Uncertainty and Expected Value
Suppose that eccentric rich aunt who asked you to decide between payouts at different times was disappointed that you realized present value analysis lets you decide the best offer, so she dreams up a new devious choice. At your option she will give you one of two gambles. You can choose to have her flip a coin or role a pair of dice. The coin flip pays $4000 if it turns up a head, but zero if it turns up tails. The dice role is more complicated. A twelve or two pays $18000, a six pays $7000, and anything else pays zero. Which would you choose?
The earlier problem was one of present value analysis. This problem is an exercise in expected value. Expected value analysis is a mathematical way of comparing alternatives under uncertainty - when one or more outcomes occur only with some chance. It uses probability which you may have studied in statistics.
Some basic facts about probability are needed before moving to expected value. Probability measures how likely one of several alternative outcomes is to occur. Outcomes which are more likely to occur have higher probabilities. Moreover, the sum of the probabilities of each of the possible outcomes is always equal to 1. Thus if there are only two outcomes, and they are equally likely, then each has a probability of 1/2. But if we flip a coin twice, our possible outcomes are HH, HT, TH and TT, where an H and T signify head and tail respectively, and the first letter is the result of the first flip, the second letter the result of the second flip. If order does not matter, then there are three possible outcomes. There can be two heads, two tails, or one of each. The probability of all heads is 1/4, of all tails is also 1/4, and the probability of a head and a tail (in any order) is 1/2. Notice that the most likely outcome (a mix of head and tail) has a higher probability than either of the pure results, and the sum of all possible outcomes is 1.
Now we can talk about expected value. Take the example from the eccentric aunt. With the coin toss there are only two possible outcomes - heads and tails. Assuming a fair coin, each outcome has a probability equal to 1/2. (In fact, the definition of a fair coin is one where the probability of both a head and a tail is 1/2.) The expected value of the game is the value of each outcome multiplied by the probability of that outcome, so in this case, the expected value is [(1/2)@$4000]+[(1/2)@0]=$2000. The term before the plus is the payoff for a head, while the term after the plus is the payoff for a tail.
Finding the expected value of the dice roll is more complicated. There is only one way to get a twelve (two 6's) and one way to get a two (two 1's). But there are five ways to get a six (1,5; 2,4; 3,3; 4,2 and 5,1 - the first and second digits in each pair are, respectively, the result of the first and second dice). It turns out a roll of the dice has 36 possible outcomes which range in value from 2 to 12. Given your aunt's offer there is a 1/18 possibility of getting $18000 (2 outcomes out of 36 earn you this amount) and a 5/36 possibility of earning $7000 (the 5 dice pairs that equal 6). Thus, the expected value is [(2/36)@$18000]+[(5/36)@$7000]+[(29/36)@$0]=$1972.22. Notice that the most likely outcome is zero dollars. With a probability of 29/36 the dice roll will come up something other than a 2, 12 or 6. You can see that the dice roll has an expected value that is smaller that the coin toss, even though the amounts offered are much higher.
Remember how present value analysis depended on the interest rate, the term, and the payoff each period. Expected value analysis also has three variables - the number of outcomes, the probability of each outcome, and the payoff of each outcome. Generally, as the payoff and probability of an outcome increase, expected value also goes up. There is, however, one caveat. Since the sum of all probabilities always equals one, increasing the probability of one outcome means the probability of another must fall. Therefore, how expanding the number of outcomes or changing the probabilities has no clear indication for the expected value.
Example 5: Switching Doors on "Let's Make a Deal"
In 1990 a puzzle started making the rounds among economists, statisticians and other professionals interested in recreational mathematics. It appeared in the Journal of Economic Perspectives and even in the column "Ask Marilyn" in Parade Magazine (December 2, 1990). The puzzle mimics how an old TV show "Let's Make a Deal" was played.
Near the end of each show a contestant is given a choice of three doors. One has a great prize, but booby prizes are behind the other two doors. After a door is chosen the host opens one of the remaining doors, always revealing a booby prize. (Since there are two booby prizes, at least one of the unchosen doors covers a booby prize.) The contestant can then choose to keep the original (not yet open) door, or switch to the other (unopened) door. The problem - does staying or switching give a better chance for the wonderful prize?
Many people are fooled by the answer, which is switching gives a better chance for the great prize. The original choice has a 1/3 probability of being correct. Revealing a loser does nothing to that probability. Since the host has inside information, that he chooses one of the remaining doors over the other remaining door raises the probability that the other remaining door has the great prize. Intuitively, there is a 2/3 chance that the great prize is behind either of the doors you did not choose. Opening a loser does nothing to change either of these odds, so switching moves you from a 1/3 chance of winning the great prize to a 2/3 chance. So you should always switch.
Risk and the return to investment
After you graduate college you will probably experience the "invasion of the credit cards." Moreover, it will become an increasing invasion as your income and time since college increase. This invasion is the number of banks willing to give you credit cards with higher and higher credit limits, often at lower and lower interest rates. What we have here is an application of expected value to risk and the return to investment.
As incomes rise and people get older they generally become better credit risks - they can afford higher payments on loans, and become more dependable about paying their bills. Banks know this, and thus expect older, better paid credit card users to be more likely to pay the bills. The probability of defaulting (not paying) on credit card goes down, and thus the expected value to the bank of the credit extended goes up. Higher incomes explain higher credit risks, but why are lower interest rates often extended to older people?
The key factor here is that the sum of all probabilities is always one. Suppose we have two people with the same income who borrow $1000 each. One is young and the other is old. Experience has shown that younger people on average default more than older people. For simplicity suppose the young default 10 percent of the time (1 in 10 young borrowers does not repay the credit) and the old default 5 percent of the time. To get the same expected value of the $1000 loan the bank will charge younger borrowers higher interest.
For every 20 young borrowers, the bank will get back $18,000 (2 of the 20 default). Unfortunately the bank does not know which of the young will not pay back the loan - otherwise it would just refuse the card to those two persons. Thus the interest charged on the 18 who pay must cover not only the opportunity cost of $20,000 dollars (see the sections on present value earlier in this chapter) but the $2000 lost to default. But 20 older borrowers pay back $19,000 so the bank must spread only a $1000 loss over the other borrowers, and spreads it over 19 instead of only 18 people. Thus, their interest rate is lower.
Suppose the opportunity cost is 5 percent, so the bank needs to get back a total of $21,000 to make a fair rate of return. It would require that the 18 young who meet the obligation pay about 16.7 percent interest ($18,000@1.167=$21,000). Older borrowers would have an interest rate of about 10.53 percent ($19,000@1.1053=$21,000 also). The different interest rates reflect the different probabilities of the default outcome.
This analysis helps to explain why different investments earn different interest rates. There are two factors - the term of the investment, and its innate riskiness. Risk, in this case, refers to the probability of the investment turning bad.
Term affects the interest rate because for most investments the longer the term the greater the probability of default. Take for example an investment that goes over two periods instead of just one. This investment has a likelihood of default of 10 percent each period. For an investor to get her money back, the investment must not default twice. There is a 90 percent chance that it makes it through the first period. But given that, there is still only a 90 percent chance of that 90 percent that it will still be good after the second period. Overall there is only an 81 percent chance that the investor will be paid back.
Putting it another way, suppose there are 100 firms to start, each with a 10 percent chance of default each year. After one year only 90 of the firms are still in business. But after the second year only 81 of those 90 are still in business. The cumulative default rate lowers the probability of investors being paid back. So an investor who would see a risk of only 10 percent for a one year investment sees a higher (overall) risk of 19 percent over two years. Clearly, the risk premium for the longer term investment needs to be higher, and this is one reason the term structure of interest rates usually requires that longer term investments pay higher rates of return.
The credit card example shows how two investments of the same term, one risky (the young) and one relatively safe (the old), must pay different rates of return. Risky investments pay higher rates of return to attract investors. Without the higher rates the expected value of the investment is too low. In earlier chapters we kept saying that profit is a signal to investors that a better than normal return could be had in that industry. When there is uncertainty investors look at the expected return and the variation (risk) to the return. Speculative investments, like oil drilling junk bonds, advertise high rates of return because often they do not pay off - the actual pay out is often zero. For every oil well that earns investors millions of dollars, hundreds are dry and those investors lose their money. On average (in an expected value sense) they give a normal or slightly above normal rate of return, and thus are able to attract investors. Safe investments like Treasury bills or insured bank accounts pay low rates because there is almost no risk of default. Any expected rate of return in excess of the equivalent risk-free return is called a risk premium, and is a payment to the lender for taking on the risk.
Take an example of a $1000 investment in Treasury bills paying 5 percent. Industrial bonds are less certain, so they might pay a risk premium of 1 percent, offering an expected return of 6 percent. Since some bonds may default, the face value return will be higher than 6 percent. Stocks are less certain still, relying on capital gains as well as dividends, so the expected rate of return is probably higher than 6 percent. However, some stocks go bust, and so some people are losers. Overall, and over long periods, stocks pay off with the best returns because of the risk premium. You want to avoid being the person who chooses the losing investment that raises the required pay off to the winners.
A digression on diversification
One of the tenets of investing is that diversification is one way to gain hefty rates of returns while lowering the risk. In folk terms, diversification means not putting all of your eggs in one basket. Then, if you "drop" the basket, not all your eggs get broken.
The value of diversification is seen in a simple example. Suppose you have $1000 to invest and your choices are to put it all in one investment or divide it into two $500 investments. Each investment has a probability of failure of 10 percent (.1) and a probability of 90 percent (.9) that it will return 20 percent interest in one year. In the case of a failure the return is zero. All investment have independent probabilities of failure.
Ignoring time (since all terms are equal), the expected value of the first investment is simply
$1200@0.9 + 0@0.1 = $1080
for an expected return of 8 percent. For the second option the expected value is somewhat more complicated. Each separate investment has an expected value of
$600@0.9 + 0@0.1 = $540
so doubling this value seems to give $1080 also. However, there are really four possible outcomes. First, both investments can be successful. Second and third, one or the other of the investments could go sour. Finally, both can be unsuccessful. Notice, only under the last scenario is the return worthless.
Using the fact that the joint probability of independent events is just the product of their separate probabilities we find that the probability of both $500 investments being successful, returning $1200, is 0.9@0.9=0.81, the probability of one being successful and the other bad, returning $600, is 0.9@0.1=0.09, times 2 for the two possibilities is 0.18, and the probability of both being bad is 0.1@0.1=0.01, in which case no money is returned to the investor. The expected value of this investment strategy is
0.81@$1200 + 0.18@$600 + 0.01@$0 = $1080
as we suspected. However, the difference is in the variation of the return. When there is less variation, there is more of a chance of getting the expected value. Variation is measured by the standard deviation, which tells how the potential outcomes are dispersed around the expected value. With the single investment the standard deviation is 360, but if the $1000 is spread over two investments the standard deviation is only about 255. In theory there is a greater likelihood of earning near the expected value.
Of course, in this example you can never earn the expected value. The discrete nature of the returns, 0, 600 or 1200 dollars makes it impossible. Diversification works by lowering the probability that all your money will be lost. The trade-off is that the probability of the biggest gain is also lower. Notice that in this example the chance of getting the full $1200 falls from 0.9 to 0.81 by diversifying. However, the chance of losing it all also falls, from 0.1 to 0.01. The rest of the chance is made up of the intermediate outcome that you lose only part of your money.
Few investment opportunities are so lumpy and all or nothing as this case. Instead, the riskiness is in the rate of return. Investors can diversify even more than by spreading money over different projects by using different types of investments that pay off differently according to the state of the economy. For example, bonds tend to appreciate poorly during periods of moderate inflation, which often favor stocks and cash (money market) accounts. Thus the standard advice of holding some of your portfolio in cash (money markets), some in bonds, and some in stocks not only diversifies by using multiple investments, but diversifies according to the economic climate. Investors give up some return, but gain much more security against the loss in the value of their portfolio.
Expected utility analysis and risk aversion
Expected utility analysis is an extension of utility analysis that enables individuals to make decisions under situations of uncertainty. People face risks all the time, and must make decisions about consumption that must hold in any of a multiple of possible outcomes. For example, a consumer may take out a loan to buy a car, predicting that she will be able to pay the monthly costs of the loan because she has a good paying job. But that job is not certain. Unemployment may loom as a (strong or weak) possibility. Likely, if she knew her job was going to end in three months her decision might have been different, or, analogously, if she knew a pay raise was coming soon, she might have bought a more expensive car.
In its simplest form expected utility compares the utility an individual can expect under different states of the world, adjusts each by the probability of its outcome, and sums them up. Mathematically, it is just like any expected value, except the values are utility.
Suppose we can measure an individual's utility as a function of his income. That is, U=f(Y) where U is utility, f(@) is the utility function, and Y is income used for consumption. By the law of diminishing marginal utility MUY=DU/DY decreases as Y gets larger. We will continue to assume more is better, so marginal utility is always positive, albeit arbitrarily small as income gets very large. Such a utility function is shown in figure 15.6. At a high income, M, utility is U(M), but at lower incomes like m utility is only U(m).
Now suppose Y can take on only these two values, M and m. The outcome is beyond the individual's control, so his income is uncertain. For simplicity assume the probability of each outcome is 1/2. Then his expected income, denoted E(I),is given by E(I)=[(1/2)@M]+[(1/2)@m]. If M=10000 and m=2000 the expected income is 6000.
Clearly the individual prefers 10000 to 2000. But with his expected income of 6000, is his expected utility U(6000)? The answer is a decided no if there is diminishing marginal utility. Instead, we must figure out the expected value of his utility, not income.
If this person's income ends up being 10000, his utility is U(10000). However, if his income ends up being only 2000, he gets utility of only U(2000) which will be clearly less than U(10000). Thus, his expected utility, denoted E(U), is E(U)=[(1/2)@U(10000)]+[(1/2)@U(2000)]. It turns out that this value will be smaller than U(6000). With diminishing marginal utility, the utility of expected income exceeds the expected utility.
Let us look at an example. A utility function that shows diminishing marginal utility is U=log(Y) where log is the natural logarithm. The log(10000) equals about 9.21, while the log(2000) is about 7.6. Thus the expected value of utility (expected utility) when there is equal probability of having 10000 or 2000 for income is (9.21+7.6)/2=8.405. But the expected income of 6000 has a utility of 8.7, larger than the expected utility.
If there are more than two states possible, we simply add the additional incomes multiplied by the probability of occurring. To keep it simple we will stay with two possible incomes. Figure 15.7 shows how to find expected utility graphically. The utility if income is 10000 or 2000 is shown. A line segment connecting the two points on f(Y) shows the possible expected utility. How certain each outcome is determines where on that line is the expected utility. For example, if there is equal likelihood of income being 2000 and 10000, the expected utility is the midpoint of the line segment, shown as point a.
If the probability of 10000 is lower than 1/2 the expected utility moves down to the left on the line segment. Obviously the opposite also holds. So if the probability of 10000 is 1/4 (leaving a probability of 2000 at 3/4) the expected utility corresponds to point b, while if the probability of 10000 is 3/4 the expected utility corresponds to point c.
Two key facts can be seen from this type of graph. Figure 15.8 shows the utility graph when U=log(Y). First, notice that it shows that E(U)=8.405 is lower than U(6000)=8.7, as we argued before. Additionally, we can see the certain income (4469) that gives the same value as the expected utility.
Attitudes towards risk
Almost any individual will prefer a certain income that gives a higher utility than some expected utility that results from two or more possible incomes. In figure 15.8, that means we should expect a person would prefer any certain income exceeding 4469 (say even 4470) to the 1/2 chance at 10000 or 2000. By default the expected income of a certain 4470 exceeds the expected income under uncertainty.
Our analysis can go further. Suppose a person is offered two choices. The first choice is between 2000 and 10000 with 1/2 probability each, and the second choice is between 907 and 22026, again with 1/2 probability each. Both choices have an expected utility of 8.405 using the logarithmic utility function. How an individual chooses between these options determines her risk attitude.
A risk neutral person will be essentially indifferent between the options. Such an individual cares only about the expected value, not at all about the risk, measured by the dispersion (standard deviation). She would choose a higher expected value over a lower one. Given the choice between the 2000, 10000 game and a 907, 22027 choice, she would choose the latter simply because the expected value (8.4051) is marginally higher.
A risk seeking person cares about risk, and loves to experience it. In the example offered, the standard deviation of the expected utility for the 2000, 10000 choice is .805, while the standard deviation of the 907, 22026 choice is 1.6, so the risk seeking person prefers the second choice, with the bigger possible win (income of 22026) but the lower low side as well. In fact, such a person may prefer a lower expected value with a larger possible up side, say 907, 22000, over a lower standard deviation with a higher expected value.
Experience shows that most people seem to be risk averse - they try to avoid risk. Given the choice between the two options they will choose the safer one, even if the expected value is lower. We can show this graphically in figure 15.9. The person faces 1/2 probabilities of incomes 10 and 20, so expected income is 15. The expected utility is E(m'), where m' is the certain income that gives the same utility as the uncertain situation.
A risk neutral person would pay at most 15-m' to avoid the risk. Called the risk premium, any payment under this amount for insurance which guarantees an income of at least 15 leaves her with a higher utility. Under such a scheme, she would give the insurer any income above 15-m', and would receive from the insurer enough to ensure an income of 15-m' if she in fact gets only 10.
A risk averse person goes further. He would pay more than m' to avoid the risk. How much of an insurance premium (an amount greater than the risk premium) a risk averse person will pay depends on how risk averse he is. Call the insurance premium i. The only restriction is that 15-m'-i must exceed 10, otherwise there is no value to the insurance. The person is ensured of at least 10 no matter what.
There are three aspects to risk aversion that can be seen from an expected utility graph. First, as an individual becomes more risk averse, the insurance premium (difference between expected income and the lower amount he will accept with certainty instead) grows. Second, the risk premium increases as the variability of incomes (high and low outcomes) gets larger, even if the expected income stays the same. Finally, because of diminishing marginal utility, the risk premium increases as the expected income grows, even if the spread between the possible high and low incomes under uncertainty grows.
Example 6: Risk Aversion and Health Insurance
We often think about health insurance as a way to make sure we can get the care we need. In fact, a more proper term may be income or wealth insurance. Health insurance does not insure health - instead it protects us against the financial burden that befalls people when they get sick.
Look at figure 15.box6.1. Ill health can have two impacts on a person. First, her ability to enjoy any income falls. If H1 is health status before becoming ill, and H2 is the health status after, we expect that the utility of income under H1 exceeds the utility of the same income under H1. Thus at an income of M1, U(M1;H1)>U(M1;H2).
But ill health can have another effect. The cost of treatment, inability to go to work, and other factors can lower income as well as health. So besides seeing the utility function fall from U(@;H1) to U(@;H2) she sees her income fall to m2. Without insurance utility falls from U1 to U2.
Health insurance helps to preserve income. By paying some or all of the cost of medical treatment health insurance keeps income at m3, and it is enjoyed according to the utility function at health level H2. On net, utility falls only to U3. Even if there were no income loss there would be lower utility because of the lower health. The health itself may or may not be restored.
The importance of perspective
Have you ever noticed that policy makers seem to approach risk differently than those affected by the policy? Think about some of the more recent problems facing our country. There is an increased need for electrical generating facilities, including nuclear power plants, disposal sites, and other obnoxious facilities. Chemical waste and by-products are found in water and food supplies. Prisons, institutions for the mentally ill, and even schools are seen as intruders into residential neighborhoods. Despite the assurances of scientists and policy makers that the enterprise is "perfectly safe" or within an acceptable risk, those who live near the site fight tooth and nail to avoid it. They picket, take legal action, and lobby policy makers to move the facility elsewhere, under the NIMBY (not in my backyard) syndrome.
Expected value, risk aversion, and individual perspective can explain why policy makers and regular people disagree on these issues. Much of the disagreement can be attributed to expected value (policy makers) differing from expected utility (those affected), disparate attitudes towards risk, and what is considered to really matter (perspective).
First take expected value. Policy makers are often forced to reduce issues to dollars and "sense." Partly because utility cannot be measured easily, they try to assess those elements of a problem that can be measured monetarily. So if one hazardous waste site is $100,000 cheaper than an alternative site which may be marginally safer, or affect fewer people, the policy analysis may come down for the first site. Of course politics (the "sense" part) will come into play, but after the decision is made policy makers always find it easier to justify on expected dollar costs and benefits than on expected utility and attitudes.
We have just seen that expected utility is a very different concept from expected value. Under diminishing marginal utility even the risk neutral person prefers a lower certain value (for example paying for a more expensive but safer disposal site) over a more risky but cheaper alternative. All that is needed is that the additional cost be less than the risk premium.
Risk attitudes also come into play. Once some scientifically acceptable threshold is met, the policy process often becomes risk neutral or mildly risk averse. Individuals, however, tend to be more strongly risk averse, plus they face the additional risk that the policy makers are not being fully truthful. On net, policy makers may be amenable to a larger risk than the general public.
Finally there is perspective. The policy process may measure risk on a different base than the general public. For example, in 1989 New Jersey trucked some radioactive waste to Hanford, WA. Policy makers felt the risk of an accident was acceptable, but measured it on a per truck mile basis. Protestors claimed such a measure clouded the real issue, which was how likely someone was to be contaminated by an accident involving one of the transport trucks. In fact, due to the large number of miles that would be driven, the chance of an accident somewhere along the transportation route was quite high. At question was if anyone would be injured when the accident occurred. Protestors felt exculpated when the very first shipment turned over and spilled in the midwest. However, policy makers were quick to point out that no one was injured. It is interesting that both sides were vindicated by appealing to the arguments first put forth by the other.
Example 7: Transporting Radioactive Waste in Colorado
The importance of perspective on assessing risk comes through in many environmental problems. One example comes from the clean-up of an old uranium processing plant in Grand Junction, Colorado. Part of the clean-up entailed transporting about 3.8 million cubic yards of radioactive material 18 miles from the old Climax Uranium Company mill site to storage at the Cheney Reservoir.
Two transportation alternatives were considered. The Truck Option moved the wastes on existing roadways in Grand Junction, mostly through and along residential areas, until it reaches existing railroads at the edge of town. At that point the waste is transferred to rail cars for the remainder of the distance. The Rail Option builds a new rail spur to the old Climax site through low population density areas. Thus, it avoids bringing the waste near residential areas, but the cost of the spur is about $12 million.
The principle difference between the two alternative modes of transportation is the risk to residential populations. Using the rail option avoided any such risk, but the truck option had an estimated probability of about 1 serious accident for every 1.2 million miles a truck goes. Policy makers looked at that figure and decided the truck accidents would be extremely rare.
There was so much waste (3.8 million cubic yards) to be moved that over the life of the project that it would take about 350,000 truckloads. Thus, an expected 5.3 (probability of an accident per mile x number of trips x the length of a trip) truck accidents would happen during the project.
Policy analysts not connected with the project estimated that an accident would harm a radius between 1/2 and 1 mile. Using the 1 mile radius a individual along the corridor of transportation had a 5.3/18=0.2944 probability of being affected by a spill sometime during the clean-up. It is this relatively high value, almost a 30 percent chance, that was relevant to the affected population.
Policy makers, on the other hand, focused on the accident probabilities. In those, the expected number of rail accidents was 1 serious accident for every 5.2 million rail miles. Since the rail option traveled about twice as far as the truck option, serious truck accidents were twice as likely as serious rail accidents, but both were very low per mile.
Policy makers looked at the alternatives and decided that the $12 million cost of the rail option was unacceptably high for the risk reduction it brought. While that might have been the correct decision, the different perspectives explains why policy makers looked at the whole episode as somewhat routine, while households on the transportation corridor were much more concerned. Policy makers saw a very low probability of an accident, while households saw an unacceptably high probability of being affected.
Combining time and uncertainty
Earlier in this chapter we saw how the higher likelihood of defaulting on a loan gave the traditional term structure of interest rates - longer term loans usually carry higher interest that shorter term loans. That example hinted that we sometimes must combine time and uncertainty into a single model. In this section we briefly take a more formal look at that idea.
In the term structure example the cumulative probability of default increased over time, thus increasing the risk premium. When individuals or policy makers face choices over time, the inherent uncertainty of the future affects calculation. A standard presentation just takes the expected value of each future period and discounts it to present value.
The mathematical formulation is notationally confusing but arithmetically simple. Let EVt be the expected value of an uncertain event at time t. Suppose t goes for 3 periods, so t is either 0 (the current period), 1 (next period), or 2 (the period after next). If the interest rate is r, the expected present value, E(PV), is
E(PV) = EV0 + EV1/(1+r) + EV2/(1+r)2
where we use the fact that (1+r)0=1. For an event that last N periods, the expected present value is 'i[EVi/(1+r)i] where i goes from 0 to N.
The next step is to just substitute the formula for the expected value in each period. So if EV1 = (1/2)@1000 + (1/2)@3000 then 2000 would be used there. The same would be done for EV0 and EV3.
Return to your rich aunt one last time. Her gambling instinct takes hold, and now she dreams up the following offer. You pay her $1000 today. Three years from now she will flip a coin. If it is heads she will pay you $1250, but if it is tails you get nothing. Then four years from now she flips 2 coins (or one coin twice). If two heads show up she will pay you $2400, but any other combination nets you zero. Is this a bet you should accept?
As you know, the present value calculation depends on your time rate of preference. Suppose that rate is 5 percent. By finding the expected value of the coin flips three and four years hence, and bringing them to present value, you can see if the expected value is more or less than the $1000 it costs to play.
The expected value of the first flip is (1/2)@$1250 or $625. Because there is only a 1 in 4 chance of winning the second flip, its expected value is (1/4)@$2400 or $600. Putting them into the present value equation gives an expected present value
E(PV) = 625/1.053 + 600/1.054 = $540 + $494 = $1034.
Because the expected (present) value exceeds the cost of the bet, it would be considered a more than fair bet. A pure expected present value maximizer - someone who is risk neutral - would accept the bet. Of course your $1000 cost is certain while the payoffs are uncertain, so depending on your level of risk aversion, you may or may not accept your aunt's offer. But by using the expected present value calculation, you can at least ascertain that she is not trying to cheat you.
Example 8: The High Cost of High Level Nuclear Waste Storage
The siting of the first United States high level nuclear waste repository (HNWR) was one of the most controversial environmental issues of the mid-1980s. Three sites; Deaf Smith County in Texas, the Hanford Nuclear Reservation in Washington, and Yucca Mountain in Nevada were considered before the Nevada site was chosen.
Among the more pressing environmental and economic issues involved in the choice was that the waste needed storage over a 10000 year time horizon. Moreover, there is no way to ensure absolutely safe storage - especially over such a long time horizon. Instead, the potential costs that would occur should the storage facility breach needed to be calculated. There were two levels of uncertainty; whether the facility would fail, and if it failed, what and how much damage would occur.
The long time horizon added some interesting turns to the analysis. Even with very low rates of discount, events occurring 100 or more years in the future contribute little to the present value. If the interest rate used for discounting is as low as even 1 percent, the present value of some future loss 100 years in the future is only 1/3 of the future value. At an interest rate of 3 percent the present value of the loss 100 years from now is only 5 percent of the future value, and at 5 percent interest it becomes less than 1 percent. You can well imagine that events 200 or 1000 or 10000 years in the future are essentially meaningless in present value calculation.
Additionally, the 10000 year period requires that values from different generations be compared. It is not clear that discounting across generations makes as much sense as within one generation since the values are enjoyed by different people.
The risk of failure was also problematic. As time passed, the probability of facility failure increases, but the extent of damage gets smaller because the radioactivity is decaying. The net effect on the probability of damage is itself uncertain.
One study comparing the three sites was performed by Robert Rosenman, Rodney Fort and William Budd at Washington State University. It found that the present expected value of the loss from siting a HNWR ranged from $1.87 billion at Hanford to $16.27 billion at the Texas site. Interestingly, the Federal government offered only $100 million compensation to the state where the selected site is situated, indicating a large welfare transfer from that state to those that produced the high level nuclear waste.
Summary
Economists handle events that occur at different times by bringing all costs and benefits to present value. Present value analysis adjusts for interest that could be earned over time and the rate of time preference by discounting. This analysis offers one way of comparing things from different time periods.
Investment opportunities often have different time frames. The internal rate of return equates the present value of the cost of an investment to the present value of the benefits that are derived from the investment. Firms generally will benefit from pursuing all projects with an internal rate of return that exceeds the market rate of interest.
Discounting also applies to consumption over time. Utility maximizers must compare the value of future utility, derived from future income and saving current income, to present utility that is bought with current income and by borrowing against future income. Lifetime consumption overall follows the normal rules for utility maximization, although future values are discounted.
Uncertainty in economic decisions forces agents to act on expected value, the probabilistic mean of potential outcomes. Expected value is found by multiplying each possible outcome by the probability of that outcome happening, and then summing over all the possible outcomes. Another term for uncertainty is risk, and returns to investments are adjusted for increased risk. In equilibrium we expect that the risk adjusted return, that is, the expected return, across investments should be approximately equal.
Expected utility analysis extends the utility maximizing model to conditions of uncertainty. Risk neutral individuals maximize expected utility, but risk averse people may accept a lower level utility to avoid some risk. The amount by which the income of some expected utility exceeds the income that gives the equivalent utility with certainty is called the risk premium. Any additional amount a person will pay to avoid risk is an insurance premium, and measures the amount of risk aversion. More risk averse individuals will pay a larger insurance premium to avoid risk.