### Statistical Techniques for Risk Analysis

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STATISTICAL TECHNIQUES FOR RISK ANALYSIS 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?

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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.

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. (215)—SENSITIVITY ANALYSIS

Sensitivity Analysis ?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. . 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.

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