Topic 9
Time Value of Money
Topic 9: Time Value of Money
• Learning Objectives
– Calculate present value and future value of single
amounts, annuities, annuities due, uneven cash
flows and serial payments.
– Calculate NPV and IRR and be able to apply the
techniques to financial planning problems.
Topic 9: Present Value
• The value today of a known amount or series
of amounts to be received in the future, given
a specified interest rate and number of
periods for discounting
• Formula for single sum
– PV = FV / (1 + i)n
Topic 9: Present Value Example
• What is the present value of $20,000 in 5
years assuming an 8% rate of return?
– It will be worth $13,612
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•
•
•
•
N=5
I=8
PMT = 0
FV = 20,000
Solve for PV
Topic 9: Future Value
• The value that will be available when a
present sum or series of periodic sums is
invested, with earnings added, most often on
a compound basis
• Formula for a single sum
– FV = PV(1 + i)n
Topic 9: Future Value Example
• What is the future value of $5,000 invested
today in a 4% CD for 3 years?
– It will be worth $5,624
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•
•
•
•
N=3
I=4
PV = 5,000
PMT = 0
Solve for FV
Topic 9: Rule of 72
• A tool for estimating the approximate length
of time it will take for a single sum of money
to double in value at a given compound
annual rate of interest
• Divide 72 by the interest rate to produce the
answer
– Example: At a 6% rate of interest it will take
approximately 12 years (72/6 = 12) for a lump sum
of money to double in value
Topic 9: Annuity
• A specified amount of money, paid or received
at a specified uniform interval, for a specified
period of time
– Ordinary annuity
• Payments or receipts are made at the end of each
period
– Annuity due
• Payments or receipts are made at the beginning of each
period
Topic 9: Annuity Example
• How much should a client pay for an
investment that will pay him $1,000 a year for
10 years given a 6% required rate of return?
– The client should pay $7,360
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•
N = 10
I=6
PMT = 1,000
FV = 0
Solve for PV
Topic 9: Net Present Value
• The present value of all cash outflows and
cash inflows
T
CFt
NPV  
 CF0
t
t 1 (1  r )
• It is how much a client should pay for an asset
Topic 9: Net Present Value Example
• How much should a client be willing to pay for an
investment with the following cash flows assuming a
7% interest rate?
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–
–
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End of Year 1 = $1,000
End of Year 2 = $2,000
End of Year 3 = ($5,000)
End of Year 4 = $10,000
• The client should be willing to pay $6,229
• Input the cash flows above starting with $0 as the first
cash flow
• Input 7% as the interest rate
• Solve for NPV
Topic 9: Internal Rate of Return (IRR)
• It is the rate of return on an investment given
the cash inflows and outflows
T
CFt
NPV  0  
 CF0
t
t 1 (1  IRR )
• The client wants the IRR to be equal to or
greater than the client’s required rate of
return
Topic 9: Internal Rate of Return
Example
• If a client pays $6,000 for the investment used in the
NPV example, what will be the client’s IRR?
• The client’s IRR will be 8.1%
– Input the following cash flows
•
•
•
•
•
($6,000)
$1,000
$2,000
($5,000)
$10,000
• Then solve for IRR which is 8.1%
• The IRR will be higher than the 7% since the client paid
less than the PV of the future cash flows (NPV>0)
Topic 9: Uneven Cash Flows
• Typically you will be solving for how much a
client should pay for the investment or how
much they will have saved at the end of a
period of time given the different cash flows
from an investment
Topic 9: Uneven Cash Flows Example
• How much should Peter pay for a bond with a 9%
coupon that matures in 7 years if comparable bonds
are yielding 10%?
• Peter should pay $951
– N = 7 x 2 = 14
• Due to semi annual coupon payments
– I = 10 / 2 = 5
• Semi annual interest rate
– PMT = 90 / 2 = 45
• Semi annual coupon payment
– FV = 1000
– Solve for PV
Topic 9: Serial Payments
• The calculation will involve an inflation adjusted
rate of return
– [(1 + nominal rate)/(1 + inflation rate) – 1] x 100
• Example
– 9% yield in a year with 2.5% inflation
– [(1 + 9%) / (1 + 2.5%) –1] x 100
– 6.34%
Topic 9: Annuity Due Serial Payment
Example
• Susan wishes to accumulate $50,000 in today’s dollars in 4
annual deposits, beginning immediately. The deposits will
earn 7% after taxes, and the deposits must also increase
each year by 3% to keep up with anticipated inflation. What
should be the size of the first deposit?
• Susan should deposit $11,354
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Set for beginning-of-period payments
N=4
I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88
PV = 0
FV = 50,000
Solve for PMT
Topic 9: Ordinary Annuity Serial
Payment Example
• Susan wishes to accumulate $50,000 in today’s dollars in 4
annual deposits. The deposits will earn 7% after taxes, and
the deposits must also increase each year by 3% to keep up
with anticipated inflation. What should be the size of the
first deposit?
• Susan should deposit $12,149
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Set for end-of-period payments
N=4
I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88
PV = 0
FV = 50,000
Solve for PMT which is 11,795
Add inflation that occurred between today and when the first
payment is made by multiplying 11,795 x 1.03 = 12,149
Topic 9: Growing Annuity Formulas
  1  g t 
 
1  
1 r  


PV  C 
 rg 




 1  r t  1  g t 
FV  C  

r

g


Topic 9: Loan Amortization Example
• Ella took out a $500,000 loan fixed at 6% interest for 30
years? How much is her monthly payment?
– N = 30 x 12 = 360, I = 6/12 = 0.50, PV = 500,000, FV = 0,
Solve for PMT which is $2,998
• How much interest and principal would Ella have paid
after 5 years of monthly payments (60 payments) and
what is her remaining loan balance?
• Without clearing your calculator from the problem
above:
– 2nd, AMORT, 1, set, ↓, 60, set, ↓
• $145,137 is the total amount of interest paid
• $34,728 is the total amount of principal paid
• $465,272 is the remaining loan balance
End of Topic 9