Measurement of Risk by Probability Distribution

The probability of an event is the chance, or odds, that the event will occur, if all possible events or outcomes are listed, and if a probability of occurrence can be assigned to each event, the listing is called a probability distribution. For example, suppose a sales manager observes that there is a 70% chance that a given customer will place a specific order within the next two weeks, versus a
30% chance that the customer will not. This situation is described by the probability distribution shown in table.

Each possible outcome is listed in column 1, and the probabilities of each outcome, expressed as decimals and percentages, appear in column 2. Notice that the probabilities sum to 1.0 or 100%, as they must if the probability distribution is to be complete (i.e. represent all possible outcomes). In this very simple example, risk can be read from the probability distribution as the 30% chance of neither the firm nor receiving the order. For most managerial decisions the relative desirability of alternative events or outcomes is not so easily computed. A more general measure of the relation between risk and the probability distribution is measure of the relation to incorporate risk considerations adequately into the decision-making process. The need for a more general measure of risk can be illustrated by the following example.

Suppose that a firm is able to choose only one of two investment projects, each calling for an outlay of $10,000. Assume also that profits earned from the two projects are related to the general level of economic activity during the coming year, as shown in table. This table is known as a payoff matrix since it illustrates the monetary outcomes associated with each possible state of profits from project B very much more as a result the state of the economy than do those from Project A. In a normal economy, both projects return $5,000 in profit. Should the economy be in a recession next year? Project B will produce nothing, whereas Project A still provides a $ 4,000 profit. If the economy is booming next year, Project B’s profit will increase to $ 12,000, but profit for Project A will increase only moderately to $6,000.

Outcomes and Probabilities for Receiving an Order
                                Event                        Probability of Occurrence
                                 (1)                                        (2)
Receive Order                                                   0.7 = 70%
Do not received order                                        0.3 = 30%
Total                                                                  1.0 = 100%

Project A is clearly more desirable if the economy is in recession, whereas Project B is superior in a boom. In a normal economy the projects offer the same profit potential, and both are equally desirable. To choose the best project, one needs to know the likelihood of a boom, a recession or normal economic conditions. If such probabilities are available, the expected profits and variability of profits for each project can be determined. These measures make it possible to evaluate each project in terms of anticipated or expected returns, and to measure the risk of such returns in terms of the difference between and expected values.

Payoff Matrix for Project A and B
State of the Economy         Project A             Project B
Recession                             4,000                     0
Normal                                   500                  5,000
Boom                                   6,000                 12,000

The expected value is the anticipated receipts from a given payoff matrix with a specified probability distribution. It is the weighted average receipt when the weights are defined by the appropriate probability distribution.

To continue with the previous example, assume that forecasts based on the current trend in economic indicators suggest a 2-in-10 chance of recession, a 6-in-10 chance of normal economy, and a 2-in-10 chance of a boom. As probabilities, the probability of recession is 0.2, or 20%, the probability of normal economic activity is 0.6 or 60% and the probability of a boom is 0.2, or 20%. These probabilities add up to 1.0 (0.2 + 0.6 + 0.2 = 1.0), or 100%, and thereby from a complete probability distribution, as shown in table.

Calculation of Expected Values
           State of the
of this State Occurring
Profit Outcome
if this State Occurs
Expected Profit
Outcomes ($)
           (1)(2)(3)(4) = (2) x (3)
Project ARecession0.24,000800
                      1.0Expected Profit A           
Project BRecession0.200
                      1.0Expected Profit B           

If each possible outcome is multiplied by its probability of occurrence, and the answers are summed, the weighted average outcomes are obtained. In this calculation, the weights are the probabilities of occurrence, and the weighted average is called the expected value. The above mentioned table illustrates the calculation of expected profits for profits for Project A and B. Each possible profit level is column 3 is multiplied by its probability of occurrence from column 2 to obtain weighted values of the possible profits. Summing column 4 of the table for each project gives a weighted average of profits under various states of the economy. This weighted average is the expected profit from the project.

Risk is a complex concept, and some controversy surrounds attempts to define and measure it. Common risk measures that are satisfactory for most purposes are based on the observation that right probability distributions imply low risk because of the correspondingly, mall chance chat actual outcomes will differ greatly from expected values. 

The standard deviation is a popular and useful measure of absolute risk. Absolute risk as measured by the standard deviation is the overall dispersion of possible payoff values. The smaller is the standard deviation, the tighter is the probability distribution and therefore the lever is the risk in absolute terms.

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