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Introduction
In fact, of all the concepts used in finance, none is more important than the time value of money, which is also called discounted cash flow (DCF) analysis.
PV : present value, or beginning amount, in your account
i : interest rate
INT : dollars of interest you earn
FV : future value
n : number of periods involved in the analysis
Time lines show timing of cash flows.
CF0 CF1 CF3CF2
0 1 2 3i%
Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of
Years 1 through 3.
100 50 75
0 1 2 3i%
-50
What’s the FV of an initial $100 after 3 years if i = 10%?
FV = ?
0 1 2 310%
Finding FVs (moving to the righton a time line) is called compounding.
100
After 1 year:
FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)= $110.00.
After 2 years:
FV2 = PV(1 + i)2
= $100(1.10)2
= $121.00.
10%
What’s the PV of $100 due in 3 years if i = 10%?
Finding PVs is discounting, and it’s the reverse of compounding.
100
0 1 2 3
PV = ?
Solve FVn = PV(1 + i )n for PV:
PV =
FV
1+ i = FV
11+ i
nn n
n
PV = $100
11.10
= $100 0.7513 = $75.13.
3
Finding the Time to Double
20%
2
0 1 2 ?
-1 FV = PV(1 + i)n
$2 = $1(1 + 0.20)n
(1.2)n = $2/$1 = 2nLN(1.2) = LN(2) n = LN(2)/LN(1.2) n = 0.693/0.182 = 3.8.
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
Ordinary Annuity
PMT PMTPMT
0 1 2 3i%
PMT PMT
0 1 2 3i%
PMT
Annuity Due
What’s the difference between an ordinary annuity and an annuity due?
PV FV
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
FV Annuity Formula
The future value of an annuity with n periods and an interest rate of i can be found with the following formula:
.33110.
100
0.10
1)0(1
i
1i)(1PMT
3
n
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
PV Annuity Formula
The present value of an annuity with n periods and an interest rate of i can be found with the following formula:
69.24810.
100
0.10)0(1
11-
ii)(1
11-
PMT
3
n
Special Function for Annuities
For ordinary annuities, this formula in cell A3 gives 248.96:
=PV(10%,3,-100)
A similar function gives the future value of 331.00:
=FV(10%,3,-100)
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
PV and FV of Annuity Due vs. Ordinary Annuity
PV of annuity due:
= (PV of ordinary annuity) (1+i)
= (248.69) (1+ 0.10) = 273.56
FV of annuity due:
= (FV of ordinary annuity) (1+i)
= (331.00) (1+ 0.10) = 364.1
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
annuity due
)i1(i
1)i1( PMT = due ANFV
n
)i1(i
i)+(11
1
PMT = due ANPVn
Excel Function for Annuities Due
Change the formula to:
=PV(10%,3,-100,0,1)
The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:
=FV(10%,3,-100,0,1)
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
Uneven Cash Flow Streams
We will use Payment (PMT) for annuity situations where the cash flows are equal amounts, and we will use the term Cash flow (CF) to denote uneven cash flows.
What is the PV of this uneven cashflow stream?
0
100
1
300
2
300
310%
-50
4
90.91247.93225.39-34.15
530.08 = PV
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
How to find PV of this uneven cash
1- We could find the PV of each individual cash flow using the numerical.
2- using NPV in excel .
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Copyright © 2002 by Harcourt, Inc. All rights reserved.
Spreadsheet Solution
Excel Formula in cell A3:
=NPV(10%,B2:E2)
A B C D E
1 0 1 2 3 4
2 100 300 300 -50
3 530.09
HOME WORK
Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1;
a. $400 per year for 10 years at 10 percent. b. $200 per year for 5 years at 5 percent. c. $400 per year for 5 years at 0 percent. d. Now rework parts a, b, and c assuming that payments are
made at the beginning of each year; that is, they are annuities due.
HOME WORK
Find the present value of the following ordinary annuities: a. $400 per year for 10 years at 10 percent. b. $200 per year for 5 years at 5 percent. c. $400 per year for 5 years at 0 percent. d. Now rework parts a, b, and c assuming that payments are
made at the beginning of each year; that is, they are annuities due.
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