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Risk Analysis Concepts and Calculations Md. Al Mamun MBA (Finance, DU) MBA (HRM, Finance, AUS) ACCA in Progress (UK) Topics - Risk and Uncertainty - General Risk Categories - Probability - Probability Distributions - Payoff Matrix - Expected Value - Variance and Standard Deviation (Measurement of Absolute Risk) - Coefficient of Variation (Measurement of Relative Risk) Topics (Con’t) - Risk Attitudes Risk Aversion Risk Neutrality Risk Loving (Taking) - Utility Theory and Risk Analysis - Risk Premium - Decision-Making Under Risk - Certainty Equivalent Definition of Risk and Uncertainty - Risk/Uncertainty: Both concepts deal with the probability of loss or the chance of adverse outcomes - Risk: All possible outcomes of managerial decisions and their probabilities are not completely known - Uncertainty: The possible outcomes and their probabilities are known General Risk Categories Business Risk – the chance of loss associated with a given managerial decision; typically a by-product of the unpredictable variation in product demand and cost conditions Market Risk – the chance that a portfolio of investments can lose money because of overall swings in financial markets Currency Risk – the chance of loss due to changes in the domestic currency value of foreign profits General Risk Categories Inflation Risk – the danger that a general increase in the price level will undermine the real economic value of corporate agreements Interest-rate Risk – another type of market risk that can affect the value of corporate investments and obligations Credit Risk – the chance that another party will fail to abide by its contractual obligations General Risk Categories (Con’t.) Liquidity Risk – the difficulty of selling corporate assets or investments that have only a few willing buyers or are otherwise not easily transferable at favorable prices under typical market conditions Derivative Risk – the chance that volatile financial derivatives such as commodity futures and index options could create losses by increasing rather than decreasing price volatility Probability - Probability: likelihood of particular outcome occurring, denoted by p. The number p is always between zero and one. - Frequency: estimate of probability, p=n/N, where n is number of times a particular outcome occurred during N trials. - Subjective probability: If we do not have frequency, we often resort to informed guesses. Subjective probabilities must follow the same rules of the probability calculus, if we are dealing with rational decision-makers. Probability Distribution A probability distribution is simply a listing of the probabilities and their Potential Outcomes associated outcomes Probability distributions are often presented Potential Outcomes graphically as in these examples Probability Distribution Event P Discrete probability State of (probabi distribution: deals with Economy “events” whose “states of lity) nature” are discrete. The Recession 0.2 “event” is the state of the Normal 0.6 economy. The “states of nature” are recession, Boom 0.2 normal, and boom. Continuous probability distribution: deals with “events” whose “states of nature” are continuous values. The “event” is profits, and the “states of nature” are various profit levels. The Expected Value The expected value of a distribution is the most likely outcome For the normal dist., the expected value E(R) is the same as the arithmetic mean N All other things being equal, we E R R t 1 t t assume that people prefer higher expected returns The Expected Return: An Example Suppose that a particular investment has the following 60% Probability probability 40% distribution: 20% 25% chance of - 0% 5% return -5% 5% 15% Rate of Return 50% chance of 5% return 25% chance of 15% return E ( Ri ) 0.25 (0.05 ) 0.50 (0.05 ) 0.25 (0.15 ) 0.05 This investment has an expected return of 5% Standard Deviation The formula for the standard deviation when analyzing population data (realized returns) is: n ( Ri ERi ) 2 i 1 n 1 Standard Deviation The formula for the standard deviation when analyzing forecast data (ex ante returns) is: n ( R ER ) i 1 i i 2 Pi it is the square root of the sum of the squared deviations away from the expected value. The Variance & Standard Deviation The variance and standard deviation Less Risky describe the Riskier dispersion (spread) of the potential outcomes around N the expected 2 R 2 t R t R value t 1 Greater dispersion generally means R 2 R greater uncertainty and Calculating variance and standard deviation Using the same example as for the expected return, we can calculate the variance and standard deviation: i2 0.25 (0.05 0.05 ) 2 0.50 (0.05 0.05 ) 2 0.25 (0.15 0.05 ) .005 i 0.25 (0.05 0.05 ) 2 0.50 (0.05 0.05 ) 2 0.25 (0.15 0.05 ) 0.071 Note: In this example, we know the probabilities. However, often we have only historical data to work with and don’t know the probabilities. In these cases, we assume that each outcome is equally likely so the probabilities for each possible outcome are 1/N or (more commonly) 1/(N-1). The Scale Problem: an Example Potential Returns Prob ABC XYZ 10% -12% -24% 15% -5% -10% 50% 2% 4% 15% 9% 18% Is XYZ really twice 10% 16% 32% as risky as ABC? E(R) 2.0% 4.0% Variance 0.00539 0.02156 Std. Dev. 7.34% 14.68% C.V. 3.6708 3.6708 No! The Coefficient of Variation The coefficient of variation (CV)provides a scale-free measure of the riskiness of a security It removes the scaling by dividing the standard deviation my the expected return (risk per unit of return): CV R E R In the previous example, the CV for XYZ and ABC are identical, indicating that they have exactly the same degree of riskiness Risk Measurement Absolute Risk: - Overall dispersion of possible payoffs - Measurement: variance, standard deviation - The smaller variance or standard deviation, the lower the absolute risk. Relative Risk - Variation in possible returns compared with the expected payoff amount - Measurement: coefficient of Variation (CV), - The lower the CV, the lower the relative risk. CV EV Risk Attitudes Risk Aversion Characterizes decision makers who seek to avoid or minimize risk. Risk Neutrality Characterizes decision makers who focus on expected returns and disregard the dispersion of returns. Risk Seeking (Loving) Characterizes decision makers who prefer risk. Risk Attitudes Scenario: A decision maker has two choices, a sure thing and a risky option, and both yield the same expected value. Risk-averse behavior: Decision maker takes the sure thing Risk-neutral behavior: Decision maker is indifferent between the two choices Risk-loving (or seeking) behavior: Decision maker takes the risky option Risk Attitudes (MU) (MU) (MU) Risk averter: diminishing MU Risk neutral: constant MU Risk lover: increasing MU Utility Functions Utility is a measure of well-being. A utility function shows the relationship between utility and return (or wealth) when the returns are risk-free. Risk-Neutral Utility Functions: Investors are indifferent to risk. They only analyze return when making investment decisions. Risk-Loving Utility Functions: For any given rate of return, investors prefer more risk. Risk-Averse Utility Functions: For any given rate of return, investors prefer less risk. Utility Functions (Continued) To illustrate the different types of utility functions, we will analyze the following risky investment for three different investors: Possible Return (%) Probability (ri) (pi) _________ _________ 10% .5 50% .5 E(ri ) .5(10%) .5(50%) 30% σ(ri ) .5(10% 30%)2 .5(50% 30%)2 20% Risk-Neutral Investor Assume the following linear utility function: ui = 10ri Return (%) Total Utility Constant (ri) (ui) Marginal Utility __________ __________ __________ 0 0 10 100 100 20 200 100 30 300 100 40 400 100 50 500 100 Risk-Neutral Investor (Continued) Expected Utility of the Risky Investment: E(u) .5 * u(10%) .5 * u(50%) E(u) .5(100) .5(500) 300 Note: The expected utility of the risky investment with an expected return of 30% (300) is equal to the utility associated with receiving 30% risk- free (300). Risk-Neutral Utility Function ui = 10ri Total Utility 600 500 400 300 200 100 0 0 10 20 30 40 50 60 Percent Return Risk-Loving Investor Assume the following quadratic utility function: ui = 0 + 5ri + .1ri2 Return (%) Total Utility Increasing (ri) (ui) Marginal Utility __________ __________ __________ 0 0 10 60 60 20 140 80 30 240 100 40 360 120 50 500 140 Risk-Loving Investor (Continued) ui = 0 + 5ri + .1ri2 Expected Utility of the Risky Investment: E(u) .5 * u(10%) .5 * u(50%) E(u) .5(60) .5(500) 280 Note: The expected utility of the risky investment with an expected return of 30% (280) is greater than the utility associated with receiving 30% risk-free (240). ) - 5 + 25 - 4(.1)(-280 C e rtainty Equivale nt : 33.5% 2(.1) That is, the investor would be indifferent between receiving 33.5% risk-free and investing in a risky asset that has E(r) = 30% and (r) = 20% Risk-Loving Utility Function ui = 0 + 5ri + .1ri2 Total Utility 600 500 280 240 60 0 0 10 30 33.5 50 60 Percent Return Risk-Averse Investor Assume the following quadratic utility function: ui = 0 + 20ri - .2ri2 Return (%) Total Utility Diminishing (ri) (ui) Marginal Utility __________ __________ __________ 0 0 10 180 180 20 320 140 30 420 100 40 480 60 50 500 20 Risk-Averse Investor (Continued) Expected Utility of the Risky Investment: E(u) .5 * u(10%) .5 * u(50%) E(u) .5(180) .5(500) 340 Note: The expected utility of the risky investment with an expected return of 30% (340) is less than the utility associated with receiving 30% risk-free (420). 0) - 20 + 400- 4(-.2)(-34 C e rtainty Equivale nt : 21.7% 2(.2) That is, the investor would be indifferent between receiving 21.7% risk-free and investing in a risky asset that has E(r) = 30% and (r) = 20%. Risk-Averse Utility Function ui = 0 + 20ri - .2ri2 Total Utility 600 500 420 340 180 0 0 10 21.7 30 50 60 Percent Return Indifference Curve Given the total utility function, an indifference curve can be generated for any given level of utility. First, for quadratic utility functions, the following equation for expected utility is derived in the text: E(u) a0 a1E(r) a 2E(r) 2 a 2σ 2 (r) Solving σ(r) : for E(u) a0 a1E(r) 2 σ(r) = E(r) a2 a2 a2 Indifference Curve (Continued) Using the previous utility function for the risk-averse investor, (ui = 0 + 20ri - .2ri2), and a given level of utility of 180: 180 20 E(r) σ(r) E(r) 2 .2 .2 Therefore, the indifference curve would be: E(r) (r) 10 0 20 26.5 30 34.6 40 38.7 50 40.0 Risk-Averse Indifference Curve When E(u) = 180, and ui = 0 + 20ri - .2ri2 Expected Return 60 50 40 30 20 10 0 0 10 20 30 40 50 Standard Deviation of Returns

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Managerial Economics, managerial accounting, challenger performance, opportunity costs, the firm, Karam Singh, Random House Business, Business Books, Emile Woolf, the challenger

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posted: | 12/29/2010 |

language: | English |

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