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1
Chapter 4: Time Value of Money
Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.
ObjectiveExplain the concept of compounding
and discounting and to provide examples of real life
applications
5
Compounding: Future Value of a Lump Sum
niPVFV )1(* FV with growths from -6% to +6%
0
500
1,000
1,500
2,000
2,500
3,000
3,500
0 2 4 6 8 10 12 14 16 18 20
Years
Fu
ture
Va
lue
of
$1
00
0
6%
4%
2%
0%
-2%
-4%-6%
10
The Frequency of Compounding
Annual Percentage Rate (APR) Effective Annual Rate (EFF): The
equivalent interest rate, if compounding were only once a year.
m
mAPR
EFF )1(1
11
Effective Annual Rates of an APR of 18%
Annual Percentage Rate
Frequency of Compounding
Annual Effective Rate
18 1 18.00 18 2 18.81 18 4 19.25 18 12 19.56 18 52 19.68 18 365 19.72
12
The Frequency of Compounding
Note that as the frequency of compounding increases, so does the annual effective rate
What occurs as the frequency of compounding rises to infinity?
111
APRm
me
mAPR
LimEFF
13
The Frequency of Compounding
11*
11
11
1
1
m
m
EFFmAPR
EFFm
APR
m
APREFF
m
14
The Frequency of Compounding
Annual Effective Rate
Compounding Frequency
Annual Percentage Rate
12 1 12.00 12 2 11.66 12 4 11.49 12 12 11.39 12 52 11.35 12 365 11.33 12 Infinity 11.33
15
Annuities
A level stream of Cash Flows or Payments
Immediate Annuity: The Cash Flows start immediately.
Ordinary Annuity: The Cash Flows start at the end of the current period.
16
Derivation of PV of Ordinary Annuity Formula
nn i
pmt
i
pmt
i
pmt
i
pmt
i
pmtPV
111
11
13
21
17
Derivation of PV of Ordinary Annuity Formula
}
1
1
1
1
1
1
1
1
1
1{*
13
21
nn iii
iipmtPV
18
PV of Ordinary Annuity Formula
n
n
iipmt
ii
pmt
PV
1
11*
}1
11{*
19
Annuity Formula: PV of Immediate Annuity
}1)1{(*
)1(*}11{*
)1(*
1 n
n
ordimm
iiipmt
iiipmt
iPVPV
20
Derivation of FV of Annuity Formula
11*
1*1
11*FV
sum) (lump 1*FV
annuity) (ord. 1
11*
n
n
n
n
n
iipmt
iii
pmt
iPV
iipmt
PV
21
Perpetual Annuities / Perpetuities
Recall the annuity formula:
niipmt
PV1
11*
• Let n -> infinity with i > 0:
ipmt
PV
22
Alternative Discounted Cash Flow Decision Rules
1. NPV rule: the NPV is the difference between the present value of all future cash inflows minus the present value of all current and future cash outflows. Accept a project if its NPV is positive.
23
DCF rules
Example: You have the opportunity to buy a piece of land for $10,000. You are sure that 5 years from now it will be worth $20,000. If you can earn 8% per year by investing your money in bank, is this investment in the land worthwhile?
24
NPV rule solution
03612$
612,13$000,10$08.1
000,20$000,10$
5
NPV
25
Alternative Discounted Cash Flow Decision Rules
2. FV rule: Invest if the future value of the investment is larger than the future value that can be obtained from the next best alternative.
26
FV rule solution
000,20$693,14$
08.1000,10$ 5
FV
27
Alternative Discounted Cash Flow Decision Rules
3. IRR rule: The IRR is the discount rate at which the NPV is zero. Invest if the IRR is greater than the opportunity cost of capital.
28
IRR rule solution
%8%87.14
)1(
000,20$000,10$
5
i
i
29
Alternative Discounted Cash Flow Decision Rules
4. Choose the investment alternative with fastest payback.
30
Payback rule solution
5908.1
000,20$000,10$
n
n
31
Loan Amortization
The process of paying a loan principal gradually over its term
Example: $100,000 mortgage loan, APR: 9%,
repaid in 3 annual installments pmt=? pmt=$39504.48
nii
pmtPV
1
11*
309.1
11*
09.0100000
pmt
32
Loan Amortization
First Year: Interest: (0.09)(100000)=9000 pmt: 39504.48 principal: 30504.48 Outstanding Balance: 69494.52
33
Loan Amortization
Second Year: Interest: (0.09)(69494.52)=6254.51 pmt: 39504.48 principal: 33250.97 Outstanding Balance: 36243.54
34
Loan Amortization
Third Year: Interest:(0.09)(36243.54)=3262 pmt: 39504.48 principal: 36244 Outstanding Balance: 0
35
Amortization of Principal
0.00
50000.00
100000.00
150000.00
200000.00
250000.00
300000.00
350000.00
400000.00
450000.00
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360
Months
Ou
tsta
nd
ing
Bal
ance
36
Percent of Interest and Principal
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360
Months
Per
cen
t
% Interest
% Principal
41
Computing NPV in Different Currencies
In any TVM calculation, the cash flows and the interest rate must be
denominated in the same currency.
42
Inflation and Future Values
Example: At age 20 you save $100 and invest it at a dollar interest rate of 8% per year, and you do not take it until age 65. If the inflation is estimated 5% per year, how much will you have accumulated in the account at that time in terms of real purchasing power?
43
Solution 1:
355$
02857.1100$
)1(
02857.11
)(1
)(.1)(1
45
FVreal
RPVFVreal
R
iInflation
rratenomRratereal
n
44
Solution 2:
355$985.8
192,3$.
985.8
05.145
192,3$
08.1100$45.
45
45
FPL
FVnomFVreal
yearsinlevelprice
yearsinFVnom
45
Inflation and Present Values
Example: Your daughter is 10 years old, and you are planning to open an account to provide for her college education. Tuition for a year of college is now $15,000. How much must you invest now in order to have enough to pay for her first year’s tuition 8 years from now, if you think you can earn a rate of interest that is 3% more than the inflation rate of 5%?
46
Solution:
Wrong:
841.11$03.1
000,15$8
PV
104.8$08.1
000,15$8
PV
47
Inflation and Present Values
Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.