Uncertainty and Risk1. Probability2. Variability3. Preferences for Risk4. Reducing Risk5. The Economics of Crime1Probability• The probability is the likelihood that a given outcome will occur.• If I flip this coin, what is the probability that the coin will turn upheads? 1/2. What is the probability that it turns up tails? 1/2.• Suppose I flip the coin twice. There are four events:head, headshead, tailstails, headstails, tailsThe probability of each of these events is 1/4.2• Suppose we make a little contract. If the coin turns up heads twice Iwill pay you $10.If the coin turns up heads once, tails once, I will pay you $5 (don’tworry about the order)If the coin turns up tails twice, I will pay you nothing.• The value associated with each possible outcome is called the payoff.• The expected value is probability weighted average of the payoffs asso-ciated with all possible outcomes.• The expected value of this contact isExpected value = Pr(heads, heads) × $10+ Pr(heads, tails) × $5+ Pr(tails, heads) × $5+ Pr(tails, tails) × $0=(1/4 × $10) + (1/4 × $5) + (1/4 × $5)+(1/4 × $0)=$5.3Variability• Often we don’t just carry about the expected payoff, we also care aboutthe variability of the payoffs.• The book defines variability as the extent to which the possible out-comes of an uncertain situation differ.• The most tradition and common measures of variability are the varianceand standard deviation.• The variance is the probability-weighted squared deviations of the pos-sible outcomes.• In general, if there are N possible outcomes, the formula for the varianceisσ2= Pr1[(X1− E(X))2]+Pr2[(X2− E(X))2]+... + PrN[(XN− E(X))2]where E(X) denotes the expected value, Priis the probability associ-ated with outcome Xi4• In our case,– first E(X)=5σ2=1/4[(10 − 5)2]+1/4[(5 − 5)2]+1/4[(5 − 5)2]+1/4[(0 − 5)2]=(1/4 × 25) + (1/4 × 0) + (1/4 × 0) + (1/4 × 25)=12.5• The standard deviation is the square root of the variance.• In our example σ =√12.5=3.54.5• Suppose I change the bet. Now if the coin turns up heads twice I willpay you $15.If the coin turns up heads once, tails once, I will pay you $5 (don’tworry about the order)If the coin turns up tails twice, you pay me $5.• The expected value of this contact isExpected value = Pr(heads, heads) × $15+Pr(heads, tails) × $5+Pr(tails, heads) × $5+Pr(tails, tails) ×−$5=(1/4 × $15) + (1/2 × $5) + (1/4 × $ − 5)=$5.Same expected value butσ2=1/4[(15 − 5)2]+1/2[(5 − 5)2]+1/4[(−5 − 5)2]=(1/4 × 100) + (1/2 × 0)+(1/4 × 100)=50the variance is four times larger.6Preferences for Risk• Image you are about to graduate and that you have two job offers.1. To join a large established company earning $54,000 per year. Thereis no risk to this job.2. To join a start-up paying $4,000 per year, but if the company be-comes profitable this year you will receive a bonus of $100,000– You believe there is a 50-50 chance the company will be profitablethis year.• Expected value of offer (1) is $54,000.• Expected value of offer (2) is (0.50 × $4000) + (0.50 × $104, 000) =$54, 000.• Same expected value, but differ considerably in risk.• So what do you do?7• When we were working with certain outcomes, we used the concept ofutility.• Now we need to work with expected utility. Expected utility is the sumof the utilities associated with all possible outcomes, weighted by theprobability that each outcome will occur.• So if there are N possible outcomes:Expected utility = Pr1× Utility1+ Pr2× Utility2+ ... + PrN× UtilityN• We usually divide people up into three groups:1. Risk averse2. Risk neutral3. Risk-loving8• A risk averse consumer prefers a sure-thing to a lottery of equal ex-pected value.– In general, a utility function that exhibits diminishing marginal util-ity implies that the utility of a sure thing will exceed the expectedutility of a lottery with the same expected value.• A risk neutral consumer compares lotteries according to their expectedvalue and is therefore indifferent between a sure thing and a lottery withthe same expected value.– utility is linear• Some people are risk loving. A risk loving consumer prefers an uncer-tain outcome to a certain one of equal expected value.• Many people have “split personalities” when it comes to risk. They arelocally risk loving and globally risk averse.– That is for big things, like their job, their home, their retirement,people generally like certainty. They will buy home owners and lifeinsurance; they will take safe jobs with steady income.– But they will still be risk-loving for small gambles. Take chances atthe blackjack table9Reducing Risk• Three main strategies for reducing risk1. diversification2. insurance3. obtaining more information• We are going to concern ourselves mainly with the first two.10Diversification• Diversification is essentially the field of finance– Don’t put all your eggs in one basket.– Reducing risk by allocating resources to a variety of activities whoseoutcomes are not closely related.• Suppose the summer in Connecticut can be either cool and rainy or hotand dry– If cool and rainy, movie theaters will make a lot of money; beachresorts will do poorly.– If hot and dry, movie theaters will do poorly, beach resorts will makea lot of money.• If you want to reduce you risk, you can invest a 1/2 of your money inmovie theaters and 1/2 of your money in beach resorts.• Then regardless of the weather you will do OK. Not as well as bettingcorrectly, but not as bad as betting on the wrong weather.• A sports betting example.• This is sometimes called a hedge.11Insurance• Suppose there are two states of the world this month1. Your house can burn down is which case2. Your house remains standing• The utility loss from your house burning down is enormous particularlyif you are like the vast majority of Americans who could not afford tolive somewhere else and rebuild through use of the savings.• If you are risk averse, you would be willing to purchase a contract thatpays off in the state of the world that you house burns down. It is aform of diversification.• We call an insurance contract actuarially fair when the premium isequal to the expected payout.• But most insurance contracts are not actuarially fair. From a purelyexpected value sense, insurance is usually not a good deal. But thereason you buy it is that it pays off in exactly