Thursday, July 29, 2010


Simulation Analysis

The sensitivity analysis and scenario analyses are quite useful to understand the uncertainty of the investment projects. But both approaches suffer from certain weakness. They do not consider the interactions between variables and also, they do not reflect on the profitability of the change in variables.

Simulation analysis considers the interactions among variables and profitability of the change in variables. It does not give the projects net present value as a single number rather it computes the profitability distribution of value. The simulation analysis is an extension of scenario analysis. In simulation analysis a computer generates a very large number of scenarios according to the profitability distributions of the variables. The analysis involves the following steps:
  • First, you should identify variables that influence cash inflows and outflows. For example, when a firm introduces a new product in the market these variables are initial investment, market size, market growth, market share, price, variable costs, fixed costs, product life cycle, and terminal variable.
  • Second, specify the formulae that relative variables. For example, revenue depends on by sales volume and price; sales volume is given by market size, market share, and market growth. Similarly, operating expenses depend on production, sales and variable and fixed costs.
  • Third, indicate the profitability distribution for each variable. Some variables will have more uncertainty than others, For example, it is quite difficult to predict price or market growth with confidence.
  • Fourth, develop a computer programme that randomly selects one variable from the profitability distinction of each variable and uses these values to calculate the projects’ net present value. The computer generates a large number of such scenarios, calculates net present values and stores them. The stored values are printed as a profitability distribution of the projects’ values along with the expected value and its standard deviation. The risk-free rate should be used as the discount rate to compute the projects’ value. Since simulation is performed to account for the risk of the projects’ cash flows, the discount rate should reflect only the time value of money.

That analysis is a very useful technique for risk analysis. Unfortunately, its practical use is limited because of a number of shortcomings. First, the model becomes quite complex to use because the variable depends are interrelated with each other, and each variable depends on its values in the previous periods as well. Identifying all possible relationships and estimating probability distribution is a difficult task; its time consuming as well as expensive. Second, the model helps to generating a profitability distribution of the projects’ net present values. But it does not indicate whether or not the project should be accepted. Third, considers the risk of any project in isolation of other projects.

Monday, July 26, 2010


Sensitivity Analysis

In the evaluation of an investment project, we work with the forecasts of cash flows. Fore casted cash flows depend on the expected revenue and costs. Further, expected revenue is a function of sales volume and unit selling price. Similarly, sales volume will depend on the market size and firms’ market share. Costs include variable costs, which depend on sales volume, and unit variable cost and fixed costs. The net present value or the internal rate of return of a project is determined by analyzing the after tax cash flows arrived at by combining forecasts of various variables. It is different to arrive at an accurate and unbiased forecast of each variable. We can’t’ be certain about the outcomes of any of these variables. The reliability of the net present value or internal rate of return of the project will depend on the reliability of the forecasts of each variable underlying the estimates of net cash flows. To determine the reliability of the projects’ net present value or internal rate of return, we can work out how much difference it makes if any of these forecasts goes wrong. We can change each of the forecasts, one at a time, to at least three values, expected, and optimistic. The net present value of the project is recalculated under these different assumptions. This method of recalculating net present value or internal rate of return by changing each forecast is called sensitivity analysis.

Sensitivity analysis is a way of analyzing change in the projects’ values for a given change in one of the variables. It indicates how sensitive a projects’ value is to changes in particular variables. The more sensitivity of the value, the more critical is the variable. The following three steps are involved in the use of sensitivity analysis:
  • Identification of all those variables, which have an influence on the projects value
  • Definition of the underlying (mathematical) relationship between the variables
  • Analysis of the impact of the change in each of the variables on the projects value

The decision maker, while performing sensitivity analysis computes the projects net present value or internal rate of return for each forecast under three assumptions.

  1. Pessimistic,
  2. Expected,
  3. Optimistic.

It all allows him to ask “what if”. For example, what if volume increase or decreases? What if variable cost or fixed cost increases or decreases? What if the selling price increase or decreases? What if the project is delayed or outlay escalates or the projects life is more or less than anticipated? A whole range of questions can be answered with the help of sensitivity analysis. It examines the sensitivity of the variables underlying the computation of net present value or internal rate of return, rater than attempting to quantify risk. It can be applied to any variable, which is an input for the after tax cash flows.

Sunday, July 25, 2010


Risk Adjusted Discount Rate

For a long time, economic theorists have assumed that, to allow for risk, the businessman required a premium over and above an alternative, which was risk-free. Accordingly, the more uncertain the returns in the future, the grater the risk and grater the premium required. Based on this reasoning, it is proposed that the risk premium be incorporated into the capital budgeting analysis through the discount rate. That is, if the time preference for money is to be recognized by discounting estimated future cash flows, at some risk free rate, to their present value, then, to allow for the riskiness, of those future cash flows a risk premium rate may be added to risk-free discount rate. Such a composite discount rate, called the risk-adjusted discount rate, will allow for both time preference and risk preference and will be a sum of the risk-free rate and risk-premium rate reflecting the investors’ attitude towards risk. The risk-adjusted discount rate method can be formally expressed as follows:

Risk-adjusted discount rate = Risk free rate + Risk premium

Under capital asset pricing model, the risk premium is the difference between the market rate of return and the risk free rate multiplied by the beta of the project.

The risk adjusted discount rate accounts for risk by varying the discount rate depending on the degree of risk of investment projects. A higher rate will be used for riskier projects and a lower rate for less risky projects. The net present value will decrease with increasing risk adjusted rate, indicating that the riskier a project is perceived, the less likely it will be accepted. If the risk free rate is assumed to be 10%, some rate would be added to it, say 5%, as compensation for the risk of the investment, and the composite 15% rate would be used to discount the cash flows.

Advantages of risk adjusted discount rate
  • It is simple and can be easily understood.
  • It has a great deal of intuitive appeal for risk-averse businessman.
  • It incorporates an attitude towards uncertainty.


This approach, however, suffers from the following limitations:

  • There is no easy way deriving a risk adjusted discount rate. Capital asset pricing model provides a basis of calculating the risk adjusted discount rate. Its use has yet to pick up in practice.
  • It does not make any risk adjusted in the numerator for the cash flows that are forecast over the future years.
  • It is based on the assumption that investor are risk-averse. Through it is generally true, there exists a category of risk seekers who do not demand premium for assuming risks; they are willing to pay premium to take risks. Accordingly, the composite discount rate would be reduced, not increased, as the level of risk increases.

Saturday, July 24, 2010


Certainty Equivalent Method for Risk Analysis

Yet another common procedure for dealing with risk in capital budgeting is to reduce the forecasts of cash flows to some conservative levels. For example, if an investor, according to his “best estimate” expects a cash flow of 60000$ next year, he will apply an intuitive correction factor and may work with 40000$ to be on safe side. There is a certainty-equivalent cash flow. In formal way, the certainty equivalent approach may be expressed as:

Net present value = (the risk adjusted factor X the forecasts of net cash flow) / (1 + Risk free rate)

The certainty equivalent coefficient, the risk adjustment factor assumes a value between zero and one, and varies inversely with risk. A lower risk adjustment rate will be used if lower risk is anticipated. The decision maker subjectively or objectively establishes the coefficients. These coefficients reflect the decision makers’ confidence in obtaining a particular cash flow in period. For example, a cash flow of 20000$ may be estimated in the next year, but if the investor feels that only 80% of it is a certain amount, then the certainty-equivalent coefficient will be 0.8. That is, he consider only 16000$ as the certain cash flow. Thus, to obtain certain cash flows, we will multiply estimated cash flows by the certainty-equivalent coefficients.

The certainty-equivalent coefficient can be determined as a relationship between the certain cash flows and the risky cash flows. That is:

Risk adjustment factor = certain net cash flow / Risky net cash flow

For example, if one expected a risky cash flow of 80000$ in period and certain cash flow of 60000$ equally desirable, then risk adjustment factor will be 0.75 = 60000/80000.
If the internal rate of return method is used, we will calculate that rate of discount, which equates the present value of certainty equivalent cash outflows. The rate so found will be compared with the minimum required risk free rate. Project will be accepted if the internal rate is higher than the minimum rate; otherwise it will be unacceptable.

Evaluation of certainty equivalent

The certainty equivalent approach explicitly recognizes risk, but the procedure for reducing the forecasts of cash flows is implicit and is likely to be inconsistent from one investment to another. Further, this method suffers from many dangers in a large enterprise. First, the forecaster, expecting the reduction that will be made in his forecasts, may inflate them in anticipation. This will no longer give forecasts according to “best estimate”. Second, if forecasts have to pass through several layers of management, the effect may be to greatly exaggerate the original forecast or to make it ultra conservative. Third, by focusing explicit attention only on the gloomy outcomes, chances are increased for passing by some good investments.


Conventional Techniques of Risk Analysis in capital budgeting

A number of techniques to handle risk are used by managers in practice. They range from simple rules of thumb to sophisticated statistical techniques. The following are the popular, non-conventional techniques of handling risk in capital budgeting.
  • Payback
  • Risk-adjusted discount rate
  • Certainty equivalent

These methods, as discussed below about payback first, but all of them are simple, familiar and partially defensible on theoretical grounds. However, they are based on highly simplified and at times, unrealistic assumptions. They fail to take account of whole range of the effect of risky factors on the investment decision-making.


Payback is one of the oldest and commonly used methods or explicitly recognizing risk associated with an investment project. This method, as applied in practice, is more an attempt to allow for risk in capital budgeting decision rather than a method to measure profitability. Business firms using this method usually prefer short payback to longer ones, and often establish guidelines that a firm should accept investments with some maximum payback period, say three or five years.

The merit of payback is its simplicity. Also payback makes an allowance for risk by focusing attention on the near term future and thereby emphasizing the liquidity of the firm through recovery of capital, and by favoring short term projects over what may be riskier, longer term projects.

It should be realized, however, that the payback period, as a method of risk analysis, is useful only in allowing for a special type of risk, the risk that a project will go exactly as planned for a certain period and will then suddenly cease altogether and be worth nothing. It is essentially suited to the assessment of risks of time nature. Once a payback period has been calculated, the decision-maker would compare it with his own assessment of the projects likely, and if the letter exceeds the former, he would accept the project. This is a useful procedure, economic only if the forecasts of cash flows associated with the project are likely to be unimpaired for a certain period. The risk that a project will suddenly cease altogether after a certain period life may arise due to reasons such as civil war in a country, closure of the business due to an indefinite strike by the workers, introduction of a new product b a competitor which captures the whole market and nature disasters such as flood or fire. Such risks undoubtedly exist but they, by no means, constitute a large proportion of the commonly encountered business risks. The usual risk in business is not that a project will go as forecast for a period and then collapse altogether; rather the normal business risk is that the forecasts of cash flows will go wrong due to lower sales, higher cost.

Tuesday, July 20, 2010


Statistical Techniques for Risk Analysis

Statistical techniques are analytical tools for handling risky investments. These techniques, drawing from the fields of mathematics, logic, economics and psychology, enable the decision-maker to make decisions under risk or uncertainty.

The concept of probability is fundamental to the use of the risk analysis techniques. Hoe is probability defined? How are probabilities estimated? How are they used in the risk analysis techniques? How do statistical techniques help in resolving the complex problem of analyzing risk in capital budgeting? We attempt to answer these questions in our posts.

Probability defined

The most crucial information for the capital budgeting decision is a forecast of future cash flows. A typical forecast is single figure for a period. This referred to as “best estimate” or “most likely” forecast. But the questions are: To what extent can one rely this single figure? How is this figure arrived at? Does it reflect risk? In fact, the decision analysis is limited in two ways by this single figure forecast. Firstly, we do not know the changes of this figure actually occurring, i.e. the uncertainty surrounding this figure. In other words, we do not know the range of the forecast and the chance or the probability estimates associated with figures within the range. Secondly, the meaning of best estimates or most likely is not very clear. It is not known whether it is mean, median or mode. For these reasons, a forecaster should not give just one estimate, but a range of associate probability- a probability distribution.

Probability may be described as a measure of someone’s option about the likelihood that an event will occur. If an event is certain to occur, we say that it has a probability of one of occurring. If an event is certain not to occur, we say that its probability of occurring is zero. Thus, probability of all events to occur lies between zero and one. A probability distribution may consist of a number of estimates. But in the simple form it may consist of only a few estimates. One commonly used form employs only the high, low and best guess estimates, or the optimistic, most likely and pessimistic estimates.

Assigning probability

The classical probability theory assumes that no statement whatsoever can be made about the probability of any single event. In fact, the classical view holds that one can talk about probability in a very long run sense, given that the occurrence or non-occurrence of the event can be repeatedly observed over a very large number of times under independent identical situations. Thus, the probability estimate, which is based on a very large number of observations, is known as an objective probability.

The classical concept of objective probability is of little use in analyzing investment decision because these decisions are non-respective and hardly made under independent identical conditions over time. As a result, some people opine that it is not very useful to express the forecaster’s estimates in terms of probability. However, in recent years another view of probability has revived, that is, the personal view, which holds that it makes a great deal of sense to talk about the probability of a single event, without reference to the repeatability, long run frequency concept. Such probability assignments that reflect the state of belief of a person rather than the objective evidence of a large number of trials are called personal or subjective probabilities.

Saturday, July 17, 2010


Risk Analysis in Capital Budgeting


In discussing the capital budgeting techniques, we have so far assumed that the proposed investment projects do not involve any risk. This assumption was made simply to facilitate the understanding of the capital budgeting techniques. In real world situation, however, the firm in general and its investment projects in particular are exposed to different of risk. What is risk? How can risk be measured and analyzed in the investment decisions?

Nature of risk

Risk exists because of the inability of the decision maker to make perfect forecasts. Forecasts cannot be made with perfection or certainty since the future events on which they depend are uncertain. An investment is not risky if, we can specify a unique sequence of cash flows for it. But whole trouble is that cash flows cannot be forecast accurately, and alternative sequences of cash flows can occur depending on the future events. Thus, risk arises in investment evaluation because we cannot anticipate the occurrence of the possible future events with certainty and consequently, cannot, make are correct prediction about the cash flow sequence. To illustrate, let us suppose that a firm is considering a proposal to commit its funds in a machine, which will help to produce a new product. The demand for this product may be very sensitive to the general economic conditions. It may be very high under favorable economic conditions and very low under unfavorable economic conditions. Thus, the investment would be profitable in the former situation and unprofitable in the later case. But, it is quite difficult to predict the future state of economic conditions, uncertainty about the cash flows associated with the investment derives
A large number of events influence forecasts. These events can be grouped in different ways. However, no particular grouping of events will be useful for all purposes. We may, for example, consider three broad categories of the events influencing the investment forecasts.

  • General economic conditions

This category includes events which influence general level of business activity. The level of business activity might be affected by such events as internal and external economic and political situations, monetary and fiscal policies, social conditions etc.

  • Industry factors

This category of events may affect all companies in an industry. For example, companies in an industry would be affected by the industrial relations in the industry, by innovations, by change in material cost etc.

  • Company factors

This category of events may affect only a company. The change in management, strike in the company, a natural disaster such as flood or fire may affect directly a particular company.

Wednesday, July 14, 2010


Summary about Complex Investment Decisions
  • A firm in practice complicated investment decisions. The most common situations include choosing among investments with different lives, deciding about the replacements of an existing asset or timing of an investment and evaluating investments under capital rationing. The NPV rule can be extended to handle such situations.
  • The choice between projects with unequal lives should be made by comparing their real annual equivalent values (AEVs). AEV is the NPV of an investment divided by the annuity factor given its life and risk-free discount rate:

AEV = NPV / Annuity factor

  • The procedure of comparing AEVs can be followed while replacing an existing asset by a new asset. The NPV rule also proves handy in resolving the limiting problem of an investment.
  • Capital rationing occurs because of either the external or internal constrain on the supply of funds. In capital rationing situations, the firm cannot accept all profitable projects. Therefore, the firm will aim at maximizing NPV subject to the funds constraint.
  • In simple one-period capital rationing situations, the profitability index (PI) rule can be used. PI rule breaks in the case of multi-period funds constraints and project indivisibility.
  • A more sophisticated approach- either linear programming or integer programming- can be used to select investment under capital rationing. However, two factors limit the use of these approaches in practice. First, they are costly; second they assume investment opportunities as known. Also, large companies in reality hardly face the real capital shortage situations. Mostly it arises on account of the internal constraints imposed by the management for control purposes.

Tuesday, July 13, 2010


Limitations of Profitability Index

The capital budgeting procedure described above does not always work. It fails in two situations:
  • Multi-period capital constraints
  • Project indivisibility
Multi-period constraints

The serious limitation in using the PI rule is caused by the multi-period constraints. In the above post example, there is a budget limit of 50000$ year 1 also and the firm is anticipating an investment opportunity 0 as in low is year 1. Thus, the decision choices today are as follows:

Project M and N have the first and second ranks in terms of PI. They together have highest NPV and also exhaust the budget in year 0; so the firm would choose them. Further, projects M and N together are expected to generate 20000$ cash flow next year. This amount with the next year’s budget (i.e. 20000$ + 50000$ = 70000$) is not sufficient to accept project O. Thus, by accepting M and N, the firm will obtain a total NPV of 15870$. However, a careful examination of the project’s cash flows reveals that if project L is accepted now it is expected to generate a cash flow of 30000$ after a year, which together with the budget of 50000$ is sufficient to undertake project O next year. Projects L and O have lower PI ranks than projects M and N, but they have higher total NPV of 19820$.

Project indivisibility

The PI rule of selecting projects under capital rationing can also fail because of project invisibility. It may be more desirable to accept many lower ranked similar projects than a single large project. The acceptance of a single large project, which may be top-ranked, excludes the possibility of accepting small projects, which may have higher total NPV. Consider the following:

Suppose that the firm has budget ceiling of 10$ million. Following the ranking by PI, the firm would choose A and C. These projects spend 850000$ of the total a budget and have a total NPV of 180000$. The next best project E needs an investment of 200000$, while the firm has only 150000$. If we examine the various combinations of projects satisfying the budget limit, we find the package of C, E and D as the best. They exhaust the entire budget and have a total NPV of 189000$. Thus, the firm can choose two lower ranked, small projects, E and D, in place of the higher ranked, large project, A. This section procedure will become very unwieldy if the firm has chosen the best package of projects from a large number of profitable projects.

  • Our discussion has shown that the profitability index can be used to choose projects under simple, one-period, capital constraint situation. It breaks down in the case of multi-period capital constraints. It will also not work when any other constraint is imposed, or when mutually exclusive projects, or dependent projects are being considered.

Sunday, July 4, 2010


Use of Profitability Index in Capital Rationing

Under capital rationing, we need a method of selecting that portfolio of projects which yields highest possible net present value (NPV) within the available funds.

Let us consider a simple situation where a firm has the following investment opportunities and has a 10% cost of capital.

If the firm has no capital rationing constraint, if should undertake all three projects because they all have possible net present values (NPVs). Suppose there is a capital constraint and the firm can spend only 50000$ in year zero, what should the firm do? If the firm strictly follows the net present value (NPV) rule and starts with the highest individual net present value (NPV), it will accept the highest net present value (NPV) project L, which will exhaust the entire budget. We can, however, see that projects M and N together have higher net present value (15870 $) than project L (12940 $) and their outlays are within the budget ceiling. The firm should, therefore, undertake M and N rather than L to obtain highest possible net present value (NPV). It should be noted that the firm couldn't select projects solely on the basis of individual net present values (NPVs) when funds are limited. The firm should intend to get the largest benefit for the available funds. That is, those projects should be selected that give the highest ratio of present value to initial outlay. This ratio is the profitability index (PI). In the example, M has the highest PI followed by N and L. If the budget limit is 50000 $, we should choose M and N following the PI rule.

The capital budgeting procedure under the simple situation of capital rationing may be summarized as follows:

  • The net present value (NPV) rule should be modified while choosing among projects under capital constraint. The objective should be to maximize NPV per rupee of capital rather than to maximize NPV. Projects should be ranked by their profitability index, and top-ranked projects should be undertaken until funds are exhausted.

Friday, July 2, 2010


Investment Decisions under Capital Rationing

Firms may have to choose among profitable investment opportunities because of the limited financial resources. In this article we shall discuss the methods of solving the capital budgeting problems under capital rationing. We shall show that the net present value (NPV) is the most valid section rule even under the capital rationing situations.

A firm should accept all investment projects with positive net present value (NPV) in order to maximize the wealth of shareholders. The net present value (NPV) rule tells us to spend funds in the projects until the net present value (NPV) of the last project is zero.

Capital rationing refers to a situation where the firm is constrained for external, or self imposed, reasons to obtain necessary funds to invest in all investment projects with positive net present value (NPV). Under capital rationing, the management has not simply to determine the profitable investment opportunities, but it has also to decide to obtain that combination of the profitable projects which yields highest net present value (NPV) within the available funds.

Why capital rationing?

Capital rationing may rise due to external factors or internal constraints imposed by the management. Thus there are two types of capital rationing.
  • External capital rationing
  • Internal capital rationing

External capital rationing

External capital rationing mainly occurs on account of the imperfections in capital markets. Imperfections may be caused by deficiencies in market information, or by rigidities of attitude that hamper the free flow of capital. The net present value (NPV) rule will not work if shareholders do not have access to the capital markets. Imperfections in capital markets alone do not invalidate use of the net present value (NPV) rule. In reality, we will have very few situations where capital markets do not exist for shareholders.

Internal capital rationing

Internal capital rationing is caused by self imposed restrictions by the management. Various types of constraints may be imposed. For example, it may be decide not to obtain additional funds by incurring debt. This may be a part of the firm’s conservative financial policy.

Management may fix an arbitrary limit to the amount of funds to be invested by the divisional managers. Sometimes management may resort to capital rationing by requiring a minimum rate of return higher than the cost of capital. Whatever, may be the type of restrictions, the implication is that some of the profitable projects will have to be forgone because of the lack of funds. However, the net present value (NPV) rule will work since shareholders can borrow or lend in the capital markets.

It is quite difficult sometimes justify the internal capital rationing. But generally it is used as a means of financial controls. In a divisional set up, the divisional managers may overstate their investment requirements. One way of forcing them to carefully assess their investment opportunities and set priorities is to put upper limits to their capital expenditures. Similarly, a company may put investment limits if it finds itself incapable of coping with the strains and organizational problems of a fast growth.