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32 Grand
33 Time S to In¬nity (n 1) to In¬nity S to n Total
34 t S 1 4.97512 0.10670 4.86842 4.86842
35 PV Factor 1.52088 1.52088 1.52088 1.52088
36 t 0 7.56654 0.16228 7.40426 7.40426
38 Assumptions:
40 S Beginning year of cash ¬‚ows (valuation at t 3.00) 2.00
41 n Ending year of cash ¬‚ows 17.00
42 r Discount rate 15.0%
43 g Growth rate in net inc/cash ¬‚ow 5.1%
44 x (1 g)/(1 r) 0.825217
45 Gordon model multiple GM [1/(r g)] 4.975124

47 Spreadsheet formulas:
49 B34: GM Gordon model for years 2.00 to in¬nity as of t 3.00
50 C34: GM*(x (n S 1)) Gordon model for years 18.00 to in¬nity as of t 3.00
51 D34: B34 C34
52 E34: GM*(1 x (n S 1)) grand total as of t S 1 3.00 years
53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0
54 Row 36: Row 34 * Row 35




CHAPTER 3 Annuity Discount Factors and the Gordon Model 75
T A B L E 3-6

ADF with Cash Flows Starting in Year 2.00 with g r: End-of-Year Formula


A B C D E F G H

5 Cash Flow t S 1 t 0

g)t S
6 t (Yrs) Growth Cash Flow (1 PV Factor PV PV Factor PV
7 2.00 NA 1.00000 1.00000 0.86957 0.86957 1.32250 1.32250
8 1.00 0.20000 1.20000 1.20000 0.75614 0.90737 0.15000 1.38000
9 0.00 0.24000 1.44000 1.44000 0.65752 0.94682 1.00000 1.44000
10 1.00 0.28800 1.72800 1.72800 0.57175 0.98799 0.86957 1.50261
11 2.00 0.34560 2.07360 2.07360 0.49718 1.03095 0.75614 1.56794
12 3.00 0.41472 2.48832 2.48832 0.43233 0.07577 0.65752 1.63611
13 4.00 0.49766 2.98598 2.98598 0.37594 1.12254 0.57175 1.70725
14 5.00 0.59720 3.58318 3.58318 0.32690 1.17135 0.49718 1.78147
15 6.00 0.71664 4.29982 4.29982 0.28426 1.22228 0.43233 1.85893
16 7.00 0.85996 5.15978 5.15978 0.24718 1.27542 0.37594 1.93975
17 8.00 1.03196 6.19174 6.19174 0.21494 1.33087 0.32690 2.02409
18 9.00 1.23835 7.43008 7.43008 0.18691 1.38874 0.28426 2.11209
19 10.00 1.48602 8.91610 8.91610 0.16253 1.44912 0.24718 2.20392
20 11.00 1.78322 10.69932 10.69932 0.14133 1.51212 0.21494 2.29974
21 12.00 2.13986 12.83918 12.83918 0.12289 1.57786 0.18691 2.39974
22 13.00 2.56784 15.40702 15.40702 0.10686 1.64647 0.16253 2.50407
23 14.00 3.08140 18.48843 18.48843 0.09293 1.71805 0.14133 2.61294
24 15.00 3.69769 22.18611 22.18611 0.08081 1.79275 0.12289 2.72655
25 16.00 4.43722 26.62333 26.62333 0.07027 1.87070 0.10686 2.84510
26 17.00 5.32467 31.94800 31.94800 0.06110 1.95203 0.09293 2.96880
27 Pres. value (t 3.00 for column F, t 0 for column H) 26.84876 40.83361
28 Pres. value factor-From S 1 (t 3.00) to 0 1.52088
29 Present Value (t 0) 40.83361
31 Calculation of PV by formulas:
32 Grand
33 Time S to In¬nity (n 1) to In¬nity S to n Total
34 t S 1 20.00000 46.84876 26.84876 26.84876
35 PV Factor 1.52088 1.52088 1.52088 1.52088
36 t 0 30.41750 71.25111 40.83361 40.83361

38 Assumptions:
40 S Beginning year of cash ¬‚ows (valuation at t 3.00) 2.00
41 n Ending year of cash ¬‚ows 17.00
42 r Discount rate 15.0%
43 g Growth rate in net inc/cash ¬‚ow 20.0%
44 x (1 g)/(1 r) 1.043478
45 Gordon model multiple GM [1/(r g)] 20.000000
47 Spreadsheet formulas:

49 B34: GM Gordon model for years 2.00 to in¬nity as of t 3.00
50 C34: GM*(x (n S 1)) Gordon model for years 18.00 to in¬nity as of t 3.00
51 D34: B34 C34
52 E34: GM*(1 x (n S 1)) grand total as of t S 1 3.00 years
53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0
54 Row 36: Row 34 * Row 35




PART 1 Forecasting Cash Flows
76
Special Case: No Growth, g 0
Setting g 0, equation (3-11) reduces to:
1 1 1 1
ADF
r)n S1
r)S 1
r (1 r (1
1 1 1
1 ADF: no growth (3-11c)
r)n S1
r)S 1
r (1 (1
This formula is useful in calculating loan amortization, as the reader can
see in the loan amortization section of the Appendix to this chapter.

Generalized Gordon Model
If we start with cash ¬‚ows at any year other than Year 1, then we have
to use a generalized Gordon model. Letting n ’ in equation (3-11), the
end-of-year formula is:
1 1
PV CF (3-11d)
r)S 1
(r g) (1
This is the formula for the PV of the reversion (the cash ¬‚ows from t
n 1 to in¬nity) that every appraiser uses in every discounted cash ¬‚ow
analysis. This is exactly what appraisers do in calculating the PV of the
reversion, i.e., the in¬nity of time that follows the discounted cash ¬‚ow
forecasts for the ¬rst n years. For example, suppose we do a ¬ve-year
forecast of cash ¬‚ows in a discounted cash ¬‚ow analysis and calculate its
PV. We must then calculate the PV of the reversion, which is the sixth-
year cash ¬‚ow multiplied by the Gordon model and then discounted ¬ve
years to t 0, or:
1 1
PV CF6 (3-11e)
r)5
r g (1
The reason we discount ¬ve years and not six is that after discount-
ing the ¬rst ¬ve years™ cash ¬‚ows to PV, we are standing at the end of
Year 5 looking at the in¬nity of cash ¬‚ows that we forecast to occur be-
ginning with Year 6. The Gordon model requires us to use the ¬rst fore-
cast year™s cash ¬‚ow, which is why we use CF6 and not CF5, but we still
must discount the cash ¬‚ows from the end of Year 5, or ¬ve years. The
¬rst two terms on the right-hand side of equation (3-11d) give us the
formula for the PV of the cash ¬‚ows from Years 6 to in¬nity as of
the end of Year 5, and the ¬nal term on the right discounts that back to
t 0.

Midyear Formula
When the starting period is not in Year 1, the midyear ADF formula is:
nS1
1 r 1 r
1 g 1
ADF
r)S 1
r g 1 r r g (1
nS1
1 r 1 g 1
1 (3-12)
r)S 1
r g 1 r (1
Note that at S 1, the term at the right”outside the brackets”becomes

CHAPTER 3 Annuity Discount Factors and the Gordon Model 77
1 and effectively drops out of the equation, which renders equation
(3-12) equivalent to equation (3-10). The midyear ADF in equation (3-12)
is identical to the end-of-year ADF in equation (3-11), except that we
replace the two Gordon model 1 r terms with the value 1 in the latter.


PERIODIC PERPETUITY FACTORS (PPFs): PERPETUITIES
FOR PERIODIC CASH FLOWS
Thus far, all ADFs and Gordon model perpetuities have been for contig-
uous cash ¬‚ows. In this section we develop perpetuities for periodic cash
¬‚ows that occur only at regular intervals or cycles. To my knowledge,
these formulas are my own creation, and I call them periodic perpetuity
factors (PPFs). PPFs are really Gordon model multiples for periodic (non-
contiguous) cash ¬‚ows and for contiguous cash ¬‚ows that have repeating
patterns.
The example we use here arose in Chapter 2 in dealing with moving
expenses. Every small to midsize company that is growing in real terms
moves periodically. We will assume a move occurs every 10 years, al-
though we will derive formulas that can handle any periodicity. To fur-
ther simplify the initial mathematics, we will assume the last move oc-
curred in the last historical year of analysis. Later we will relax that
assumption to handle different timing of the cash ¬‚ows.
Suppose our subject company moved last year, and the move cost
$20,000. We expect to move every 10 years, and moving costs increase at
g 5% per year. The PPFs are the present values of these periodic cash
¬‚ows for both midyear and end-of-year assumptions.


The Mathematical Formulas
For every $1.00 of forecast moving costs in Year 10, the PV of the lifetime
expected moving costs would be as follows in equation (3-13):
g)10
(1 (1 g)
1
PV (3-13)
r)10 r)20
(1 (1 (1 r)
The $1.00 grows at rate g for 10 years, and we discount it back to PV for
10 years. We follow the same pattern at 20 years, 30 years, etc. to in¬nity.
r)]10, we get:
Multiplying equation (3-13) by [(1 g)/(1
10
g)10 g)20
1 g (1 (1 (1 g)
PV (3-14)
r)20 r)30
1 r (1 (1 (1 r)
Subtracting equation (3-14) from equation (3-13), we get:
10
1 g 1
1 PV (3-15)
r)10
1 r (1
The left-hand side of equation (3-15) simpli¬es to
r)10 g)10
(1 (1
PV
r)10
(1
Multiplying both sides of equation (3-15) by the inverse,



PART 1 Forecasting Cash Flows
78
r)10
(1
r)10 g)10
(1 (1
we come to:
r)10
(1 1
PV (3-16)
r)10 g)10 (1 r)10
(1 (1
r)10 in the numerator and denominator, the so-
Canceling out (1
lution is:
1
PV (3-17)
r)10 g)10
(1 (1
We can generalize this formula to other periods of cash ¬‚ows by
letting cash ¬‚ows occur every j years. The PV of the cash ¬‚ows is the
same, except that we replace each 10 in equation (3-17) with a j in equa-
tion (3-18). Additionally, we rename the term PV as PPF, the periodic
perpetuity factor. Therefore, the PPF for $1 of payment, ¬rst occurring in
year j, is:
1
PPF PPF”end-of-year (3-18)
r) j g) j
(1 (1
The midyear PPF is again our familiar result of 1 r times the
end-of-year PPF, or:
1 r
PPF PPF”midyear (3-19)
r) j g) j
(1 (1
Note that for j 1, equations (3-18) and (3-19) reduce to the Gordon
model. As you will see further below, the above two formulas only work
if the last cash ¬‚ow occurred in the immediate prior year, i.e., t 1. In
the section on other starting years, we generalize these two formulas to
equations (3-18a) and (3-18b) to be able to handle different starting times.


Tables 3-7 and 3-8: Examples of Equations
(3-18) and (3-19)
We begin in Table 3-7 with $1.00 (B5) of moving expenses11 that we fore-
cast to occur in the next move, 10 years from now. The second move,
g)10
which we expect to occur in 20 years, should cost (1 $1.62889
(B6), assuming a 5% (D26) constant growth rate (g) in the cost. We dis-
count cash ¬‚ows at a 20% discount rate (D25).
Column A shows time in 10-year increments going up to 100 years.
Cells B5 to B14 contain the forecast cash ¬‚ows and are equal to (1 g)t j,
where t 10, 20, 30, . . . , 100 years and j 10. Actually, time should
continue to t , but at a 20% discount rate and 5% growth rate, the


11. Another common periodic expense that is less predictable than moving expenses is losses from
lawsuits. Rather than use the actual loss from the last lawsuit, one should use a base-level,
long-run average loss, which will grow at a rate of g.




CHAPTER 3 Annuity Discount Factors and the Gordon Model 79
T A B L E 3-7

Periodic Perpetuity Factor (PPF): End-of-Year Formula


A B C D E F

Cash Flow PV Factor
g)t j r)t
4 t(Yrs) (1 1/(1 PV % PV Cum % PV
5 10 1.00000 0.16151 0.16151 74% 74%
6 20 1.62889 0.02608 0.04249 19% 93%
7 30 2.65330 0.00421 0.01118 5% 98%
8 40 4.32194 0.00068 0.00294 1% 100%
9 50 7.03999 0.00011 0.00077 0% 100%
10 60 11.46740 0.00002 0.00020 0% 100%
11 70 18.67919 0.00000 0.00005 0% 100%
12 80 30.42643 0.00000 0.00001 0% 100%
13 90 49.56144 0.00000 0.00000 0% 100%
14 100 80.73037 0.00000 0.00000 0% 100%
15 Totals 0.21916 100%
17 Calculation of PPF by formula:
19 PPF
20 0.21916
22 Assumptions:
24 j Number of years between moves 10
25 r Discount rate 20.0%
26 g Growth rate in moving costs 5.0%
28 Spreadsheet formulas:
30 A20: 1/((1 r) j (1 g) j) Equation (3-18)




T A B L E 3-8

Periodic Perpetuity Factor (PPF): Midyear Formula


A B C D E F

Cash Flow V Factor
g)t j r)t 0.5)
4 t (Yrs) (1 1/(1 PV % PV Cum % PV

5 10 1.00000 0.17692 0.17692 74% 74%
6 20 1.62889 0.02857 0.04654 19% 93%
7 30 2.65330 0.00461 0.01224 5% 98%
8 40 4.32194 0.00075 0.00322 1% 100%
9 50 7.03999 0.00012 0.00085 0% 100%
10 60 11.46740 0.00002 0.00022 0% 100%
11 70 18.67919 0.00000 0.00006 0% 100%
12 80 30.42643 0.00000 0.00002 0% 100%
13 90 49.56144 0.00000 0.00000 0% 100%
14 100 80.73037 0.00000 0.00000 0% 100%

15 Totals 0.24008 100%
17 Calculation of PPF by formula:
19 PPF
20 0.24008
22 Assumptions:
24 j Number of years between moves 10
25 r Discount rate 20.0%
26 g Growth rate in moving costs 5.0%
28 Spreadsheet formulas:
30 A20: SQRT(1 r)/((1 r) j (1 g) j) Equation (3-19)




PART 1 Forecasting Cash Flows
80
present value factors nullify all cash ¬‚ows after year 40.12 Column C con-
tains a standard present value factor, where

1
PV
r)t
(1

Column D, the present value of the cash ¬‚ows, equals Column B
Column C. Cell D15, the total PV, equals $0.21916 for every $1.00 of mov-
ing expenses in the next move. This is the ¬nal result using the ˜˜brute
force™™ method of scheduling all the cash ¬‚ows and discounting them to
PV. Cell A20 contains the formula for equation (3-18), and the result is
$0.21916, which demonstrates the accuracy of the formula. Note that the
formula for A20 appears at A30.
To calculate the PV of $20,000 of the previous year™s moving expense
growing at 5% per year and occurring every 10 years, we forecast the
cost of the next move by multiplying the $20,000 by 1.0510 $32,577.89.
We then multiply the cost of the next move by the PPF, i.e., $32,577.89
0.21916 (A20) $7,139.83 before corporate taxes. Assuming a 40% tax
rate, that rounds to $4,284 after tax. Since this is an expense, we must
remember to subtract it from”not add it to”the value we calculated
before moving expenses.13 For example, suppose we calculated a mar-
ketable minority interest FMV of $1,004,284 before moving expenses. The
¬nal marketable minority FMV would be $1 million.
Column E shows the percentage of the PV contributed by each move.
Seventy-four percent (E5) of the PV comes from the ¬rst move (Year 10),
and 19% from the second move (Year 20, at E6). Column F shows the
cumulative PV. The ¬rst two moves cumulatively account for 93% (F6) of
the entire PV generated by all moves, and the ¬rst three moves account
for 98% (F7) of the PV. Thus, in most circumstances we need not worry
about the argument that after attaining a certain size a company tends to
not move anymore. As long as it moves at least twice, the PPF will be
accurate.
Table 3-8 is identical to Table 3-7, except that it is testing equation
(3-19), the midyear formula, instead of the end-of-year formula, equation
(3-18). Again C20 D15, which veri¬es the formula.




Other Starting Years
Another question to address is what happens when the periodic expense
occurred before the prior year. Using our moving expense every 10 years
example, suppose the subject company last moved 4 years ago. It will be
another 6 years, not 10 years, to the next move. The easiest way to handle
this situation is ¬rst to value the cash ¬‚ows from a point in time where


12. Of course, at a higher growth rate and the same discount rate, it will take longer for the
present value factors to nullify the growth. The converse is also true.
13. We accomplish this by removing moving expenses from historical costs before developing our
forecast of expenses (see Chapter 2).


CHAPTER 3 Annuity Discount Factors and the Gordon Model 81
we can use the ADF equations in (3-18) and (3-19) and then adjust. Thus,
if we choose t 4 as our temporary valuation date, all cash ¬‚ows will
be spaced every 10 years, and the ADF formulas (3-18) and (3-19) apply.
We then roll forward to t 0 by multiplying the preliminary PPF by
b

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