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The Time Value of Money
Introduction to Time Value of Money, TVM
Future Value, FV
Lump-sum amount
Annuity
Uneven cash flow
Present Value, PV
Lump-sum amount
Annuity
Uneven cash flow
FV and PV Comparison
Solving for r and n
Intra-year Interest Compounding
Amortization
1
Time Value of Money
Why is it important to understand and apply time
value of money concepts?
What is the difference between a present value
amount and a future value amount?
What is an annuity?
What is the difference between the Annual
Percentage Rate and the Effective Annual Rate?
What is an amortized loan?
How is the return on an investment determined?
2
The Time Value of Money
Time value of money is considered the
most important concept in finance
Mathematics of finance
“Nuts & Bolts” of financial analysis—
apply of TVM concepts to determine
value
Interest = Rate of return = r = i = k =
Y
3
The Time Value of Money
“$1 received today is more valuable than $1
received in one year.” Why?
Because if you have the opportunity to earn a
positive return, investing the $1 today will cause
it to grow to greater than $1 in one year.
For example, $1 invested at 5 percent will grow
to $1.05 in one year because 5¢ of interest will
be earned.
4
Future Value and Present Value
Future Value (FV)—determine to what amount an
investment will grow over a particular time period
re-invested interest (earned in previous periods) earns
interest
compounding—interest compounds or grows the investment
Present value (PV)—determine the current value of
an amount that will be paid, or received, at some
time in the future
PV is the future amount restated in current dollars; future
interest has not been earned, thus it is not included in the
PV
discounting—deflate, or discount, the future amount by
future interest that can be earned (“deinterest” the FV)
5
Lump-Sum Amounts, Annuities, and Uneven
Cash Flow Streams
Lump-sum amount—a single amount invested
(received) today or in the future; growth in value is
the result of interest only
Annuity—equal payments made (received) at equal
intervals; growth in value is the result of additional
payments as well as interest
ordinary annuity—end of period payments
annuity due—beginning of period payments
Uneven Cash Flows—payments that are not all equal
that are generally made (received) at equal intervals;
growth in value is the result of additional payments as
well as interest
6
Cash Flow Time Lines
Helps you to visualize the timing of the
cash flows associated with a particular
situation
Constructing a cash flow time line is easy:
Time 0 1 2 3 4
r = 10%
Cash Flows -500 FVn = ?
7
Approaches to TVM Solutions
Time line solution
Solve using a cash flow time line
Equation (numerical) solution
Use equations to solve the problem
Financial calculator solution
Financial calculators are programmed to solve time value of
money problems using the numerical solution
Spreadsheet solution
Spreadsheets contain functions that can be used to solve
time value of money problems using the numerical solution
Interest tables
Obsolete
8
Future Value
Determine to what amount an
investment will grow over a particular
time period if it is invested at a positive
rate of return.
Compounding
Lump-sum amount
Annuity
Uneven cash flow stream
9
Future Value, FV, of a Lump-Sum Amount
Example: If you invest $500 today at 10%, what
will the investment be worth in four years if interest
is paid annually?
Time 0 1 2 3 4
r = 10%
Cash Flows -500 FVn = ?
10
Future Value
Graphically, these computations are:
0 1 2 3 4
10%
×
-500 1.10 × 1.10 × 1.10 × 1.10
End of year = 550.00 = 605.00 = 665.50 = 732.05
amount
FV4 = 500(1.10 x 1.10 x 1.10 x 1.10)
= 500(1.10)4
= 732.05
11
Future Value
The future value of an amount invested today for n
years, FVn, can be found using the following
equation:
FVn = PV(1 + r)n = PV(interest multiple)
FVn = future value in period n
PV = present, or current, value
r = interest rate per period
n = number of periods interest is earned
12
Equation (Numerical) Solution
Determined by applying the appropriate
equation:
FVn
4 = PV
500 x (1 + r)n = 732.05
(1.10)4n
In our example: PV = $500, r = 10.0%, n = 4
13
Financial Calculator Solution
In our example: PV = $500, r = 10.0%, n = 4
4 10 -500 0 ?
N I/Y PV PMT FV
732.05
14
Future Value of an Annuity
Annuity—a series of equal payments that are
made at equal intervals
Ordinary annuity—end of the period
Annuity due—beginning of the period
The future value of an annuity, FVA, can be
computed by solving for the future value of a
lump-sum amount
15
Future Value of an Annuity, FVA
Time 0 1 2 3
7%
Cash Flows -100 -100 -100
x (1.07)0
100.00
x (1.07)1
107.00
x (1.07)2
114.49
FVA = 321.49
FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49
= 100[(1.07)2 + (1.07)1 + (1.07)2] = 100(3.2149) = 321.49
16
FVA—
Equation (Numerical) Solution
n1
(1 r)n - 1
FVA n PMT (1 r)t PMT
0 r
In our example: PMT = $100, r = 7%, n = 3
(1.07)3 - 1
FVA n 100
0.07
100(3.2149 321.49
)
17
FVA—Annuity Due
Annuity due is an annuity with cash flows
that occur at the beginning of the period.
When compared to an ordinary annuity,
which has end-of-period cash flows, the cash
flows of an annuity due earn one additional
period of interest.
18
FVA—Annuity Due
Time 0 1 2 3
7%
Cash Flows -100 -100
-100 -100 -100
x (1.07) x
107.00
(1.07)0 100.00
x (1.07) x (1.07)1
107.00
114.49
x (1.07) x (1.07)2
114.49
122.50
FVA = 343.99
FVA(DUE) = 321.49
19
FVA(DUE)—Equation (Numerical) Solution
(1 r)n - 1
FVA(DUE)n = PMT
FVA (1 r)
r
In our example: PMT = $100, r = 7%, n = 3
(1.07)3 - 1
FVA(DUE)n 100 1.07
0.07
100(3.4399 343.99
)
20
FVA—Financial Calculator Solution
In our example: n = 3, r = 7%, PMT = $100
3 7.0 0 -100 ?
N I/Y PV PMT FV
FVA = 321.49
BEGIN
3 7.0 0 -100 ?
N I/Y PV PMT FV
FVA(DUE) = 343.99
21
Solutions: Future value computations
FV of $25,000 lump-sum amount: N = 5, I/Y = 5,
31,907
PV = -25,000, PMT = 0, FV = ?
FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0,
31,907
PMT = -5,499.40, FV = ?
Uneven Cash Flow Streams
Uneven cash flow stream—cash flows
that are not all the same (equal)
Simplifying techniques (that is, using a
single equation) used to compute FVA
cannot be used
23
FV—Uneven Cash Flow Streams
0 1 2 3
4%
-600 -400 -200
(1.04) 0
200.00
1
(1.04)
416.00
(1.04) 2
648.96
_______
FV 600(1.04)2 400(1.04)1 200(1.04)0 1,264.96
24
FV of Uneven Cash Flow Streams—
Equation (Numerical) Solution
1 2 n
FVn CF (1 r) CF2 (1 r) CFn (1 r)
1
n
CFt (1 r) t
t 1
2 1 0
FVn 600(1.04) 400(1.04) 200(1.04)
600(1.0816 400(1.0400 200(1.0000
) ) )
1,264.96
25
FV of Uneven CF Streams—Calculator Solution
Input the cash flows, find the present
value, PV, and then compute the future
value, FV, of PV.
Discussed in the next section.
26
Present Value
Determine the current value of an amount
that will be paid, or received, at a particular
time in the future.
Finding the present value (PV), or
discounting, an amount to be received
(paid) in the future is the reverse of
compounding, or determining the future
value of an amount invested today.
We find the PV by “de-interesting” the FV.
27
Present Value—Lump-Sum Amount
What is the PV of $800 to be received in four
years if your opportunity cost is 8 percent?
Stated differently: How much would you be
willing to pay today for an investment that
will pay $800 in four years if you have the
opportunity to invest at 8 percent per year?
28
Present Value—Lump-Sum Amount
Time 0 1 2 3 4
8%
Cash Flows PV = ? 800
29
Present Value—Equation (Numerical) Solution
Remember that FV is computed as follows:
FV = PV x (1 + r)n
800 (1.08)n
4
800 = PV x 1.36049
In our example, FV4 = 800, n = 4, r = 8.0%
PV = 800/1.36049 = 588.02
30
Present Value—Equation (Numerical) Solution
1
PV Equation: PV FV
n
(1r)
1
800
4
(1.08)
800(0.7350 3) 588.02
In our example: FV = $800, r = 8.0%, and N = 4
31
Present Value—Time Line Solution
Graphically, this computation is:
0 1 2 3 4
8%
588.02 1 635.07 1
685.87 1
740.74 1
800.00
End of year
1.08 1.08 1.08 1.08
amount
32
PV Lump-Sum Amount—Financial
Calculator Solution
In our example: FV = $800, r = 8.0%, n = 4
4 8.0 ? 0 800
N I/Y PV PMT FV
-588.02
33
Present Value of an Annuity, PVA
0 1 2 3
7%
1 100 100 100
1
(1.07) 1
93.46 (1.07)2
1
87.34
(1.07)3
81.63
262.43 = PVA 1 1 1
100 1
100 2
100
(1.07) (1.07) (1.07)3
1 1 1
262.43 100 1
2
3
100(2.6243)
(1.07) (1.07) (1.07)
34
PVA—Equation (Numerical) Solution
n
1 1 - (1r)n
1
PVA PMT t
PMT
1 (1 r) r
In our example: PMT = $100, r = 7%, n = 3
1 - (1.07) 3
1
PVA 100
0.7
100(2.6243) 262.43
35
PVA—Annuity Due
Annuity due is an annuity with cash flows
that occur at the beginning of the period.
36
PVA—Annuity Due
0 7% 1 2 3
1 1
100 1 1
(1.07) 100 100 100
100
(1.07)
(1.07)
93.46
100.00
1
(1.07)
(1.07)2
87.34
93.46
1
(1.07) 3
(1.07)
81.63
87.34
262.43
280.80 = PVA (DUE)
37
PVA(DUE)—Equation (Numerical) Solution
1 - (1r)n
1
PVA(DUE)n = PMT
PVA r
x (1 + r)
In our example: PMT = $100, r = 7%, n = 3
1 - (1.07) 3
1
PVA(DUE)3 100 1.07
0.7
100(2.8080 280.80
)
38
PVA—Financial Calculator Solution
In our example: n = 3, r = 7%, PMT = $100
3 7.0 ? 100 0
N I/Y PV PMT FV
-262.43 = PVA
BEGIN
3 7.0 ? 100 0
N I/Y PV PMT FV
-280.80 = PVA(DUE)
39
Calculator Solution:
N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = ?
25,000
Numerical Solution:
1
1
(1.05 )5 x(1.05 ) 5,499 .40( 4.54595 )
PVA 5,499 .40
0.05
25,000
Uneven Cash Flow Streams
Uneven cash flow stream—cash flows
that are not all the same (equal)
Simplifying techniques (that is, using a
single equation) used to compute PVA
cannot be used
41
Present Value of an
Uneven Cash Flow Stream
0 1 2 3
4%
1 600 400 200
(1.04)1 1
576.92 (1.04)2
1
369.82 3
(1.04)
177.80
1 1 1
1,124.54 600 1
400 2
200 3
(1.04) (1.04) (1.04)
42
PV of Uneven Cash Flows— Equation
(Numerical) Solution
1 1 1
PV CF
1 1
CF2 (1 r)2 CFn (1 r)n
(1 r)
n
1
CFt
1 (1 r)t
1 1 1
600 1
400 2
200
(1.04) (1.04) (1.04)3
1,124.54
43
PV of Uneven Cash Flows—Financial
Calculator Solution
Use the cash flow (CF) register (see calculator
instructions)
Input CFs in the order they occur—that is,
first input CF1, then input CF2, and so on
CF0—most calculators require you to input a
value for before entering any other cash flows
Enter the value for I
NPV = PV of uneven cash flows
44
PV of Uneven Cash Flows—Financial
Calculator Solution
CF0 = 0
CF1 = 600
CF2 = 400
CF3 = 200
r = 4%
Compute NPV = 1,124.54
45
FV of Uneven CF Streams—Calculator Solution
Input the cash flows, find the present
value, PV, and then compute the future
value, FV, of PV
In our example, PV = $1,124.54, so the
future value is:
FV = $1,124.54(1.04)3 = $1,264.95
46
Numerical solution:
PV = ($2 million)/(1.06)1 + ($4 million)/(1.06)2
+ ($5 million)/(1.06)3
= ($2 million)(0.943396) + ($4 million)(0.889996)
+($5 million)(0.839619)
= 1.8868 + 3.5600 + 4.1981 = 9.6449
Calculator solution:
CF0 = 0 I=6
CF1 = 2 million NPV = 9.6448
CF2 = 4 million
CF3 = 5 million
Comparison of PVA, FVA,
and Lump-Sum Amount
PMT = $100; r = 7%; n = 3
FVA = $321.49
PVA = $262.43
0 1 C 2 3
7%
A 100 100 100 B
PVA = 262.43 FVA = 321.49
FV = 262.43 x (1.07)3 = 321.49
PV = 321.49/(1.07)3 = 262.43
48
PVA, FVA, and Lump-Sum Amount
PMT = $100; r = 7%; n = 3; PVA = $262.43
Beginning Interest Ending Payment/
Year Balance @ 7% Balance Withdrawal
1 $262.43 $18.37 $280.80 $100.00
2 180.80 12.66 193.46 100.00
3 93.46 6.54 100.00 100.00
FVA = 321.49
49
Solving for Time (n) and Interest Rates
(r)—Lump Sums
The computations for lump-sum amounts
included four variables: n, r, PV, and FV.
If three of the four variables are known, then
we can solve for the unknown variable—e.g., if
n, PV, and FV are known, we can solve for r.
50
Solving for Interest Rates, r—Lump-Sum
Amount
If $200 that was invested three years ago is now
worth $245, what rate of return (r) did the
investment earn?
Time 0 r=? 1 2 3
Cash Flows -200 245
51
Solving for r for a Lump-Sum—Equation
(Numerical) Solution
FV = PV (1+r)n
245 = 200 (1+r)3
(1 r)3 245
200
245
1/3
(1 r)
200
r 2.45
0.333 - 1.0 = 7.0%
52
Solving for r for a Lump-Sum—Financial
Calculator Solution
In our example: PV = $200, FV = $245, n = 3
3 ? -200 0 245
N I/Y PV PMT FV
7.00
53
Solving for Number of Years, n—Lump-
Sum Amount
If $712 is invested at 6 percent, how long will it
take to grow to $848?
Time 0 r = 6% 1 2 n=?
…
Cash Flows -712 848
54
Solving for n for a Lump-Sum—Equation
(Numerical) Solution
FV PV(1 r)n
848 712(1.06)n
(1.06)n 848
712
55
Solving for n for a Lump-Sum—Financial
Calculator Solution
In our example: PV = $712, FV = $848, r = 6%
? 6.0 -712 0 848
N I/Y PV PMT FV
3.00
56
Solving for Time (n) and Interest Rates
(r)—Annuities
The computations for annuities included
four variables: n, r, PMT, and PVA or FVA.
If three of the four variables are known,
then we can solve for the unknown
variable—e.g., if n, PVA (or FVA), and PMT
are known, we can solve for r.
57
Solving for Interest Rates, r, for Annuities
The current value of an investment that will pay
$300 each year for three years is $817. What rate
of return (r) will the investment earn?
0 1 2 3
r=?
PVA = -817 300 300 300
58
Solving for r for Annuities—Equation
(Numerical) Solution
1 - (1r)n
1
PVA PMT
r
1 - (1r)3
1
817 300
r
To solve, use a trial-and-error process
Solution = 5.0%
59
Solving for r for Annuities—Financial
Calculator Solution
In our example: PMT = $300, PVA = $817, n = 3
3 ? -817 300 0
N I/Y PV PMT FV
5.00
60
Solving for Number of Years, n, for
Annuities
If $480 is invested each year at 8 percent,
how long will take to grow to $2,816?
Time 0 r = 8% 1 2 n=?
…
Cash Flows -480 -480 -480
FVA = 2,816
61
Solving for n for Annuities—Equation
(Numerical) Solution
(1 r)n - 1
FVA PMT
r
(1.08)n - 1
2,816 480
0.08
To solve, use a trial-and-error process
Solution = 5 years
62
Solving for n for Annuities—Financial
Calculator Solution
In our example: PMT = $480, FVA = $2,816, r = 8%
? 8.0 0 -480 2,816
N I/Y PV PMT FV
5.00
63
Solving for r for Uneven Cash Flows
Internal Rate of Return (IRR)—average rate
of return an investment earns
Capital budgeting decisions—decisions
concerning what investments a firm should
purchase
64
Intra-Year Interest Compounding
Interest is compounded more than once per
year—quarterly, monthly, or daily
Adjustments to computations:
Use the interest rate per compounding period and
the number of interest compounding periods
during the life of the investment, or
Use the effective annual rate, EAR, and the
number of years to maturity
65
Intra-Year Interest Compounding—
Example
How much will an amount invested today grow to in two years if
interest is paid quarterly? PV = $200 and r = 8%
Quarterly interest = 8%/4 = 2% = r/m
Number of interest payments = 2 years x 4 = 8 = n
Year 0 1 2
Quarter 0 1 2 3 4 8
r = 2%
…
x 1.02 x 1.02 x 1.02 x 1.02
200.00 204.00 208.08 212.24 216.49 234.33
FVn = PV(1 + r)n = 200(1.02)8 = 200(1.17166) = 234.33
66
Intra-Year Interest Compounding
Financial calculator solution:
8 2.0 -200 0 ?
N I/Y PV PMT FV
234.33
67
Intra-Year Interest Compounding—Effective
Annual Rate
Interest = 8%, compounded quarterly
r = 8% per year is the simple, or non-compounded rate
r/m = 2% per quarter is the effective rate per
compounding period (each quarter; m=4)
Effective Annual Rate (EAR), rEAR
rEAR = (1 + r/m)m – 1.0
m = number of interest payment periods per year
rEAR = (1.02)4 – 1.0 = 0.08243 = 8.243%
68
Intra-Year Interest Compounding—Using
Effective Annual Rate
In our example:
PV = $200
n = 2 years
rEAR = 8.243%
FVn = PV(1 + rEAR)n
= 200(1.08243)2
= 234.33
= 200(1.02)8
69
Annual Percentage Rate, APR, Versus
Effective Annual Rate, EAR
EAR—the rate of return per year considering
interest compounding
rEAR = (1 + r/m)m – 1.0
APR—simple rate of return; does not consider
compounding
APR = r/m x m = r = simple interest
rEAR = APR only if interest is paid once per year—
that is, annual compounding
rEAR > APR if interest is paid more than once per
year
70
FVA—Financial Calculator Solution
In our example: n = 365, PV = -$1, PMT = 0, FV
= $1.06168
365 ? -1 0 1.06168
N I PV PMT FV
0.016399
APR = 0.016399 x 365 = 5.986
Advertise: 5.986% compounded daily
Amortized Loans
Loan agreement requires equal periodic
payments
A portion of the payment represents interest
on the debt and the remainder is applied to
the repayment of the debt
Amortization schedule—used to determine
what portion of the total payment is interest
and what portion is repayment of principal
Mortgage payment—only the interest portion
is considered an expense for tax purposes
72
Amortization Schedule—Example
Example: $6,655 home-equity loan; r = 6%; n = 2
years; payments are made quarterly
The constant payments per quarter represent an annuity
and the amount of the loan ($6,655) represents the
present value of the loan payments
PVA = $6,655; (n x m) = 2 x 4 = 8 payments; r/m =
6%/4 = 1.5%
Financial calculator solution:
8 1.5 6,655 ? 0
N I/Y PV PMT FV
-889
73
Amortization Schedule—Example
Payment = $889
Begin. Payment Interest (I) Loan Repay.
Year Pmt # Balance (Pmt) [1.5% x Beg Bal] = Pmt – I
1 1 $6,655.00 $889 $99.83 $789.17
2 5,865.83 889 87.99 801.01
3 5,064.82 889 75.97 813.03
4 4,251.79 889 63.78 825.22
2 5 3,426.57 889 51.40 837.60
6 2,588.97 889 38.83 850.17
7 1,738.80 889 26.08 862.92
8 875.88* 889 13.14 875.86*
* Rounding difference
74
Answers to TVM Questions
Why is it important to understand and apply
time value of money concepts?
To be able to compare various investments.
What is the difference between a present
value amount and a future value amount?
Future value adds interest (compounds); present
value subtracts interest (“de-interests”).
What is an annuity?
A series of equal payments that occur at equal
time intervals.
75
Answers to TVM Questions
What is the difference between the Annual
Percentage Rate and the Effective Annual Rate?
APR is a simple interest rate quoted on
loans/investments, whereas EAR is the actual interest or
rate of return.
What is an amortized loan?
A loan paid off in equal payments over a specified period,
which include payments of interest and principal
How is the return on an investment determined?
Compute the annual rate based on the amount to which
an investment will grow in the future.
76
