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 .
• 6. Determine the present value of a perpetuity.
Power of time of value of money
• History of Interest Rates
$1000 ( 1 + .08)400 = ?
Power of time value of money
• Money Angles: by Andrew Tobias.
1. Chessboard with the King
2. Manhattan
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
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
Future value
1.Simple compounding
2.Complex compounding
k mn
FV  PV  (1  )
m
Future value
i n
FV  100(1  )
t
•
•
•
•
FV1=PV (1+i)
=$100(1+0.06)
=$100(1.06)
=$106
Compound twice a year
0.06 2
FV  100(1 
)  106.09
2
Compound four times a year
0.06 4
FV  100(1 
)  106.14
4
Compound 12 times a year
0.06 12
FV  100(1 
)  106.17
12
Compound 360 times a year
0.06 360
FV  100(1 
)  106.18
360
Continuous compounding
100  e
n ( 0.061)
 106.1836
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
0.08 1
K eff  (1 
) 1
1
1
 (1  0.08)  1
 1.08  1
 8%
0.08 2
K eff  (1 
) 1
2
2
 (1  0.04)  1
 1.0816  1
 8.16%
0.08 4
K eff  (1 
) 1
4
4
 (1  0.02)  1
 1.0824  1
8.24%
Future value and future value interest factor
FVn  PV (1  i )
n
 $1,000(1  0.05)
10
 $1,000(1.62889)
 $1,628.89
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
12
1.127
1.268 1.426 1.601 1.796 2.012
2.252
2.518 2.813 3.138
2.853
PV=$300,
Vn=$774;
i=9 %
FVn  pv (1  i )
N= ？
n
$774  $300(1  0.09)
2.58  (1  0.09)
n
n
PV=$100;
FVn=$179.10;
n=10 years.
FVn  PV (1  i )
n
$179.10  $100(1  i )
1.791  (1  i )
10
I= ?
10
PRESENT VALUE
PV  FVn

1
(1 i ) n

FV10=$500,
n=10,
i=6 %

PV  $500
PV = ?
1
(1 0.06)1 0
1
 $500( 1.791
)
 $500(0.558)
 $279

(PVIF i, n)
• present-value interest factor for I and n
(PVIF i, n),
(PVIF i, n)
= 1/(1+i)
Present value
• FV10 =$1,500
• N= 10 years
• discount rate= 8 %
•
PV=$1500(0.463)
=$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
ANNUITIES
• Annuity: equal annual cash flows.
• Ordinary annuity: at the end of each period.
• Annuity due: at the beginning of each eriod.
Table 5-4
Illustration of a Five-Year $500 Annuity Compounded at 6 percent
YEAR
0
DOLLAR DEPOSITS AT END OF YEAR
1
2
500
500
3
500
4
5
500 500
$500.00
530.00
562.00
595.50
631.00
Future value of the annuity
$2,818.50
FV5  $500(1  0.06) 4  $500(1  0.06) 3  $500(1  0.06) 2
 $500(1  0.06)  $500
 $500(1.262)  $500(1.191)  $500(1.124)  $500(1.060)
 $500
 $631.00  $595.50  $562.00  $530.00  $500.00
 $2,818.50
Ordinary annuity

t
FVn  PMT  (1  i ) 
 t 0

n 1
FVIFAk,n = [(1/k) ( (1+ k)n – 1)]
Present value of an Annuity





PV  $500 (1 01.06)  $500 (1 01.06) 2  $500 (1 01.06)3



 $500 (1 01.06) 4  $500 (1 01.06)5


 $500(0.943)  $500(0.890)  $500(0.840)  $500(0.792)  $500(0.747)
 $2,106
 n 1 
PV  PMT  (1i ) 
 t 1

Table 5-6
Illustration of a Five-Year $500 Annuity Discounted to the
Present at 6 percent
YEAR
0
Dollars received at the
the end of year
$471.50
445.00
420.00
396.00
373.50
PV annuity
$2,106.00
1
2
500
500
3
4
500 500
5
500


1
PV  PMT  (1i ) 
 t 1

n
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


1
PV  $1,000 (10.05)t   $1,000( PVIFA5%,10 yr )


PV= $1,000(7.722)
= $7,722
PMT
Annuity: $5,000, n =5 years, i=8 percent,
PMT:?
$5,000 = PMT (3.993)
$1,252.19=PMT
AMORTIZED LOANS


1
$6,000  PMT  (1 0.15)t 
 t 1

$6,000  PMT ( PVIFA15%, 4 yr )
4
$6,000  PMT (2.855)
$2,101.58  PMT
Loan Amortization Schedule Involving a $6,000 Loan at 15 Percent
to Be Repaid in Four Years
Year
Annuity
Interest Portion
Of The Annuity1
Portion Of The
Annuity2
The Principal
Repayment of
Outstanding
Loan Balance
After The Annuity 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
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
End
year
Loan
payment
(1)
1
$1892.74
2
$1892.7
4
3
Beginning
principal
(2)
payments
$6000.00
Interest(3
)
[0.1 ×
(2)]
$600.00
Princip
al(4)
[(1) －
(3)]
$1292.
74
$
4707.26
$470.73
$
$
1892.74
$
3285.25
$
328.53
$
$
1892.74
$
1721.04
$
172.10
$
4
1422.0
1
1564.2
1
1720.6
4
End of
year
principal(
5)
[(2) －
(4)]
$4707.26
$3285.25
$
1721.04
The Value of $100 Compounded at Various Intervals
FOR 1 YEAR AT i
PERCENT
I = 2%
5%
10%
15%
$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
Compounded annually
PRESENT VALUE OF AN
UNEVEN STREAM
YEAR
1
2
3
4
5
CASH FLOW
$500
200
-400
500
500
YEAR
6
7
8
9
10
CASH FLOW
500
500
500
500
500
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
Quiz 1
Warm up Quiz.
Terms:
: n = 5, m = 4, I =12 percent, and PV =$100 solve for fv
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?
Quiz 3
1
2
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).
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).
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 25-30/3040/40-60 year old with the same rate and
the compounding rate.
PERPETUITIES
$500 perpetuity discounted back to the present
at 8 percent?
PV = $500/0.08 = $6,250
Power of time of value of money
• History of Interest Rates
$1000 ( 1 + .08)400 = ?
Power of time value of money
• Money Angles: by Andrew Tobias.
Chessboard with the King
Manhattan