# The Time Value of Money

```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
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
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
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
r = 10%
Cash Flows -500
1
2
3
4
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
 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
r = 10%
Cash Flows -500
1
2
3
4
FVn = ?
10
Future Value
Graphically, these computations are:
0
10%
-500× 1.10
End of year
amount
2
1
× 1.10
= 550.00
3
× 1.10
= 605.00
4
× 1.10
= 665.50
= 732.05
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
PV
r
n
=
=
=
=
future value in period n
present, or current, value
interest rate per period
number of periods interest is earned
12
Equation (Numerical) Solution
Determined by applying the appropriate
equation:
FV4n
=
PV
500
x
(1 + r)4nn = 732.05
(1.10)
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
 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
Cash Flows
0
7%
1
2
3
-100
-100
-100
x
x (1.07)2
x (1.07)0
(1.07)1
100.00
107.00
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
n

(1

r)
-1
t
FVAn  PMT  (1  r)  PMT 

r
0


n1
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
Cash Flows
0
-100
7%
1
2
3
-100
-100
-100
-100
x (1.07)
x (1.07) x (1.07)1
x (1.07) x (1.07)2
x
107.00
(1.07)0 100.00
107.00
114.49
114.49
122.50
FVA =
= 321.49
343.99
FVA(DUE)
19
FVA(DUE)—Equation (Numerical) Solution
 (1  r)n - 1 

 (1  r)



FVA
FVA(DUE)
=
PMT
n
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
3
7.0
0
N
I/Y
PV
BEGIN
-100
PMT
?
FV
FVA(DUE) = 343.99
21
Solutions: Future value computations
FV of \$25,000 lump-sum amount: N = 5, I/Y = 5,
PV = -25,000, PMT = 0, FV = ?
31,907
FV of \$5,499.40 annuity: N = 5, I/Y = 5, PV = 0,
PMT = -5,499.40, FV = ?
31,907
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
4%
1
2
3
-600
-400
-200
 (1.04)0
1
 (1.04)
 (1.04)2
FV  600(1.04) 2  400(1.04) 1  200(1.04) 0 
200.00
416.00
648.96
_______
1,264.96
24
FV of Uneven Cash Flow Streams—
Equation (Numerical) Solution
1
2
n
FVn  CF1 (1  r)  CF2 (1  r)    CFn (1  r)
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
Cash Flows
0
8%
PV = ?
1
2
3
4
800
29
Present Value—Equation (Numerical) Solution
Remember that FV is computed as follows:
FV = PV x (1.08)
(1 + r)4nn
800
800 = PV x 1.36049
In our example,
r = 8.0%
PV =FV
800/1.36049
588.02
4 = 800, n ==4,
30
Present Value—Equation (Numerical) Solution








1
PV Equation: PV  FV
n
(1r)








 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
8%
588.02 1
End of year
amount

1.08
2
1

635.07
1


1.08
3
685.87
1


1.08
740.74
4
800.00
1


1.08
32
PV Lump-Sum Amount—Financial
Calculator Solution
In our example: FV = \$800, r = 8.0%, n = 4
4
8.0
?
0
N
I/Y
PV
PMT
800
FV
-588.02
33
Present Value of an Annuity, PVA
0
7%
1
100
1

1
(1.07)
93.46
87.34
81.63
262.43 = PVA
2
3
100
100
1

(1.07) 2
1

(1.07) 3
 1 
 1 
 1 
 100 
 100 
 100 
1
2
3
(1.07)
(1.07)
(1.07)






 1
1
1 


 100(2.6243 )
262.43  100 
1
2
3
(1.07)
(1.07) 
 (1.07)
34
PVA—Equation (Numerical) Solution
1
1


1
(1 r) n
PVA  PMT 
 PMT 

t
1 (1  r)
 r 
n
In our example: PMT = \$100, r = 7%, n = 3
1
1 - (1.07)
3 

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%
100
100.00
93.46

1 1

(1.07)
1
1
(1.07)
(1.07)
(1.07)
1

(1.07) 2
(1.07)
1

3
(1.07)
93.46
87.34
1
2
3
100
100
100
100
87.34
81.63
280.80
262.43 = PVA (DUE)
37
PVA(DUE)—Equation (Numerical) Solution
1 - (1 1r) n

PVA
PVA(DUE)
n = PMT
r


 x (1 + r)

In our example: PMT = \$100, r = 7%, n = 3
1


1

3 


(1.07)
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
N
I/Y
?
100
0
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
PVA  5,499.40
 0.05

 25,000




 x(1.05)  5,499.40( 4.54595)





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
576.92
369.82
177.80
4%
1
600
1

(1.04) 1
2
3
400
200
1

2
(1.04)
1

3
(1.04)
 1 
 1 
 1 
 400 
 200 
1,124.54  600 
1
2
3
 (1.04) 
 (1.04) 
 (1.04) 
42
PV of Uneven Cash Flows— Equation
(Numerical) Solution
 1 
PV  CF1 
 CF2
1
 (1  r) 
 1 
 (1  r)2     CFn


 1 
 (1  r)n 


 1 
  CFt 
t 
(1

r)
1


 1 
 1 
 1 
 600
 400
 200
1
2
3
(1.04)
(1.04)
(1.04)






n
 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
CF1 = 2 million
CF2 = 4 million
CF3 = 5 million
I=6
NPV = 9.6448
Comparison of PVA, FVA,
and Lump-Sum Amount
 PMT = \$100; r = 7%; n = 3
 FVA = \$321.49
 PVA = \$262.43
C
0
1
7%
A
PVA = 262.43
100
2
100
FV = 262.43 x (1.07)3 = 321.49
3
100 B
FVA = 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
Year
1
2
3
Beginning Interest
Balance
@ 7%
\$262.43
180.80
93.46
\$18.37
12.66
6.54
Ending
Balance
Payment/
Withdrawal
\$280.80
193.46
100.00
\$100.00
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
Cash Flows
0
-200
r=?
1
2
3
245
51
Solving for r for a Lump-Sum—Equation
(Numerical) Solution
FV
= PV (1+r)n
245 = 200 (1+r)3
245
3
(1 r) 
200
1/3
 245 

(1 r)  

 200 


0.333

r  2.45
- 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—LumpSum Amount
If \$712 is invested at 6 percent, how long will it
take to grow to \$848?
Time
Cash Flows
0 r = 6% 1
-712
2
…
n=?
848
54
Solving for n for a Lump-Sum—Equation
(Numerical) Solution
FV  PV(1 r)n
848  712(1.06)n
848
n
(1.06) 
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
PVA = -817
r=?
1
2
3
300
300
300
58
Solving for r for Annuities—Equation
(Numerical) Solution
1 - (1 1r)n 
PVA  PMT 

 r 
1 - (1 1r) 3 
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
?
N
I/Y
-817
PV
300
0
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
Cash Flows
0 r = 8% 1
-480
2
…
-480
n=?
-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
N
I/Y
0
PV
-480
2,816
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
 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
Quarter
0
r = 2%
200.00
x 1.02
1
3
2
x 1.02
204.00
x 1.02
208.08
x 1.02
212.24
1
2
4
8
…
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
?
N
I
-1
PV
0
PMT
0.016399
APR = 0.016399 x 365 = 5.986
1.06168
FV
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
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
N
I/Y
PV
?
PMT
0
FV
-889
73
Amortization Schedule—Example
Payment = \$889
Begin.
Payment
Year Pmt # Balance
(Pmt)
1
1
\$6,655.00
\$889
2
5,865.83
889
3
5,064.82
889
4
4,251.79
889
2
5
3,426.57
889
6
2,588.97
889
7
1,738.80
889
8
875.88*
889
Interest (I)
Loan Repay.
[1.5% x Beg Bal]
= Pmt – I
\$99.83
\$789.17
87.99
801.01
75.97
813.03
63.78
825.22
51.40
837.60
38.83
850.17
26.08
862.92
13.14
875.86*
* Rounding difference
74
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
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
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