Time Value of Money
Introduction
TVM Preferences
• More vs. Less
• Sooner vs. Later
• More Now vs. Less Later
• Less Now vs. More Later ????
TVM Questions
• What will my investment grow to?
• How much do I need today?
• How fast must my investment grow?
• How long will it take?
Compare and Contrast
1970
2011
TVM
Cost of a first-class stamp:
$ 0.06
$
0.44
4.98%
Cost of a gallon of gas:
$ 0.36
$
2.98
5.29%
Cost of a dozen eggs:
$ 0.62
$
2.20
3.14%
Cost of a gallon of Milk:
$ 1.15
$
3.69
2.88%
TVM Basic Concepts
• Simple vs. Compound Interest
Simple Interest = interest earned only on
principal (amount loaned)
Compound Interest = interest earned on
principal and any unpaid interest earned
in an earlier time period
Simple Interest Calculation
Future Value  Principal x Interest Rate  x Periods 
 Principal
FV  PV * i * n   PV
Interest Example
• Principal
$1,000
• Interest Rate
10%
• Term
5 years
Interest Example
FV = (1,000 x .10 x 5) + 1,000
FV = 500 +1,000
FV = 1,500
Simple Interest Example
• Principal
• Total Interest
• Ending Balance
$1,000
500
$1,500
Compound Interest Calculation
 n

Future Value    Principal * Interest Rate
 Periods1

 n -1

   Interesti * Interest Rate
 Periods  1

 Principal
FV  PV * (1  i) n
Compound Interest Example
5
FV  1,000 x (1  .10)
5
FV  1000 x (1.10)
FV  1,000 x 1.611
FV  1,611
Compound Interest Example
• Principal
• Total Interest Rate
• Ending Balance
$1,000
611
$1,611
Time Value of Money
Calculator Tips
• Set Calculator to 4 decimal points
• Set P/Yr to 1 and do not change
• Clear calculator before calculation
• Use recommended format
• Learn to use special features
• Read carefully
• Know the concepts of TVM
TVM Concepts
• Use a time line
• Use + or - to indicate cash flow
• Periodic Cash flows can be at
Beginning or End of Period
• Calculators use Percentages
• Excel uses decimals
Lump Sum vs. Periodic Pmts
• Lump Sum
– Single Payment
– At time zero
– Present Value
OR
– Single Payment
– At end of time
– Future Value
• Periodic Payment
– Ordinary Annuity
• Pmt at end of periods
• For life of investment
– Annuity Due
• Pmt at beg. of periods
• For life of investment
– PMT
Annuities
• Must be
– Equal Amounts
– Occurring in every compounding
period
– Ordinary Annuity – End of Period
– Annuity Due – Beginning of Period
Annuity?
0
1
2
3
4
5
100
100
100
100
100
Annuity?
0
1
2
3
4
5
100
100
100
100
100
Annuity?
0
1
2
3
4
5
100
100
100
Annuity?
0
1
2
3
4
5
100
200
300
400
500
Lump Sum & Periodic Payment
• Combination
– Single Payment
– With periodic payments for life of
investment
– PV & PMT
Recommended Structure
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of Lump Sum
If you invest $1,000 in a savings
account earning 10% compounded
annually, how much will you have
after 5 years?
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(1,000.00)
10.00%
5
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
1,610.51
(1,000.00)
10.00%
5
Future Value of Lump Sum
If you invest $10,000 in a mutual
fund that is expected to earn a
12% compound after-tax return,
how much will you have at the
end of 50 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(10,000.00)
12.00%
50
Future Value
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
2,890,021.90
(10,000.00)
12.00%
50
Future Value of an Annuity
If you invest $10,000 at the end of
each year in a mutual fund that is
expected to earn a 12%
compound after-tax return, how
much will you have at the end of 5
years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(10,000.00)
12.00%
5
End
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
63,528.47
(10,000.00)
12.00%
5
End
Future Value of an Annuity
If you invest $10,000 at the beginning
of each year in a mutual fund that is
expected to earn a 12% compound
after-tax return, how much will you
have at the end of 5 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(10,000.00)
12.00%
5
Beg
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
71,151.89
(10,000.00)
12.00%
5
Beg
Ordinary Annuity
Time
Payment
Return
FV
0
1
10,000
12% / 4 yrs 15,735.19
2
10,000
12% / 3 yrs 14,049.28
3
10,000
12% / 2 yrs 12,544.00
4
10,000
12% / 1 yr
5
10,000
12% / 0 yrs 10,000.00
Total
11,200.00
63,528.47
Annuity Due
Time
Payment
Return
FV
0
10,000
12% / 5 yrs 17623.42
1
10,000
12% / 4 yrs 15,735.19
2
10,000
12% / 3 yrs 14,049.28
3
10,000
12% / 2 yrs 12,544.00
4
10,000
12% / 1 yr
11,200.00
5
Total
71,151.89
Future Value of a Combination
If you invest $10,000 today and $1,000
at the end of each year in a mutual
fund that is expected to earn a 12%
compound after-tax return, how
much will you have at the end of 5
years?
Future Value of a Combination
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(10,000.00)
(1,000.00)
12.00%
5
End
Future Value
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
23,976.26
(10,000.00)
(1,000.00)
12.00%
5
End
Combination Investment
Time
Payment
Return
FV
0
10,000
12% / 5 yrs
17,623.42
1
1,000
12% / 4 yrs
1,573.52
2
1,000
12% / 3 yrs
1,404.93
3
1,000
12% / 2 yrs
1,254.40
4
1,000
12% / 1 yr
1,120.00
5
1,000
12% / 0 yrs
1,000.00
Total
23,976.26
Annual Rate of Return
TVM can also solve for the
rate of return required for a
PV to reach a FV in n years.
Annual Rate of Return
What rate of return is required
for $10,000 to grow to $16,000
in 5 years?
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
16,000.00
(10,000.00)
?
5
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
16,000.00
(10,000.00)
9.86%
5
Annual Rate of Return
If you invest $2,000 at the end of
each year for 5 years, what rate of
return must your investment earn
for you to have $16,000 at the end of
that period?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
16,000.00
(2,000.00)
?
5
End
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
16,000.00
(2,000.00)
23.69%
5
End
Annual Rate of Return
If you invest $10,000 today and $500
at the end of each year for the next 5
years, what rate of return must you
earn to have $16,000 at the end of
that period?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
16,000.00
(10,000.00)
(500.00)
?
5
End
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
16,000.00
(10,000.00)
(500.00)
5.71%
5
End
Number of Periods
TVM can also solve for the
holding period required for a
PV, a series of Payments or a
combination of PV and
Payments to reach a FV given
a specific rate of return
Number of Periods
How long will it take for a
$10,000 investment to grow to
$24,000 if it earns 11.25%
compounded annually?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
24,000.00
(10,000.00)
11.25%
?
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
24,000.00
(10,000.00)
8.21
Number of Periods
If you deposit $3,000 at the
beginning of each year in a
savings account earning 9.75%,
how long will it take for you to
save for a $20,000 down payment
for a house?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
20,000.00
(3,000.00)
9.75%
?
Beg
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
20,000.00
(3,000.00)
9.75%
5
Beg
Present Value
TVM can also solve for the price
you would pay for a FV, a series
of Payments, or a combination of
a series of Payments and a FV
given a specific rate of return and
holding period.
Present Value of a Future Amount
What would you pay for the right
to collect $8,000 in 7 years, if your
required return is 8.75%?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Present Value of a Future Amount
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
8,000.00
?
8.75%
7
Present Value of a Future Amount
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
8,000.00
(4,447.18)
8.75%
7
Stop
Present Value of Periodic Payments
What would you pay for the right
to collect $8,000 at the beginning
of each year for 7 years, if your
required return is 8.75%?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Present Value of Periodic Payment
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
8,000.00
8.75%
7
Beg
Present Value of Periodic Payment
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
(44,156.42)
8,000.00
8.75%
7
Beg
Present Value of a Combination
What would you pay for the right
to collect $800 at the end of each
year for 7 years and an additional
$10,000 at the end of the period, if
your required return is 7.25%?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Present Value of a Combination
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
10,000.00
?
800.00
7.25%
7
End
Present Value of a Combination
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
10,000.00
(10,400.70)
800.00
7.25%
7
End
Time Value of Money
Compounding Periods
Shorter than One Year
Compounding Periods
• Cash Flows are often more
frequent than annually
– Monthly, quarterly, semi-annually
• If Compound periods < annual
– Effective Interest Rate is higher
– FV is higher and PV is lower
Compound Interest Formula
with Compounding Periods less
than 1 Year

i 

FV  PV *  1  
m


n*m



Where m = the number of compounding periods
within the year.
Adjustments for Compounding
Periods < Annual
• Compounding Periods = m
• Divide Annual rate by m
i/m
• Multiply Years by m
nxm
• Input i/m for I/Y
• Input (n x m) for N
Future Value of Lump Sum
If you invest $6,000 in a savings
account earning 10% compounded
quarterly, how much will you have
after 5 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(6,000.00)
10.00%
5
4
2.50%
20
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
$9,831.70
(6,000.00)
10.00%
5
4
2.50%
20
Future Value of Lump Sum
If you invest $1,000 in a savings
account earning 10% compounded
daily, how much will you have
after 5 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(1,000.00)
10.00%
5
365
0.0274%
1,825
Future Value of Lump Sum
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
$1,648.61
(1,000.00)
10.00%
5
365
0.0274%
1,825
Future Value of an Annuity
If you invest $1,000 at the end of
each month in a mutual fund that
is expected to earn a 12% aftertax return, how much will you
have at the end of 5 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(1,000.00)
12.00%
5
End
12
1.00%
60
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
$81,669.67
(1,000.00)
12.00%
5
End
12
1.00%
60
Future Value of an Annuity
If you invest $1,000 at the beginning
of each month in a mutual fund that
is expected to earn a 12% after-tax
return, how much will you have at
the end of 5 years?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
?
(1,000.00)
12.00%
5
Beg
12
1.00%
60
Future Value of an Annuity
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
$82,486.37
(1,000.00)
12.00%
5
Beg
12
1.00%
60
Annual Rate of Return
If you invest $2,000 at the end of
each quarter for 5 years, what rate
of return must your investment earn
for you to have $60,000 at the end of
that period?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(2,000.00)
?
5
End
4
?
20
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(2,000.00)
?
5
End
4
4.07%
20
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(2,000.00)
16.29%
5
End
4
4.07%
20
Annual Rate of Return
If you invest $10,000 today and $500
at the end of each month for the
next 5 years, what rate of return
must you earn to have $60,000 at the
end of that period?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(10,000.00)
(500.00)
?
5
End
12
60
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(10,000.00)
(500.00)
?
5
End
12
1.04%
60
Annual Rate of Return
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
60,000.00
(10,000.00)
(500.00)
12.50%
5
End
12
1.04%
60
Number of Periods
If you deposit $300 at the
beginning of each month in a
savings account earning 9.75%,
how long will it take for you to
save for a $20,000 down payment
for a house?
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
20,000.00
(300.00)
9.75%
?
Beg
12
0.81%
?
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
20,000.00
(300.00)
9.75%
?
Beg
12
0.81%
53
Number of Periods
Future Value
Present Value
Payment
Annual Rate
Years
Beg / End
Compounding Periods
Rate / Period
Years * Periods
20,000.00
(300.00)
9.75%
4.43
Beg
12
0.81%
53
Uneven Cash Flows
• How do you calculate Present
Value when your required
return is 9.0% and you expect to
receive the following cash flows:
Year 1
2,000
Year 2
3,000
Year 5
1,000
Uneven Cash Flows
•
Alternative One – The Hard Way
1. Draw a Time Line
2. Calculate the PV of each cash flow
3. Total the Present Values
Uneven Cash Flows
FV
I/Y
N
PV
CF 1
2,000
9.00%
1
(1,834.86)
Uneven Cash Flows
FV
I/Y
N
PV
CF 1
CF 2
2,000
3,000
9.00%
9.00%
1
2
(1,834.86) (2,525.04)
Uneven Cash Flows
FV
I/Y
N
PV
Total PV
CF 1
2,000
9.00%
1
(1,834.86)
CF 2
3,000
9.00%
2
(2,525.04)
(5,009.83)
CF 3
1,000
9.00%
5
(649.93)
Uneven Cash Flows
•
Alternative Two – Use the CF Register
1. Draw Time Line
2. Input Cash Flows into CF Register
3. Go to NPV Register
1.
Input Rate of Return
2.
Compute NPV
Uneven Cash Flows Example 1 –
Alternative Two
1. Draw Time Line
2. Push CF button
3. Clear CF register
2nd CLR Work
4. Input Cash Flows
Cash Flow Register
• Inputs
– CF0 = Investment, Price, Cost at Time 0
We are solving for PV so CF0 should be 0
Since CF0 already = 0,

– C01 = Cash Flow at the end of Period 1
– F01 = Frequency of C01
The number of times that C01 occurred consecutively
Uneven Cash Flows Example 1
1. Draw Time Line
2. Clear the CF Register
3. Input Cash Flows
a.
b.
c.
d.
e.
CF0 = 0, 
C01 = 2,000; F01 = 1, 
C02 = 3,000; F02 =1, 
C03 = 0;
F03 = 2, 
C04 = 1,000; F04 = 1, 
Uneven Cash Flows Example 1
1. Check Inputs
2. Go To NPV Register
3. Input I
9 ENTER, 
4. CPT
NPV = 5,009.83
Uneven Cash Flows Example 2
What would you be willing to pay for a real
estate investment that has the following
expected cash flows: Yr. 1 $500, Yrs. 2-6
$1,000, Yr. 7-10 $1,500, and Yr. 11 $30,000?
Assume your required return for this type of
investment is 17.0%.
Uneven Cash Flows Example 2
1. Draw Time Line
2. Input Cash Flows
a.
CF0 = 0
b. C01 = 500;
c.
F01 = 1
C02 = 1,000; F02 = 5
d. C03 = 1,500; F03 = 4
e.
C04 = 30,000; F04 = 1
Uneven Cash Flows Example 2
3. Check your Inputs
4. Go to “NPV” Register
1. Enter I = 17.0; 
2. Press “CPT”
NPV = ?
Uneven Cash Flows Example 2
NPV = 10,100.25
Uneven Cash Flows
• The CF Register can also be used to
find the rate of return associated
with uneven cash flows.
• This cannot be done easily any
other way.
Uneven Cash Flows
• Inputs
– CF Register Steps are the same
– Go to IRR Register
CPT IRR
IRR = the Internal Rate of Return
IRR = the rate of return on the investment
Effective Interest Rate
Calculation
Effective Interest Rate
Calculation
The annual rate of return
actually earned when
compounding or payment
periods are less than 1 year.
Effective Interest Rate
• Nominal rate = i
– The nominal rate is the rate
“named” in the information.
– “The credit card rate is for 18.0%
compounded monthly.”
• 18.0% is the nominal rate
EIR Calculations
What is the Effective Interest Rate
for a credit card with an 18%
nominal interest rate if the card is
not paid off each month?
Effective Interest Rate with
Compounding Periods < 1 Year
m


i 
Eff. Int. Rate   1    1 * 100
m


Where m = the number of compounding periods
within the year.
EIR Calculations
12


.18 
EIR   1 
  1 * 100
12 


EIR Calculations


EIR  1.015  1 * 100
12
EIR Calculations
EIR  1.1956  1* 100
EIR Calculations
EIR  19.56%
EIR CALCULATIONS
Use “I Conv” Register for
easy Effective Interest Rate
calculations.
I Conv Register
•
Steps
2nd I Conv
Input Nominal Rate, ENTER
Arrow Down Twice
Input C/Y
• (Compounding Periods per Year)
5. Arrow Up
6. CPT EFF (Effective Interest Rate)
1.
2.
3.
4.