Chapter 15 Trade-offs Involving Time and Risk

Chapter 15 Outline 15 Trade-offs Involving Time and Risk 15.1 Modeling Time and Risk 15.2 The Time Value of Money 15.3 Time Preferences 15.4 Probability and Risk 15.5 Risk Preferences Key Ideas Interest is the payment received for temporarily giving up the use of money. Economists have developed tools to calculate the present value of payments received at different points in the future. 3. Economists have developed tools to calculate the value of risky payments.

In the future = less value 15.1 Modeling Time and Risk In the future = less value We may incur costs today and don’t reap the benefits until some time in the future – for example, getting an education now leads to higher income later. Other times, the opposite is the case—we get the benefits today and defer the costs to another time – for example, buying something on credit. In both cases, the event that happens in the future is more difficult to value than something that happens today. As we will see, this difficulty is one reason why things that occur in the future have less value today. To compare costs/benefits of current events with costs/benefits of future events, use factors to weight time and risk. We handle the problem of how to compare current and future events by weighting the future events to take into account the fact that they are either worth less (in the case of time), or that they may or may not occur (in the case of risk).

When is $1 not worth $1? 15.2 The Time Value of Money The natural preference—now rather than later. In order to get people to wait for a year (or whatever length of time), we need to give them an incentive to do so. That’s part of the role of financial markets—moving money from “now” to “later,” but at a cost. 15.2 The Time Value of Money Future Value and the Compounding of Interest Principal --- the amount of an original investment Interest --- The payment received for temporarily giving up the use of money (or payment for the opportunity to temporarily use someone else’s money)

Starting Principal: $100: assume an interest rate of 8%. 15.2 The Time Value of Money: Future Value and the Compounding of Interest Starting Principal: $100: assume an interest rate of 8%. After One Year: $100 + $100 x (0.08) = $100 x (1 + 0.08) = $108.00 After Two Years: $108 + $108 x (0.08) = $108 x (1 + 0.08) = $116.64 = [$100 x (1 + 0.08)](1 + 0.08) = $100 x (1 + 0.08)2 After Three Years: $116.64 + $116.64 x (0.08) = $116.64x (1 + 0.08) = $125.97 = [$100x(1 + 0.08)2] (1 + 0.08) = $100 x (1 + 0.08)3 After T Years: = $100 x (1 + 0.08)T Future value - the sum of principal and interest In general, Compound interest formula Future value = Principal x (1 + r)T

Starts saving $2,000 per year beginning at age 18 15.2 The Time Value of Money: Future Value and the Compounding of Interest Cousin It: Starts saving $2,000 per year beginning at age 18 Saves at this rate up to and including age 25, then stops saving and lets the account sit Earns 10% per year Cousin Id: Picks up where It left off and starts saving $2,000 per year beginning at age 26 Saves at this rate up to and including age 62, then stops saving Cousin It saves for 8 years; Cousin Id saves for 37 years at the same interest rate. How much more money they think Cousin Id will have than Cousin It. Cousin It will have $855,504; Cousin Id will have $726,087. More importantly, only $16,000 of that $855,504 is Cousin It’s money. The rest, $839,504 is accumulated interest. Cousin Id’s total includes $74,000 of his (her?) own money, so only $652,087 was earned in interest.

Exhibit 15.1 Value of a $1 Investment over the Next 50 Years 15.2 The Time Value of Money: Future Value and the Compounding of Interest The difference between Cousin It and Cousin Id was so striking because the interest rate was 10%. If the interest rate were lower, the difference wouldn’t be so much Exhibit 15.1 Value of a $1 Investment over the Next 50 Years

Exhibit 15.2 The Mechanics of Lending and Borrowing 15.2 The Time Value of Money: Future Value and the Compounding of Interest Exhibit 15.2 The Mechanics of Lending and Borrowing Interest payments are a two-way street. When you are a lender (when you put money in the bank) you are lending your money to the bank so that it can loan your money out to other people; in return, you get paid interest. But when you are a borrower, you are the one paying interest -- compound interest is a great thing…if you’re a saver. If you’re a borrower, it’s another story…

Credit card has a balance of $897.30 15.2 The Time Value of Money: Future Value and the Compounding of Interest Assume: Credit card has a balance of $897.30 Pay monthly minimum of $15 per month 18% APR = 1.5% per month How long will it take to pay off the balance if you do not add new purchases?

Amt. Applied to Interest Amt. Applied to Principle 15.2 The Time Value of Money: Future Value and the Compounding of Interest Payment Balance Amt. Applied to Interest Amt. Applied to Principle New Balance 1 $897.30 $15 $13.46 $1.54 $895.76 2 895.76 15 13.44 1.56 894.20 3 13.41 1.59 892.61 4 13.39 1.61 891.00 5 13.37 1.63 889.37 6 13.34 1.66 887.71 7 13.32 1.68 886.03 8 13.29 1.71 884.32 9 13.26 1.74 882.58 10 13.24 1.76 880.82 11 13.21 1.79 879.03 12 13.19 1.81 877.22 Total $180 Point out that paying $15 per month reduces the principal by $20.08 at the end of a year. At this rate, it will take 7 years to pay off the principal. © 2015 Pearson Education, Inc.

Future value = principal x (1 + r)T $3,000 = principal x (1.08)3 15.2 The Time Value of Money: Present Value and Discounting Planning a trip after graduation. Estimate cost is $3,000 after they do some research on flights and hotels and factor in some inflation. Assume that you’ll take this trip in 3 years. Your parents have offered to give you money for the trip. The question is, how much do you parents need to put into an investment today to yield $3,000 in three years? Future value = principal x (1 + r)T $3,000 = principal x (1.08)3 $3,000/(1.08)3 = principal Or $2,381.50 = principal Assume an interest rate of 8%.

15.2 The Time Value of Money: Present Value and Discounting Present value -- the present value of a future payment is the amount of money that would need to be invested today to produce that future payment. Also called the discounted value of a future payment. The present value and future value formulas are just opposites of each other and can be derived by simply changing the unknown in one and solving. This formula assumes that there are only payments that occur, but that sometimes costs occur at different times as well. They are handled the same way as payments. General present value formula: Present value = Payment T periods from now (1 + interest rate)T Invest $10,000 today to get $20,000 in 20 years. Good deal? Present value = $20,000/(1.05)20 = $7,538

15.2 The Time Value of Money: Present Value and Discounting $20,000 in 20 years is worth $7,538 today. Why would you pay $10,000 for something worth $7,538? That’s $2,462 too much! Net present value = [Present value of benefits - present value of costs] Does it matter how you get the $20,000? What if you got $10,000 of it in ten years and $10,000 five years later? Good deal? $10,000 in 10 years: Present value = $10,000/(1.05)10 = $6,139 $10,000 in 15 years: Present value = $10,000/(1.05)15 = $4,810 $6,139 + $4,810 = $10,949 - $10,000 = $949 Yes, this is a good deal. You’re getting a greater return than your opportunity cost (assuming that’s 5%).

What does time preference have to do with marshmallows? 15.3 Time Preferences: Time Discounting What does time preference have to do with marshmallows? Everything you need to know about time preference and time discounting is found on the video: http://www.youtube.com/watch?v=QX_oy9614HQ Eating the marshmallows was not about money, but clearly some of the kids valued future marshmallows less than present marshmallows. And just as with money, we “discount” future activities--either costs or benefits—to compare them to present ones.

15.3 Time Preferences: Time Discounting Utils – fictitious individual measures of utility or happiness Discount weight --multiplies future utils to translate them into current utils Utils can be discounted just as money can so that we can compare the future satisfaction we get from something to the current cost of delaying gratification. Choose between one marshmallow now or 2 marshmallows in 10 minutes. Suppose each marshmallow has a utility of 6 utils. But the discount rate for waiting 10 minutes is 1/3. One marshmallow now = 6 utils Two marshmallows in 10 minutes = 4 utils Choose between one marshmallow now or 2 marshmallows in 10 minutes Suppose each marshmallow has a utility of 6 utils. If you are better at waiting, the discount rate could be 2/3. Two marshmallows in 10 minutes = 8 utils If you were better at waiting, you would not discount the future as much—the discount factor would be larger, and you would wait.

15.3 Time Preferences: Time Discounting What about starting a diet? Should you start a diet today or wait until tomorrow? Starting today: Benefit is become healthier more quickly Cost is giving up food you enjoy earlier Starting tomorrow: Benefit is you get to eat what you want for another day Cost is delaying becoming healthier Problem: The benefit of starting today is in the future, while its cost is immediate. The benefit of starting tomorrow is immediate, while the cost is in the future. Result: diets that always start “tomorrow”

Probability --Frequency with which something occurs 15.4 Probability and Risk: Roulette Wheels and Probabilities Probability --Frequency with which something occurs Play the 100-slot wheel and ask the probabilities of getting a certain number. Ask what happens if you bet on 10 numbers, etc. 15.4 Probability and Risk: Independence and the Gambler’s Fallacy Your lucky number is 27 and you’ve won 10 times in a row! What number should you bet next?

15.4 Probability and Risk: Independence and the Gambler’s Fallacy What number would you bet. Many, if not most, will probably say #27. Ask what the probability was of getting 27 on the first spin. Should say 1/100. Ask them what the probability of getting 27 is on the next spin, assuming the first spin never happened. Again, they should say 1/100. What effect the existence of the first spin has on the outcome of the second spin. Does the wheel get into a rut, causing the same # to come up again? Does the marble develop a liking for a certain slot? What about the first spin would influence the second spin? The answer, of course, is nothing. That betting on #27 for the next spin is perfectly fine—and has the same odds as any other number. Your lucky number is 27 and you’ve won 10 times in a row! What number should you bet next?

Expected value -- a probability-weighted value 15.4 Probability and Risk: Expected Value Expected value -- a probability-weighted value Present and future values are weighted by time. Expected values are weighted by probability of occurrence, or risk. Bet on number 64. If win, get $100 If ball goes on number 15, lose $200 If ball goes on another number, nothing Expected value = sum of payoffs x probability of occurring How many would take that bet?

15.4 Probability and Risk :Expected Value Payoff Probability of Occurring Expected Value (payoff x probability) $100 1/100 = .01 $1 -$200 -$2 $0 98/100 = .98 Sum 100/100 -$1 When computing expected values, you must take into consideration all possibilities, so the probabilities need to add up to 1. So the expected value of this game is -$1—not a good bet. Remember that the expected value of this bet is -$1 over a large number of times playing the game. Someone might win (or lose) several times in a row, but spread out over a large number of repetitions, the average loss would be $1. If number is 50 or below, win $200 If number is 51 or higher, lose $100 How many would take this bet?

15.4 Probability and Risk: Expected Value Payoff Probability of Occurring Expected Value (payoff x probability) $200 50/100 = .5 $100 -$100 -$50 Sum 100/100 $50 Since the expected value here is positive, this would be a good bet—again, over a large number of games. 15.4 Probability and Risk: Extended Warranties You buy a $300 TV with a 1-year warranty included. You can buy an extended warranty for years 2 and 3 for another $75. Should you do it?

15.4 Probability and Risk :Extended Warranties Two components: risk and present value Risk—assume the probability of breakdown is 10% per year. Present value—if TV breaks in year 2, could replace it for $250 without a warranty; in year 3, could replace for $200 Have you ever purchased an extended warranty on something, like a computer, a car, or TV? Most will probably say they, or someone they know, has. Extended warranties are a perfect way to combine both concepts in this chapter. People will say that they wanted to be covered in case the item broke. This situation is about two things: the risk associated with the product breaking, and the present value of the costs and benefits of the warranty. BUT technology improves such that you can replace a TV that’s a couple of years old with the exact same specifications for less money than you originally bought it. So, the cost of the warranty is incurred now, but the benefit is delayed until year 2 or year 3—and we don’t know for sure that we will get the benefit.

15.5. Risk Preferences 15.4 Probability and Risk: Extended Warranties First, present value (assume 10%): Present value = - $75 + $250/(1.1)2 + $200/(1.1)3 = -$75 + $206.61 + $150.26 But these benefits do not occur with certainty, so to get expected value: -$75 + $206.61(10/100) + $150.26(10/100) = -$39.31 Since the net present value is negative, in strictly monetary terms, this is not a good deal. 15.5. Risk Preferences Given that, in general, extended warranties are not a good deal, why do people buy them? Some people would still buy them anyway because they just like knowing that they are protected from future financial outlays.

15.5. Risk Preferences Loss aversion--psychologically weighting a loss more heavily than weighting a gain. Choose between: Option 1: getting $0, OR Option 2: $200 gain if heads or $100 loss if tails Expected value of Option 1: $0 Expected value of Option 2: $200(1/2) + (-$100)(1/2) = $50 Option 2 clearly has the higher expected value, so should be preferred -- unless someone has loss aversion, when losses are weighted more heavily than gains (then Option 1)

What if you’re this person? 15.5. Risk Preferences What if you’re this person? Would you pick Option 2 if you have this amount of loss aversion? Expected value = $200(1/2) + 2 x (-$100)(1/2) = $0 If she weights losses twice as much as gains, she would be indifferent between Options 1 and 2.

15.5. Risk Preferences Option 1: 1 Possible Outcome: 6% Expected Rate of Return = 6% Option 2: 3 Possible Outcomes: 5%, 6%, and 7% (all equally likely) Option 3: 5 Possible Outcomes: -4%, 1%, 6%, 11%, and 16% (all equally likely) You have 3 investment options. Option 1 has only 1 possible outcome and it’s a 6% return. Option 2 could go 3 different directions, as shown. Option 3 has 5 possible outcomes, but still the expected return is 6%. Choice of Option 1 --- it is safe and has the same expected return as the others. Choice of Option 3 --- because even though the expected return is the same as the other options, there is a chance of getting a very high return.

15.5. Risk Preferences Risk averse Prefer less risk (Option 1) Risk seeking Prefer more risk (Option 3) Risk neutral Don’t care about risk (any option)