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Chapter 1
AppendixTime Value of
Money:The Basics
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
• 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?”
Time Value of Money
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 FV = PV(1 + i)n
Annuity
Present Value
Single Amount
Annuity
i
iPMT n 1)1(
nn iFVPV
i
FVPV
)1(
)1(
ii
PMTn)1(
11
App 1-10
TVM Calculator SolutionsTexas Instruments BA-II Plus
• FV = future value
• PV = present value
• PMT = periodic payment
• I/Y = interest rate
• N = number of periods
One of these MUST be negative
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
• Calculating interest earned:– Principal = dollar amount of savings– Annual rate of interest – Length of time money on deposit (in years)
• Simple interest:
Time Value of MoneyInterest Rate Basics
Amt in Svgs X
Annual Interest
Rate
Time Period InterestX =
App 1-14
You borrow $1,000 at 5% annual interest for 1 year:
Principal = $1,000
Interest rate = 5% = .05
Time period = 1
Interest Rate BasicsExample A
$50$1,000 X .05 1X =
App 1-15
You deposit $750 at 8% per year for 9 months:
Principal = $750Interest rate = 8%Time period = 9/12 = .75
Interest Rate BasicsExample B
$45$750 X .08 0.75X =
App 1-16
Principal = $750
Interest rate = 8%
Time period = 9/12 = .75
Interest Rate BasicsExample B – Calculator*
Calculator Solution Keystrokes .75 N 8 I/Y -750 PV 0 PMT CPT 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 BasicsExample B – Calculator
Calculator Solution .75 N 8 I/Y -750 PV ** 0 PMT CPT 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:
• Table Solution:
• Calculator Solution: N I/Y PV PMT CPT FV
• Excel Function: FV(Rate,Nper,Pmt,PV )
niPVFV )1(
Factor) Table(PVFV
App 1-19
Future Value of a Single AmountFormula & TVM Table Solutions
Example C
• Suppose you invest $1 for 3 years at 10%• How much would you have?
Formula Solution:
FV =PV(1+i)n
=1(1.10)3
=1(1.331)
=1.331
TVM Tables Solution:
Exhibit 1-A
Periods = 3
Rate = 10%
Factor = 1.331
FV = PV(Factor)
FV = 1(1.331)
FV = 1.331App 1-20
Future Value of a Single Amount
Calculator SolutionExample 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
App 1-21
Excel Function:=FV(.10,3,0,-1)=1.331
Future Value of a Single AmountFormula & 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
=400(1.01)18
=400(1.196)
=478.46
TVM Tables Solution:
Exhibit 1-A
Periods = 18
Rate = 1%
Factor = 1.196
FV = 400(1.196)
FV = 478.40
App 1-22
Future Value of a Single Amount
Calculator SolutionExample 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
App 1-23
Excel Function:=FV(.01,18,0,-400)=478.46
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:
• Table Solution:
• Calculator Solution: N I/Y PV PMT CPT FV
• Excel Function: FV(Rate,Nper,Pmt,PV)
i
)i(PMTFVA
n 11
Factor Table)( PMTAnnuityFVA
App 1-24
Future Value of a Series of Equal Amounts
Formula & TVM TablesExample E
• What is the future value of three $1 deposits made at the end of the next three years, earning 10% interest?
Formula Solution:
TVM Tables Solution:
Exhibit 1-B
Periods = 3
Rate = 10%
Factor = 3.31
FV = 1(3.31)
FV = 3.313.31
(3.31)1
10
11011
1)1(
3
.
).(
i
iPMTFVA
n
App 1-25
Future Value of a Series of Equal Amounts
Calculator SolutionExample E
Calculator Solution 3 N 10 I/Y 0 PV -1 PMT* CPT FV = 3.31
* Note that the PMT value is negative since it is an outflow/deposit.
App 1-26
Excel Function:=FV(.10,3,-1,0)=3.31
Future Value of a Series of Equal AmountsFormula & TVM Tables
Example F
• What is the future value of ten $40 deposits earning 8% compounded annually?
Formula Solution:
TVM Tables Solution:
Exhibit 1-B
Periods = 10
Rate = 8%
Factor = 14.487
FV = 40(14.487)
FV = 579.48579.46
(14.487)40
08
108140
1)1(
10
.
).(
i
iPMTFVA
n
App 1-27
Future Value of a Series of Equal Amounts
Calculator SolutionExample F
Calculator Solution 10 N 8 I/Y 0 PV -40 PMT CPT FV = 579.46
App 1-28
Excel Function:=FV(.08,10,-40,0)=579.46
Present ValueSingle Amount - Basic Equation
FV = PV(1 + i)n
• Rearrange to solve for PV
• “Discounting” = finding the present value of one or more future amounts
n
n
)i(FVPV
)i(
FVPV
1
1
App 1-29
Present Value of a Single Amount
• Formula Solution:
• Table Solution:
• Calculator Solution: N I/Y PMT FV CPT PV
• Excel Function: =PV(Rate,Nper,Pmt,FV)
ni)(1
FV
)1(
niFVPV
Factor) Table(FVPV
App 1-30
Present Value of a Single AmountFormula & TVM Tables Example
Example G
• What is the present value of $1 to be received in 3 years at a 10% interest rate?
Formula Solution:
PV =FV/(1+i)n
=1/(1.10)3
=1*(.7513)
=0.7513
TVM Tables Solution:
Exhibit 1-C
Periods = 3
Rate = 10%
Factor = .751
PV = FV*(Factor)
PV = 1*(0.751)
PV = 0.751
App 1-31
Present Value of a Single Amount
Example GFormula 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
PV = FV(Factor)
PV = 1(0.751)
PV = 0.751
Calculator Solution 3 N 10 I/Y CPT PV = -.7513 0 PMT 1 FV
App 1-32
Excel Function:=PV(.10,3,0,1)= -0.75
Present Value of a Single Amount
Example H
Formula Solution: PV=FV/(1+i)n
=300/(1.05)14
=300/(1.9799)=151.52
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
Calculator Solution 14 N 5 I/Y CPT PV = -151.52 0 PMT 300 FV
You want to have $300 seven years from now. Your savings earns 10% compounded semiannually. How much must you deposit today?
App 1-33
Excel Function:=PV(.05,14,0,300) = -151.52
Present Value of a Series of Equal Amounts
• Formula Solution:
• Table Solution:
• Calculator Solution: N I/Y PMT FV CPT PV
• Excel Function: =PV(Rate,Nper,Pmt,FV)
i)i1(
11
AnnuityPVn
Factor) Table(AnnuityPV
• Annuity• Table Factors = Exhibit 1-D
App 1-34
Present Value of an AnnuityExample 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?
49.2$10.
)10.1(1
11
3
PV
3 N; 10 I/Y; 1 PMT; CPT PV = -2.48685FV 0
Exhibit 1-D: Row 3, column 10%
Factor = 2.487
PV = PMT*(Factor) = 1*(2.487)
PV = $2.49
App 1-35
Excel Function:=PV(.10,3,1,0) = -2.49
Present Value of an AnnuityExample 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?
61.521$14.
)14.1(1
11
10
PV
10 N; 14 I/Y; 100 PMT; CPT PV = -521.61FV 0
Exhibit 1-D:
Factor = 5.216
PV = PMT*(Factor) = 100*(5.216)
PV = $521.60App 1-36
Excel Function:=PV(.14,10,100,0) = -521.61
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?
• Table Solution:
11.374$2.673
$1,000Factor TablePVA
BorrowedAmount
PMT
PMT
Calculator Solution:3 N; 6 I/Y; CPT PMT = -374.10981 PV = 1000FV 0
App 1-37
Excel Function:=PMT(.06,3,1000,0) = -374.11
