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5 6 1.00000 0.33490 0.33490 74% 74%
6 16 1.62889 0.05409 0.08810 19% 93%
7 26 2.65330 0.00874 0.02318 5% 98%
8 36 4.32194 0.00141 0.00610 1% 100%
9 46 7.03999 0.00023 0.00160 0% 100%
10 56 11.46740 0.00004 0.00042 0% 100%
11 66 18.67919 0.00001 0.00011 0% 100%
12 76 30.42643 0.00000 0.00003 0% 100%
13 86 49.56144 0.00000 0.00001 0% 100%
14 96 80.73037 0.00000 0.00000 0% 100%

15 Totals 0.45445 100%
17 Calculation of PPF by formula:
19 PPF
20 0.45445
22 Assumptions:
24 j Number of years between moves [1] 10
25 r Discount rate 20.0%
26 g Growth rate in net inc/cash ¬‚ow 5.0%
27 b Number of years from last cash ¬‚ow 4
29 Spreadsheet formulas:
31 A20: (1 r) b/((1 r) j (1 g) j) Equation (3-18a)

[1] As j decreases, the PV Factors and the PV increase. It is possible that you will have to add additional rows above Row 15 to
capture all the PV of the cash ¬‚ows. Otherwise, the PV in C20 will appear to be higher than the total of the cash ¬‚ows in D15.




PPFs in New versus Used Equipment Decisions
Another important use of PPFs is in new versus used equipment deci-
sions and in valuing used income-producing equipment. Let™s use a taxi-
cab as an example. The cab company can buy a new car or a used car.
Suppose a new car would last six years. It costs $20,000 to buy a new
one today, and we can model the cash ¬‚ows for its six-year expected life.
The cash ¬‚ows will consist of the purchase of the cab, income, gas-
oline, maintenance, insurance, etc. Each expense category has its own
pattern. Gas consumption is a variable expense that increases in dollars
over time with the rate of increase in gas prices. Maintenance is probably
low for the ¬rst two years and then begins increasing rapidly in Year 3
or 4.
We can then take the NPV of the cash ¬‚ows, and that represents the
NPV of operating a new cab for six years. It would be nice to compare
that with the NPV of operating a one-year-old cab for ¬ve years (or any
other term desired). The problem is that these are different time periods.
We could use the lowest common multiple of 30 years (6 years 5 years)
and run the new cab cash ¬‚ows ¬ve times and the used cab cycle six
times, but that is a lot of work. It is a far more elegant solution to use a
PPF for the new and the used equipment. The result of those computa-




CHAPTER 3 Annuity Discount Factors and the Gordon Model 83
tions will be the present value of keeping one new cab and one used cab
in service forever. We can then choose the one with the superior NPV.
Even though the cash ¬‚ows are contiguous, which is not true in the
periodic expense example, the cycle and the NPV of the cash ¬‚ows are
periodic. Every six years the operator buys a new cab. We can measure
the NPV of the ¬rst cab as of t 0. The operator buys the second cab
and uses it from Years 7“12. Its NPV as of the end of Year 6 (t 6) should
be the same as the NPV at t 0 of the ¬rst six years™ cash ¬‚ows, with a
growth rate for the rise in prices. If there are substantial difference in the
growth rates of income versus expenses or of the different categories of
expenses, then we can break the expenses into two or more subcategories
and apply a PPF to each subcategory, then add the NPVs together. Buying
a new cab every six years would then generate a series of NPVs with
constant growth at t 0, 6, 12, . . . . That repeating pattern is what enables
us to use a PPF to value the cash ¬‚ows.
We could perform this procedure for each different vintage of used
equipment, e.g., buying one-year-old cabs, two-year old cabs, etc. Our
¬nal comparison would be the NPV of buying and operating a single cab
of each age (a new cab, one year old, two years old, etc.) forever. We then
simply choose the cab life with the highest NPV.
If equipment is not income producing, we can still the PPF to value
the periodic costs in perpetuity. Then the NPV would be negative.

ADFs IN LOAN MATHEMATICS
There are four related topics that should ideally all be together dealing
with the use of ADFs in loan mathematics to create formulas to calculate:
loan payments, principal amortization, the after-tax cost of a loan, and
the PV of a loan when the nominal and market rates differ. We will deal
with the ¬rst and the last topics in this section. Calculating the amorti-
zation of principal is mathematically very complex. To maintain read-
ability, it will be explained, along with the related problem of calculating
the after-tax cost of a loan, in the Appendix.

Calculating Loan Payments
We can use our earlier ADF results to easily create a formula to calculate
loan payments. We know that in the case of a ¬xed rate amortizing loan,
the principal must be equal to the PV of the payments when discounted
by the nominal rate of the loan. We can calculate the PV of the payments
using equation (3-6d) and the following de¬nitions:
ADFNominal ADF at the nominal interest rate of the loan
ADFMkt ADF at the market interest rate of the loan
The nominal ADF is simply an end-of-year ADF with no growth.
Repeating equation (3-6d), the ADF is:
1
1
r)n
(1
ADFNominal
r
where r in this case is the nominal interest of the loan. If we use the

PART 1 Forecasting Cash Flows
84
market interest rate instead of the nominal rate, we get ADFMkt. We know
that the loan payment multiplied by the nominal ADF equals the prin-
cipal of the loan. Stating that as an equation:
Loan Payment ADFNominal Principal (3-20)
Dividing both sides of the equation by ADFNominal, we get:
Principal 1
Loan Payment Principal (3-21)
ADFNominal ADFNominal


Present Value of a Loan
The PV of a loan is the loan payment multiplied by the market rate ADF,
or:
PV Loan Payment ADFMkt (3-22)
From equation (3-21), the loan payment is the principal divided by the
nominal ADF. Substituting this into equation (3-22) gives us:
ADFMkt
PV of Loan Principal (3-23)
ADFNominal
The intuition behind this is the Principal 1/ADFNominal is the amount
of the loan payment. When we then multiply that by the ADFMkt, this
gives us the PV of the loan.

Table 3-10: Example of Equation (3-23)
Table 3-10 is an example of calculating the present value of a loan. The
assumptions appear in Table 3-10 in E77 to E82. We assume a $1 million
principal on a ¬ve-year loan. The loan payment, calculated using Excel™s
spreadsheet function, is $20,276.39 (E78) for 60 months. The annual loan
rate is 8% (E79), and the monthly rate is 0.667% (E80 E79/12). The
annual market rate of interest (the discount rate) on this loan is assumed
at 14% (I81), and the monthly market interest rate is 1.167% (I82
I81/12).
Column A shows the 60 months of payments. Column B shows the
monthly payment of $20,276.39 for 60 months. Columns C and D show
the PV factor and the PV of each month™s payment at the nominal 8%
annual interest rate (0.667% monthly rate), while Columns E and F show
the same calculations at the market rate of 14% (1.167% monthly rate).
The present value factors in C6 to C65 total 49.31843, and present
value factors in E6 to E65 total 42.97702. Note also that the PV of the loan
at the nominal interest rate adds to the $1 million principal (D66), as it
should.
E70 is the ADF at 8% according to equation (3-6d). We show the
spreadsheet formula for E70 in A86. E71 is 1/ADFNominal $0.02027639,
the amount of loan payment for each $1 of principal. We multiply that
by the $1 million principal to obtain the loan payment of $20,276.39 in
F71, which matches E78, as it should. In E72 we calculate the ADF at the
market rate of interest, the formula for which is also equation (3-6d),
merely using the 1.167% monthly interest rate in the formula, which we
show in A88. In E73 we calculate the ratio of the market ADF to the

CHAPTER 3 Annuity Discount Factors and the Gordon Model 85
T A B L E 3-10

PV of Loan with Market Rate Nominal Rate: ADF, End-of-Year


A B C D E F

4 r 8% r 14%

5 Month Cash Flow PV Factor Present Value PV Factor Present Value
6 1 $20,276.39 0.99338 $ 20,142 0.98847 $ 20,043
7 2 $20,276.39 0.98680 $ 20,009 0.97707 $ 19,811
8 3 $20,276.39 0.98026 $ 19,876 0.96580 $ 19,583
9 4 $20,276.39 0.97377 $ 19,745 0.95466 $ 19,357
10 5 $20,276.39 0.96732 $ 19,614 0.94365 $ 19,134
11 6 $20,276.39 0.96092 $ 19,484 0.93277 $ 18,913
12 7 $20,276.39 0.95455 $ 19,355 0.92201 $ 18,695
13 8 $20,276.39 0.94823 $ 19,227 0.91138 $ 18,480
14 9 $20,276.39 0.94195 $ 19,099 0.90087 $ 18,266
15 10 $20,276.39 0.93571 $ 18,973 0.89048 $ 18,056
16 11 $20,276.39 0.92952 $ 18,847 0.88021 $ 17,848
17 12 $20,276.39 0.92336 $ 18,722 0.87006 $ 17,642
18 13 $20,276.39 0.91725 $ 18,598 0.86003 $ 17,438
19 14 $20,276.39 0.91117 $ 18,475 0.85011 $ 17,237
20 15 $20,276.39 0.90514 $ 18,353 0.84031 $ 17,038
21 16 $20,276.39 0.89914 $ 18,231 0.83062 $ 16,842
22 17 $20,276.39 0.89319 $ 18,111 0.82104 $ 16,648
23 18 $20,276.39 0.88727 $ 17,991 0.81157 $ 16,456
24 19 $20,276.39 0.88140 $ 17,872 0.80221 $ 16,266
25 20 $20,276.39 0.87556 $ 17,753 0.79296 $ 16,078
26 21 $20,276.39 0.86976 $ 17,636 0.78382 $ 15,893
27 22 $20,276.39 0.86400 $ 17,519 0.77478 $ 15,710
28 23 $20,276.39 0.85828 $ 17,403 0.76584 $ 15,529
29 24 $20,276.39 0.85260 $ 17,288 0.75701 $ 15,349
30 25 $20,276.39 0.84695 $ 17,173 0.74828 $ 15,172
31 26 $20,276.39 0.84134 $ 17,059 0.73965 $ 14,997
32 27 $20,276.39 0.83577 $ 16,946 0.73112 $ 14,824
33 28 $20,276.39 0.83023 $ 16,834 0.72269 $ 14,654
34 29 $20,276.39 0.82474 $ 16,723 0.71436 $ 14,485
35 30 $20,276.39 0.81927 $ 16,612 0.70612 $ 14,318
36 31 $20,276.39 0.81385 $ 16,502 0.69797 $ 14,152
37 32 $20,276.39 0.80846 $ 16,393 0.68993 $ 13,989
38 33 $20,276.39 0.80310 $ 16,284 0.68197 $ 13,828
39 34 $20,276.39 0.79779 $ 16,176 0.67410 $ 13,668
40 35 $20,276.39 0.79250 $ 16,069 0.66633 $ 13,511
41 36 $20,276.39 0.78725 $ 15,963 0.65865 $ 13,355
42 37 $20,276.39 0.78204 $ 15,857 0.65105 $ 13,201
43 38 $20,276.39 0.77686 $ 15,752 0.64354 $ 13,049
44 39 $20,276.39 0.77172 $ 15,648 0.63612 $ 12,898
45 40 $20,276.39 0.76661 $ 15,544 0.62879 $ 12,749
46 41 $20,276.39 0.76153 $ 15,441 0.62153 $ 12,602
47 42 $20,276.39 0.75649 $ 15,339 0.61437 $ 12,457
48 43 $20,276.39 0.75148 $ 15,237 0.60728 $ 12,313
49 44 $20,276.39 0.74650 $ 15,136 0.60028 $ 12,171
50 45 $20,276.39 0.74156 $ 15,036 0.59336 $ 12,031
51 46 $20,276.39 0.73665 $ 14,937 0.58651 $ 11,892
52 47 $20,276.39 0.73177 $ 14,838 0.57975 $ 11,755
53 48 $20,276.39 0.72692 $ 14,739 0.57306 $ 11,620
54 49 $20,276.39 0.72211 $ 14,642 0.56645 $ 11,486
55 50 $20,276.39 0.71732 $ 14,545 0.55992 $ 11,353
56 51 $20,276.39 0.71257 $ 14,448 0.55347 $ 11,222
57 52 $20,276.39 0.70785 $ 14,353 0.54708 $ 11,093
58 53 $20,276.39 0.70317 $ 14,258 0.54077 $ 10,965
59 54 $20,276.39 0.69851 $ 14,163 0.53454 $ 10,838
60 55 $20,276.39 0.69388 $ 14,069 0.52837 $ 10,714
61 56 $20,276.39 0.68929 $ 13,976 0.52228 $ 10,590



PART 1 Forecasting Cash Flows
86
T A B L E 3-10 (continued)

PV of Loan with Market Rate Nominal Rate: ADF, End-of-Year


A B C D E F

4 r 8% r 14%

5 Month Cash Flow PV Factor Present Value PV Factor Present Value

62 57 $20,276.39 0.68472 $ 13,884 0.51626 $ 10,468
63 58 $20,276.39 0.68019 $ 13,792 0.51030 $ 10,347
64 59 $20,276.39 0.67569 $ 13,700 0.50442 $ 10,228
65 60 $20,276.39 0.67121 $ 13,610 0.49860 $ 10,110

66 Totals $1,216,584 49.31843 $1,000,000 42.97702 $871,419

68 X Principal
69 Per $1 of $1 Million

70 ADF @ 8% C66 49.318433
71 Formula for payment 1/ADF 0.02027639 $20,276.39
72 ADF @ 14% E66 42.977016
73 ADF @ 14%/ADF @ 8% F66 0.871419 $871,419
75 Assumptions:
77 Principal $1,000,000
78 Loan payment $20,276.39
79 r Nominal discount rate-annual 8.0%
80 r1 Nominal discount rate-monthly 0.667%
81 r2 Market discount rate 14.0%
82 r3 Market discount rate 1.167%
84 Spreadsheet formulas:
86 E70: (1 1/(1 E80) 60)/E80
87 E71: 1/E70
88 E72: (1 1/(1 E82) 60)/E82
89 E73: E72/E70



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