The Time Value of Money
description
Transcript of The Time Value of Money
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The Time Value of Money
Chapter 5
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LEARNING OBJECTIVES
• 1. Explain the mechanics of compounding when invested.
• 2. Present value and future value.• 3. Ordinary annuity and its future value.• 4. An ordinary annuity and an annuity due.• 5. Non-annual future or present value of a sum .
• 6. Determine the present value of a perpetuity.
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Power of time of value of money
• History of Interest Rates
$1000 ( 1 + .08)400 = ?
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Power of time value of money
• Money Angles: by Andrew Tobias.
1. Chessboard with the King
2. Manhattan
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Terms
• Compound Interest• Future value and Present Value• Annuities• Annuities Due• Amortized Loans • Compound Interest with Non-annual Periods• Present Value of an Uneven Stream·• Perpetuities
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COMPOUND INTEREST
• FV1=PV (1+i) (5-1)• Where FV1=the future value of the investment
at the end of one year• i=the annual interest (or discount) rate• PV=the present value, or original amount
invested at the beginning of the first year
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Future value
1.Simple compounding
2.Complex compounding
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nm
m
kPVFV )1(
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n
t
iFV )1(100
Future value
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• FV1=PV (1+i)
• =$100(1+0.06)
• =$100(1.06)
• =$106
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09.106)2
06.01(100 2 FV
Compound twice a year
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14.106)4
06.01(100 4 FV
Compound four times a year
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17.106)12
06.01(100 12 FV
Compound 12 times a year
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18.106)360
06.01(100 360 FV
Compound 360 times a year
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1836.106100 )106.0( ne
Continuous compounding
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Illustration of Compound Interest Calculations
Year Beginning Value Interest Earned Ending Value
1 $100.00 $6.00 $106.00
2 106.00 6.36 112.36
3 112.36 6.74 119.10
4 119.10 7.15 126.25
5 126.25 7.57 133.82
6 133.82 8.03 141.85
7 141.85 8.51 150.36
8 150.36 9.02 159.38
9 159.38 9.57 168.95
10 168.95 10.13 179.08
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%8
108.1
1)08.01(
1)1
08.01(
1
1
effK
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%16.8
10816.1
1)04.01(
1)2
08.01(
2
2
effK
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%24.8
10824.1
1)02.01(
1)4
08.01(
4
4
effK
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89.628,1$
)62889.1(000,1$
)05.01(000,1$
)1(10
nn iPVFV
Future value and future value interest factor
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FVn=PV(FVIFi,n)
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Table 5-2
FVIFi,n or the Compound Sum of $1
N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100
2 1.020 1.040 1.061 1.082 1.102 1.124 1.145 1.166 1.188 1.210
3 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331
4 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464
5 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611
6 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772
7 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949
8 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.815 1.993 2.144
9 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358
10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594
11 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853
12 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138
13 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452
14 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797
15 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177
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n
n
nn ipvFV
)09.01(58.2
)09.01(300$774$
)1(
PV=$300, Vn=$774; i=9 % N= ?
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10
10
)1(791.1
)1(100$10.179$
)1(
i
i
iPVFV nn
PV=$100; FVn=$179.10; n=10 years. I= ?
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ninFVPV)1(
1
PRESENT VALUE
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FV10=$500, n=10, i=6 % PV = ?
279$
)558.0(500$
)(500$
500$
791.11
)06.01(1
10
PV
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(PVIF i, n)
• present-value interest factor for I and n (PVIF i, n),
(PVIF i, n) = 1/(1+i)
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Present value
• FV10 =$1,500
• N= 10 years
• discount rate= 8 %
• PV=$1500(0.463)
=$694.50
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Table 5-3
PVIFi,n or the Present Value of $1
N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909
2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826
3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751
4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683
5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621
6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564
7 0.933 0.871 0.813 0.760 0.711 0.655 0.623 0.583 0.547 0.513
8 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424
9 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386
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ANNUITIES
• Annuity: equal annual cash flows.
• Ordinary annuity: at the end of each period.
• Annuity due: at the beginning of each eriod.
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Table 5-4
Illustration of a Five-Year $500 Annuity Compounded at 6 percent
YEAR 0 1 2 3 4 5
DOLLAR DEPOSITS AT END OF YEAR 500 500 500 500 500
$500.00
530.00
562.00
595.50
631.00
Future value of the annuity $2,818.50
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50.818,2$
00.500$00.530$00.562$50.595$00.631$
500$
)060.1(500$)124.1(500$)191.1(500$)262.1(500$
500$)06.01(500$
)06.01(500$)06.01(500$)06.01(500$ 2345
FV
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1
0
)1(n
t
tn iPMTFV
FVIFAk,n = [(1/k) ( (1+ k)n – 1)]
Ordinary annuity
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106,2$
)747.0(500$)792.0(500$)840.0(500$)890.0(500$)943.0(500$
500$500$
500$500$500$
54
32
)06.01(1
)06.01(1
)06.01(1
)06.01(1
)06.01(1
PV
Present value of an Annuity
n
tiPMTPV
1)1(
1
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Table 5-6
Illustration of a Five-Year $500 Annuity Discounted to the Present at 6 percent
YEAR 0 1 2 3 4 5
Dollars received at the 500 500 500 500 500
the end of year $471.50
445.00
420.00
396.00
373.50
PV annuity $2,106.00
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n
tiPMTPV
1)1(
1
PVIFAK,n = (1/k) [( 1 – 1/(1+k)n]
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Table 5-7
PVIFi,n or the Present Value of an Annuity of $1
N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909
2 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736
3 2.941 2.884 2.829 2.775 2.723 2.673 2.642 2.577 2.531 2.487
4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170
5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.003 3.890 3.791
6 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355
7 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868
8 7.652 7.326 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335
9 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759
10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145
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)(000,1$000,1$ 10%,5)05.01(1
yrPVIFAPV t
PV= $1,000(7.722)
= $7,722
n=10 years, I=5 percent, and current PMT=$1,000
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PMT
Annuity: $5,000, n =5 years, i=8 percent, PMT:?
$5,000 = PMT (3.993)
$1,252.19=PMT
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PMT
PMT
PVIFAPMT
PMT
yr
tt
58.101,2$
)855.2(000,6$
)(000,6$
000,6$
4%,15
4
1)15.01(
1
AMORTIZED LOANS
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Loan Amortization Schedule Involving a $6,000 Loan at 15 Percent to Be Repaid in Four Years
Year Annuity Interest Portion Repayment of Outstanding
Of The Annuity1 The Principal Loan Balance
Portion Of The After The An-
Annuity2 nuity Payment
1 $2,101.58 $900.00 $1,201.58 $4,798.42
2 2,101.58 719.76 1,381.82 3,416.60
3 2,101.58 512.49 1,589.09 1,827.51
4 2,101.58 274.07 1,827.51
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ANNUITIES DUE
• FVn (annuity due)=PMT(FVIFA I,n)(1+I) (5-10)
FV5=$500(FVIFA5%,5)(1+0.06)
=$500(5.637)(1.06)
=$2,987.61
from $2,106 to $2,232.36,
PV=$500(PVIFA6%,5)(1+0.06)
=$500(4.212)(1.06)
=$2,232.36
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End year
Loan payment
(1)
Beginning principal
(2)
payments End of year
principal(5)
[(2) -(4)]
Interest(3)
[0.1 × (2)]
Principal(4)
[(1) - (3)]
1 $1892.74 $6000.00 $600.00 $1292.74
$4707.26
2 $1892.74
$4707.26 $470.73 $1422.01
$3285.25
3
$1892.74 $3285.25 $328.53 $1564.21
$1721.04
4
$1892.74 $1721.04 $172.10 $1720.64
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The Value of $100 Compounded at Various Intervals FOR 1 YEAR AT i PERCE
NT
I = 2% 5% 10% 15%
Compounded annually $102.00 $105.00 $110.00 $115.00
Compounded semiannually 102.01 105.06 110.25 115.56
Compounded quarterly 102.02 105.09 110.38 115.87
Compounded monthly 102.02 105.12 110.47 116.08
Compounded weekly (52) 102.02 105.12 110.51 116.16
Compounded daily (365) 102.02 105.13 110.52 116.18
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PRESENT VALUE OF AN UNEVEN STREAM
YEAR CASH FLOW YEAR CASH FLOW
1 $500 6 500
2 200 7 500
3 -400 8 500
4 500 9 500
5 500 10 500
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1. Present value of $500 received at the end of one year
= $500(0.943) = $471.50
2. Present value of $200 received at the end of tree years
= $200(0.890) = 178.00
3. Present value of a $400 outflow at the end of three years
= -400(0.840) = -336.00
4. (a) Value at the end of year 3 and a $500 annuity, years 4 through 10
= $500 (5.582) = $2,791.00
(b) Present value of $2,791.00 received at the end of year 3
= 2,791(0.840) = 2,344.44
5. Total present value = $2,657.94
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Quiz 1
Warm up Quiz.
Terms:
: n = 5, m = 4, I =12 percent, and PV =$100 solve for fv
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Quiz 2
What is the present value of an investment involving $200 received at the end of years 1 through 4, a $300 cash outflow at the end of year 5 to 8, and $500 received at the end of years 9 through 10, given a 5 percent discount rate?
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Quiz 3
1 A 25 year-old graduate has his $50,000 salary a year. How much will he get when he reaches to 60 (35 years later)year-old with a value rate of 8%(annual compounding).
2 The graduate will have his $80,000 salary at age of 30. How much will he get when he reaches to his age of 60(30 years later) with the value rate of 8%(semi-annual compounding).
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Quiz 4
3. The graduate will have his $100,000 salary at age of 40. How much will he get when he reaches to his age of 60(20 years later) with the value rate of 12%(quarterly-annual compounding).
4. Compute the future value from 25-30/30-40/40-60 year old with the same rate and the compounding rate.
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PERPETUITIES
$500 perpetuity discounted back to the present at 8 percent?
PV = $500/0.08 = $6,250
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Power of time of value of money
• History of Interest Rates
$1000 ( 1 + .08)400 = ?
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Power of time value of money
• Money Angles: by Andrew Tobias.
Chessboard with the King
Manhattan