1 chapter 4: time value of money copyright, 2000 prentice hall ©author nick bagley, bdellasoft,...
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Chapter 4: Time Value of Chapter 4: Time Value of MoneyMoney
Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.
ObjectiveExplain the concept of compounding
and discounting and to provide examples of real life
applications
2
Value of Investing $1Value of Investing $1
1 Year $1.1
2 Years $1.21
3 Years $1.331
4 Years $1.4641
– Continuing in this manner you will find Continuing in this manner you will find that the following amounts will be that the following amounts will be earnt:earnt:
3
Value of $5 InvestedValue of $5 Invested
1 Year $5*(1+0.10) $5.5
2 years $5.5*(1+0.10) $6.05
3 years $6.05*(1+0.10) $6.655
4 Years $6.655*(1+0.10) $7.3205
• More generally, with an investment More generally, with an investment of $5 at 10% we obtainof $5 at 10% we obtain
4
Future Value of a Lump Future Value of a Lump SumSum
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%
5
Example: Future Value of Example: Future Value of a Lump Suma Lump Sum• Your bank offers a
CD with an interest rate of 3% for a 5 year investments.
• You wish to invest $1,500 for 5 years, how much will your investment be worth?
1111145.1738$
)03.01(*1500$
)1(*5
niPVFV
n 5i 3%PV 1,500FV ?Result 1738.911111
6
Present Value of a Lump Present Value of a Lump SumSum
nn
n
n
iFVi
FVPV
i
iPVFV
)1(*)1(
:obtain to)1(by sidesboth Divide
)1(*
7
Example: Present Value of Example: Present Value of a Lump Suma Lump Sum
• You have been You have been offered $40,000 for offered $40,000 for your printing your printing business, payable business, payable in 2 years. Given in 2 years. Given the risk, you require the risk, you require a return of 8%. a return of 8%. What is the present What is the present value of the offer?value of the offer? today55.293,34$
55281.34293
)08.01(
000,40
)1(
2
ni
FVPV
8
Solving Lump Sum Cash Solving Lump Sum Cash Flow for Interest RateFlow for Interest Rate
1
)1(
)1(
)1(*
n
n
n
n
PVFV
i
PVFV
i
iPVFV
iPVFV
9
Example: Interest Rate on Example: Interest Rate on a Lump Sum Investmenta Lump Sum Investment
• If you invest If you invest $15,000 for ten $15,000 for ten years, you receive years, you receive $30,000. What is $30,000. What is your annual your annual return?return?
point) basisnearest the(to %18.7
071773463.0
121211500030000
1
101
1010
n
PVFV
i
10
Review of LogarithmsReview of Logarithms
• The basic properties of logarithms that The basic properties of logarithms that are used by finance are:are used by finance are:
)ln()ln(
)ln()ln()*ln(
)ln(
0,)ln(
xyx
yxyx
xe
xxe
y
x
x
11
Review of LogarithmsReview of Logarithms
• The following properties are easy to prove The following properties are easy to prove from the last ones, and are useful in financefrom the last ones, and are useful in finance
)ln(*)ln()ln(
)ln()ln()ln()**ln(
)ln()ln()/ln(
yxyx
zyxzyx
yxyx
12
Solving Lump Sum Cash Solving Lump Sum Cash Flow for Number of Flow for Number of PeriodsPeriods
iPVFV
iPVFV
n
iniPV
FV
iPV
FV
iPVFV
n
n
n
1ln
lnln
1ln
ln
1ln*)1(lnln
)1(
)1(*
13
Effective Annual Rates of Effective Annual Rates of an APR of 18%an APR of 18%
AnnualPercentagerate
Frequency ofCompounding
AnnualEffective 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
14
The Frequency of The Frequency of CompoundingCompounding• Note that as the frequency of Note that as the frequency of
compounding increases, so does the compounding increases, so does the annual effective rateannual effective rate
• What occurs as the frequency of What occurs as the frequency of compounding rises to infinity?compounding rises to infinity?
111
km
m
me
m
kLimEFF
15
The Frequency of The Frequency of CompoundingCompounding
11*
11
11
1
1
m
m
EFFmk
EFFm
k
m
kEFF
m
m
m
m
16
The Frequency of The Frequency of CompoundingCompounding
AnnualEffective Rate
CompoundingFrequency
AnnualPercentageRate
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
17
Derivation of PV of Derivation of PV of Annuity Formula: Algebra. Annuity Formula: Algebra. 1 of 51 of 5
nn i
pmt
i
pmt
i
pmt
i
pmt
i
pmtPV
111
11
13
21
18
Derivation of PV of Derivation of PV of Annuity Formula: Algebra. Annuity Formula: Algebra. 2 of 52 of 5
}
1
1
1
1
1
1
1
1
1
1{*
13
21
nn iii
iipmtPV
19
Derivation of PV of Derivation of PV of Annuity Formula: Algebra. Annuity Formula: Algebra. 3 of 53 of 5
}
1
1
1
1
1
1
1
1
1
1{*)1(*)1(*
13
21
nn iii
iiipmtiPV
20
Derivation of PV of Derivation of PV of Annuity Formula: Algebra. Annuity Formula: Algebra. 4 of 54 of 5
nnnn
nnnn
ipmt
iiii
ipmt
ipmt
iiiii
iipmtiPV
1
1}
1
1
1
1
1
1
1
1
1
1{*
1
1*
]}1
1
1
1[
1
1
1
1
1
1
1
1
1
1{*)1(*
122
10
122
10
21
Derivation of PV of Derivation of PV of Annuity Formula: Algebra. Annuity Formula: Algebra. 5 of 55 of 5
n
n
n
n
ii
pmt
ii
pmt
PV
ipmtpmtPViPV
ipmtPV
ipmtiPV
1
11*
}1
11{*
1
1)1(*
1
1
1
1*)1(* 0
23
PV Annuity Formula: PV Annuity Formula: PaymentPayment
n
n
n
i
iPVpmt
iipmt
iipmt
PV
11
*
11*
1
11*
24
PV Annuity Formula: PV Annuity Formula: Number of PaymentsNumber of Payments
ipmt
iPV
npmt
iPVi
pmtiPV
inpmt
iPVi
ipmt
iPVi
ipmt
PV
n
n
nn
1ln
*1ln
;*
11
*1ln1ln*;
*11
11*
;11*
25
Annuity Formula: PV Annuity Formula: PV Annuity DueAnnuity Due
}1)1{(*
)1(*}11{*
)1(*
1 n
n
regdue
iiipmt
iiipmt
iPVPV
26
Derivation of FV of Derivation of FV of Annuity Formula: AlgebraAnnuity Formula: Algebra
11*
1*1
11*FV
sum) (lump 1*FV
annuity) (reg. 1
11*
n
n
n
n
n
iipmt
iii
pmt
iPV
iipmt
PV
28
FV Annuity Formula: FV Annuity Formula: Number of PaymentsNumber of Payments
ipmt
iFV
n
pmtiFV
ini
ipmt
iFV
iipmt
FV
n
n
n
1ln
*1ln
*1ln1ln*1ln
1*
1
11*
29
Perpetual Annuities / Perpetual Annuities / PerpetuitiesPerpetuities
• Recall the annuity formula:Recall the annuity formula:
niipmt
PV1
11*
• Let n -> infinity with i > 0:
ipmt
PV
30
Mortgage: The paymentMortgage: The payment
• We will examine this problem using a We will examine this problem using a financial calculatorfinancial calculator
• The first quantity to determine is the The first quantity to determine is the amount of the loan and the pointsamount of the loan and the points
500,13$
03.0*)1.01(*500000$Points
000,450$
)1.01(*500000$Loan
31
Calculator SolutionCalculator Solution
n i PV FV PMT Result
360 0.5% 450,000 0 ? -2,697.98
This is the monthly repayment
32
Calculator SolutionCalculator Solution
n i PV FV PMT Result
360 .5% 450,000 0 ? -2,697.98
300 .5% ? 0 -2,697.98 418,745
Outstanding @ 60 Months
33
Summary of PaymentsSummary of Payments
• The family has made 60 payments = The family has made 60 payments = $2687.98*12*5 = $161,878.64$2687.98*12*5 = $161,878.64
• Their mortgage repayment =Their mortgage repayment = 450,000 - 418,744.61 = $31,255.39450,000 - 418,744.61 = $31,255.39
• Interest = payments - principle Interest = payments - principle reduction = 161,878.64 - 31,255.39 = reduction = 161,878.64 - 31,255.39 = $130,623.25$130,623.25
34
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
35
After Tax Cash Flow
$1,500
$1,700
$1,900
$2,100
$2,300
$2,500
$2,700
$2,900
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360
Monthly Cash Flow
Mo
nth
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
37
10% Aditional Payments
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360
Months
Pri
nci
pal
Ou
tsta
nd
ing
38
$10,000
$11,000 ¥
1,000,000¥
1,030,000¥
Time
10% $/$ (direct)
0.01 $/¥
3% ¥ / ¥
? $/¥
U.S.A.
Japan
39
$10,000
$11,124$11,000 ¥
1,000,000¥
1,030,000¥
Time
10% $/$ (direct)
0.01 $/¥
3% ¥/¥
0.0108 $/¥
U.S.A.
Japan
40
$10,000
$10,918 ¥$11,000 ¥
1,000,000¥
1,030,000¥
Time
10% $/$ (direct)
0.01 $/¥
3% ¥ / ¥
0.0106 $/¥
U.S.A.
Japan
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