# Risk & Uncertainty in Decision Making

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Risk & Uncertainty in Decision Making
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Topics to be discussed Describing & Differentiating Risk from Uncertainty Role of Risk in Crime Deterrence Preferences Toward Risk Mechanisms for Reducing Risk Demand for Risky Assets (optional) 2

Describing & Interpreting Risk
To measure risk one must know: 1) All of the possible outcomes. 2) The likelihood that each outcome will occur (its probability). In case of uncertainty, either (1) or even both (1) and (2) are unknown. Objective Interpretation Based on the observed frequency of past events Subjective Interpretation Based on perception or experience with or without an observed frequency Different information or different abilities to process the same information can influence the subjective probability 4

Describing Risk: Expected Value
The weighted average of the payoffs or values resulting from all possible outcomes. The probabilities of each outcome are used as weights Expected value measures the central tendency; the payoff or value expected on average E(X) = p1x1 + p2x2 + …..+ pnxn 7

Standard deviation measuring risk
While the expected values are the same, the variability is not. Greater variability from expected values signals greater risk. Deviations or differences between expected payoff and actual payoff are important The standard deviation measures the square root of the average of the squares of the deviations of the payoffs associated with each outcome from their expected value. 16

Why SD important? f(X), g(Y) = prob. densities X, Y
Which distribution would you prefer, if you are risk-averse? X, Y E(X)=E(Y), but sd(X) >sd(Y)

Example of a risky situation
Job 1 is a job in which the income ranges from \$1000 to \$2000 in increments of \$100 that are all equally likely. Job 2 is a job in which the income ranges from \$1300 to \$1700 in increments of \$100 that, also, are all equally likely. Job 1: greater spread & standard deviation Job 2: peaked distribution - extreme payoffs are less likely Decision Making A risk avoider would choose Job 2: same expected income as Job 1, but with less risk. 24

Role of Risk in Crime Deterrence
Suppose, Extent of crime=r Gain per unit of crime=g Probability of detection & conviction =P(r) Cost to guilty if detected & convicted=C Hence, optimizing exercise of the criminal is to Maximize E(net gain) = g.r – P(r).C w.r.t. r FOC=> g – P’(r).C = 0 => g = P’(r).C => r = r (g, C, P) (+) (-) (-) SOC=> -P’’(r)<0 => P’’(r)>0 => you need some curb to keep crime rate r at finite level (check C(r) with C’’(r)>0 and/or g(r) with g’’(r)<0 can serve similar purpose) Observations: (i) A rise in P and drop in C or a drop in P with rise in C can serve the same purpose; (ii) With very high C, E(net gain) < 0 even with low P(r) close to zero; (iii) Societal cost to make P(r) = 1 or even close to 1 is very high; (iv) If P(r) = 0 then E(net gain) >0 for all values of r & r tends to be infinite. Why then capital punishment sometimes doesn’t work as deterrent to even heinous crimes? (look into text of example 6.2 for answer)

Role of Risk in Crime Deterrence: Figure
g.r, P.C P(r).C with P’(r)>0 & P’’(r)>0 g.r r r*=equilibrium level of crime

Preferences Toward Risk
Assumptions for choosing among risky alternatives Consumption of a single commodity The consumer knows all probabilities Payoffs measured in terms of utility Utility function given Defining risk averseness – i.e., preferences toward risk A person who prefers a certain given income to a risky income with the same expected value. A person is considered risk averse if he has a diminishing marginal utility of income The use of insurance demonstrates risk aversive behavior If an individual gets more utility from his present lower-paid job than a higher-paid risky job, he is said to be risk averse 34

Risk Averse Preferences Toward Risk
Utility The consumer is risk averse because she would prefer a certain income of \$20,000 to a gamble with a 0.5 probability of \$10,000 and a 0.5 probability of \$30,000. E 10 15 20 13 14 16 18 30 A B C D Risk averseness means gain valued less than loss at the margin Income (\$1,000) 46

Risk Neutral Preferences Toward Risk
6 A E C 12 18 The consumer is risk neutral as he is indifferent between certain events and uncertain events with the same expected income. Utility A person is said to be risk neutral if he shows no preference between a certain income, and an uncertain one with the same expected value. Risk neutrality means gain valued same as loss at the margin Income (\$1,000) 10 20 30 49

Risk Loving Preferences
A person is said to be risk loving if he shows a preference toward an uncertain income over a certain income with the same expected value. Examples: gambling, criminal activity Utility 3 10 20 30 A E C 8 18 The consumer is risk loving because she would prefer the gamble to a certain income. Read example 6.3 carefully to understand why the US is becoming a nation of gamblers & poor nations too are prone to the same virus! Risk loving behavior means gain valued more than loss at the margin Income (\$1,000) 52

Notion of Risk Premium The risk premium is the amount of money that a risk-averse person would pay to avoid taking a risk. R = U[E(Y)] – E[U(Y)] in utility terms It can also be measured along the horizontal axis in terms of money The greater the variability – i.e., the spread of values of the random variable around mean, the greater the risk premium. The greater the concavity of the utility curve, the greater the risk premium – it means a much higher weight is assigned to a decline in income as compared to an equal amount of rise in income 53

10 16 Here , the risk premium is \$4,000 because a certain income of \$16,000 gives the person the same expected utility as the uncertain income that has an expected value of \$20,000. 18 30 40 20 14 A C E G F Risk Premium in money terms Utility Risk premium in utility terms Income (\$1,000) 57

Meaning of High Degree of Risk Aversion
Here increase in Standard deviation requires a large increase in income to maintain satisfaction. U1 U2 U3 Expected Income Standard Deviation of Income

Meaning of Low Degree of Risk Aversion
Expected Income Slightly Risk Averse => A large increase in standard deviation requires only a small increase in income to maintain satisfaction. U1 U2 U3 Standard Deviation of Income

Indifference curves of risk-neutral & risk-loving people
Risk-neutrality means horizontal indifference curve showing zero premium the consumer is willing to pay to buy hedge against risk (i.e., he doesn’t mind bearing more risk) Risk-lover willing to pay (i.e., make sacrifice in terms expected income) to enjoy the thrill of greater risk-bearing – thus making indifference curves usual downward sloping

Utility function of a person who buys insurance and gambles too!
Total utility TU curve Near point of inflection, A (MU at its minimum), a rise averter will spend small amount to buy insurance against small chance of large loss, while as a risk seeker spend a small amount (on lottery) to seize small chance of a large gain. Convex to horizontal axis A Concave to horizontal axis Income

Risk Reduction Strategies
Three ways consumers attempt to reduce risk are: 1) Diversification 2) Insurance 3) Obtaining more information Firms can reduce risk by diversifying among a variety of activities that are not closely related. 67

Reducing Risk thro’ Diversification
Reducing risk by allocating resources to a variety of activities whose outcomes are not closely related Example: Suppose a firm has a choice of selling air conditioners, heaters, or both The probability of it being hot or cold is 0.5 How does a firm decide what to sell? 68

Income from Sales of Appliances
Hot Weather Cold Weather Air conditioner sales \$30,000 \$12,000 Heater sales 12,000 30,000 69

Diversification – Example
If the firm sells only heaters or air conditioners their income will be either \$12,000 or \$30,000 Their expected income would be: 1/2(\$12,000) + 1/2(\$30,000) = \$21,000 If the firm divides their time evenly between appliances, their air conditioning and heating sales would be half their original values If it were hot, their expected income would be \$15,000 from air conditioners and \$6,000 from heaters, or \$21,000 If it were cold, their expected income would be \$6,000 from air conditioners and \$15,000 from heaters, or \$21,000 With diversification, expected income is \$21,000 with no risk Better off diversifying to minimize risk Firms can reduce risk by diversifying among a variety of activities that are not closely related 70

Reducing Risk – The Stock Market
If invest all money in one stock, then take on a lot of risk If that stock loses value, you lose all your investment value Can spread risk out by investing in many different stocks or investments Ex: Mutual funds 73

Choosing an Investment Portfolio
Rate of return(%) Highest possible indifference curve Risk-return frontier F=high return, high risk investment C=optimum portfolio choice E= less return, less risk investment Risk

Reducing Risk – Insurance
Risk averse are willing to pay to avoid risk If the cost of insurance equals the expected loss, risk averse people will buy enough insurance to recover fully from a potential financial loss For the risk averse consumer, guarantee of same income regardless of outcome has higher utility than facing the probability of risk Expected utility with insurance is higher than without Purchases of insurance transfers wealth and increases expected utility Actual Fairness: When the insurance premium = expected payout 74

Value of Title Insurance When Buying a House : An Example
In the absence of title insurance, a risk averse buyer would pay much less for the house By reducing risk, title insurance thus increases the value of the house by an amount far greater than the premium. 77

The Decision to Insure – An example
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How does insurance work? The Law of Large Numbers
Insurance companies know that although single events are random and largely unpredictable, the average outcome of many similar events can be predicted When insurance companies sell many policies, they face relatively little risk Insurance companies can be sure total premiums paid will equal total money paid out Companies set the premiums so money received will be enough to pay expected losses 77

Why no insurance against certain events?
Some events with very little probability of occurrence such as floods and earthquakes are no longer insured privately Cannot calculate true expected values and expected losses Governments have had to create insurance for these types of events Ex: National Flood Insurance Program 77

The Value of Information
Risk often exists because we don’t know all the information surrounding a decision Because of this, information is valuable and people are willing to pay for it The value of complete information The difference between the expected value of a choice with complete information and the expected value when information is incomplete 79

The Value of Information – Example
Per capita milk consumption has fallen over the years The milk producers engaged in market research to develop new sales strategies to encourage the consumption of milk Milk advertising increases sales most in the spring Allocating advertising based on this information in New York increased profits by 9% or \$14 million The cost of the information was relatively low, while the value was substantial (increased profits) Findings Milk demand is seasonal with the greatest demand in the spring Price elasticity of demand is negative and small Income elasticity is positive and large 85

Demand for Risky Assets
Assets: Something that provides a flow of money or services to its owner. However, the flow of money or services can be explicit (dividends) or implicit (capital gain). Capital Gain: An increase in the value of an asset, while a decrease is a capital loss. Risky Asset: Provides an uncertain flow of money or services to its owner. Examples: apartment rent, capital gains, corporate bonds, stock prices Risk-free Asset: Provides a flow of money or services that is known with certainty. Examples: short-term government bonds, short-term certificates of deposit Return on an Asset: The total monetary flow of an asset as a fraction of its price. Real Return of an Asset: The simple (or nominal) return less the rate of inflation. Asset Return: Return = Money flow / bond price 86

Higher returns are associated with greater risk. The risk-averse investor must balance risk relative to return An investor is choosing between T-Bills and stocks: T-bills (risk-free) versus Stocks (risky) Rf = the return on risk free T-bills Expected return equals actual return when there is no risk Rm = the expected return on stocks rm = the actual returns on stock The risky asset will have a higher expected return than the risk free asset (Rm > Rf). Otherwise, risk-averse investors would buy only T-bills. 94

The Investment Portfolio
How to allocate savings: b = fraction of savings in stock market 1 - b = fraction in T-bills Expected Return: Rp: weighted average of the expected return on the two assets Rp = bRm + (1-b)Rf If Rm = 12%, Rf = 4%, and b = ½, then Rp = 1/2(.12) + 1/2(.04) = 8% 97

The Investment Portfolio
Risk (standard deviation) of the portfolio is the fraction of the portfolio invested in the risky asset times the standard deviation of that asset: σp = bσm Rp = bRm + (1-b)Rf => Rp = Rf + b(Rm-Rf) => Rp = Rf + (σp/σm)(Rm-Rf), which is a budget line describing the trade-off between risk (σp) and expected return (Rp). Observations: This is a straight line with Rp, Rf and σm as constants; Its slope= (Rm-Rf)/σm; Expected return, RP, increases as σp - i.e., risk increases. 99

Choosing between Risk and Return
U2 is the optimal choice of those obtainable, since it gives the highest return for a given risk and is tangent to the budget line. Expected Return,Rp Rf Budget Line Rm R* U2 U1 U3 109

The Choices of Two Different Investors
Can equilibrium occur at a point to the right of point M? Under what circumstances? Expected Return,Rp Given the same budget line, investor A chooses low return-low risk, while investor B chooses high return- high risk. UA RA UB RB Rm Budget line How will this figure change for a risk-lover? Indifference curves will have the usual shape for two goods, rather than one good and one bad. M Rf 113

Hedging against Foreign Exchange Risk
Read the example in pp for details about Forward contract & Futures contract (standardized forward contract for predetermined quantities & selected calendar dates) What happens if US\$ & Yen continues to depreciate vis-à-vis Indian rupee?

Investing in the Stock Market: Example
Observations Percent of American families who had directly or indirectly invested in the stock market 1989 = 32% 1995 = 41% Share of wealth in the stock market 1989 = 26% 1995 = 40% Participation in the stock market by age Less than 35 1989 = 23% 1995 = 29% More than 35 Small increase

Behavioral Economics Sometimes individuals’ behavior contradicts basic assumptions of consumer choice More information about human behavior might lead to better understanding This is the objective of behavioral economics Behavioral economics is a developing field to help explain and elaborate on situations not well explained by the basic consumer model Improving understanding of consumer choice by incorporating more realistic and detailed assumptions regarding human behavior

Behavioral Economics There are a number of examples of consumer choice contradictions You take at trip and stop at a restaurant that you will most likely never stop at again. You still think it fair to leave a 15% tip rewarding the good service. You choose to buy a lottery ticket even though the expected value is less than the price of the ticket

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