What will you learn in this chapter?
• Why money is worth more now than in the future.
• How compounding works over time.
• How to calculate the present value of a future sum.
• What the costs and benefits are of a choice using expected value.
• How risk aversion makes a market for insurance possible.
• What the importance is of pooling and diversification for managing risk.
• What challenges adverse selection and moral hazard pose for insurance.

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1© 2014 by McGraw-Hill Education 1
Chapter 11
Time and Uncertainty
© 2014 by McGraw-Hill Education 2
What will you learn in this chapter?
• Why money is worth more now than in the future.
• How compounding works over time.
• How to calculate the present value of a future sum.
• What the costs and benefits are of a choice using
expected value.
• How risk aversion makes a market for insurance
possible.
• What the importance is of pooling and diversification
for managing risk.
• What challenges adverse selection and moral hazard
pose for insurance.
© 2014 by McGraw-Hill Education 3
Value over time
• When a decision requires weighing uncertain
future costs and benefits, two complications
are faced:
– The value of money changes over time, causing an
inaccurate direct comparison of current costs and
benefits to future costs and benefits.
– The future is uncertain, causing future benefits and
costs to be only approximate estimates.
2© 2014 by McGraw-Hill Education 4
Timing matters
• When costs and benefits of a choice occur at
different times, this profoundly affects the
choice.
• Consider the following scenario: You have won
a competition and can choose one of the
following prizes:
Option A: $100,000 now.
Option B: $105,000 ten years from now.
• Which would you choose and why?
© 2014 by McGraw-Hill Education 5
Interest rates
• When considering money today versus future
money, individuals consider the opportunity cost
of waiting until the future to receive the money.
– The interest rate tells how much today’s money is
worth in the future.
– Depositing $100,000 in a bank at a 5% annual interest
rate is worth in one year:
$100,000 + ($100,000*5%) = $105,000
• Future money can be equated to the present. In
the above example, $105,000 in one year is worth
$100,000 today.
© 2014 by McGraw-Hill Education 6
Compounding
• When analyzing the value of money over a time
period longer than one year, compounding the
interest payments is necessary.
– This causes a process of accumulation, as interest is
paid on interest that has already been earned.
• The future value of depositing $100,000 in a bank
at a 5% annual interest rate for two years earns:
$100,000(1 + .05) = $105,000 in 1 year
$105,000(1.05) = $100,000(1.05)2 = $110,250 in 2 years
3© 2014 by McGraw-Hill Education 7
Active Learning: Computing future value
What is the future value of depositing $100,000 in a bank
at a 5% annual interest rate for ten years?
© 2014 by McGraw-Hill Education 8
Present value
• Interest rates are used to compare the present
value and future value of a sum.
• Individuals have a preferred interest rate that
reflects their opportunity cost of waiting for
money in the future versus receiving it today.
• Suppose that an individual has a preferred
interest rate of 8% annually.
– The future value of $100,000 in 10 years is $215,893.
• The present value of $215,893 in 10 years is
$100,000.
© 2014 by McGraw-Hill Education 9
Present value
• If the future value is known, then given an
individual’s preferred interest rate, his or her
present value of any sum can be determined.
• Rearranging the earlier formula:
ܸܲ = ிଵା
• Present value translates future costs or benefits
into the equivalent amount of value today.
• This information enables us to directly compare
future amounts with the present sums.
4© 2014 by McGraw-Hill Education 10
Active Learning: Comparing present and
future values
Suppose you have a preferred interest rate of 9%
annually. You just won the lottery and have two
options:
– Option A: Take $1,000,000 today.
– Option B: Take $5,000,000 in 20 years.
• What is the present value of Option B? Which
option will you prefer?
© 2014 by McGraw-Hill Education 11
Present value
• Sometimes benefits and costs accrue over
several years.
• To calculate the present value of a flow of
money in the future, add up the present value
of each amount in the future.
© 2014 by McGraw-Hill Education 12
Present value
• Consider that many people expect to earn additional
income every year after earning a college degree.
• Suppose an individual expects to earn an additional
$20,000 each year after starting their first job in 5 years
and working for 30 years. Their preferred annual
interest rate is 5%.
• What is the present value of this future flow of income?
ܸܲ = $ଶ(ଵ.ହ)ఱା $ଶ(ଵ.ହ)లା⋯ା $ଶଵ.ହ యరୀ$ଶହଶ,ଽଷଽ
• The present value of an extra $600,000 spaced out
evenly over 30 years is $252,939.
• Thus, if the present value of the cost of attending
college is less than $252,939, the individual will attend.
5© 2014 by McGraw-Hill Education 13
Present value
• Knowing how to calculate present value can be
useful in making other decisions when the
benefits and opportunity cost occur at different
times.
– If you want a certain level of income when you retire,
how much should you save into your retirement fund
now?
– If you run a business, what value of future sales would
be needed to make it worthwhile to invest in a new
piece of machinery?
• Comparing the present value of costs and benefits
leads to informed decision making.
© 2014 by McGraw-Hill Education 14
Risk and uncertainty
• The previous examples assumed certain future
costs and benefits.
• Many decisions are based on weighing
uncertain future costs and benefits against
today’s costs and benefits.
– Risk is a special class of uncertainty in which the
costs or benefits of an event or choice are
uncertain, but calculable.
• Evaluating risk requires analysis of different
possible outcomes.
© 2014 by McGraw-Hill Education 15
Expected value
• Even when future events are uncertain, often the
set of outcomes are known.
• There are costs and benefits as well as a likelihood
that the outcome will be realized.
• By combining outcomes with likelihoods, a single
cost or benefit estimate can be calculated.
ܧܸ = ଵܲ ଵܵ + ଶܲܵଶ++ ܲܵ
• The expected value of a choice, EV, is equal to the
sum of each possible event, S, weighted by its
probability of occurring, P.
6© 2014 by McGraw-Hill Education 16
Expected value
Our previous analysis of the decision to attend
higher education can be extended by assuming
that additional future earnings is uncertain.
• Using the probabilities and outcomes, the
expected value of attending college and not
attending college can be calculated.
Lifetime earnings by
education level
College degree 25% 25% 50%
No college degree 50% 0%50%
$0.9 million $1.5 million $2.4 million
© 2014 by McGraw-Hill Education 17
Expected value
• The value of not attending college is:
ܧܸ = .5 × $0.9ܯ + .5 × $1.5ܯ + (0 × $2.4ܯ)
= $1.2ܯ
• The value of attending college is:
ܧܸ = .25 × $0.9ܯ + .25 × $1.5ܯ + (.5 × $2.4ܯ
= $1.8ܯ
• Unlike the earlier estimates, these expected values
incorporate the risk of lower income.
• Using these estimates, one can make a choice based on
expected future income.
• The expected benefit of attending college is $600,000,
the difference in the expected values.
© 2014 by McGraw-Hill Education 18
Propensity for risk
• Although individuals have varying tastes for taking on
risks, people are generally risk-averse with their
financial decisions. (Someone who does have a high
tolerance for risk is risk-seeking.)
• When faced with two options, with equal expected
value, individuals typically prefer the one with lower
risk.
• Imagine you are given two options based on a coin toss:
– Option A: Heads, receive $100,001. Tails, receive $99,999.
– Option B: Heads, receive $200,000. Tails, receive $0.
• While both have an estimated value of $100,000, most
people prefer Option A because it has less risk.
7© 2014 by McGraw-Hill Education 19
Propensity for risk
• The previous example suggests that many worry
about worst-case outcome.
• Even if two possible outcomes have the same
expected value, the one with lower risk will
typically be chosen.
• This implies that individuals must be compensated
for taking on risk.
– The expected value of Option B would have to be
greater than $100,000 before most individuals would
accept the risk of winning nothing.
• How much higher would depend on individuals’
personal taste for risk.
© 2014 by McGraw-Hill Education 20
Insurance and managing risk
• Risk averse individuals cope with risk in many
ways.
– If possible, avoid the risk altogether.
– If unavoidable, buy insurance.
• An insurance policy is a product that lets risk
averse individuals (or companies) pay to reduce
some uncertainty.
• Insurance is an agreement in which:
– An individual pays a regular fee.
– An insurance company covers costs associated with a
specific event occurring.
© 2014 by McGraw-Hill Education 21
Insurance and managing risk
• The cost of insurance is typically greater than
its expected value.
– Most people are risk-averse enough to find
insurance worth the extra expense.
• Individuals are generally willing to pay for
insurance because the costs of the worst-case
events are typically quite large.
• If the cost of insurance was equal to its
expected value, then insurance companies
would not make any profits.
8© 2014 by McGraw-Hill Education 22
Pooling and diversifying risk
• Insurance does not reduce risk.
• Insurance reallocates costs from individuals to
insurance companies.
• Why are insurance companies better able to
handle the same risk?
– Insurance companies pool individuals together,
called risk pooling.
– Insurance companies use risk diversification in
which risks are shared across many different assets
or people.
© 2014 by McGraw-Hill Education 23
Problems with insurance
• There are two big inherent problems with
insurance: adverse selection and moral hazard.
• Both occur due to individuals and insurance
companies having different information
(asymmetric information sets).
• Adverse selection occurs when higher risk
individuals are drawn towards insurance.
• Moral hazard occurs when individuals behave
riskier once they become insured.
• If insurance companies had the same information
set as their clients, adverse selection and moral
hazard would be eliminated.
© 2014 by McGraw-Hill Education 24
Summary
• The present value formula provides a way of
comparing current costs and benefits to future
costs and benefits.
– An interest rate links the future value to the present.
• Sometimes these costs and benefits are known
and other times they are uncertain, but
calculable.
– When they are calculable, an expected value of each
outcome can be determined.
• When worse-case events are high cost, individuals
tend to avoid these activities or buy insurance to
reduce risk.