The Time Value of Money
description
Transcript of The Time Value of Money
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The Time Value of MoneyThe Time Value of Money
Chapter 9
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The Time Value of MoneyThe Time Value of Money
Which would you rather have ? $100 today - or $100 one year from today Sooner is better !
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The Time Value of MoneyThe Time Value of Money
How about $100 today or $105 one year from today?
We revalue current dollars and future dollars using the time value of money
Cash flow time line graphically shows the timing of cash flows
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Time 0 is today; Time 1 is one period from today
Time 0 1 2 3 4 5
Cash Flows-100
5%
Outflow
Interest rate
105Inflow
Cash Flow Time LinesCash Flow Time Lines
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Future ValueFuture Value
Compounding the process of determining the value of a
cash flow or series of cash flows some time in the future when compound interest is applied
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Future ValueFuture Value PV = present value or starting amount,
say, $100 i = interest rate, say, 5% per year would
be shown as 0.05 INT = dollars of interest you earn during
the year $100 0.05 = $5 FVn = future value after n periods or
$100 + $5 = $105 after one year = $100 (1 + 0.05) = $100(1.05) = $105
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Future ValueFuture Value
i)PV(1 PV(i)PV
INTPVFV
1
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Future ValueFuture Value
The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate
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Compounded InterestCompounded Interest
Interest earned on interest
nn i)PV(1FV
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Cash Flow Time LinesCash Flow Time Lines
Time 0 1 2 3 4 5
Interest
-100
5%
5.00
Total Value 105.00
5.25
110.25
5.51
115.76
5.79
121.55
6.08
127.63
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Future Value Interest Factor Future Value Interest Factor for i and n (FVIFfor i and n (FVIFi,ni,n))
The future value of $1 left on deposit for n periods at a rate of i percent per period
The multiple by which an initial investment grows because of the interest earned
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Future Value Interest Factor Future Value Interest Factor for i and n (FVIFfor i and n (FVIFi,ni,n))
FVn = PV(1 + i)n = PV(FVIFi,n)Period (n) 4% 5% 6%
1 1.0400 1.0500 1.0600 2 1.0816 1.1025 1.1236 3 1.1249 1.1576 1.1910 4 1.1699 1.2155 1.2625 5 1.2167 1.2763 1.3382 6 1.2653 1.3401 1.4185
For $100 at i = 5% and n = 5 periods
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FVn = PV(1 + i)n = PV(FVIFi,n)Period (n) 4% 5% 6%
1 1.0400 1.0500 1.0600 2 1.0816 1.1025 1.1236 3 1.1249 1.1576 1.1910 4 1.1699 1.2155 1.2625 5 1.2167 1.2763 1.3382 6 1.2653 1.3401 1.4185
For $100 at i = 5% and n = 5 periods
$100 (1.2763) = $127.63
Future Value Interest Factor Future Value Interest Factor for i and n (FVIFfor i and n (FVIFi,ni,n))
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Financial Calculator Financial Calculator SolutionSolution
Five keys for variable input N = the number of periods I = interest rate per period
may be I, INT, or I/Y PV = present value PMT = annuity payment FV = future value
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Find the future value of $100 at 5% interest per year for five years
1. Numerical Solution:
Two SolutionsTwo Solutions
Time 0 1 2 3 4 5
Cash Flows
-100
5%
5.00
Total Value 105.00
5.25
110.25
5.51
115.76
5.79
121.55
6.08
127.63
FV5 = $100(1.05)5 = $100(1.2763) = $127.63
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Two SolutionsTwo Solutions
2. Financial Calculator Solution:
Inputs: N = 5 I = 5 PV = -100 PMT = 0 FV = ?
Output: = 127.63
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Graphic View of the Compounding Graphic View of the Compounding Process: GrowthProcess: Growth
Relationship among Future Value, Growth or Interest Rates, and Time
0
1
2
3
4
5
1
i= 15%
i= 10%
i= 5%
i= 0%
2 4 6 8 10Periods
Future Value of $1
0
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Present ValuePresent Value
Opportunity cost the rate of return on the best available
alternative investment of equal risk
If you can have $100 today or $127.63 at the end of five years, your choice will depend on your opportunity cost
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Present ValuePresent Value
The present value is the value today of a future cash flow or series of cash flows
The process of finding the present value is discounting, and is the reverse of compounding
Opportunity cost becomes a factor in discounting
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0 1 2 3 4 5
PV = ?
5%
127.63
Cash Flow Time LinesCash Flow Time Lines
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Present ValuePresent Value
Start with future value: FVn = PV(1 + i)n
nnnn
i1
1FV
i)(1
FVPV
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Find the present value of $127.63 in five years when the opportunity cost rate is 5%
1. Numerical Solution:
110.25 115.76 121.55
5%0 1 2 3 4 5
PV = ? 127.63
105.00
÷ 1.05÷ 1.05÷ 1.05÷ 1.05
-100.00
Two SolutionsTwo Solutions
100$)7835.0(63.127$
2763.1
$127.63
05.1
$127.63PV 5
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Find the present value of $127.63 in five years when the opportunity cost rate is 5%
2. Financial Calculator Solution:
Inputs: N = 5 I = 5 PMT = 0 FV = 127.63 PV = ?
Output: = -100
Two SolutionsTwo Solutions
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Relationship among Present Value, Interest Rates, and Time
Graphic View of the Graphic View of the Discounting ProcessDiscounting Process
0
0.2
0.4
0.6
0.8
1
i= 15%
i= 10%
i= 5%
i= 0%
2 4 6 8 10Periods
Present Value of $1
12 14 16 18 20
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Solving for Time and Solving for Time and Interest RatesInterest Rates
Compounding and discounting are reciprocals
FVn = PV(1 + i)n
nnnn
i1
1FV
i)(1
FVPV
Four variables: PV, FV, i and n
If you know any three, you can solve for the fourth
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For $78.35 you can buy a security that will pay you $100 after five years
We know PV, FV, and n, but we do not know i
FVn = PV(1 + i)n
$100 = $78.35(1 + i)5 Solve for i
Solving for iSolving for i
0 1 2 3 4 5
-78.35
i = ?
100
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FVn = PV(1 + i)n
$100 = $78.35(1 + i)5
0.051-1.05i
..i1
..$
$i1
05127631
276313578
100
51
5
Numerical SolutionNumerical Solution
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Inputs: N = 5 PV = -78.35 PMT = 0 FV = 100 I = ?
Output: = 5 This procedure can be used for any rate
or value of n, including fractions
Financial Calculator Financial Calculator SolutionSolution
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Solving for nSolving for n
Suppose you know that the security will provide a return of 10 percent per year, that it will cost $68.30, and that you will receive $100 at maturity, but you do not know when the security matures. You know PV, FV, and i, but you do not know n - the number of periods.
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Solving for nSolving for n
FVn = PV(1 + i)n $100 = $68.30(1.10)n By trial and error you could substitute
for n and find that n = 4
0 1 2 n-1 n=?
-68.30
10%
100
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Financial Calculator Financial Calculator SolutionSolution
Inputs: I = 10 PV = -68.30 PMT = 0 FV = 100 N = ?
Output: = 4.0
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AnnuityAnnuity
An annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods
Ordinary (deferred) annuity has payments at the end of each period
Annuity due has payments at the beginning of each period
FVAn is the future value of an annuity over n periods
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Future Value of an AnnuityFuture Value of an Annuity
The future value of an annuity is the amount received over time plus the interest earned on the payments from the time received until the future date being valued
The future value of each payment can be calculated separately and then the total summed
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If you deposit $100 at the end of each year for three years in a savings account that pays 5% interest per year, how much will you have at the end of three years?
100 100.00 = 100 (1.05)0
105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
315.25
Future Value of an AnnuityFuture Value of an Annuity
0 1 2 35%
100
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1-n
0t
t1-n10n i1 PMTi)PMT(1i)PMT(1i)PMT(1FVA
i
i1PMTi1PMT
nn
t
tn 11
2531515253100
1100 .$).($
0.05
1.05$FVA
3
3
Future Value of an AnnuityFuture Value of an Annuity
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Future Value of an AnnuityFuture Value of an Annuity
Financial calculator solution: Inputs: N = 3 I = 5 PV = 0 PMT = -
100 FV = ? Output: = 315.25 To solve the same problem, but for the
present value instead of the future value, change the final input from FV to PV
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Annuities DueAnnuities Due
If the three $100 payments had been made at the beginning of each year, the annuity would have been an annuity due.
Each payment would shift to the left one year and each payment would earn interest for an additional year (period).
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$100 at the end of each year
0 1 2 35%
100 100 100.00 = 100 (1.05)0
105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
315.25
Future Value of an AnnuityFuture Value of an Annuity
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$100 at the start of each year
0 1 2 35%
100 100100105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
331.0125115.7625 = 100 (1.05)3
Future Value of Future Value of an Annuity Duean Annuity Due
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Numerical solution:
i1i
1i1PMT
i1i1PMT
i1PMTFVA(DUE)
n
n
1t
t-n
n
1t
tn
Future Value of Future Value of an Annuity Duean Annuity Due
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Numerical solution:
0125331
05115253100
0511051
100
.$
..$
.0.05
.$FVA(DUE)
3
n
Future Value of Future Value of an Annuity Duean Annuity Due
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Future Value of Future Value of an Annuity Duean Annuity Due
Financial calculator solution: Inputs: N = 3 I = 5 PV = 0 PMT = -
100 FV = ? Output: = 331.0125
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Present Value of an AnnuityPresent Value of an Annuity
If you were offered a three-year annuity with payments of $100 at the end of each year
Or a lump sum payment today that you could put in a savings account paying 5% interest per year
How large must the lump sum payment be to make it equivalent to the annuity?
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0 1 2 35%
100 100 100
384.86
05.1
100
703.9005.1
100
238.9505.1
100
3
2
1
272.325
Present Value of an AnnuityPresent Value of an Annuity
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Present Value of an AnnuityPresent Value of an Annuity
Numerical solution:
n
1t t
n21n
iPMT
iPMT
iPMT
iPMTPVA
1
1
1
1
1
1
1
1
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3227272322100
1
11
1
11
1
1
1
1
1
1
1
1
.$).($0.05
.05 $100
ii
PMTi
PMT
iPMT
iPMT
iPMTPVA
3
nn
1t t
n21n
Present Value of an AnnuityPresent Value of an Annuity
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Present Value of an AnnuityPresent Value of an Annuity
Financial calculator solution: Inputs: N = 3 I = 5 PMT = -100 FV
= 0 PV = ? Output: = 272.325
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Present Value of Present Value of an Annuity Duean Annuity Due
Payments at the beginning of each year Payments all come one year sooner Each payment would be discounted for
one less year Present value of annuity due will exceed
the value of the ordinary annuity by one year’s interest on the present value of the ordinary annuity
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0 1 2 35%
100 100
70390
051
100051
051
100
23895051
100051
051
100
000100051
100051
051
100
22
11
01
. .
..
. .
..
..
..
285.941
Present Value of Present Value of an Annuity Duean Annuity Due
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Numerical solution:
ii
iPMT
ii
PMTi
PMTPVA(DUE)
n
n
1t t
1-n
0t 1n
11
11
11
1
1
1
Present Value of Present Value of an Annuity Duean Annuity Due
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$285.941
(2.85941) $100
(1.05)][(2.72325) $100
...
$PV(DUE)3
051
050051
11
1003
Present Value of Present Value of an Annuity Duean Annuity Due
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Present Value of Present Value of an Annuity Duean Annuity Due
Financial calculator solution: Switch to the beginning-of-period mode,
then enter Inputs: N = 3 I = 5 PMT = -100 FV
= 0 PV = ? Output: = 285.94 Then switch back to the END mode
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-846.80
i = ?0 1 2 3 4
250 250 250 250
Solving for Interest Rates Solving for Interest Rates with Annuitieswith Annuities
Suppose you pay $846.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year
ii
$250 $846.8041
11
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Solving for Interest Rates Solving for Interest Rates with Annuitieswith Annuities
Numerical solution: Trial and error using different values for
i using until you find i where the present value of the four-year, $250 annuity equals $846.80. The solution is 7%.
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Solving for Interest Rates Solving for Interest Rates with Annuitieswith Annuities
Financial calculator solution: Inputs: N = 4 PV = -846.8 PMT = 250
FV = 0 I = ? Output: = 7.0
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PerpetuitiesPerpetuities
Perpetuity - a stream of equal payments expected to continue forever
Consol - a perpetual bond issued by the British government to consolidate past debts; in general, and perpetual bond
i
PMT
RateInterest
PaymentPVP
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Uneven Cash Flow StreamsUneven Cash Flow Streams
Uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next
Payment (PMT) designates constant cash flows
Cash Flow (CF) designates cash flows in general, including uneven cash flows
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Present Value of Present Value of Uneven Cash Flow StreamsUneven Cash Flow Streams
PV of uneven cash flow stream is the sum of the PVs of the individual cash flows of the stream
n
1t tt
nn21
iCF
iCF
iCF
iCFPV
1
1
1
1
1
1
1
121
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Future Value of Future Value of Uneven Cash Flow StreamsUneven Cash Flow Streams
Terminal value is the future value of an uneven cash flow stream
n
1t
t-nt
0n
2-n2
1-n1n
iCF
iCFiCFiCFFV
1
111
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Solving for i with Solving for i with Uneven Cash Flow StreamsUneven Cash Flow Streams
Using a financial calculator, input the CF values into the cash flow register and then press the IRR key for the Internal Rate of Return, which is the return on the investment.
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Compounding PeriodsCompounding Periods
Annual compounding interest is added once a year
Semiannual compounding interest is added twice a year 10% annual interest compounded semiannually
would pay 5% every six months adjust the periodic rate and number of periods
before calculating
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Interest RatesInterest Rates
Simple (Quoted) Interest Rate rate used to compute the interest payment paid
per period
Effective Annual Rate (EAR) annual rate of interest actually being earned,
considering the compounding of interest
01.m
i1 EAR
msimple
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Interest RatesInterest Rates
Annual Percentage Rate (APR) the periodic rate multiplied by the number of
periods per year this is not adjusted for compounding
More frequent compounding:
nmsimple
n m
i1PV FV
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Amortized LoansAmortized Loans
Loans that are repaid in equal payments over its life
Borrow $15,000 to repay in three equal payments at the end of the next three years, with 8% interest due on the outstanding loan balance at the beginning of each year
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0 1 2 38%
PMT PMT PMT15,000
3
3
000151t t
1t t
3213
1.08
PMT,$
i1
PMTi1
PMT
i1
PMT
i1
PMTPVA
Amortized LoansAmortized Loans
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Numerical Solution:
$5,820.502.5771
$15,000PMT
2.5771PMT$15,000
0.081.08
1-1
PMT1.08
1PMT
1.08
PMT$15,000
33
1tt
3
1tt
Amortized LoansAmortized Loans
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Amortized LoansAmortized Loans
Financial calculator solution: Inputs: N = 3 I = 8 PV = 15000 FV =
0 PMT = ? Output: = -5820.5
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Amortized LoansAmortized Loans
Amortization Schedule shows how a loan will be repaid with a breakdown of interest and principle on each payment date
YearBeginning Amount (1)
Payment (2) Interesta (3)Repayment of Principalb (2)-
(3)=(4)
Remaining Balance (1)-
(4)=(5)1 15,000.00$ 5,820.50$ 1,200.00$ 4,620.50$ 10,379.50$ 2 10,379.50 5,820.50 830.36 4,990.14 5,389.36 3 5,389.36 5,820.50 431.15 5,389.35 0.01 c
aInterest is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefor, interest in Year 1 is $15,000(0.08) = $1,200; in Year 2, it is $10,379.50(0.08)=$830.36; and in Year 3, it is $5,389.36(0.08) = $431.15 (rounded).
bRepayment of principal is equal to the payment of $5,820.50 minus the interest charge for each year.
cThe $0.01 remaining balance at the end of Year 3 results from rounding differences.
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Comparing Interest RatesComparing Interest Rates
1. Simple, or quoted, rate, (isimple) rates compare only if instruments have the
same number of compounding periods per year
2. Periodic rate (iPER) APR represents the periodic rate on an
annual basis without considering interest compounding
APR is never used in actual calculations
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Comparing Interest RatesComparing Interest Rates
3. Effective annual rate, EAR the rate that with annual compounding (m=1)
would obtain the same results as if we had used the periodic rate with m compounding periods per year
1
mPER
mSIMPLE
i1
1.0m
i1EAR
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End of Chapter 9End of Chapter 9
The Time Value of Money