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Hirschey Chapter 14 RISK ANALYSIS In the real world, most future events are not known with any degree of certainty. Managers must make decisions relying on estimates that involve some uncertainty. Risk and uncertainty will be a part of all analyses we will carry out throughout the semester. Risk vs Uncertainty Risk and uncertainty are used interchangeably in economics and finance. Most future events are not known with certainty and some of these events can be assigned probabilities. We are talking about risk when future events can be defined and probabilities can be assigned. The Measures of Risk When there is a number of alternative outcomes, each outcome will have a probability attached to it. A probability distribution describes the chances of all possible occurrences in percentage terms. Probability Distributions Cash Inflow Probability 3,000 0.10 4,000 0.20 5,000 0.40 6,000 0.20 7,000 0.10 Probability Distributions Expected Value: Average of all possible outcomes weighted by their respective probabilities. E(Cash Flow) = (3,000 x 0.1) + (4,000 x 0.2) + (5,000 x 0.4) + (6,000 x 0.2) + (7,000 x 0.1) = 5,000 Expected Value n E(X) X ipi i1 Probability Distribution Risk is the dispersion of possible outcomes around the expected value. Risk is higher when the potential differences from the average are high. Standard Deviation: Square root of the weighted average of the squared deviations of all possible outcomes from the expected value. Standard Deviation (Cash Flow) = [(3,000 - 5,000)2(0.1) + (4,000 - 5,000)2(0.2) + (5,000 - 5,000)2(0.4) + (6,000 - 5,000)2(0.2) + (7,000 - 5,000)2(0.1)]1/2 = 1,095 Standard Deviation n σ (Xi X) pi i1 2 Standard Deviation as a Measure of Risk When the expected values of two alternatives are equal or close to one another, the standard deviation is a proper measure of risk. In such a case, the standard deviation shows the amount of dispersion (risk) around the expected value. Coefficient of Variation If two alternatives have divergent expected values, standard deviation will not be an appropriate measure of risk. E.g. Exp Value Std Dev Project A 100 30 Project B 50 20 Coefficient of Variation Project A has both the larger expected value and the larger standard deviation. We need a “relative” measure of risk to compare these two projects. Coefficient of Variation measures risk relative to expected value. It shows the amount of risk per unit of return. Coefficient of Variation CV σ X Coefficient of Variation From the example: CVA = 30/100 = 0.30 CVB = 20/50 = 0.40 The CV is greater for Project B. Project A has a larger absolute risk () but its relative risk (CV) is lower. A is preferred over B since its expected value is higher and its relative risk is lower. Standard Normal Distribution When the probability distribution of outcomes is known, the risk of a given course of action can be measured by using the expected value and the standard deviation of the distribution. Normal Distribution: A special case probability distribution where the dispersion about the expected value is symmetrical. Since it is symmetrical, it is possible to measure the probability of a certain outcome in a standardized manner: The actual outcome of the decision will lie Within ± 1 standard deviation of the mean about 68% of the time Within ± 2 standard deviations of the mean about 95% of the time Within ± 3 standard deviations of the mean about 99% of the time Probability Ranges for a Normal Distribution 68.26% 95.46% 99.74% -3σ -2σ -1σ +1σ +2σ +3σ Mean or expected value The Standardized Variable A standardized variable has a mean of 0, a standard deviation of 1. Any distribution can be standardized in the following way: x μ z σ z : standardized variable x : outcome of interest μ : mean of the distribution σ : standard deviation of the distribution Interpretation of the Standardized Variable When the outcome is 1 σ away from the mean, x – μ = σ. This means z = 1. So, when z = 1, the point of interest is 1 σ away from the mean. Or, when z = 2, the point of interest is 2 σ away from the mean. Utility Theory Assumptions about risk is central to many decisions in managerial economics. When analyzing human behavior, we make assumptions about attitudes towards risk. Theoretically, there are three different attitudes: Risk Aversion Risk Neutrality Risk Seeking Risk Attitudes Risk Aversion: Individuals seek to avoid or minimize risk. Risk Neutrality: Individuals focus on expected returns and disregard the dispersion of returns (risk). Risk Seeking: Individuals prefer risk. Risk Attitudes Choice between more risky and less risky investments with identical returns: Risk averter chooses the one with less risk Risk seeker chooses the one with high risk Risk neutral is indifferent between the choices Risk Attitudes Business managers and investors are mostly risk averters. Some individuals prefer risk when small amounts of money are involved (entrepreneurs, innovators, inventors, speculators, lottery ticket buyers). However, the assumption of risk aversion generally holds. Why should risk aversion generally hold? Utility theory provides a satisfying answer. Utility Theory In order to understand risk aversion, we have to look at the relation between money and its utility. Diminishing Marginal Utility When additional increments of money brings ever smaller increments of added benefit, diminishing marginal utility is observed. For risk averters, money has diminishing marginal utility. For such an individual, a less than proportional relation holds between total utility and money. The utility of a doubled quantity of money is less than twice the utility of the original level. For a risk neutral individual, there is a strictly proportional relationship between total utility and money. For a risk seeker, there is a more than proportional relationship between total utility and money. Total Utility Risk Averter Income or Wealth Total Utility Indifferent to Risk Income or Wealth Total Utility Risk Seeker Income or Wealth Risk Aversion Example Alternative 1: Riskless government securities offer a 9% return on an annual basis. If you invest $10,000 in government securities, you will earn $900 at year-end and your total wealth will become $10,900. Alternative 2: If you invest $10,000 in a risky oil- drilling venture, with 60% chance, oil will be discovered and you will earn $10,000 and your total wealth will become $20,000. With 40% chance, there will be a dry hole and you will recover only $5,000 of your initial investment and your total wealth will be $5,000. Which alternative should you choose? Expected Value from Alternative 1: E(R) = $10,900 (no calculation necessary since there is no risk) Expected Value from Alternative 2: E(R) = 0.6($20,000) + 0.4($5,000) = $14,000 E(R) suggests that we should choose Alternative 2. How about the risk? Assume the following schedule for utilities: 40 35 Total Utility 30 25 20 15 10 5 0 0 5000 10000 15000 20000 Income or Wealth Diminishing MU Constant MU Increasing MU Reading from the schedule, for a diminishing marginal utility (risk averter): TU (government bond) = 10.7 TU (the oil drill) = 0.6(13) + 0.4(6) = 10.2. The risk averse person will choose the government bond since it has the larger expected total utility. Firm Value Under Risk The valuation model N CFt V t t 1 (1 i) needs to be adjusted for risk. The valuation model can be adjusted by reflecting risk either in CFs or in the discount rate (i). Adjustment to CFs Certainty Equivalent Adjustment Specify the certain sum that is comparable to the expected value of a risky investment alternative. The certainty equivalent of an expected value has the same total utility but it differs in dollar/lira terms. Certainty Equivalent Adjustment E.g. Invest $100,000 in a risky project. If successful, earn $1,000,000 (50% chance) If unsuccessful, earn $0 (50% chance) Expected Value = 0.5($1,000,000) + 0.5($0) = $500,000 If you do not invest, you keep the $100,000. Certainty Equivalent Adjustment If you are indifferent between investing and not investing, then Risky expected return = $500,000 Certainty equivalent = $100,000 and these two sums have the same total utility. Note that a much smaller certain sum ($100,000) has the same total utility as the much larger risky sum ($500,000). Certainty Equivalent Adjustment Equivalent certain sum < Expected risky sum Risk aversion Equivalent certain sum = Expected risky sum Risk neutrality (indifference) Equivalent certain sum > Expected risky sum Risk preference Certainty Equivalent Adjustment Certainty Equivalent Adjustment Factor = = equivalent certain sum / expected risky sum From the example, = $100,000 / $500,000 = 0.20. Risk-Adjusted Valuation Model The valuation model becomes N E(CFt ) V t t 1 (1 i) where we convert the expected value of cash flows into their certainty equivalent and use the riskless discount factor i in the denominator. Adjusting the Discount Rate for Risk The valuation model can be adjusted for risk by reflecting the risk in the discount rate while using the expected cash flows in the numerator: N E(CFt ) V t t 1 (1 k) where k = risk-free rate (i) + risk premium.

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