9
The Time Value of
Money
Block, Hirt, and Danielsen
Foundations of Financial Management
18th edition
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Learning Objectives
• Explain the time value of money and how a dollar received
today is worth more than a dollar received in the future.
• Recognize that the future value is based on the number of
periods over which the funds are to be compounded at a
given interest rate.
• Recognize that the present value is based on the current value
of funds to be received.
• Calculate yield on investment (rate of return), annuity due,
and payments required.
• Recall that compounding or discounting may take place on a
less-than-annual basis such as semiannually or monthly.
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9-2
Relationship to the Capital Outlay
Decision
• Time value of money used to determine
whether future benefits sufficiently large to
justify current outlays
• Mathematical tools of time value of money
used in making capital allocation decisions
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9-3
Future Value—Single Amount
• Measuring value of amount allowed to grow
at given interest rate over period of time
• An investor has $1,000 and needs to calculate its
worth after four years at 10 percent interest per
year
1st year……$1,000 × 1.10 = $1,100
2nd year.....$1,100 × 1.10 = $1,210
3rd year……$1,210 × 1.10 = $1,331
4th year……$1,331 × 1.10 = $1,464.10
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9-4
Future Value—Single Amount
Continued
A generalized formula for Future Value:
FV = PV(1 + i)n
where,
FV = Future value
PV = Present value
i = Interest rate
n = Number of periods
In the previous case, PV = $1,000, i = 10 percent, n = 4, hence;
FV = $1,000V(1.10)4 = $1,000 × 1.464 = $1,464.10
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9-5
Table 9A-1 Future Value of
$1, FVIF = (1 + i)n
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9-6
Present Value—Single Amount
• A sum payable in the future is worth less
today than the stated amount
• The formula for the present value is derived from
the original formula for future value:
FV = PV(1 + i)n
PV = FV[1/(1 + i)n]
• Present value can be determined by solving for
the formula above, restating the formula as:
PV = FV × PV IF
• Assuming
n = 4 and i = 10%
PV = $1,464 × 0.683 = $1,000
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9-7
Table 9A-2 Present Value of
$1, PVIF = 1(1 + i)n
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9-8
Figure 9-1 Relationship of
Present Value to Future Value
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9-9
Interest Rate—Single Amount
• Measures the return on investment
• Determine unknown variable, i, given the
following variables
FV/PV: Future/Present value of money
n: number of years
i = (FV/PV)1/n – 1
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9-10
Number of Periods—Single Amount
• Determine how many years it would take an
investment to grow to a certain dollar amount
• Find the number of periods, n, given the
following
FV/PV: Future/Present value of money
i: interest rate
n = ln(FV/PV)
ln(1 + i)
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9-11
Future Value—Annuity
• Annuity
• A series of consecutive payments or receipts of equal
amount
• Ordinary annuity payments assumed at end of each
period
• Future value of an annuity
• Calculated by compounding each individual payment
into the future and then adding up all of these
payments
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9-12
Figure 9-2 Compounding
Process for Annuity
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9-13
Table 9A-3 Future Value of an Annuity of
$1, FVIFA = [(1 + i)n – 1]/i
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9-14
Future Value—Annuity Continued
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9-15
Present Value—Annuity
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9-16
Table 9A-4 Present Value of an Annuity of
$1, PVIFA = {1 – [1/(1 + i)n]}/i
• Assuming A = $1,000, n = 4, and i = 10%
PVA = $1,000 × 3.16987 = $3,169.87
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9-17
The Relationship between Present
Value and Future Value
• Future value and present value of a single
amount
• Mirror images of each other (inversely related)
• Figure 9-3 Future Value of $0.68 at 10%
• Figure 9-4 Present Value of $1.00 at 10%
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9-18
Figure 9-3 Future Value of $0.68 at 10%
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9-19
Figure 9-4 Present Value of $1.00 at 10%
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9-20
The Relationship between the Present Value of a
Single Amount and the Present Value of an Annuity
• Present value of annuity is sum of present
values of single amounts payable at end of
each period
• Assumption that $1.00 is received at end of each
period
• Same concept as lottery winnings
• Figure 9-5 Present Value of $1.00 at 10%
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9-21
Figure 9-5 Present Value of $1.00 at 10%
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9-22
Future Value Related to the
Future Value of an Annuity
• Future value of annuity is sum of future values
of single amounts receivable at end of each
period
• Future value of $1.00 assumes the $1.00 is
invested at the beginning of the period and grows
to the end of the period
• Future value of ordinary annuity assumes $1.00 is
invested at end of period and grows to end of next
period
• Figure 9-6 Future Value of $1.00 at 10%
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9-23
Figure 9-6 Future Value of $1.00 at 10%
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9-24
Determining the Annuity Value
•
Review of variables involved in time value of
money
FV/PV: Future/present value of money
n: Number of years
i: Interest or yield
A: Annuity value/payment per period in annuity
•
Given first three variables and determining
fourth variable, A (unknown)
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9-25
Annuity Equaling a Future Value
• Assume that at a 10 percent interest rate,
after 4 years an amount of $4,641 needs to be
accumulated
FVA = A × FVIFA
A = FVA / FVIFA
• Assuming n = 4 and i = 10%
A = $4,641 = $1,000
4.641
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9-26
Annuity Equaling a Present Value
• Determining what size of an annuity can be
equated to a given amount
PVA = A × PVIFA
A = PVA / PVIFA
• Assuming n = 4 and i = 6%
A = $10,000 = $2,886
3.465
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9-27
Table 9-1 Relationship of Present
Value to Annuity
• Annual interest based on beginning balance
for each year
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9-28
Annuity Equaling a Present Value
Continued
• Same process used to indicate necessary loan
repayment
• Mortgage loan of $80,000 to be repaid over
20 years at 8 percent interest
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9-29
Table 9-2 Payoff Table for Loan
(Amortization Table)
• Part of payments to mortgage company for
interest payment, remainder applied to debt
reduction
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9-30
Compounding over Additional Periods
• Compounding frequency
• Certain contractual agreements may require
semiannual, quarterly, or monthly compounding
periods
• In such cases, adjust the normal formula
n = No. of years × No. of compounding periods
during year
i = Quoted annual interest rate / No. of
compounding periods during year
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9-31
Compounding over Additional Periods
Continued
• Case 1: compounded semiannually
• Find the future value of a $1,000 investment after
five years at 8 percent annual interest
• Where n = 5 × 2 = 10; i = 8%/2 = 4%
FV = PV × (1 + i)n
FV = $1,000 × (1.04) 10 = $1,480.24
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9-32
Compounding over Additional Periods
Concluded
• Case 2: compounded semiannually
• Find the present value of 20 quarterly payments of
$2,000 each to be received over the next five years
at 8 percent annual interest
• Where A = $2,000; n = 20, i = 8%/2 = 4%
PVA = A × {1 – [1/(1 + i)n]}/i
PVA = $2,000 × {1 – [1/(1.02)20]}/.02
PVA = $32,702.87
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9-33
Patterns of Payment with a Deferred
Annuity
• Patterns of Payment
• Problems may evolve around number of different
payment or receipt patterns
• Not every situation involves single amount or
annuity
• Contract may call for payment of different amount
each year overstated period or period of annuity
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9-34
Patterns of Payment with a
Deferred Annuity Continued 1
• Assume a contract involving payments of different
amounts each year for a three-year period
• To determine the present value, each payment is
discounted to the present and then totaled
(Assuming 8% discount rate)
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9-35
Patterns of Payment with a
Deferred Annuity Continued 2
• More complicated problem is when situations
include a combination of single amounts and
an annuity
• When annuity is paid at some time in future, it
is referred to as a deferred annuity
• Requires special treatment
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9-36
Patterns of Payment with a
Deferred Annuity Continued 3
• Assuming a contract involving payments of different
amounts each year for three-year period
• Annuity of $1,000 paid at end of each year from fourth
through eighth year
• With an 8 percent discount rate:
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9-37
Patterns of Payment with a
Deferred Annuity Continued 4
• Know the present value of the first three
payments ($5,022) from previous calculation
• Diagram the five annuity payments
• Annuity begins at end of year 4
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9-38
Patterns of Payment with a
Deferred Annuity Continued 5
• To discount $3,993 back to present, which falls at beginning of
fourth period, discount back three periods at 8 percent
interest rate
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9-39
Patterns of Payment with a
Deferred Annuity Continued 6
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9-40
Patterns of Payment with a
Deferred Annuity Concluded
• The present value of the five-year annuity may
now be added to the present value of inflows
over the first three years to arrive at the total
value
$5,022
Present value of the first three period flows
+ 3,170
$8,192
Present value of the five-year annuity
Total present value
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9-41
Appendix 9A
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9-42