# Time Value of Money

```Chapter 5
Time Value of
Money
Learning Goals
1. Discuss the role of time value in finance, the use of
computational aids, and the basic patterns of cash
flow.
2. Understand the concept of future value and present
value, their calculation for single amounts, and the
relationship between them.
3. Find the future value and the present value of both an
ordinary annuity and an annuity due, and the present
value of a perpetuity.
4-2
Learning Goals (cont.)
4. Calculate both the future value and the present value
of a mixed stream of cash flows.
5. Understand the effect that compounding interest more
frequently than annually has on future value and the
effective annual rate of interest.
6. Describe the procedures involved in (1) determining
deposits needed to accumulate to a future sum, (2)
loan amortization, (3) finding interest or growth rates,
and (4) finding an unknown number of periods.
4-3
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
• Time value of money allows comparison of cash
flows from different periods.
• Question: Your father has offered to give you some
money and asks that you choose one of the following
two alternatives:
– \$1,000 today, or
– \$1,100 one year from now.
• What do you do?
4-4
The Role of Time Value in Finance
(cont.)
• The answer depends on what rate of interest
you could earn on any money you receive
today.
• For example, if you could deposit the \$1,000
today at 12% per year, you would prefer to be
paid today.
• Alternatively, if you could only earn 5% on
deposited funds, you would be better off if you
chose the \$1,100 in one year.
4-5
Basic Concepts
• Future Value: compounding or growth over
time
• Present Value: discounting to today’s value
• Single cash flows & series of cash flows can be
considered
• Time lines are used to illustrate these
relationships
4-6
Computational Aids
• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
4-7
Computational Aids (cont.)
Figure 4.1 Time Line
4-8
Computational Aids (cont.)
Figure 4.2 Compounding and
Discounting
4-9
Computational Aids (cont.)
Figure 4.4 Financial Tables
4-10
Basic Patterns of Cash Flow
• The cash inflows and outflows of a firm can be described by its general
pattern.
• The three basic patterns include a single amount, an annuity, or a mixed
stream:
4-11
Simple Interest
• With simple interest, you don’t earn interest on interest.
• Year 1: 5% of \$100
=
\$5 + \$100 = \$105
• Year 2: 5% of \$100
=
\$5 + \$105 = \$110
• Year 3: 5% of \$100
=
\$5 + \$110 = \$115
• Year 4: 5% of \$100
=
\$5 + \$115 = \$120
• Year 5: 5% of \$100
=
\$5 + \$120 = \$125
4-12
Compound Interest
• With compound interest, a depositor earns interest on interest!
• Year 1: 5% of \$100.00
= \$5.00 + \$100.00
= \$105.00
• Year 2: 5% of \$105.00
= \$5.25 + \$105.00
= \$110.25
• Year 3: 5% of \$110.25
= \$5 .51+ \$110.25
= \$115.76
• Year 4: 5% of \$115.76
= \$5.79 + \$115.76
= \$121.55
• Year 5: 5% of \$121.55
= \$6.08 + \$121.55
= \$127.63
4-13
Time Value Terms
• PV0 = present value or beginning amount
• i
= interest rate
• FVn = future value at end of “n” periods
• n
= number of compounding periods
• A
= an annuity (series of equal payments
or receipts)
4-14
Four Basic Models
• FVn
=
PV0(1+i)n
= PV x (FVIFi,n)
• PV0
=
FVn[1/(1+i)n]
= FV x (PVIFi,n)
A (1+i)n - 1
= A x (FVIFAi,n)
• FVAn =
i
• PVA0
= A 1 - [1/(1+i)n] = A x (PVIFAi,n)
i
4-15
Future Value of a Single Amount
• Future Value techniques typically measure cash flows at the end
of a project’s life.
• Future value is cash you will receive at a given future date.
• The future value technique uses compounding to find the future
value of each cash flow at the end of an investment’s life and
then sums these values to find the investment’s future value.
• We speak of compound interest to indicate that the amount of
interest earned on a given deposit has become part of the
principal at the end of the period.
4-16
Future Value of a Single Amount: Using
FVIF Tables
• If Fred Moreno places \$100 in a savings account
paying 8% interest compounded annually, how
much will he have in the account at the end of
one year?
\$100 x (1.08)1
\$100 x 1.08
= \$100 x FVIF8%,1
= \$108
4-17
Future Value of a Single Amount: The
Equation for Future Value
• Jane Farber places \$800 in a savings account
paying 6% interest compounded annually. She
wants to know how much money will be in the
account at the end of five years.
FV5 = \$800 X (1 + 0.06)5 = \$800 X (1.338) = \$1,070.40
4-18
Future Value of a Single Amount:
A Graphical View of Future Value
Figure 4.5
Future Value
Relationship
4-19
Present Value of a Single Amount
• Present value is the current dollar value of a future amount of
money.
• It is based on the idea that a dollar today is worth more than a
dollar tomorrow.
• It is the amount today that must be invested at a given rate to
reach a future amount.
• Calculating present value is also known as discounting.
• The discount rate is often also referred to as the opportunity cost,
the discount rate, the required return, or the cost of capital.
4-20
Present Value of a Single Amount: Using
PVIF Tables
• Paul Shorter has an opportunity to receive \$300
one year from now. If he can earn 6% on his
investments, what is the most he should pay now
for this opportunity?
\$300 x [1/(1.06)1] = \$300 x PVIF6%,1
\$300 x 0.9434
= \$283.02
4-21
Present Value of a Single Amount: The
Equation for Future Value
• Pam Valenti wishes to find the present value of
\$1,700 that will be received 8 years from now.
Pam’s opportunity cost is 8%.
PV = \$1,700/(1 + 0.08)8 = \$1,700/1.851 = \$918.42
4-22
Present Value of a Single Amount: A
Graphical View of Present Value
Figure 4.6
Present
Value
Relationship
4-23
Annuities
• Annuities are equally-spaced cash flows of equal size.
• Annuities can be either inflows or outflows.
• An ordinary (deferred) annuity has cash flows that
occur at the end of each period.
• An annuity due has cash flows that occur at the
beginning of each period.
• An annuity due will always be greater than an
otherwise equivalent ordinary annuity because interest
will compound for an additional period.
4-24
Types of Annuities
• Fran Abrams is choosing which of two annuities to
receive. Both are 5-year \$1,000 annuities; annuity A is
an ordinary annuity, and annuity B is an annuity due.
Fran has listed the cash flows for both annuities as
shown in Table 4.1 on the following slide.
Note that the amount of both annuities total \$5,000.
4-25
Table 4.1 Comparison of Ordinary
Annuity and Annuity Due Cash Flows
(\$1,000, 5 Years)
4-26
Finding the Future Value of an Ordinary
Annuity
• Fran Abrams wishes to determine how much money
she will have at the end of 5 years if he chooses annuity
A, the ordinary annuity and it earns 7% annually.
Annuity a is depicted graphically below:
4-27
Future Value of an Ordinary Annuity:
Using the FVIFA Tables
FVA = \$1,000 (FVIFA,7%,5)
= \$1,000 (5.751)
= \$5,751
4-28
Present Value of an Ordinary Annuity
• Braden Company, a small producer of plastic toys, wants to
determine the most it should pay to purchase a particular
annuity. The annuity consists of cash flows of \$700 at the end
of each year for 5 years. The required return is 8%.
4-29
Present Value of an Ordinary Annuity:
The Long Method
Table 4.2 Long Method for Finding the Present
Value of an Ordinary Annuity
4-30
Present Value of an Ordinary Annuity:
Using PVIFA Tables
PVA = \$700 (PVIFA,8%,5)
= \$700 (3.993)
= \$2,795.10
4-31
Future Value of an Annuity Due:
Using the FVIFA Tables
• Fran Abrams now wishes to calculate the future
value of an annuity due for annuity B in Table
4.1. Recall that annuity B was a 5 period
annuity with the first annuity beginning
immediately.
FVA
= \$1,000(FVIFA,7%,5)(1+.07)
= \$1,000 (5.751) (1.07)
= \$6,154
4-32
Present Value of an Annuity Due:
Using PVIFA Tables
• In the earlier example, we found that the value of
Braden Company’s \$700, 5 year ordinary annuity
discounted at 8% to be about \$2,795. If we now
assume that the cash flows occur at the beginning of the
year, we can find the PV of the annuity due.
PVA
= \$700 (PVIFA,8%,5) (1.08)
= \$700 (3.993) (1.08)
= \$3,018.40
4-33
Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or cash
flow stream continues forever.
PV = Annuity/Interest Rate
• For example, how much would I have to deposit
today in order to withdraw \$1,000 each year
forever if I can earn 8% on my deposit?
PV = \$1,000/.08 = \$12,500
4-34
Future Value of a Mixed Stream
4-35
Future Value of a Mixed Stream (cont.)
Table 4.3 Future Value of a Mixed Stream of
Cash Flows
4-36
Present Value of a Mixed Stream
• Frey Company, a shoe manufacturer, has been
offered an opportunity to receive the following
mixed stream of cash flows over the next 5
years.
4-37
Present Value of a Mixed Stream
• If the firm must earn at least 9% on its investments,
what is the most it should pay for this opportunity?
• This situation is depicted on the following time line.
4-38
Present Value of a Mixed Stream
Table 4.4 Present Value of a Mixed Stream of Cash
Flows
4-39
Compounding Interest More Frequently
Than Annually
• Compounding more frequently than once a year results
in a higher effective interest rate because you are
earning on interest on interest more frequently.
• As a result, the effective interest rate is greater than the
nominal (annual) interest rate.
• Furthermore, the effective rate of interest will increase
the more frequently interest is compounded.
4-40
Compounding Interest More Frequently
Than Annually (cont.)
• Fred Moreno has found an institution that will pay him 8%
annual interest, compounded quarterly. If he leaves the
money in the account for 24 months (2 years), he will be paid
2% interest compounded over eight periods.
Table 4.5 Future Value from Investing \$100 at 8% Interest
Compounded Semiannually over 24 Months (2 Years)
4-41
Compounding Interest More Frequently
Than Annually (cont.)
Table 4.6 Future Value from Investing \$100 at 8%
Interest Compounded Quarterly over 24 Months (2 Years)
4-42
Compounding Interest More Frequently
Than Annually (cont.)
Table 4.7 Future Value at the End of Years 1 and
2 from Investing \$100 at 8% Interest, Given
Various Compounding Periods
4-43
Compounding Interest More Frequently
Than Annually (cont.)
• A General Equation for Compounding More
Frequently than Annually
4-44
Compounding Interest More Frequently
Than Annually (cont.)
• A General Equation for Compounding More Frequently
than Annually
– Recalculate the example for the Fred Moreno example
assuming (1) semiannual compounding and (2) quarterly
compounding.
4-45
Continuous Compounding
• With continuous compounding the number of compounding
periods per year approaches infinity.
• Through the use of calculus, the equation thus becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
• Continuing with the previous example, find the Future value of
the \$100 deposit after 5 years if interest is compounded
continuously.
4-46
Continuous Compounding (cont.)
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
FVn
= 100 x (2.7183).08x2 = \$117.35
4-47
Nominal & Effective Annual Rates of
Interest
• The nominal interest rate is the stated or contractual
rate of interest charged by a lender or promised by a
borrower.
• The effective interest rate is the rate actually paid or
earned.
• In general, the effective rate > nominal rate whenever
compounding occurs more than once per year
4-48
Nominal & Effective Annual Rates of
Interest (cont.)
• Fred Moreno wishes to find the effective annual rate associated
with an 8% nominal annual rate (I = .08) when interest is
compounded (1) annually (m=1); (2) semiannually (m=2); and
(3) quarterly (m=4).
4-49
Special Applications of Time Value:
Deposits Needed to Accumulate to a Future Sum
4-50
Special Applications of Time Value: Deposits
Needed to Accumulate to a Future Sum (cont.)
• Suppose you want to buy a house 5 years from now and
you estimate that the down payment needed will be
\$30,000. How much would you need to deposit at the
end of each year for the next 5 years to accumulate
\$30,000 if you can earn 6% on your deposits?
PMT = \$30,000/5.637 = \$5,321.98
4-51
Special Applications of Time Value:
Loan Amortization
Table 4.8 Loan Amortization Schedule
(\$6,000 Principal, 10% Interest, 4-Year Repayment Period)
4-52
Special Applications of Time Value:
Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied
by a series of cash flows.
Ray Noble wishes to find the rate of interest or growth
reflected in the stream of cash flows he received from a
real estate investment over the period from 2002 through
2006 as shown in the table on the following slide.
4-53
Special Applications of Time Value:
Interest or Growth Rates (cont.)
PVIFi,5yrs = PV/FV = (\$1,250/\$1,520) = 0.822
PVIFi,5yrs = approximately 5%
4-54
Special Applications of Time Value:
Finding an Unknown Number of Periods
• At times, it may be desirable to determine the number
of time periods needed to generate a given amount of
cash flow from an initial amount.
Ann Bates wishes to determine the number of years it
will take for her initial \$1,000 deposit, earning 8% annual
interest, to grow to equal \$2,500. Simply stated, at an
8% annual rate of interest, how many years, n, will it
take for Ann’s \$1,000 (PVn) to grow to \$2,500 (FVn)?
4-55
Special Applications of Time Value:
Finding an Unknown Number of Periods (cont.)
PVIF8%,n = PV/FV = (\$1,000/\$2,500) = .400
PVIF8%,n = approximately 12 years