Chapter 15
Trade-offs
Involving Time
and Risk
15 Trade-offs Involving Time and Risk
Chapter 15 Outline
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
1. Interest is the payment received for temporarily giving up the use of money.
2. 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.
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).
15.2 The Time Value of Money
When is $1 not worth $1?
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)
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
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
 Earns 10% per year
 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.
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
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…
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?
15.2 The Time Value of Money: Future Value and the Compounding of Interest
Payment
1
2
3
4
5
6
7
8
9
10
11
12
Balance
$897.30
895.76
894.20
892.61
891.00
889.37
887.71
886.03
884.32
882.58
880.82
879.03
Payment
$15
15
15
15
15
15
15
15
15
15
15
15
Total
$180
Amt. Applied
to Interest
$13.46
13.44
13.41
13.39
13.37
13.34
13.32
13.29
13.26
13.24
13.21
13.19
© 2015 Pearson
Amt. Applied
to Principle
$1.54
1.56
1.59
1.61
1.63
1.66
1.68
1.71
1.74
1.76
1.79
1.81
New Balance
$895.76
894.20
892.61
891.00
889.37
887.71
886.03
884.32
882.58
880.82
879.03
877.22
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%).
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.
One marshmallow now = 6 utils
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”
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?
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
1/100 = .01
-$2
$0
98/100 = .98
$0
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
50/100 = .5
-$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.
15.4 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)
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)
Expected Rate of Return = 6%
Option 3: 5 Possible Outcomes:
-4%, 1%, 6%, 11%, and 16% (all equally likely)
Expected Rate of Return = 6%
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)