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Table 3-2: Example of Equation (3-10) through (3-10b)
Table 3-2 is identical to Table 3-1, except that here we use the midyear
rather than end-of-year ADF. Note that the Gordon model multiple (GM)
in B20 and F28 is 10.83213 versus 10.101010 in Table 3-1. The GM in Table
3-2 is exactly 1 r times the GM in Table 3-1, i.e., 10.1010 1.15
10.83213. This demonstrates the validity of equations (3-10) through
(3-10b), the midyear ADF.



Special Cases for Midyear Cash Flows: No Growth, g 0
Letting g 0 in the equation above, we obtain the following ADF for
midyear cash ¬‚ows with no growth:

PART 1 Forecasting Cash Flows
68
T A B L E 3-2

ADF: Midyear Formula


A B C D E F

g)t 1
4 t (Yrs) Cash Flow (CF) Growth in CF (1 PV Factor NPV
5 1 1.00000 0.00000 1.00000 0.93250 0.93250
6 2 1.05100 0.05100 1.05100 0.81087 0.85223
7 3 1.10460 0.05360 1.10460 0.70511 0.77886
8 4 1.16094 0.05633 1.16094 0.61314 0.71181
9 5 1.22014 0.05921 1.22014 0.53316 0.65053
10 6 1.28237 0.06223 1.28237 0.46362 0.59453
11 7 1.34777 0.06540 1.34777 0.40315 0.54335
12 8 1.41651 0.06874 1.41651 0.35056 0.49658
13 9 1.48875 0.07224 1.48875 0.30484 0.45383
14 10 1.56468 0.07593 1.56468 0.26508 0.41476
15 Totals 6.42899
17 Calculation of NPV by formulas:
18 Grand
19 Time 1 to In¬nity (n 1) to In¬nity 1 to n Total
20 NPV 10.83213 4.40314 6.42899 6.42899
22 Assumptions:
24 n Number of years of cash ¬‚ows 10
25 r Discount rate 15.0%
26 g Growth rate in net inc/cash ¬‚ow 5.1%
27 x (1 g)/(1 r) 0.9139
28 Gordon model multiple GM SQRT(1 r)/(r g) 10.83213
30 Spreadsheet formulas:

32 B20: GM SQRT(1 r)/(r G)
33 C20: GM*x n
34 D20 B20 C20
35 E20 GM * (1 x n) This is equation (3-10b)




1 r 1 r
1
ADF midyear ADF, no growth (3-10c)
r)n
r (1 r
This follows the same type of logic as equation (3-6), with modi¬-
cation for growth being zero. The ¬rst and third terms on the RHS of
equation (3-10c) are midyear Gordon models for a constant $1 cash ¬‚ow.
g)n
Since there is no growth of cash ¬‚ows in this special case, the (1
in equation (3-10) simpli¬es to 1 and drops out of the equation. The
r)n discounts the second Gordon model term from t
1/(1 n back to
t 0, i.e., it reduces the PV of the perpetuity to time zero. Again, the
ADF is the difference of two perpetuities: the ¬rst one with cash ¬‚ows
from 1 to in¬nity, less the second one with cash ¬‚ows from n 1 to
in¬nity, the difference being cash ¬‚ows from 1 to n.
We can rewrite equation (3-10c) as equation (3-10d) by factoring out
the 1 r/r.
1 r 1
ADF 1 alternate expression for (3-10c),
r)n
r (1
midyear, no growth (3-10d)

CHAPTER 3 Annuity Discount Factors and the Gordon Model 69
Gordon Model
Letting n ’ in equation (3-10) leads us to the Gordon model.
1 r
PV CF Gordon model”midyear (3-10e)
(r g)
This can be split into the following terms:
1 r
CF
(r g)
The ¬rst term is the forecast net income for the ¬rst year, and the second
term is the Gordon model multiple for a midyear cash ¬‚ow.


STARTING PERIODS OTHER THAN YEAR 1
When cash ¬‚ows begin in any year other than 1, it is necessary to use a
more general (and complicated) ADF formula. We will present formulas
for both the end-of-year and midyear cash ¬‚ows when this occurs.


End-of-Year Formulas
In the following equations, S is the starting year of the cash ¬‚ows. The
end-of-year ADF is:
nS1
1 g
1 1 1
ADF
r)S 1
r g 1 r r g (1
generalized end-of-year ADF (3-11)
Note that when S 1, n S 1 n, and equation (3-11) reduces to
equation (3-6).
The intuition behind this formula is that if we are standing at point
t S 1 looking at the cash ¬‚ows that begin at S and end at n, they
would appear the same as if we were at t 0 looking at a normal series
of cash ¬‚ows that begin at t 1. The only difference is that there are n
cash ¬‚ows in the latter case and n (S 1) n S 1 cash ¬‚ows in
the former case.
Therefore, the term in square brackets, which is the PV of the cash
¬‚ows at t S 1, is the usual ADF formula, except that the exponent
of the second term in square brackets changes from n in equation (3-6)
to n S 1 in equation (3-11). If the cash ¬‚ows begin in a year later
than Year 1, S 1 and there are fewer years of cash ¬‚ows from S to n
than there are from 1 to n.7 From the end of Year S 1 to the end of
Year n, there are n (S 1) n S 1 years.
In order to calculate the PV as of t 0, it is necessary to discount
r)S 1. Note that at S
the cash ¬‚ows S 1 years using the term 1/(1
1, the term at the right”outside the brackets”becomes 1 and effectively


7. The converse is true for cash ¬‚ows beginning in the past, where S is less than 1.




PART 1 Forecasting Cash Flows
70
drops out of the equation. The exponent within the square brackets, n
S 1, simpli¬es to n, and (3-11) simpli¬es to (3-6).
An alternative form of (3-11) with the Gordon model speci¬cally fac-
tored out is:
nS1
1 g
1 1
ADF 1
r)S 1
r g 1 t (1
generalized end-of-year ADF”alternative form (3-11a)


Valuation Date 0
If the valuation date is different than t 0, then we do not discount by
the entire S 1 years. Letting the valuation date v, then we discount
back to t S v 1, the reason being that normally we discount S
1 years, but in this case we will discount only to v, not to zero. Therefore,
we discount S 1 v years, which we restate as S v 1. For example,
if we want to value cash ¬‚ows from t 23 months to 34 months as of t
8
10 months, then we discount 23 10 1 12 months, or 1 year.
This formula is important in calculating the reduction in principal for an
amortizing loan. The formula is:
nS1
1 g
1 1 1
ADF generalized ADF:
r)S v1
r g 1 r r g (1
(3-11b)
end-of-year
where v valuation date. We will demonstrate the accuracy of this for-
mula in Sections 2 and 3 of Table A3-3 in the Appendix.


Table 3-3: Example of Equation (3-11)
In Table 3-3, we begin with $1 of cash ¬‚ows (C7) at t 3.25 years, i.e.,
S 3.25 (G40). The discount rate is 15% (G42), and cash ¬‚ows grow at
5.1% (G43). In Year 4.25, cash ¬‚ow grows 5.1% $1.00 $0.051 (B8),
which is equal to the prior year cash ¬‚ow of $1.00 in C7 plus the growth
in the current year, for a total of $1.051 in C8. We continue in the same
fashion to calculate growth in cash ¬‚ows and the actual cash ¬‚ows
through the last year n 22.25.
g)t S, which
In Column D, we use the formula Cash Flow (1
duplicates the results in Column C. Thus, the formula in Column D is a
general formula for cash ¬‚ow in any period.9
Next, we discount the cash ¬‚ows to present value. In this table we
show both a two-step and a single-step discounting process.


8. We actually do this in Table A3-3 in the Appendix. In the context of loan payments, cash ¬‚ows
are ¬xed, which means g 0. Also, with loan payments we generally deal with time
measured in months, not years. To remain consistent, the discount rates must also be
monthly, not annual.
g)t S g)t 1, which is the formula that
9. Note that when cash ¬‚ows begin at t 1, then (1 (1
g)t S is truly a
describes the cash ¬‚ows in Column D in Tables 3-1 and 3-2. Thus, (1
general formula for the cash ¬‚ow.




CHAPTER 3 Annuity Discount Factors and the Gordon Model 71
T A B L E 3-3

ADF with Cash Flows Starting in Year 3.25: 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 3.25 NA 1.00000 1.00000 0.86957 0.86957 0.63494 0.63494
8 4.25 0.05100 1.05100 1.05100 0.75614 0.79471 0.55212 0.58028
9 5.25 0.05360 1.10460 1.10460 0.65752 0.72629 0.48011 0.53032
10 6.25 0.05633 1.16094 1.16094 0.57175 0.66377 0.41748 0.48467
11 7.25 0.05921 1.22014 1.22014 0.49718 0.60663 0.36303 0.44295
12 8.25 0.06223 1.28237 1.28237 0.43233 0.55440 0.31568 0.40481
13 9.25 0.06540 1.34777 1.34777 0.37594 0.50668 0.27450 0.36997
14 10.25 0.06874 1.41651 1.41651 0.32690 0.46306 0.23870 0.33812
15 11.25 0.07224 1.48875 1.48875 0.28426 0.42320 0.20756 0.30901
16 12.25 0.07593 1.56468 1.56468 0.24718 0.38676 0.18049 0.28241
17 13.25 0.07980 1.64447 1.64447 0.21494 0.35347 0.15695 0.25810
18 14.25 0.08387 1.72834 1.72834 0.18691 0.32304 0.13648 0.23588
19 15.25 0.08815 1.81649 1.81649 0.16253 0.29523 0.11867 0.21557
20 16.25 0.09264 1.90913 1.90913 0.14133 0.26981 0.10320 0.19701
21 17.25 0.09737 2.00649 2.00649 0.12289 0.24659 0.08974 0.18005
22 18.25 0.10233 2.10883 2.10883 0.10686 0.22536 0.07803 0.16455
23 19.25 0.10755 2.21638 2.21638 0.09293 0.20596 0.06785 0.15039
24 20.25 0.11304 2.32941 2.32941 0.08081 0.18823 0.05900 0.13744
25 21.25 0.11880 2.44821 2.44821 0.07027 0.17202 0.05131 0.12561
26 22.25 0.12486 2.57307 2.57307 0.06110 0.15722 0.04461 0.11480
27 Pres. value (t 2.25 for column F, t 0 for column H) 8.43199 6.15687
28 Pres. value factor-discount from S 1 (t 2.25) to 0 0.73018
29 Present value (t 0) 6.15687
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 10.10101 1.66902 8.43199 8.43199
35 PV Factor 0.73018 0.73018 0.73018 0.73018
36 t 0 7.37555 1.21869 6.15687 6.15687
38 Assumptions:
40 S Beginning year of cash ¬‚ows (valuation at t 2.25) 3.25
41 n Ending year of cash ¬‚ows 22.25
42 r Discount rate 15.0%
43 g Growth rate in net inc/cash ¬‚ow 5.1%
44 x (1 g)/(1 r) 0.913913
45 Gordon model multiple GM [1/(r g)] 10.101010

47 Spreadsheet formulas:
49 B34: GM Gordon model for years 3.25 to in¬nity as of t 2.25
50 C34: GM*(x (n S 1)) Gordon model for years 23.25 to in¬nity as of t 2.25
51 D34: B34 C34
52 E34: GM*(1 x (n S 1)) grand total as of t S 1 2.25 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
72
First, we demonstrate two-step discounting in Columns E and F. Col-
umn E contains the present value (PV) factors to discount the cash ¬‚ows
to t S 1, the formula for which is 1/(1 r)t S 1. Column F is the PV
as of t 2.25 Years. The present value of the cash ¬‚ows total $8.43199
(F27). F28 is the PV factor, 0.73018, to discount that result back to t 0
by multiplying it by F27, or $8.43199 0.73018 $6.15687 (F29).
In Columns G and H, we perform the same procedures, the only
difference being that Column G contains the PV factors to discount back
to t 0. Column H is the PV of the cash ¬‚ows, which totals the same
$6.15687 (H27), which is the same result as F29. This demonstrates that
the two-step and the one-step present value calculation lead to the same
results, as long as they are done properly.
Cell B34 contains the Gordon model multiple 10.10101 for cash ¬‚ows
from t S (3.25) to in¬nity, which we can see calculated in G45. C34 is
the Gordon model multiple for t n 1 to in¬nity, discounted to t
S 1. Subtracting C34 from B34, we get the cash ¬‚ows from S to n in
D34, or $8.43199, which also equals F27. Row 35 is the PV factor 0.73018,
and Row 34 Row 35 Row 36, the PV as of t 0. The total for cash
¬‚ows from S 3.25 to n appears in D36 as $6.15687.
In E34 we show the grand total cash ¬‚ows, as per equation (3-11).
The spreadsheet formula for E34 is in A52, where GM is the Gordon
model multiple. The $8.43199 is the total of the cash ¬‚ows from 3.25 to
22.25 as of t 2.25 and corresponds to the term in equation (3-11) in
square brackets. The PV factor 0.73018 is the term in equation (3-11) to
the right of the square brackets, and the one multiplied by the other is
the entirety of equation (3-11). Note that E36 D36 F29 H27, which
demonstrates the validity of equation (3-11).


Tables 3-4 through 3-6: Variations of Table 3-3 with S 0,
Negative Growth, and r g
Tables 3-4 through 3-6 are identical to Table 3-3. The only difference is
that Tables 3-4 through 3-6 have cash ¬‚ows that begin in Year 2, (S
2.00 in G40). Additionally, in Table 3-5 growth is a negative 5.1% (G43),
instead of the usual positive 5.1% in the other tables.
In Table 3-6, r g, so the discount rate is less than the growth rate,
which is impossible for a perpetuity but acceptable for a ¬nite annuity.
Note that the Gordon model multiple is 20 (B34 and G45), which by
itself would be a nonsense result. Nevertheless, it still works for a ¬nite
annuity, as the term for the cash ¬‚ows from n 1 to in¬nity is positive
and greater than the negative Gordon model multiple.10
In all cases, equation (3-11) performs perfectly, with D36 E36
F29 H27.


r)]n
10. This is so because [(1 g)/(1 1, so when we multiply that term by the GM”which is
negative”the resulting term is negative and of greater magnitude than the GM itself. Since
we are subtracting a larger negative from the negative GM, the overall result is a positive
number.




CHAPTER 3 Annuity Discount Factors and the Gordon Model 73
T A B L E 3-4

ADF with Cash Flows Starting in Year 2.00: 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 0.05100 1.05100 1.05100 0.75614 0.79471 0.15000 1.20865
9 0.00 0.05360 1.10460 1.10460 0.65752 0.72629 1.00000 1.10460
10 1.00 0.05633 1.16094 1.16094 0.57175 0.66377 0.86957 1.00951
11 2.00 0.05921 1.22014 1.22014 0.49718 0.60663 0.75614 0.92260
12 3.00 0.06223 1.28237 1.28237 0.43233 0.55440 0.65752 0.84318
13 4.00 0.06540 1.34777 1.34777 0.37594 0.50668 0.57175 0.77059
14 5.00 0.06874 1.41651 1.41651 0.32690 0.46306 0.49718 0.70425
15 6.00 0.07224 1.48875 1.48875 0.28426 0.42320 0.43233 0.64363
16 7.00 0.07593 1.56468 1.56468 0.24718 0.38676 0.37594 0.58822
17 8.00 0.07980 1.64447 1.64447 0.21494 0.35347 0.32690 0.53758
18 9.00 0.08387 1.72834 1.72834 0.18691 0.32304 0.28426 0.49130
19 10.00 0.08815 1.81649 1.81649 0.16253 0.29523 0.24718 0.44901
20 11.00 0.09264 1.90913 1.90913 0.14133 0.26981 0.21494 0.41035
21 12.00 0.09737 2.00649 2.00649 0.12289 0.24659 0.18691 0.37503
22 13.00 0.10233 2.10883 2.10883 0.10686 0.22536 0.16253 0.34274
23 14.00 0.10755 2.21638 2.21638 0.09293 0.20596 0.14133 0.31324
24 15.00 0.11304 2.32941 2.32941 0.08081 0.18823 0.12289 0.28627
25 16.00 0.11880 2.44821 2.44821 0.07027 0.17202 0.10686 0.26163
26 17.00 0.12486 2.57307 2.57307 0.06110 0.15722 0.09293 0.23910
27 Pres. value (t 2.25 for column F, t 0 for column H) 8.43199 12.8240
28 Pres. value factor-from S 1 (t 3.00) to 0 1.52088
29 Present value (t 0) 12.82400
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 10.10101 1.66902 8.43199 8.43199
35 PV factor 1.52088 1.52088 1.52088 0.73018
36 t 0 15.36237 2.53838 12.82400 12.82400
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) 0.913913
45 Gordon model multiple GM [1/(r g)] 10.101010

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
74
T A B L E 3-5

ADF with Cash Flows Starting in Year 2.00 with Negative Growth: 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 0.05100 0.94900 0.94900 0.75614 0.71758 0.15000 1.09135
9 0.00 0.04840 0.90060 0.90060 0.65752 0.59216 1.00000 1.90060
10 1.00 0.04593 0.85467 0.85467 0.57175 0.48866 0.86957 0.74319
11 2.00 0.04359 0.81108 0.81108 0.49718 0.40325 0.75614 0.61329
12 3.00 0.04137 0.76972 0.76972 0.43233 0.33277 0.65752 0.50610
13 4.00 0.03926 0.73046 0.73046 0.37594 0.27461 0.57175 0.41764
14 5.00 0.03725 0.69321 0.69321 0.32690 0.22661 0.49718 0.34465
15 6.00 0.03535 0.65785 0.65785 0.28426 0.18700 0.43233 0.28441
16 7.00 0.03355 0.62430 0.62430 0.24718 0.15432 0.37594 0.23470
17 8.00 0.03184 0.59246 0.59246 0.21494 0.12735 0.32690 0.19368
18 9.00 0.03022 0.56225 0.56225 0.18691 0.10509 0.28426 0.15983
19 10.00 0.02867 0.53357 0.53357 0.16253 0.08672 0.24718 0.13189
20 11.00 0.02721 0.50636 0.50636 0.14133 0.07156 0.21494 0.10884
21 12.00 0.02582 0.48054 0.48054 0.12289 0.05906 0.18691 0.08982
22 13.00 0.02451 0.45603 0.45603 0.10686 0.04873 0.16253 0.07412
23 14.00 0.02326 0.43277 0.43277 0.09293 0.04022 0.14133 0.06116
24 15.00 0.02207 0.41070 0.41070 0.08081 0.03319 0.12289 0.05047
25 16.00 0.02095 0.38976 0.38976 0.07027 0.02739 0.10686 0.04165
26 17.00 0.01988 0.36988 0.36988 0.06110 0.02260 0.09293 0.03437
27 Pres. value (t 2.25 for column F, t 0 for column H) 4.86842 7.40426
28 Pres. value factor-from S 1 (t 3.00) to 0 1.52088
29 Present value (t 0) 7.40426
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 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
(1 r) .
The generalized PPF formulas are:

r)b
(1
PPF generalized PPF”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 PPF”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 ¬‚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
have already discussed.
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 ¬‚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 ¬‚ows are
3-7 the cash ¬‚ows are equal to (1
g)t j b because the cash ¬‚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 ¬‚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 ¬rst cash ¬‚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-Year”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 ¬‚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




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 mistakes”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 de¬nitions that will aid in developing the mathematics. All other
de¬nitions retain their same meaning as in the rest of the chapter.

De¬nitions
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 ¬‚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 ¬rm is its fair market value, and that is equal to its PE ratio times
the previous year™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 ¬‚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 ¬‚ow (see Chapter 1). Now we have an ex-
pression for the FMV of the ¬rm15 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
growth rate times the midyear Gordon model multiple. In reality, inves-


14. I wish to thank Larry Kasper for pointing out the need for this.
15. Assuming the present value of the cash ¬‚ows of the ¬rm is its FMV. This ignores valuation
discounts, an acceptable simpli¬cation in this limited context.




PART 1 Forecasting Cash Flows
88
tors do not expect constant growth to perpetuity. They usually have ex-
pectations of uneven growth for a few years and a vague, long-run ex-
pectation of growth thereafter that they approximate as being constant.
Therefore, we should look at g, the perpetual growth rate in cash ¬‚ow,
as an average growth rate over the in¬nite period of time that we are
modeling.
We should be very clear that the earnings base in the PE multiple
and the Gordon model are different. The former is the immediate prior
year and the latter is the ¬rst forecast year. When an appraiser develops
PE multiples from guideline companies, whether publicly or privately
owned, he should multiply the PE multiple from the guideline companies
(after appropriate adjustments) by the subject company™s prior year earn-
ings. When using a discounted cash ¬‚ow approach, the appraiser should
multiply the Gordon model by the ¬rst forecast year™s earnings. Using the
wrong earnings will cause an error in the valuation by a factor of one
plus the forecast one-year growth rate.


CONCLUSIONS
We can see that there is a family of annuity discount factors (ADFs), from
the simplest case of an ordinary annuity to the most complicated case of
an annuity with stub periods (fractional years), as discussed in the Ap-
pendix. The elements that determine which formula to use are:
— Whether the cash ¬‚ows are midyear versus end-of-year.
— When the cash ¬‚ows begin (Year 1 versus any other time).
— If they occur every year or at regular, skipped intervals (or have
repeating cycles).
— Whether or not the constant growth is zero.
— Whether there is a stub period.
For cash ¬‚ows without a stub period, the ADF is the difference of
two Gordon model perpetuities. The ¬rst term is the perpetuity from S
to in¬nity, where S is the starting year of the cash ¬‚ow. The second term
is the perpetuity starting at n 1 (where n is the ¬nal cash ¬‚ow in the
annuity) going to in¬nity. For cash ¬‚ows with a stub period, the preced-
ing statement is true with the addition of a third term for the single cash
¬‚ow of the stub period itself, discounted to PV.
While this chapter contains some complex algebra, the focus has been
on the intuitive explanation of each ADF. The most dif¬cult mathematics
have been moved to the Appendix, which contains the formulas for ADF
with stub periods and some advanced material on the use of ADFs in
calculating loan amortization. ADFs are also used for practical applica-
tions in Chris Mercer™s quantitative marketability discount model (see
Chapter 7), periodic expenses such as moving costs and losses from law-
suits, ESOP valuation, in reducing a seller-subsidized loan to its cash
equivalent price in Chapter 10, and to calculate loan payments.
We have performed a rigorous derivation of the PE multiple and the
Gordon model. This derivation demonstrates that the PE multiple equals
one minus the earnings retention rate times one plus the one-year growth



CHAPTER 3 Annuity Discount Factors and the Gordon Model 89
T A B L E 3-11

ADF Equation Numbers


With Growth No Growth

Formulas in the Chapter End-of-Year Midyear End-of-Year Midyear

Ordinary ADF (3-6) to (3-6b) (3-10) to (3-10b) (3-6d) (3-10c) & (3-10d)
Gordon model (3-7) (3-10e)
Starting cash ¬‚ow not t 1 (3-11) & (3-11a) (3-12) (3-11c)
Valuation date v (3-11b)
Gordon model for starting CF not 1 (3-11d)
Periodic expenses (3-18) (3-19)
Periodic expenses-¬‚exible timing (3-18a) (3-19a)
Loan payment (3-21)
Relationship of Gordon model to PE (3-28)
Formulas in the Appendix
ADF with stub period (A3-3) (A3-4)
Amortization of loan principal (A3-10)
PV of loan after-tax (A3-24) & (A3-25)




rate times the midyear Gordon model multiple. Furthermore, we showed
how the former uses the prior year™s earnings, while the latter uses the
¬rst forecast year™s earnings. Many appraisers have found that confusing,
and hopefully this section of the chapter will do much to eliminate that
confusion.
Because there are so many ADFs for different purposes and assump-
tions, we include Table 3-11 to point the reader to the correct ADF equa-
tion.


BIBLIOGRAPHY
Gordon, M. J., and E. Shapiro. 1956. ˜˜Capital Equipment Analysis: The Required Rate of
Pro¬t,™™ Management Science 3: 102“110.
Gordon, M. J. 1962. The Investment, Financing, and Valuation of the Corporation, 2d ed.
Homewood, Ill.: R. D. Irwin.
Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing and Supporting
Marketability Discounts in the Appraisal of Closely Held Business Interests. Memphis,
Tenn.: Peabody.
Williams, J. B. The Theory of Investment Value. 1938. Cambridge, Mass.: Harvard University
Press.



APPENDIX
INTRODUCTION
This appendix is an extension of the material developed in the chapter.
The topics that we cover are:
— Developing ADFs for cash ¬‚ows that end on a fractional year
(stub period).
— Developing ADFs for loan mathematics, consisting of calculating
the amortization of principal in loans and the net after-tax cost of
a loan.

PART 1 Forecasting Cash Flows
90
This appendix is truly for the mathematically brave. The topics cov-
ered and formulas developed are esoteric and less practically useful than
the formulas in the chapter, though the formula for the after-tax cost of
a loan may be useful to some practitioners. The material in this appendix
is included primarily for reference. Nevertheless, even those not com-
pletely comfortable with the dif¬cult mathematics can bene¬t from fo-
cusing on the verbal explanations before the equations and the develop-
ment of the ¬rst one or two equations in the derivation of each of the
formulas. The rest is just the tedious math, which can be skipped.


THE ADF WITH STUB PERIODS (FRACTIONAL YEARS)
We will now develop a formula to handle annuities that have stub peri-
ods, constant growth in cash ¬‚ows, and cash ¬‚ows that start at any time.
To the best of my knowledge, I invented this formula. In this section we
will assume midyear cash ¬‚ows and later present the formula for end-
of-year cash ¬‚ows.
Let™s begin with constructing a timeline of the cash ¬‚ows in Figure
A3-1, using the following de¬nitions and assumptions:


De¬nitions
S time (in years) of the ¬rst cash ¬‚ow for end-of-year cash
¬‚ows. For midyear cash ¬‚ows, S end of the year in which the
¬rst cash ¬‚ow occurs 3.25 years in this example, which means
the cash ¬‚ow for that year begins at t 2.25 years and we assume
the cash ¬‚ow occurs in the middle of the year, or S 0.5
3.25 0.5 2.75 years.
n end of the last whole year™s cash ¬‚ows 12.25 years in this
example
z end of the stub period 12.60 years.
p proportion of a full year represented by the stub period
z n 12.60 12.25 0.35 years
g constant growth rate in cash ¬‚ows 5.1%
t point in time, measured in years


The Cash Flows
We assume the ¬rst cash ¬‚ow of $1.00 (Figure A3-1, cell C4) occurs during
year S (S is for starting cash ¬‚ow), where t 2.25 to t 3.25 years. For


F I G U R E A3-1

Timeline of Cash Flows


Row \ Col. B C D E F G H
1 Year (numeric) 3.25 4.25 5.25 ¦ 12.25 12.60
2 Year (symbolic) S S+1 S+2 ¦ n z
g(1+g)n-S-1
3 Growth (in $) 0 g g(1+g) ¦ NA
(1+g)2 (1+g)n-S p(1+g)n-S+1
4 Cash Flow 1 1+g ¦


CHAPTER 3 Annuity Discount Factors and the Gordon Model 91
simplicity, we denote that the cash ¬‚ow is for the year ending at t 3.25
years (cell C1). Note that for Year 3.25, there is no growth in the cash
¬‚ow, i.e., cell B3 0.
The following year is 4.25 (cell D1), or S 1 (cell D2). The $1.00
grows at a rate of g (cell D3), so the ending cash ¬‚ow is 1 g (cell D4).
tS
g)4.25 3.25.
Note that the ending cash ¬‚ow is equal to (1 g) (1
For Year 5.25, or S 2 (cell E2), growth in cash ¬‚ows is g times the
prior year™s cash ¬‚ow of (1 g), or g (1 g) (cell E3), which leads to a
cash ¬‚ow equal to the prior year™s cash ¬‚ow plus this year™s growth, or
(1 g) g(1 g) (1 g) (1 g) (1 g)2 [cell E4]. Again, the cash
g)t S (1 g)5.25 3.25.
¬‚ow equals (1
For the year 6.25, or S 3, which is not shown in Figure A3-1, cash
g)2, so cash ¬‚ows are (1 g)2 g)2 g)2
¬‚ows grow g(1 g(1 (1
g)3 (1 g)t S g)6.25 3.25.
(1 g) (1 (1
We continue in this fashion through the last whole year of cash ¬‚ows,
which we call Year n (Column G). In our example, n 12.25 years (cell
nS
G1). The cash ¬‚ows during Year n are equal to (1 g) [cell G4].
Had we completed one more full year, the cash ¬‚ows would have
extended to Year 13.25, or Year n 1. If so, the cash ¬‚ow would have
nS1
been (1 g) . However, since the stub year™s cash ¬‚ow is only for a
partial year, the ending cash ¬‚ow is multiplied by p”the fractional por-
g)n S 1.
tion of the year”leading to an ending cash ¬‚ow of p(1
It is important to recognize that there may be other ways of speci-
fying how the partial year affects the cash ¬‚ows. For example, it is pos-
sible, but very unlikely, that the cash ¬‚ows can be based on a legal doc-
ument that speci¬es that only the growth rate itself will be fractional, but
the corpus of the cash ¬‚ow will not diminish for the partial year. We
could calculate a solution to this ADF, but we will not, as it is very un-
likely to be of any practical use and we have already demonstrated how
to model the most likely method of splitting the cash ¬‚ows in the frac-
tional year. The point is that modeling the fractional year cash ¬‚ows de-
pends on the agreement and/or the underlying scenario, and one should
not blindly charge off into the sunset applying a formula developed un-
der an assumption that does not apply in another case.


Discounting Periods
The ¬rst cash ¬‚ow occurs during the year that spans from
t 2.25 to t 3.25. We assume the cash ¬‚ows occur evenly throughout
the year, which is tantamount to assuming all cash ¬‚ows occur on average
halfway through the year, i.e., at Year 2.75. Therefore as of time zero,
de¬ned as t 0, the ¬rst $1 cash ¬‚ow has a present value of
1 1
r)2.75 r)S 0.5
(1 (1
We will be discounting the cash ¬‚ows in two stages because that will
later enable us to provide a more intuitive explanation of our results. Our
¬rst discounting of cash ¬‚ows will be to t S 1, the beginning of
the ¬rst year of cash ¬‚ows. The ¬rst year™s cash ¬‚ow then receives a dis-



PART 1 Forecasting Cash Flows
92
r)0.5, the second year™s cash ¬‚ows receive a discount
count of 1/(1
r)1.5, etc. Thus, the denominators here are identical to those
of 1/(1
for cash ¬‚ows that would begin in Year 1 instead of S.


The Equations
The PV of our series of cash ¬‚ows as of t S 1 is:
(1 g)
1
PV
r)0.5 r)1.5
(1 (1
g)n S
g)n S 1

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