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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:

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