the square root of variance. The variance can be calculated by finding out the difference between each event's i.e., possible cash flow that can occur and their expected value. This difference is squared and then multiplied by respective probability of events.
Here, σ2 = variance; σ = standard deviation, NCFj = Net cash flow of jth event; ENCF = Expected net present value; and Pj = Probability of jth event.
Q.9 Describe the decision tree approach with the help of an example. How is this
A.9 When the sequential decision making is involved, to resolve the risk, decision tree
analysis is much useful. The decision tree provides a way to represent different possibilities so that we can be sure that the decision we make today, take proper account of what we can do in the future. To draw a decision tree, branches from points marked with squares are used to denote different possible decisions, and branches from points marked with circles denote different possible outcomes. In a decision tree analysis, one has to work out the best decisions at the second stage before one can choose the best first stage decision. Decision trees are valuable because they display links between today's and tomorrow's decisions. Further, the decision maker explicitly considers various assumptions underlying the decision. For example, to construct a decision tree to build small plant or large plant following steps are required:
1. Define Investment - Here, investment proposal should be defined - e.g. to
2. Identify decision alternatives - The decision alternatives should be clearly
identified, e.g. to construct small plant initially and expand it later on or to construct large plant or construct no plant.
3. Draw a decision tree - Decision tree to be graphed indicating the decision
points, chance events and other data e.g. relevant data such as project cash flows, probability, and expected net present value should be located on decision tree branches.
4. Analyze data: Analyze the data for each alternative, workout ENPV for each
alternative at each decision tree branch level, and select the best alternative.
Q.10 How can utility theory be incorporated in the capital budgeting decision to
account for the risk preferences of the decision maker?
A. 10 Utility theory aims at including a decision makers' risk preferences explicitly in to
the capital expenditure decision. The underlying principle is that an investor prefers a higher return to a lower return, and that each successive identical increment of money is worth less to him than the preceding one. The decision maker's utility function is derived to determine the decision's utility value. A rational decision maker would maximize his utility, by accepting the investment project which yields maximum utility to him. The risk-averse investors attach lower utility to increasing wealth while risk-seeking investors attach more utility to increasing wealth. Risk neutral investors attach same utility to increasing or decreasing wealth. It is very difficult to derive utility function; it does not remain constant over time. Problems are also encountered when decision is taken by a group of people. Individuals differ in their risk preferences.

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