Chapter 3

Time Value of
Money

1
 The Time Value of Money

 The Interest Rate

 Simple Interest
 Compound Interest
 Amortizing a Loan

2
 The Interest Rate

Which would you prefer -- $10,000

today or $10,000 in 5 years?
years

Obviously, $10,000 today.

today

You already recognize that there is

TIME VALUE TO MONEY!!
MONEY

3
 Why TIME?

Why is TIME such an important

element in your decision?

TIME allows you the opportunity to

postpone consumption and earn
INTEREST.
INTEREST

4
 Types of Interest
 Simple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).
 Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).

5
 Simple Interest Formula

Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
6
 Simple Interest Example
 Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

 SI = P0(i)(n)
= $1,000(.07)(2)
= $140
7
 Simple Interest (FV)
 What is the Future Value (FV)
FV of the
deposit?
FV = P0 + SI
= $1,000 + $140
= $1,140
 Future Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
8
interest rate.
 Simple Interest (PV)
 What is the Present Value (PV)
PV of the
previous problem?
The Present Value is simply the
$1,000 you originally deposited.
That is the value today!
 Present Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
9
rate.
 Why Compound Interest?
Future Value of a Single $1,000 Deposit
Future Value (U.S. Dollars)

20000
10% Simple
15000 Interest
10000 7% Compound
Interest
5000 10% Compound
Interest
0
1st Year 10th 20th 30th
Year Year Year
10
 Future Value
Single Deposit (Graphic)
Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
years
0 1 2
7%
$1,000
FV2
11
 Future Value
Single Deposit (Formula)
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
12
 Future Value
Single Deposit (Formula)
FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070
FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) =
$1,000(1.07)(1.07)
$1,000 = P0 (1+i)2 =
$1,000(1.07)
$1,000 2
= $1,144.90
You earned an EXTRA $4.90 in Year 2 with
compound over simple interest.

13
 General Future
Value Formula
FV1 = P0(1+i)1
FV2 = P0(1+i)2
etc.

General Future Value Formula:

FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
14
 Valuation Using Table I
FVIFi,n is found on Table I at the end
of the book or on the card insert.
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
15
5 1.338 1.403 1.469
 Using Future Value Tables
FV2 = $1,000 (FVIF7%,2) = $1,000
(1.145) = $1,145 [Due to Rounding]

Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
16
 Story Problem Example
Julie Miller wants to know how large her deposit
of $10,000 today will become at a compound
annual interest rate of 10% for 5 years.
years

0 1 2 3 4 5
10%
$10,000
FV5
17
 Story Problem Solution
 Calculation based on general formula:
FVn = P0 (1+i)n FV5 =
$10,000 (1+ 0.10)5 = $16,105.10

 Calculation based on Table I: FV5 =

$10,000 (FVIF10%, 5) = $10,000 (1.611) =
$16,110 [Due to Rounding]

18
 Double Your Money!!!

Quick! How long does it take to double

$5,000 at a compound rate of 12% per
year (approx.)?

We will use the “Rule-of-72”.

19
 The “Rule-of-72”

Quick! How long does it take to double

$5,000 at a compound rate of 12% per
year (approx.)?

Approx. Years to Double = 72 / i%

72 / 12% = 6 Years
[Actual Time is 6.12 Years]
20
 Present Value
Single Deposit (Graphic)
Assume that you need $1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded annually.
0 1 2
7%
$1,000
PV0 PV1
21
 Present Value
Single Deposit (Formula)
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2 =
FV2 / (1+i)2 = $873.44

0 1 2
7%
$1,000
PV0
22
 General Present
Value Formula
PV0 = FV1 / (1+i)1

PV0 = FV2 / (1+i)2

etc.

General Present Value Formula:

PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
23
 Valuation Using Table II
PVIFi,n is found on Table II at the end
of the book or on the card insert.
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
24
 Using Present Value Tables
PV2 = $1,000 (PVIF7%,2)= $1,000
(.873) = $873 [Due to Rounding]

Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
25
5 .747 .713 .681
 Story Problem Example
Julie Miller wants to know how large of a
deposit to make so that the money will grow
to $10,000 in 5 years at a discount rate of
10%.
0 1 2 3 4 5
10%
$10,000
PV0
26
 Story Problem Solution
 Calculation based on general formula:
PV0 = FVn / (1+i)n PV0 =
$10,000 / (1+ 0.10)5 = $6,209.21
 Calculation based on Table I: PV0
= $10,000 (PVIF10%, 5) = $10,000 (.621)
= $6,210.00 [Due to Rounding]

27
 Types of Annuities
 An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
 Ordinary Annuity:
Annuity Payments or receipts
occur at the end of each period.
 Annuity Due:
Due Payments or receipts
occur at the beginning of each period.

28
 Examples of Annuities

 Student Loan Payments

 Car Loan Payments
 Insurance Premiums
 Mortgage Payments
 Retirement Savings
29
 Parts of an Annuity

(Ordinary Annuity)
End of End of End of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today Equal Cash Flows

30
Each 1 Period Apart
 Parts of an Annuity

(Annuity Due)
Beginning of Beginning of Beginning of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today Equal Cash Flows

Each 1 Period Apart
31
 Overview of an
Ordinary Annuity -- FVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .
R R R
R = Periodic
Cash Flow

FVAn = R(1+i)n-1 + R(1+i)n-2 + FVAn

... + R(1+i)1 + R(1+i)0

32
 Example of an
Ordinary Annuity -- FVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000
$1,070
$1,145
FVA3 = $1,000(1.07)2 +
$3,215 = FVA3
$1,000(1.07) + $1,000(1.07)
1 0

= $1,145 + $1,070 + $1,000

= $3,215
33
 Hint on Annuity Valuation
The future value of an ordinary
annuity can be viewed as
occurring at the end of the last
cash flow period, whereas the
future value of an annuity due
can be viewed as occurring at
the beginning of the last cash
flow period.
34
 Valuation Using Table III
FVAn = R (FVIFAi%,n) FVA3 =
$1,000 (FVIFA7%,3) = $1,000 (3.215) =
$3,215
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
35
 Solving the FVA Problem
Inputs 3 7 0 -1,000
N I/Y PV PMT FV
Compute 3,214.90

N:3 periods (enter as 3 year-end deposits)

I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Not relevant in this situation (no beg value)
PMT: $1,000 (negative as you deposit annually)
FV: Compute (Resulting answer is positive)

36
 Overview View of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 n-1 n
i% . . .
R R R R R

FVADn = R(1+i)n + R(1+i)n-1 + FVADn

... + R(1+i)2 + R(1+i)1
= FVAn (1+i)
37
 Example of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000 $1,070
$1,145
$1,225

FVAD3 = $1,000(1.07)3 + $3,440 = FVAD3

$1,000(1.07)2 + $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440
38
 Valuation Using Table III
FVADn = R (FVIFAi%,n)(1+i)
FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000
(3.215)(1.07) = $3,440

Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
39
 Overview of an
Ordinary Annuity -- PVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .
R R R

R = Periodic
Cash Flow
PVAn
PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
40
 Example of an
Ordinary Annuity -- PVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000
$ 934.58
$ 873.44
$ 816.30
$2,624.32 = PVA3 PVA3 = $1,000/(1.07)1 +
$1,000/(1.07)2 + $1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32

41
 Hint on Annuity Valuation
The present value of an ordinary
annuity can be viewed as
occurring at the beginning of the
first cash flow period, whereas
the present value of an annuity
due can be viewed as occurring
at the end of the first cash flow
period.
42
 Valuation Using Table IV
PVAn = R (PVIFAi%,n) PVA3 =
$1,000 (PVIFA7%,3) = $1,000 (2.624) =
$2,624
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
43
 Overview of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2 n-1 n
i% . . .
R R R R

R: Periodic
PVADn Cash Flow

PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1

= PVAn (1+i)
44
 Example of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%

$1,000.00 $1,000 $1,000

$ 934.58
$ 873.44
$2,808.02 = PVADn

PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +

$1,000/(1.07)2 = $2,808.02
45
 Valuation Using Table IV
PVADn = R (PVIFAi%,n)(1+i)
PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000
(2.624)(1.07) = $2,808

Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
46
 Steps to Solve Time Value
of Money Problems
1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single CF,
annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
47
 Mixed Flows Example
Julie Miller will receive the set of cash
flows below. What is the Present Value
at a discount rate of 10%?
10%

0 1 2 3 4 5
10%
$600 $600 $400 $400 $100
PV0
48
 How to Solve?

1. Solve a “piece-at-a-time”
piece-at-a-time by
discounting each piece back to
t=0.
2. Solve a “group-at-a-time”
group-at-a-time by first
breaking problem into groups of
annuity streams and any single cash
flow group. Then discount each
group back to t=0.
49
 “Piece-At-A-Time”

0 1 2 3 4 5
10%
$600 $600 $400 $400 $100
$545.45
$495.87
$300.53
$273.21
$ 62.09
$1677.15 = PV0 of the Mixed Flow
50
 “Group-At-A-Time” (#1)
0 1 2 3 4 5
10%
$600 $600 $400 $400 $100
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60

$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10
51
 “Group-At-A-Time” (#2)
0 1 2 3 4

$400 $400 $400 $400

$1,268.00
0 1 2 PV0 equals
Plus
$200 $200 $1677.30.
$347.20
0 1 2 3 4 5
Plus
$100
$62.10
52
 Frequency of
Compounding
General Formula:
FVn = PV0(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per
Yeari: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
53
 Impact of Frequency
Julie Miller has $1,000 to invest for 2
years at an annual interest rate of
12%.
Annual FV2 = 1,000(1+
1,000 [.12/1])(1)(2)
= 1,254.40
Semi FV2 = 1,000(1+
1,000 [.12/2])(2)(2)
= 1,262.48
54
 Impact of Frequency
Qrtly FV2 = 1,000(1+
1,000 [.12/4])(4)(2)
= 1,266.77
Monthly FV2 = 1,000(1+
1,000 [.12/12])(12)(2)
= 1,269.73
Daily FV2 = 1,000(1+
1,000 [.12/365])(365)
(2)
= 1,271.20
55
 Effective Annual
Interest Rate
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.

(1 + [ i / m ] )m - 1

56
 BW’s Effective
Annual Interest Rate
Basket Wonders (BW) has a $1,000
CD at the bank. The interest rate
is 6% compounded quarterly for 1
year. What is the Effective Annual
Interest Rate (EAR)?
EAR
EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 -
1 = .0614 or 6.14%!
57
 Steps to Amortizing a Loan
1.Calculate the payment per period.
2.Determine the interest in Period t. (Loan
balance at t-1) x (i% / m)
3.Compute principal payment in Period t.
(Payment - interest from Step 2)
4.Determine ending balance in Period t.
(Balance - principal payment from Step 3)
5.Start again at Step 2 and repeat.
58
 Amortizing a Loan Example
Julie Miller is borrowing $10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
59
 Amortizing a Loan Example
End of Payment Interest Principal Ending
Year Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000

[Last Payment Slightly Higher Due to Rounding]

60
 Usefulness of Amortization

1. Determine Interest Expense --

Interest expenses may reduce
taxable income of the firm.
2. Calculate Debt Outstanding --
The quantity of outstanding debt
may be used in financing
the day-to-day activities of the firm.

61