Chapter 5
Time Value of Money
Instructor’s Resources
Chapter Overview
This chapter introduces a key—indeed, perhaps the most important—concept in finance: the time
value of money. The present and future value of a sum and an annuity are explained. Special
applications include intra-year compounding, mixed cash-flow streams, mixed cash flows with an
embedded annuity, perpetuities, loan amortization, and deposits necessary to accumulate a future
sum. The discussion employs numerous business and personal examples to stress all applications as
variations on the same theme—sums received at different points in time are worth different amounts
to the recipient, with differences traceable to when the sums are received and how frequently interest
is compounded. The chapter drives home the need to understand the time value of money to analyze
project profitability as a finance professional and achieve personal-finance goals as an individual.
Suggested Answer to Opener-in-Review
A lottery winner can choose between two options: (i) $480 million now or (ii) mixed stream of 30
payments, with an immediate payment of $11.42 million and 29 additional annual payments growing
at 5% per year. [So the second payment will be $11.42 million (1.05), the third $11.42 million
(1.05)2, …, and the 30th $11.42 million (1.05)29.] If the winner could earn 2% on cash invested
today, should she take the lump sum or mixed stream? What if the rate of return is 3%? What
general principle do those calculations illustrate?
The lottery winner should choose the payment option with the higher present value. With a 2%
discount rate, the present value of the mixed stream is $538.2 million—significantly higher than
immediate payment of $480 million—so the winner should opt for the mixed-stream payment.
Answers to Review Questions
5-1 Future value (FV) is the sum an amount today will grow to equal by a future date with
compound interest. Present value (PV) is the dollar value today of an amount
promised at a specific future date for a given interest rate. Viewed another way, it is
5-2 A single amount cash flow refers to a single payment or cash receipt, occurring at one point in time.
stream is an unequal series of cash flows that take place over multiple periods.
5-3 Compounding occurs when interest earned in the initial period is added to the original principal and
that sum grows by the interest rate in the second period, and so on. With compounding, interest is
5-4 A fall in the interest rate reduces the future value of a deposit for a given holding period because of
the decline in the amount of interest on which additional interest is subsequently paid (i.e., the
5-5. Present value (PV) is the current dollar value of a future amount (FVn)—that is, the amount today
5-6. Increasing required rate of return (discount rate) reduces the present value of a sum promised in the
future. Mathematically, a higher interest rate (r) increases the denominator in the present-value
5-7 The same equation links present and future value for a given interest rate and compounding period;
the only difference is the given information. Consider, for example, the present-value equation in
5-8 The value at retirement of a sum invested today may be obtained with the future-value equation for a
simple sum, FVn PV(1 r)n, assuming FVn is value at retirement, PV the sum invested today, n
5-9 The amount needed for investment today to cover future college expenses may be obtained with
present-value equation for a simple sum. Assuming, for simplicity, expenses for all four years must
be paid at the beginning of the child’s freshmen year, PV FVn (1 r)n, FVn is the amount needed
5-10 Annuities offer equal per-period payments for a given number of periods. Ordinary annuities make
payments at the end of each period while payments for annuities due occur at the beginning of the
period. For the same interest rate and number of payments, annuities due are more valuable—that is,
5-11 The most efficient ways to calculate the present value of an ordinary annuity are with an algebraic
equation, a financial calculator, or a spreadsheet program like Excel.
5-12 The future value (FV) of an ordinary annuity, where CF1 is the first payment made at end of period
1, n the number of payments, and r the interest rate, is given by:
Assuming the same cash flows (i.e., CF0 = CF1), interest rate and number of payments, the only
difference is that each payment on the annuity due arrives one period sooner than the payment on an
5-14 A perpetuity is an infinite-lived annuity; the present value of an ordinary annuity is given by:
where CF1 is the first payment received at the end of the first period, n is the number of periods, and
If the number of payments (n) goes to infinity in the equations for the future value of ordinary
annuities and annuities due (meaning the instrument is a perpetuity rather than a regular annuity),
5-15 The future value (at retirement) of equal annual IRA contributions at the end of every year can
be determined using the formula for the future value of an ordinary annuity:
where CF1 is the equal contribution made at the end of each year, r the interest rate, and n the number
5-16 Specific numerical answers will vary because values are algorithmically generated in
MyFinanceLab.
5-17 The future value (at retirement) of equal annual IRA contributions at the beginning of every year can
be determined using the formula for the future value of an annuity due:
where CF0 is the equal contribution made at the beginning of each year, r the interest rate, and n the
5-18 The future value of a mixed stream of cash flows equals the sum of the future values of the individual
cash flows—that is, each cash flow should be treated like a single payment in the “future value of a
5-19 Numerical answers will vary because values are algorithmically generated in MyFinanceLab. But the
graduation at per period interest rate r and sum the individual future values.
5-20 For a given interest rate and holding period, as the number of compounding periods per year rises,
on interest.
5-21 Continuous compounding means interest is compounded an infinite number of times per year—that
value compared with any other compounding period.
5-22 The nominal annual rate is the contractual annual rate of interest charged by the lender or promised
to the borrower, while the effective annual rate is the rate actually charged or paid after accounting
for the number of compounding periods per year. When interest is compounded annually, the two
5-23 Numerical answers will vary because values are algorithmically generated in MyFinanceLab. But the
value of investment opportunities.
5-24 Numerical answers will vary because values are algorithmically generated in MyFinanceLab. But the
when interest is compounded more than once per year.
5-25. Numerical answers will vary because values are algorithmically generated in MyFinanceLab. In
compounded annually to a maximum (relative to the nominal rate) when compounding is continuous.
5-26 The equal annual end-of-year deposits (CF1) needed to accumulate a given amount over a certain
number of periods (n) for a specific rate of interest per period (r) can be determined using the
5-27 Amortizing a loan over equal annual payments involves finding the sequence of payments with a
present value at the loan interest rate equal to the initial principal borrowed. In other words, start with
5-28 If the present value, future value, and interest rate are given, and the goal is finding the number of
periods needed for a sum today to grow to the future sum, then insert given information in the
The same approach will yield the number of periods it will take an ordinary annuity to grow to a
An alternative approach is to use trial and error, plugging in various values for the number of periods
5-29 Numerical answers will vary because values are algorithmically generated in MyFinanceLab. The
5-30 Numerical answers will vary because values are algorithmically generated in MyFinanceLab. The
approach is to treat retirement age as n months from now in the formula for future-value of an
Suggested Answer to Focus on Practice Box:
“New Century Brings Trouble for Subprime Mortgages”
As a reaction to problems in the subprime area, lenders tightened lending standards. What effect do you
think this change had on the housing market?
When mortgage lenders tightened underwriting standards, fewer people qualified for home loans, and
Suggested Answer to Focus on Ethics Box:
“Was the Deal for Manhattan a Swindle?
How much responsibility do lenders have to educate borrowers? The federal government requires
disclosure statements with standardized examples illustrating the time value of money; does this change
your answer?
This question is an excellent springboard to class discussion because there is no “correct” answer.
Conservative-leaning students may argue the borrower has a responsibility to educate herself—that is, “let
customers. A recent Freakonomics podcast summarizing research on payday loans, along with interviews
Answers to Warm-Up Exercises
E5-1 Future value of a lump-sum investment (LG 2)
Answer: FV $2,500 (1 0.007) $2,517.50, or in Excel, write the bracketed formula
E5-2 Finding future value (LG 2 and LG 5)
Answer: Because interest is compounded monthly, the number of periods is 4 (years) 12 (months) 48
and the monthly interest rate is 1/12th of the annual rate = 0.00166667:
FV48 PV (1 r)48 where r is the monthly interest rate
E5-3 Comparing a lump sum with an annuity (LG 3)
Answer: Note the 25-year payout option is an ordinary annuity. So the answer turns on whether the
present value of an ordinary annuity offering $100,000 annual cash flows for 25 years (given a
This problem can also be solved with a financial calculator or spreadsheet program like Excel.
Gabrielle should take the 25-year payout because the present value ($1.41 million) exceeds the
E5-4 Comparing the present value of two alternatives (LG 4)
Answer: You have the option of investing a sum today in software that offers the company a mixed
stream of savings over the next five years. This option will be profitable if the present value of
Year Savings Estimate Discount Factor (1+r)nPresent Value of Savings
1 $35,000 (1.09)1$32,110
2 50,000 (1.09)242,084
of the software (marginal cost), the firm should invest in the new software.
E5-5 Compounding more frequently than annually (LG 5)
Answer: The future value of $12,000 invested for one year in Partners’ Savings Bank at 3%, compounded
semi-annually (where m is compounding periods per year, and n is years, so r/m is the interest
FV = PV , where the value of e is approximately 2.7183
FV = $12,000 = $12,000 2.71860.0275 = $12,000 1.027882 = $12,334.58
E5-6 Determining deposits needed to accumulate a future sum (LG 6)
Answer: Jack and Jill want to make equal, end-of-year contributions for 18 years to accumulate $150,000
for their child’s college education. Equal, end-of-year contributions means ordinary annuity.
This problem can also be solved with a financial calculator or spreadsheet program like Excel.