Chapter 1
Appendix
Time Value of Money:
The Basics
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Time Value of Money
• Answers the questions:
– “If I deposit $10,000 today, how much will I
have for a down payment on a house in 5
years?”
– “Will $2,000 saved each year give me
enough money when I retire?”
– “How much must I save today to have
enough for my children’s education?”
App 1-2
Time Value of Money
Basic Principles
– A dollar received today is worth more than
a dollar received a year from today
– A dollar that will be received in the future
is worth less than a dollar today
– Why?
• A dollar today could be saved or invested
• A dollar in the future is uncertain
App 1-3
Time Value of Money
• Definitions
• Solving TVM Problems
– Types of Problems
•
•
•
•
Interest rate basics - Simple interest
Future value - Single amount & Annuity
Present value - Single amount & Annuity
Calculating Loan payments
– Solutions Methods
•
•
•
•
Formulas
TVM Tables
Financial Calculator
Excel Functions
App 1-4
Basic TVM Definitions
• Future Value (FV)
– The increased value of money from interest
earned
– The amount to which a current sum will grow
given a certain interest rate and time period
– “Compounding”
• Present Value (PV)
– The current value of a future amount given a
certain interest rate and time period
– “Discounting”
App 1-5
Basic TVM Definitions
• Payment (PMT or annuity)
– Amount of annuity deposit or withdrawal
• Sign Convention:
– Applies to PV, PMT and FV
– Positive = inflow to YOU
• Money received as a loan is an inflow
– Negative = outflow from YOU
• Deposit to an account is an outflow
App 1-6
Basic TVM Definitions
• Interest rate (i or I/Y)
– Stated as a percent per year
– Also called “discount rate”
– 12% =
• “0.12” in formulas & in Excel
• “12” in financial calculators
App 1-7
Basic TVM Definitions
• Time Periods (n or t)
– Expressed in years
• 3 months = “0.25” years
• 2 ½ years = “2.5” years
– Interest rate and time period must match
• Annual periods  annual rate
• Monthly periods  monthly rate
App 1-8
Single Amount & Annuities
• Single Amount:
– A single payment made or received at one
time
– Calculator: PMT=0
• Annuity:
– Finite series of equal payments that occur at
regular intervals
– PMT key used
– Sign convention is important
App 1-9
Basic TVM Formulas
Simple Interest:
Principal x Rate x Time
Future Value:
Single Amount
Annuity
FV = PV(1 + i)n
PMT (1  i )n  1
i
Present Value
Single Amount
Annuity
FV
PV 
(1  i )n
 PV  FV (1  i ) n
1

1


n
(
1

i
)
PMT 

i








App 1-10
TVM Calculator Solutions
Texas Instruments BA-II Plus
•
•
•
•
•
FV = future value
One of these MUST
be negative
PV = present value
PMT = periodic payment
I/Y = interest rate
N = number of periods
N
I/Y
PV
PMT
FV
App 1-11
Texas Instruments BA-II Plus
• I/Y = period interest rate (i)
– P/Y must = 1
– Interest is entered as a percent, not a
decimal
• 5% interest = “5”, not “0.05”
• Clear the registers before each problem
– [2nd] [CLR TVM]
– Or reenter each field
App 1-12
TVM with Excel
Spreadsheet Functions
=FV(Rate,Nper,Pmt,PV)
=PV(Rate,Nper,Pmt,FV)
=RATE(Nper,Pmt,PV,FV)
=NPER(Rate,Pmt,PV,FV)
=PMT(Rate,Nper,PV,FV)
• Use the formula icon (ƒx)
when you can’t remember
the exact formula
Time Value of Money
Interest Rate Basics
• Calculating interest earned:
– Principal = dollar amount of savings
– Annual rate of interest
– Length of time money on deposit (in years)
• Simple interest:
Amt in
Svgs
X
Annual
Interest
Rate
X
Time
Period
=
Interest
App 1-14
Interest Rate Basics
Example A
You borrow $1,000 at 5% annual interest
for 1 year:
Principal = $1,000
Interest rate = 5% = .05
Time period = 1
$1,000
X
.05
X
1
=
$50
App 1-15
Interest Rate Basics
Example B
You deposit $750 at 8% per year for 9
months:
Principal = $750
Interest rate = 8%
Time period = 9/12 = .75
$750
X
.08
X
0.75
=
$45
App 1-16
Interest Rate Basics
Example B – Calculator*
Principal = $750
Interest rate = 8%
Time period = 9/12 = .75
Calculator Solution Keystrokes





.75
8
-750
0
CPT
N
I/Y
PV
PMT
FV
= 794.56 – 750 = 44.56 ≈ 45
*Calculator solutions match the TI Business Analyst II+. Keystroke
adjustments may need to be made for other financial calculators
App 1-17
Interest Rate Basics
Example B – Calculator
Calculator Solution





.75
8
-750
0
CPT
N
I/Y
PV **
PMT
FV
= 794.56 – 750 = 44.56 ≈ 45
** Remember: when using a financial calculator, either
PV or FV must be negative.
• Outflows (from you) are negative
• Inflows (to you) are positive
• Depositing money in an account is an outflow
App 1-18
Future Value of a Single Amount
• Amount to which current savings will increase
• = Original amount + compounded interest
• = Compounding
• Formula Solution:
FV  PV (1  i )n
• Table Solution:
FV  PV (Table Factor)
• Calculator Solution:
N I/Y PV PMT CPT FV
• Excel Function:
FV(Rate,Nper,Pmt,PV )
App 1-19
Future Value of a Single Amount
Formula & TVM Table Solutions
Example C
• Suppose you invest $1 for 3 years at 10%
• How much would you have?
Formula Solution:
FV
TVM Tables Solution:
=PV(1+i)n
Exhibit 1-A
=1(1.10)3
Periods = 3
=1(1.331)
Rate = 10%
=1.331
Factor = 1.331
FV = PV(Factor)
FV = 1(1.331)
FV = 1.331
App 1-20
Future Value of a Single Amount
Calculator Solution
Example C
• Suppose you invest $1 for 3 years at 10%.
How much would you have?
Calculator Solution
 3
N
 10
I/Y
 -1
PV
 0
PMT
 CPT
FV
= 1.331
Excel Function:
=FV(.10,3,0,-1)
=1.331
App 1-21
Future Value of a Single Amount
Formula & TVM Tables
Example D
• Your savings of $400 earns 12% compounded
monthly (=1% per month)
• How much would you have after 18 months?
• Table Hint: Use 1% and 18 periods
Formula Solution:
FV=PV(1+i)n
TVM Tables Solution:
Exhibit 1-A
Periods = 18
=400(1.01)18
Rate = 1%
=400(1.196)
Factor = 1.196
=478.46
FV = 400(1.196)
FV = 478.40
App 1-22
Future Value of a Single Amount
Calculator Solution
Example D
• Suppose you invest $400 for 18 months at
12% compounded monthly. How much
would you have?
Calculator Solution
 18
N
 1
I/Y
 -400
PV
 0
PMT
 CPT
FV
= 478.46
Excel Function:
=FV(.01,18,0,-400)
=478.46
App 1-23
Future Value of a Series of Equal
Amounts
• “Annuity” = series of equal deposits at equal intervals
earning a constant rate
– Equal annuity deposit amounts = PMT
• Formula Solution:
 ( 1  i )n  1

FVA  PMT 
i


• Table Solution:
FVA  Annuity (PMT )  Table Factor
• Calculator Solution:
N I/Y PV PMT CPT FV
• Excel Function:
FV(Rate,Nper,Pmt,PV)
App 1-24
Future Value of a Series of Equal Amounts
Formula & TVM Tables
Example E
• What is the future value of three $1 deposits
made at the end of the next three years, earning
10% interest?
Formula Solution:
 (1  i )n  1 

FVA  PMT 
i


TVM Tables Solution:
Exhibit 1-B
Periods = 3
 ( 1.10 ) 3  1 

 1 
.10


 1 (3.31)
Rate = 10%
 3.31
FV = 3.31
Factor = 3.31
FV = 1(3.31)
App 1-25
Future Value of a Series of Equal Amounts
Calculator Solution
Example E
Calculator Solution
 3
N
 10
I/Y
 0
PV
 -1
PMT*
 CPT
FV
= 3.31
Excel Function:
=FV(.10,3,-1,0)
=3.31
* Note that the PMT value is negative since it is an
outflow/deposit.
App 1-26
Future Value of a Series of Equal Amounts
Formula & TVM Tables
Example F
• What is the future value of ten $40
deposits earning 8% compounded
annually?
Formula Solution:
 (1  i )  1 

FVA  PMT 
i


n
TVM Tables Solution:
Exhibit 1-B
Periods = 10
 ( 1.08 )10  1 

 40  
.08


 40  (14.487)
Rate = 8%
 579.46
FV = 579.48
Factor = 14.487
FV = 40(14.487)
App 1-27
Future Value of a Series of Equal Amounts
Calculator Solution
Example F
Calculator Solution
 10
N
 8
I/Y
 0
PV
 -40
PMT
 CPT
FV
= 579.46
Excel Function:
=FV(.08,10,-40,0)
=579.46
App 1-28
Present Value
Single Amount - Basic Equation
FV = PV(1 + i)n
• Rearrange to solve for PV
FV
PV 
( 1  i )n
PV  FV ( 1  i )
n
• “Discounting” = finding the present value
of one or more future amounts
App 1-29
Present Value of a Single Amount
• Formula Solution:
• Table Solution:
PV  FV (1  i )n
FV

(1  i)n
PV  FV  (Table Factor)
• Calculator Solution: N I/Y PMT FV CPT PV
• Excel Function:
=PV(Rate,Nper,Pmt,FV)
App 1-30
Present Value of a Single Amount
Formula & TVM Tables Example
Example G
• What is the present value of $1 to be received
in 3 years at a 10% interest rate?
TVM Tables Solution:
Formula Solution:
PV
Exhibit 1-C
=FV/(1+i)n
Periods = 3
=1/(1.10)3
Rate = 10%
=1*(.7513)
Factor = .751
=0.7513
PV = FV*(Factor)
PV = 1*(0.751)
PV = 0.751
App 1-31
Present Value of a Single Amount
Example G
Formula Solution:
PV
=FV/(1+i)n
=1/(1.10)3
=1*(0.7513)
=0.7513
TVM Tables Solution:
Exhibit 1-C
Periods = 3 (down left
column)
Rate = 10% (across top)
Factor = .751
Calculator Solution





3
10
CPT
0
1
N
I/Y
PV
= -.7513
PMT
FV
PV = FV(Factor)
PV = 1(0.751)
PV = 0.751
Excel Function:
=PV(.10,3,0,1)
= -0.75
App 1-32
Present Value of a Single Amount
Example H
You want to have $300 seven years from now. Your
savings earns 10% compounded semiannually. How
much must you deposit today?
Formula Solution:
PV
=FV/(1+i)n
=300/(1.05)14
=300/(1.9799)
=151.52
Calculator Solution
 14
N
 5
I/Y
 CPT
PV = -151.52
 0
PMT
 300
FV
TVM Tables Solution:
Exhibit 1-C
Periods = 14 (down left
column)
Rate = 5% (across top)
Factor = .505
PV = FV(Factor)
PV = 300 x (0.505)
PV = $151.50
Excel Function:
=PV(.05,14,0,300) = -151.52
App 1-33
Present Value of a Series of Equal
Amounts
• Annuity
• Table Factors = Exhibit 1-D
• Formula Solution:
1
1
(1  i )n
PV  Annuity 
i
• Table Solution: PV  Annuity  (Table Factor)
• Calculator Solution: N I/Y PMT FV CPT PV
• Excel Function:
=PV(Rate,Nper,Pmt,FV)
App 1-34
Present Value of an Annuity
Example I
• You wish to withdraw $1 at the end of each of the next
3 years. (= an Inflow)
• The account earns 10% compounded annually.
• How much do you need to deposit today to be able to
make these withdrawals?
1

1


(1.10)3
PV  1 
.10




  $2.49


Exhibit 1-D: Row 3, column 10%
Factor = 2.487
PV = PMT*(Factor) = 1*(2.487)
PV = $2.49
3 N;
10 I/Y;
1 PMT;
CPT PV = -2.48685
FV 0
Excel Function:
=PV(.10,3,1,0) = -2.49
App 1-35
Present Value of an Annuity
Example J
• You wish to withdraw $100 at the end of each of the next
10 years. (Inflow)
• The account earns 14% compounded annually.
• How much do you need to deposit today to be able to
make these withdrawals?
1

1


(1.14)10
PV  1 
.14




  $521.61


Exhibit 1-D:
Factor = 5.216
PV = PMT*(Factor) = 100*(5.216)
PV = $521.60
10 N;
14 I/Y;
100 PMT;
CPT PV = -521.61
FV 0
Excel Function:
=PV(.14,10,100,0) = -521.61
App 1-36
Using Present Value to Determine
Loan Payments
Example K
If you borrow $1,000 with a 6% interest rate to be
repaid in three equal payments at the end of the next
three years, what will the annual payment be?
Amount Borrowed
PMT 
• Table Solution:
PVA Table Factor
$1,000
PMT 
 $374.11
2.673
Calculator Solution:
3 N;
6 I/Y;
CPT PMT = -374.10981
PV = 1000
FV 0
Excel Function:
=PMT(.06,3,1000,0) = -374.11
App 1-37