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(1 r) .
The generalized PPF formulas are:

r)b
(1
PPF generalized PPFвЂ”end-of-year (3-18a)
r) j g) j
(1 (1

The midyear generalized PPF is again our familiar result of 1 r
times the end-of-year PPF, or:

r)b
1 r (1
PPF generalized PPFвЂ”midyear (3-19a)
r) j g) j
(1 (1

Note that for j 1 and b 0, equations (3-18a) and (3-19a) reduce to the
Gordon model.
It is important to roll forward the cash п¬‚ow properly. With the
\$20,000 move occurring 4 years ago, our forecast of the next move is still
1.0510
\$20,000 \$32,577.89. Whether the last move occurred 4 years
ago or yesterday, the forecast cost of the next move is the same 10 years
growth. The present value, and therefore the PPF, is different for the two
different moves, and that is captured in the numerator of the PPF, as we
It is also important to recognize that the valuation date is at t 0,
which is the end of the prior year. Thus, if the valuation date is January
1, 1998, the end of the prior year is December 31, 1997. If the move oc-
curred, for example, in December 1995, then that is 2 years ago and b
2. We would use an end-of-year assumption, which means using the for-
mula in equation (3-18a). If the move occurred in June 1995, we use the
formula in equation (3-19a), and b still equals 2.
Table 3-9 is identical to Table 3-7, except that the expenses occur in
Years 6, 16, . . . instead of 10, 20, . . . . The nominal cash п¬‚ows are identical
to Table 3-7, but the formula that generates them is different. In Table
g)t j. In Table 3-9 the cash п¬‚ows are
3-7 the cash п¬‚ows are equal to (1
g)t j b because the cash п¬‚ows still grow at the rate g for 10
equal to (1
years from the last move, not just the 6 years to the next move. However,
the cash п¬‚ows in Table 3-9 are discounted 6 years instead of 10 years. The
PPF is \$0.45445. The calculation by formula in A20 matches the brute
force calculation in D15, which demonstrates the validity of equation
(3-18a).
Modifying the moving expense example in Table 3-7, the PV of all
moving costs throughout time equals \$20,000 1.62889 \$0.45445
\$14,805.14. Assuming a 40% tax rate, the after-tax present value of the
perpetuity of moving costs is \$8,883, compared to the \$4,284 we calcu-
lated in the discussion of Table 3-7. The present value of moving costs is
higher in this example, because the п¬Ѓrst cash п¬‚ow occurs in Year 6 instead
of Year 10.

PART 1 Forecasting Cash Flows
82
T A B L E 3-9

Periodic Perpetuity Factor (PPF): End-of-YearвЂ”Cash Flows Begin Year 6

A B C D E F

Cash Flow PV Factor
g)t j b r)t
4 t (Yrs) (1 1/(1 PV % PV Cum % PV

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 п¬‚ow 5.0%
27 b Number of years from last cash п¬‚ow 4
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 п¬‚ows. Otherwise, the PV in C20 will appear to be higher than the total of the cash п¬‚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. LetвЂ™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 п¬‚ows for its six-year expected life.
The cash п¬‚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 п¬Ѓrst two years and then begins increasing rapidly in Year 3
or 4.
We can then take the NPV of the cash п¬‚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 п¬Ѓve 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 п¬‚ows п¬Ѓve 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 п¬‚ows are contiguous, which is not true in the
periodic expense example, the cycle and the NPV of the cash п¬‚ows are
periodic. Every six years the operator buys a new cab. We can measure
the NPV of the п¬Ѓrst cab as of t 0. The operator buys the second cab
and uses it from Years 7вЂ“12. Its NPV as of the end of Year 6 (t 6) should
be the same as the NPV at t 0 of the п¬Ѓrst six yearsвЂ™ cash п¬‚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 п¬‚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
п¬Ѓnal 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.

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 п¬Ѓrst 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 п¬Ѓxed 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 deп¬Ѓnitions:
Repeating equation (3-6d), the ADF is:
1
1
r)n
(1
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:
Dividing both sides of the equation by ADFNominal, we get:
Principal 1
Loan Payment Principal (3-21)

Present Value of a Loan
The PV of a loan is the loan payment multiplied by the market rate ADF,
or:
From equation (3-21), the loan payment is the principal divided by the
nominal ADF. Substituting this into equation (3-22) gives us:
PV of Loan Principal (3-23)
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 п¬Ѓve-year loan. The loan payment, calculated using ExcelвЂ™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 monthвЂ™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
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
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%
86 E70: (1 1/(1 E80) 60)/E80
87 E71: 1/E70
88 E72: (1 1/(1 E82) 60)/E82
89 E73: E72/E70

nominal ADF, which is E72 divided by E70 and equals 0.871419. In F73
we multiply E73 by the \$1 million principal to obtain the present value
of the loan of \$871,419. Note that this matches our brute force calculation
in F66, as it should.

RELATIONSHIP OF THE GORDON MODEL
TO THE PRICE/EARNINGS RATIO
In this section, we will mathematically derive the relationship between
the price/earnings (PE) ratio and the Gordon model. The confusion be-
tween the two leads to possibly more mistakes by appraisers than any
other single source of mistakesвЂ”I have seen numerous reports in which
the appraiser used the wrong earnings base. Understanding this section
should clear the potential confusion that exists. First, we will begin with
some deп¬Ѓnitions that will aid in developing the mathematics. All other
deп¬Ѓnitions retain their same meaning as in the rest of the chapter.

Deп¬Ѓnitions
Pt stock price at time t
Et historical earnings in the prior year (usually the prior 12
months)

CHAPTER 3 Annuity Discount Factors and the Gordon Model 87
Et 1 forecast earnings in the upcoming year
b earnings retention rate. Thus, cash п¬‚ow to shareholders equals
(1 b) earnings.
g1 one-year forecast growth rate in earnings, i.e., E t 1/E t 1
PE price/earnings ratio Pt/E t

Mathematical Derivation
We begin with the statement that the market capitalization of a publicly
held п¬Ѓrm is its fair market value, and that is equal to its PE ratio times
the previous yearвЂ™s historical earnings:
Pt
FMV * Et (3-24)
Et
We repeat equation (3-10e) below as equation (3-25), with one
change. We will assume that forecast cash п¬‚ow to shareholders, CFt 1, is
E t 1, where b is the earnings retention rate.14 The
equal to (1 b)
earnings retention rate is the sum total of all the reconciling items be-
tween net income and cash п¬‚ow (see Chapter 1). Now we have an ex-
pression for the FMV of the п¬Ѓrm15 according to the midyear Gordon
model.
1 r
FMV (1 b) E t midyear Gordon model (3-25)
1
(r g)
Substituting Et E t (1 g1) into equation (3-25), we come to:
1

1 r
FMV (1 b) E t (1 g1) (3-26)
(r g)
The left-hand sides of equations (3-24) and (3-26) are the same. There-
fore, we can equate the right-hand sides of those equations.
1 r
Pt
* Et (1 b) E t (1 g1) (3-27)
Et (r g)
E t cancels out on both sides of the equation. Additionally, we use the
simpler notation PE for the price-earnings multiple. Thus, equation (3-27)
reduces to:
1 r
PE (1 b) (1 g1)
r g
relationship of PE to Gordon model multiple (3-28)
The left-hand term is the price-earnings multiple and the right-hand
term is one minus the earnings retention rate times one plus the one-year
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