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The Dow Jones Industrial Average Companies since March 18, 2015
Company Symbol Industry Added
3M MMM Conglomerate 1976 American Express AXP Consumer finance 1982 Apple AAPL Consumer electronics 2015 Boeing BA Aerospace and defense 1987
Caterpillar CAT
Construction and mining equipment
1991
Chevron CVX Oil & gas 2008 Cisco Systems CSCO Computer networking 2009 Coca-Cola KO Beverages 1987 DuPont DD Chemical industry 1935 ExxonMobil XOM Oil & gas 1928 General Electric GE Conglomerate 1907 Goldman Sachs GS Banking, Financial services 2013 The Home Depot HD Home improvement retailer 1999 Intel INTC Semiconductors 1999 IBM IBM Computers and technology 1979 Johnson & Johnson JNJ Pharmaceuticals 1997 JPMorgan Chase JPM Banking 1991 McDonald's MCD Fast food 1985 Merck MRK Pharmaceuticals 1979 Microsoft MSFT Consumer electronics 1999 Nike NKE Apparel 2013 Pfizer PFE Pharmaceuticals 2004 Procter & Gamble PG Consumer goods 1932 Travelers TRV Insurance 2009 UnitedHealth Group UNH Managed health care 2012 United Technologies UTX Conglomerate 1939 Verizon VZ Telecommunication 2004 Visa V Consumer banking 2013 Wal-Mart WMT Retail 1997 Walt Disney DIS Broadcasting and entertainment 1991
Since the four factors – FVF, PVF, FVFA, and PVFA – from the four equations
above are each combinations of r, t, n, we can create Tables of them. These four Tables
appear at the back of this Appendix. Each Table assumes that n = 1. If (r, t, n) is not =
1, then divide r by n and multiply t x n.
Let’s try a few:
TVM Factor Tables
Fut
ure
Val
ue o
f $1
Tab
le o
f Fu
ture
Val
ue F
acto
rs (
"FV
F")
= (
1 +
r) t
Inte
res
t = r
1.00
%2.
00%
3.00
%4.
00%
5.00
%6.
00%
7.00
%8.
00%
9.00
%10
.00%
Per
iods
=
t
11.
0100
1.02
001.
0300
1.04
001.
0500
1.06
001.
0700
1.08
001.
0900
1.10
00
21.
0201
1.04
041.
0609
1.08
161.
1025
1.12
361.
1449
1.16
641.
1881
1.21
00
31.
0303
1.06
121.
0927
1.12
491.
1576
1.19
101.
2250
1.25
971.
2950
1.33
10
41.
0406
1.08
241.
1255
1.16
991.
2155
1.26
251.
3108
1.36
051.
4116
1.46
41
51.
0510
1.10
411.
1593
1.21
671.
2763
1.33
821.
4026
1.46
931.
5386
1.61
05
61.
0615
1.12
621.
1941
1.26
531.
3401
1.41
851.
5007
1.58
691.
6771
1.77
16
71.
0721
1.14
871.
2299
1.31
591.
4071
1.50
361.
6058
1.71
381.
8280
1.94
87
81.
0829
1.17
171.
2668
1.36
861.
4775
1.59
381.
7182
1.85
091.
9926
2.14
36
91.
0937
1.19
511.
3048
1.42
331.
5513
1.68
951.
8385
1.99
902.
1719
2.35
79
101.
1046
1.21
901.
3439
1.48
021.
6289
1.79
081.
9672
2.15
892.
3674
2.59
37
151.
1610
1.34
591.
5580
1.80
092.
0789
2.39
662.
7590
3.17
223.
6425
4.17
72
201.
2202
1.48
591.
8061
2.19
112.
6533
3.20
713.
8697
4.66
105.
6044
6.72
75
251.
2824
1.64
062.
0938
2.66
583.
3864
4.29
195.
4274
6.84
858.
6231
10.8
347
301.
3478
1.81
142.
4273
3.24
344.
3219
5.74
357.
6123
10.0
627
13.2
677
17.4
494
Tab
le 1
FV
Fs
Pre
sent
Val
ue o
f $1
Tab
le o
f P
rese
nt V
alue
Fac
tors
("P
VF
") =
(1
+ r
) -t
Inte
res
t = r
1.00
%2.
00%
3.00
%4.
00%
5.00
%6.
00%
7.00
%8.
00%
9.00
%10
.00%
Per
iods
=
t
10.
9901
0.98
040.
9709
0.96
150.
9524
0.94
340.
9346
0.92
590.
9174
0.90
91
20.
9803
0.96
120.
9426
0.92
460.
9070
0.89
000.
8734
0.85
730.
8417
0.82
64
30.
9706
0.94
230.
9151
0.88
900.
8638
0.83
960.
8163
0.79
380.
7722
0.75
13
40.
9610
0.92
380.
8885
0.85
480.
8227
0.79
210.
7629
0.73
500.
7084
0.68
30
50.
9515
0.90
570.
8626
0.82
190.
7835
0.74
730.
7130
0.68
060.
6499
0.62
09
60.
9420
0.88
800.
8375
0.79
030.
7462
0.70
500.
6663
0.63
020.
5963
0.56
45
70.
9327
0.87
060.
8131
0.75
990.
7107
0.66
510.
6227
0.58
350.
5470
0.51
32
80.
9235
0.85
350.
7894
0.73
070.
6768
0.62
740.
5820
0.54
030.
5019
0.46
65
90.
9143
0.83
680.
7664
0.70
260.
6446
0.59
190.
5439
0.50
020.
4604
0.42
41
100.
9053
0.82
030.
7441
0.67
560.
6139
0.55
840.
5083
0.46
320.
4224
0.38
55
150.
8613
0.74
300.
6419
0.55
530.
4810
0.41
730.
3624
0.31
520.
2745
0.23
94
200.
8195
0.67
300.
5537
0.45
640.
3769
0.31
180.
2584
0.21
450.
1784
0.14
86
250.
7798
0.60
950.
4776
0.37
510.
2953
0.23
300.
1842
0.14
600.
1160
0.09
23
300.
7419
0.55
210.
4120
0.30
830.
2314
0.17
410.
1314
0.09
940.
0754
0.05
73
Tab
le 2
PV
Fs
FV
of
$1 "
ordi
nary
" A
nnui
ty T
able
: ("F
VF
A")
= [
(1 +
r) t
- 1] x
1/ r
Inte
res
t = r
1.00
%2.
00%
3.00
%4.
00%
5.00
%6.
00%
7.00
%8.
00%
9.00
%10
.00%
Per
iods
=
t
11.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
22.
0100
2.02
002.
0300
2.04
002.
0500
2.06
002.
0700
2.08
002.
0900
2.10
00
33.
0301
3.06
043.
0909
3.12
163.
1525
3.18
363.
2149
3.24
643.
2781
3.31
00
44.
0604
4.12
164.
1836
4.24
654.
3101
4.37
464.
4399
4.50
614.
5731
4.64
10
55.
1010
5.20
405.
3091
5.41
635.
5256
5.63
715.
7507
5.86
665.
9847
6.10
51
66.
1520
6.30
816.
4684
6.63
306.
8019
6.97
537.
1533
7.33
597.
5233
7.71
56
77.
2135
7.43
437.
6625
7.89
838.
1420
8.39
388.
6540
8.92
289.
2004
9.48
72
88.
2857
8.58
308.
8923
9.21
429.
5491
9.89
7510
.259
810
.636
611
.028
511
.435
9
99.
3685
9.75
4610
.159
110
.582
811
.026
611
.491
311
.978
012
.487
613
.021
013
.579
5
1010
.462
210
.949
711
.463
912
.006
112
.577
913
.180
813
.816
414
.486
615
.192
915
.937
4
1516
.096
917
.293
418
.598
920
.023
621
.578
623
.276
025
.129
027
.152
129
.360
931
.772
5
2022
.019
024
.297
426
.870
429
.778
133
.066
036
.785
640
.995
545
.762
051
.160
157
.275
0
2528
.243
232
.030
336
.459
341
.645
947
.727
154
.864
563
.249
073
.105
984
.700
998
.347
1
3034
.784
940
.568
147
.575
456
.084
966
.438
879
.058
294
.460
8##
####
###
####
###
####
#
Tab
le 3
FV
FA
s
PV
of a
$1
"ord
inar
y" A
nnui
ty T
able
:Pre
sent
Val
ue A
nnui
ty F
acto
rs (
"PV
FA
") =
[1 -
(1
+ r
) -t ] x
1/ r
Inte
res
t = r
1.00
%2.
00%
3.00
%4.
00%
5.00
%6.
00%
7.00
%8.
00%
9.00
%10
.00%
Per
iods
=
t
10.
9901
0.98
040.
9709
0.96
150.
9524
0.94
340.
9346
0.92
590.
9174
0.90
91
21.
9704
1.94
161.
9135
1.88
611.
8594
1.83
341.
8080
1.78
331.
7591
1.73
55
32.
9410
2.88
392.
8286
2.77
512.
7232
2.67
302.
6243
2.57
712.
5313
2.48
69
43.
9020
3.80
773.
7171
3.62
993.
5460
3.46
513.
3872
3.31
213.
2397
3.16
99
54.
8534
4.71
354.
5797
4.45
184.
3295
4.21
244.
1002
3.99
273.
8897
3.79
08
65.
7955
5.60
145.
4172
5.24
215.
0757
4.91
734.
7665
4.62
294.
4859
4.35
53
76.
7282
6.47
206.
2303
6.00
215.
7864
5.58
245.
3893
5.20
645.
0330
4.86
84
87.
6517
7.32
557.
0197
6.73
276.
4632
6.20
985.
9713
5.74
665.
5348
5.33
49
98.
5660
8.16
227.
7861
7.43
537.
1078
6.80
176.
5152
6.24
695.
9952
5.75
90
109.
4713
8.98
268.
5302
8.11
097.
7217
7.36
017.
0236
6.71
016.
4177
6.14
46
1513
.865
112
.849
311
.937
911
.118
410
.379
79.
7122
9.10
798.
5595
8.06
077.
6061
2018
.045
616
.351
414
.877
513
.590
312
.462
211
.469
910
.594
09.
8181
9.12
858.
5136
2522
.023
219
.523
517
.413
115
.622
114
.093
912
.783
411
.653
610
.674
89.
8226
9.07
70
3025
.807
722
.396
519
.600
417
.292
015
.372
513
.764
812
.409
011
.257
810
.273
79.
4269
Tab
le 4
PV
FA
s
Practical Problems
Practical Problem #1
Each of us faces a generic economic life cycle where, in general, we first consume,
then we save & consume, and lastly we consume. These phases are roughly correlated
with our early life (as children), our adult years (working), and our retirement years. To
be financially secure, we must save enough during the middle “working” phase to
finance spending in retirement. Divide the process into three phases: (1) Saving, (2)
Investing, and (3) Spending.
i. Starting at age 25, save and invest on an annual annuity basis;
ii. From age 45 to 65, no additional saving, just invest the accumulation from
age 25 to 45;
iii. In retirement, age 65 to 85, spend on an annual annuity basis.
If you believe that you need $100,000/ year to be comfortable in retirement, then
without investing you will need $2M in savings at age 65. To achieve this while
working and not investing, you will need to save $ 50,000 per year. This is a daunting
task, especially in the presence of taxes, to say nothing of children and bad habits like
sleeping in a bed and eating hot food a few times a day.
Today, Defined Benefit Plans (“DBP”), financed by employers, are a
progressively rare manner of retirement funds. Instead, most individuals will rely on
Defined Contribution Plans (“DCP”) which are self‐financed such as the 401k, 403b,
Roth and SEP IRA’s.
The question is how much does one need to save, for how long, and earn what
rate‐of‐return to fund retirement spending. – financed by employers Even a modest
annual rate of return (“RoR”) can reduce the required savings necessary to finance a
modest retirement annuity if individuals start early.
Here is an exercise to examine the effect that an investment returns can have on
the retirement saving problem. Imagine that you want to spend $100,000 per year from
age 65 to 85 and that you don’t believe that you will be able to save anything from age
45 to 65, so all of your savings need to be made from age 25 to 45. You believe that
reasonable expected annual rates‐of‐return are as follows:
Age 25‐45 8 percent
Age 45‐65 6 percent
Age 65‐85 4 percent
How much must you save and invest from age 25 to 45 so that you can spend
$100,000 per year from age 65 to 85 assuming zero savings for the 20 years from age 45
to 65 but investing what accumulated from age 25 to 45.
It is easier if we sketch this problem in its three phases listing the parameters –
amounts and rates. Before we solving it, write down your best guess as to how much
you think you will need to save per year for those initial 20 years in order to spend
$100,000 per year for the last 20 years:
Save $_______________ per year from age 25 to 45. OK. Now let’s do the
calculation.
Practical Problem #2
You want to buy a Tesla S4. Assume that this car’s cost, including options, fees,
and taxes is $100,000. Calculate the monthly loan payments on a $100,000 loan over 6‐
years at 5 percent. You are borrowing $100,000 in present value. You plan to repay this
present value with 72 future, monthly payments. Thus, the present value, at 5 percent,
of these 72 future, monthly payments must equal $100,000.
PV(PAYMENTs) = $100,000 = PAYMENT (“$A”) x PVFA(r=5%, t=6, n=1)
Calculated on monthly, not an annual, basis.
Let’s start by looking at the annual compounding PVFA:
PVFA| T, r = = 1
… and since we want the monthly PVFA, we make some adjustments to our parameters. We have T x 12 = 72 periods and must apply only 12th the annual rate as the discount rate:
% 0.00417 0.417% 47.1 per month, so the PVFA calculation,
in detail, is:
PVFA =
5%
12
5%
12
= .
1.
PVFA = .
. = 62.043
Returning to the payment calculation:
$100,000 = 62.043 x $A, and
$A = $ 1,611.78 per month for 72 months.
Total dollars paid for the car will be $ 116,048.16, comprised of $100,000 in loan
principal plus $ 16,048 in interest. The present value of the loan payments is exactly
equal to $ 100,000. The loan payments fully amortize the loan ‐meaning that the
payments “kill‐off the amount owed including interest”.