The Time Value of Money. Chapter 5. 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 .
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The Time Value of Money
Chapter 5
$1000 ( 1 + .08)400 = ?
1.Chessboard with the King
2.Manhattan
1.Simple compounding
2.Complex compounding
Future value
Compound twice a year
Compound four times a year
Compound 12 times a year
Compound 360 times a year
Continuous compounding
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
Future value and future value interest factor
FVn=PV(FVIFi,n)
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
PV=$300, Vn=$774; i=9 % N= ？
PV=$100; FVn=$179.10; n=10 years. I= ?
PRESENT VALUE
FV10=$500, n=10, i=6 % PV = ?
(PVIF i, n) = 1/(1+i)
=$694.50
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
Ordinary annuity
FVIFAk,n = [(1/k) ( (1+ k)n – 1)]
Present value of an Annuity
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
PVIFAK,n = (1/k) [( 1 – 1/(1+k)n]
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
n=10 years, I=5 percent, and current PMT=$1,000
PV= $1,000(7.722)
= $7,722
Annuity: $5,000, n =5 years, i=8 percent, PMT:?
$5,000 = PMT (3.993)
$1,252.19=PMT
AMORTIZED LOANS
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
22,101.58 719.76 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
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
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
The Value of $100 Compounded at Various IntervalsFOR 1 YEAR AT i PERCENT
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
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
Warm up Quiz.
Terms:
: n = 5, m = 4, I =12 percent, and PV =$100 solve for fv
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?
$500 perpetuity discounted back to the present at 8 percent?
PV = $500/0.08 = $6,250
$1000 ( 1 + .08)400 = ?
Chessboard with the King
Manhattan