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. 5
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(1 p(1
... (A3-1)
r)n S 0.5
r)n S 1 0.5p
(1 (1
Note that the exponent in the denominator of the last term (the frac-
tional year) is equal to the one before it (the last whole year) plus 1„2 year
to bring us to the end of Year n, plus 1„2 of the fractional year, thus main-
taining a midyear assumption.
We already have a solution to the PV of the whole years in the body
of the chapter”equation (3-10). Thus, the PV of the entire series of cash
¬‚ows as of t S 1 is equation (3-10) plus the ¬nal term in equation
(A3-1), or:
nS1
g)n S 1
1 r 1 r
1 g p(1
NPV (A3-2)
r)n S 1 0.5p
r g 1 r r g (1
The next step is to discount the PV from t S 1 to t 0. We do
S1
this by multiplying by 1/(1 r) . The result is our annuity discount
factor for midyear cash ¬‚ows with a stub period.
nS1
1 r 1 r
1 g
NPV
r g 1 r r g
g)n S 1
p(1 1
(A3-3)
r)n S 1 0.5p r)S 1
(1 (1
The ADF formula for end-of-year cash ¬‚ows with a stub period is:
nS1
1 g
1 1
ADF
r g 1 r r g
g)n S 1
p(1 1
(A3-4)
r)(z S 1) r)S 1
(1 (1
The individual terms in equation (A3-4) have the same meaning as
in the midyear cash ¬‚ows of equation (A3-3). To easily see the derivation
of the end-of-year (EOY) model from the midyear, note that an EOY
model in equation (A3-1) would require the exponent in each denomi-
nator to be 0.5 years larger, which changes the 1 r term in equation
(A3-3) to 1. 1/(r g) is the EOY Gordon model formula. The only other
difference is the discount factor in the rightmost term in the braces
of equations (A3-3) and (A3-4). In the former, we discount the stub pe-



CHAPTER 3 Annuity Discount Factors and the Gordon Model 93
r)n S 1 0.5p
riod cash ¬‚ow by (1 , while in the latter we discount by
r)(z S 1).
(1

Tables A3-1 and A3-2: Example of Equations [A3-3]
and [A3-4]
Table A3-1 is an example of the midyear ADF with a fractional year cash
¬‚ow, and Table A3-2 is an example using end-of-year cash ¬‚ows. Table
A3-2 has the identical structure and meaning as Table A3-1”merely us-
ing end-of-year formulas rather than midyear. Therefore, we will explain
only Table A3-1.
In the ¬rst part of Table A3-1, we will use a ˜˜brute force™™ method of
scheduling out the cash ¬‚ows, calculating their present values, and then
summing them. Later we will directly test the formulas and demonstrate
they produce the same result as the brute force method.

Brute Force Method of Calculating PV of Cash Flows
Rows 7 through 17 in Table A3-1 are a detailed listing of the cash ¬‚ows
and their present values each year. The ¬rst cash ¬‚ows begin in Row 7
at Year 2.25 and ¬nish at t 3.25, with Year 2.75 as the midpoint from
which we discount. We will refer to the years by the ending year, i.e., the
cash ¬‚ow in Row 7 is for the year ending at t 3.25. Assumptions of the
model begin in Row 33.
We begin with $1.00 of cash ¬‚ow for the year ending at t 3.25 (C7).
Column B shows the growth in cash ¬‚ows and is equal to g 5.1%
multiplied by the previous period™s cash ¬‚ow. In B8 the calculation is
$1.00 5.1% $0.051. The cash ¬‚ow in C8 is C7 B8, or $1.00 $.051
$1.051. We repeat this pattern through Row 16, the last whole year™s
cash ¬‚ow.
Column D replicates Column C using the formula cash ¬‚ow
g)t S for all cells except D17, which is the fractional year cash ¬‚ow.
(1
g)n S 1, where multiplying by p 0.35
The formula for that cell is p(1
years converts what would have been the cash ¬‚ow for the whole year
n 1 (and would have been $1.64447) into the fractional year cash ¬‚ow
of $0.57557.16 Note that in that formula, n 12.25 years, the last whole
year.
We show the present values of the cash ¬‚ows as of t S 1 in
Columns E and F and the present values as of t 0 in Columns G and
H. The discount rate is 15% (G36).
Column E contains the present value factors (PVFs), and its formula
17
is
1
PVF
r)t S 0.5
(1
Column F is Column C (or Column D, as the results are identical) times

16. See cell A45 for the formula in the spreadsheet.
17. The intuition behind the exponent is that we are discounting from t to S 1, which is equal to
1 years. Using a midyear convention, we always discount from 1„2
t (S 1) t S
year earlier than end-of-year, which reduces the exponent to t S 0.5. The 0.5 reverts to
1 in the end-of-year formula.




PART 1 Forecasting Cash Flows
94
T A B L E A3-1

ADF with Fractional Year: Midyear Formula


A B C D E F G H

5 Cash Flows t S 1 t 0

g)t S
r)t S 0.5
r)t 0.5
6 t (Yrs) Growth Cash Flow (1 PVF 1/(1 PV PVF 1/(1 PV
7 3.25 NA 1.00000 1.00000 0.93250 0.93250 0.68090 0.68090
8 4.25 0.05100 1.05100 1.05100 0.81087 0.85223 0.59208 0.62228
9 5.25 0.05360 1.10460 1.10460 0.70511 0.77886 0.51486 0.56871
10 6.25 0.05633 1.16094 1.16094 0.61314 0.71181 0.44770 0.51975
11 7.25 0.05921 1.22014 1.22014 0.53316 0.65053 0.38930 0.47501
12 8.25 0.06223 1.28237 1.28237 0.46362 0.59453 0.33853 0.43412
13 9.25 0.06540 1.34777 1.34777 0.40315 0.54335 0.29437 0.39674
14 10.25 0.06874 1.41651 1.41651 0.35056 0.49658 0.25597 0.36259
15 11.25 0.07224 1.48875 1.48875 0.30484 0.45383 0.22259 0.33138
16 12.25 0.07593 1.56468 1.56468 0.26508 0.41476 0.19355 0.30285
17 12.60 NA 0.57557 0.57557 0.24121 0.13883 0.17613 0.10137
18 Totals for whole years 3.25 12.25 6.42899 4.69432
19 Add fractional year 12.60 0.13833 0.10137
20 Grand total (t S 1 in Column G and t 0 in Column I) 6.56782 4.79469
21 Present value factor-discount from S 1 (t 2.25) to 0 0.73018
22 Grand total (t 0) 4.79569
24 Calculation of PV by formulas:
25 Grand
26 Whole Yrs Frac Yr Total Total
27 t S 1 6.42899 0.13883 6.56782
28 PV Factor 0.73018 0.73018
29 t 0 4.69432 0.10137 4.79469 4.79569

31 Assumptions:
33 S Beginning year of cash ¬‚ows (valuation at t 2.25) 3.25
34 n Ending year of cash ¬‚ows-whole year 12.25
35 z Ending year of cash ¬‚ows-stub year 12.60
36 r Discount rate 15.0%
37 g growth rate in cash ¬‚ow 5.1%
38 p proportion of year in the stub period 0.35
39 Midpoint n 0.5 p midpoint of the fractional year 12.425
40 x (1 g)/(1 r) 0.913913
41 Gordon model multiple GM Sqrt (1 r)/(r g) 10.832127
43 Spreadsheet Formulas:
45 C17, D17: p*(1 g) (n s 1) stub period cash ¬‚ow
46 E17: 1/(1 r) (n S 1 0.5*p) stub period present value factor at t 2.25
47 G17: 1/(1 r) (n 0.5*p) stub period present value factor for t 0
48 B27: GM*(1 x (n S 1)) ADF for years 3.25 to 32.25 at t 2.25
49 C27: p*(1 g) (n S 1)/(1 r) (n S 1 0.5*p) PV of stub period CF at t 2.25
50 B28, C28: 1/(1 r) (S 1) present value factor at t S 1 2.25
51 E29: (GM*(1 x (n S 1)) p*(1 G) (n S 1)/(1 r) (n S 1 0.5*p))*(1/(1 r) (S 1))

Note: E29 is the formula for the Grand Total




CHAPTER 3 Annuity Discount Factors and the Gordon Model 95
T A B L E A3-2

ADF with Fractional Year: Midyear Formula


A B C D E F G H

5 Cash Flows t S 1 t 0

g)t S
r)t S1
r)t
6 t (Yrs) Growth Cash Flow (1 PVF 1/(1 PV PVF 1/(1 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 12.60 NA 0.57557 0.57557 0.23538 0.13548 0.17187 0.09892
18 Totals for whole years 3.25 22.25 5.99506 4.37747
19 Add fractional year 22.60 0.13548 0.09892
20 Grand total (t S 1 in Column G and t 0 in Column H) 6.13054 4.47640
21 Present value factor-discount from S 1 (t 2.25) to 0 0.73018
22 Grand total (t 0) 4.47640
24 Calculation of PV by formulas:
25 Grand
26 Whole Yrs Frac Yr Total Total
27 t S 1 5.99506 0.13548 6.13054
28 PV Factor 0.73018 0.73018
29 t 0 4.33747 0.09892 4.447640 4.47640

31 Assumptions:
33 S Beginning year of cash ¬‚ows (valuation at t 2.25) 3.25
34 n Ending year of cash ¬‚ows-whole year 12.25
35 z Ending year of cash ¬‚ows-stub year 12.60
36 r Discount rate 15.0%
37 g Growth rate in cash ¬‚ow 5.1%
38 p Proportion of year in the stub period 0.35
39 This row is not used
40 x (1 g)/(1 r) 0.913913
41 Gordon model multiple GM 1/(r g) 10.101010
43 Spreadsheet formulas:
45 C17, D17: p*(1 g) (n s 1) stub period cash ¬‚ow
46 E17: 1/(1 r) (z S 1) stub period present value factor at t 2.25
47 G17: 1/(1 r) z stub period present value factor for t 0
48 B27: GM*(1 x (n S 1)) ADF for years 3.25 to 32.25 at t 2.25
49 C27: p*(1 g) (n S 1)/(1 r) (z S 1) PV of stub period CF at t 2.25
50 B28, C28: 1/(1 r) (S 1) present value factor at t S 1 2.25
51 E29: (GM*(1 x (n S 1)) p*(1 g) (n S 1)/(1 r) (z S 1))/(1 r) (S 1)

Note: E29 is the formula for the Grand Total




PART 1 Forecasting Cash Flows
96
Column E. The only exception to the PVF formula is cell E17, the frac-
tional year. Its formula is

1
PVF n S 1 0.5p
(1 r)

(in the EOY formula, the exponent is z S 1). This formula appears
in the spreadsheet at A46. The total present value at t 2.25 of the cash
¬‚ows from t 3.25 through t 12.25 is $6.42899 (F18). The present value
of the fractional year cash ¬‚ow is $0.13883 (F19), for a total of $6.56782
(F20). In F21 we show the present value factor of 0.73018 to discount from
0.18 Multiplying G20 by G21, we come to the PV of the
t 2.25 to t
cash ¬‚ows in F22 at t 0 of $4.79569 for each $1.00 of starting cash ¬‚ows.
Thus, if our annuity were actually $100,000 at the beginning, with all
other assumptions remaining the same, the PV would be $479,569.
Column G contains the present value factors for t 0, the formula
of which is the more usual

1
PVF
r)t 0.5
(1

When we multiply Column D by Column G to get Column H, the latter
is the PV of the cash ¬‚ows as of time zero. Note that the ¬nal sum in
H20 is identical to F22, as it should be.
So far we have come to the PV of the cash ¬‚ows using the brute force
method. In the next section we will test the formulas in the preceding
pages to see if they produce the same result.

Testing Equations (A3-3) and (A3-4)
Cell B27 contains the formula for the PV of the ¬rst 10 whole years of
cash ¬‚ows (see A48 for the spreadsheet formula). It is the same as equa-
tion (A3-2) without the rightmost term.19 The result of $6.42899 in B27
matches F18, thereby demonstrating the accuracy of that portion of equa-
tion (A3-2).
Cell C27 is calculated using the rightmost term in equation (A3-2)
and comes to $0.13883 (see A49 for the spreadsheet formula), which
matches F19, thus proving that portion of the formula. The sum of the
two is $6.56782 (D27), which matches F20.
Row 29 is the result of multiplying Row 27 by Row 28, the latter of
which is the present value factor to discount the cash ¬‚ows from t 2.25
to t 0 (it is the same as F21). We total B29 and C29 to $4.79569 (D29),
which matches F22 and H20. Finally, in E29 we use the complete formula
in equation (A3-3) to produce the same result of $4.79569 (see cell A51
for the spreadsheet formula). Thus, we have demonstrated the accuracy


r)S 1 1/1.152.25
18. This is 1/(1 0.73018 (see formulas in cell A50).
19. The formulas are the same; however, in the spreadsheet we have substituted GM (Gordon
multiple) for 1 r/(r g) and x for (1 g)/(1 r). Additionally, we have factored out
the GM.




CHAPTER 3 Annuity Discount Factors and the Gordon Model 97
of equation (A3-3) as a whole as well as showing how we can calculate
the parts.
Table A3-2 is identical to Table A3-1, except that we use end-of-year
present values, and equation (A3-4) is the relevant ADF formula. The end-
of-year formula gives a grand total of 4.47640 (F22, H20, D29, and E29).


TABLE A3-3: LOAN AMORTIZATION
In the chapter we demonstrated how ADFs are useful in calculating loan
payments and the present value of a loan. This section on loan amorti-
zation complements the material we presented in the chapter.
The amortization of loan principal in any time period is the PV of
the loan at the beginning of the period, less the PV at the end of the
period. While this is conceptually easy, it is a cumbersome procedure.
Let™s develop some preliminary results that will lead us to a more ef¬cient
way to calculate loan amortization.


Section 1: Traditional Loan Amortization Schedule
Table A3-3 is a loan amortization schedule that is divided into three sec-
tions. Section 1 is a traditional amortization schedule for a $1 million loan
at 10% for 5 years. The loan begins on February 28, 1998 (B7), and the
¬rst payment is on March 31, 1998 (B8). During the calendar year 1998
there will be 10 payments, leaving 50 more. There will be 12 monthly
payments in each of the years 1999“2002, and the ¬nal two payments are
in the beginning of 2003, with February 28, 2003 (B67), being the ¬nal
payment.
Column A is the payment number. There are 60 months of the loan,
hence 60 payments. Columns D and E are the interest and principal for
the particular payment, while columns G and H are interest and principal
cumulated in calendar year totals. Because the loan payments begin on
March 31, 1998, the ¬rst year™s totals in columns G and H are totals for
the ¬rst 10 payments only. Column I is the present value factor (PVF) at
10%, and column J is the present value of each loan payment. Column K
is the sum of the present values of the loan payments by calendar year.
Note that the PV of the loan payments sum to $1 million (J68).


Section 2: Present Values of Yearly Loan Payment
In Section 2 we calculate the present value of each year™s loan payment
using the ADF equation for no growth, no stub period, and end-of-year
cash ¬‚ows. We could use equation (3-11b) from the chapter, but ¬rst we
will simplify it further by setting g 0, so equation (3-11b) reduces to:
1 1 1 1
ADF
r)n S1
r)S v1
r (1 r (1
1 1 1
1 (A3-5)
r)n S1
r)S v1
r (1 (1
Cells D77 through D82 list the PV of the various calendar years™ cash



PART 1 Forecasting Cash Flows
98
¬‚ows discounted to the inception of the loan, February 28, 1998. Note
that these amounts exactly match those in column K of Section 1, and the
total is exactly $1 million”the principal of the loan”as it should be. This
demonstrates the accuracy of equation (A3-5), as all amounts calculated
in D77 through D82 use that equation (note that v, the valuation date in
months, appears in Row 86).
In Column E we are viewing the cash ¬‚ows from January 1, 1999,
i.e., immediately after the last payment in 1998 and one month before the
¬rst payment in 1999. Therefore, the 1998 cash ¬‚ows drop out entirely
and the PV of the 1999“2003 cash ¬‚ows increase relative to column D
because we discount the cash ¬‚ows 10 months less. The difference be-
tween the sum of the 1998 PVs discounted to February 28, 1998, and the
1999 payments discounted to January 1, 1999,20 is $1 million (D84)
$865,911 (E84) $134,089 (E85). We follow the same procedure each year
to calculate the difference in the PVs (Row 85), and ¬nally we come to a
total of the reductions in PV of $1 million, in K85, which is identical with
the original principal of the loan.
There are some signi¬cant numbers that repeat in southeasterly-
sloped diagonals in Section 2. The PV $241,675 appears in cells E78, F79,
G80, and H81. This means that the 1999 payments as seen from the be-
ginning of 1999 have the same PV as the 2000 payments as seen from the
beginning of 2000, etc. through 2002. Similarly, the PV of $218,767 repeats
in cells E79, F80, and G81. The interpretation of this series is the same as
before, except everything is moved back one year, i.e., the 2000 payments
as seen from the beginning of 1999 have the same PV as the 2001 pay-
ments as seen from the beginning of 2000 and the 2002 payments as seen
from the beginning of 2001.
This downward-sloping pattern gives us a clue to a more direct for-
mula for loan amortization. At the start of the loan, we have 60 payments
of $21,247. In the ¬rst calendar year, 10 payments will be made, for a total
of $212,470. At the end of the ¬rst year, which effectively is the same as
January 1, 1999, 50 payments will remain. The PV of the ¬nal 50 payments
discounted to January 1, 1999, is the same as the PV of the ¬rst 50 pay-
ments discounted to March 1, 1998 (using March 1 synonymously with
February 28 in a present value sense), because the entire time line will
have shifted by 10 months (10 payments). Therefore, the ¬rst calendar
year™s loan amortization can be represented by the PV of the ¬nal 10
payments discounted to March 1, 1998, as that would make up the only
difference in the two series of cash ¬‚ows as perceived from their different
points in time. This is illustrated graphically in Figure A3-2.
Figure A3-2 is a time line of payments on the ¬ve-year (60-month)
loan. The top portion of the ¬gure, labeled A, graphically represents the
entire payment schedule. In the bottom ¬gure the loan is split into several
pieces: payments 1“10, which are not labeled;21 payments 1“50, labeled



20. Technically, we discount to the end of December 31, 1998, but in PV terms it is easier to think
of January 1, 1999.
21. In all cases the zero is there only as a valuation date. There are no loan payments (cash ¬‚ows)
that occur at zero.




CHAPTER 3 Annuity Discount Factors and the Gordon Model 99
F I G U R E A3-2

Payment Schedule

A


0 11 20 30 40 50 60
±PV = C


B
0 50 60
D
11 60
3/1/98 1/31/99
PV = PV0(A) - PV10(D) = PV0(A) - PV0(B) = PV0(C)
$134,089 = $1 Million - $865,911 = $1 Million - $865,911 $134,089



B; payments 11“60, labeled D; and payments 51“60, labeled C (t 50 is
the end of B, not the beginning of C).
The equation at the bottom of Figure A3-2, which we explain below
in 1“3, is: PV PV0(A) PV10(D) PV0(A) PV0(B) PV0(C). The
amortization of the loan principal during any year is the change in the
present value of the loan between years. That is equal to each of the
following three expressions:
1. PV0(A) PV10(D): The PV at t 0 of A (all 60 months of the
loan) minus the PV at t 10 of D, the last 50 payments of the
loan. Notice that the valuation dates are different, t 0 versus
t 10. The PV at t 0 of A is the principal, $1 million (Table
A3-3, Section 2, D84). The PV at t 10 of D is $865,911 (E84).
The difference of the two is the amortization of $134,089 (E85).
2. PV0(A) PV0(B): The PV at t 0 of A (all 60 months of the
loan), which is $1 million, minus the PV at t 0 of the ¬rst 50
months of the loan. The latter calculation does not appear
directly in Table A3-3. However, using equation (3-6d) from the
chapter with g 0, r 0.83333%, and n 50 periods leads to
the ADF of 40.75442. Multiplying the ADF by the monthly
payment of $21,247.04 gives us the PV of B, which is $865,911.
The difference of the two PVs is $134,089, the same as above.
3. PV0(C): The PV at t 0 of C, payments 51“60. This is the most
important of the expressions because it is the most compact and
the easiest to use. The other expressions are the difference of
two formulas, while this one requires only a single formula. It is
stated in mathematical terms below in equation (A3-10). The
reduction in the principal is the PV of the opposite or mirror-
image series of cash ¬‚ows working backward from the end of
the loan.


Section 3: A Better Way to Calculate Loan Amortization
In Section 3 we calculate the principal reduction using equation (A3-10).
Let™s look ¬rst at the 1998 cash ¬‚ows in Row 93. The amortization of
principal in 1998 is equal to the PV of the last 10 payments of the loan.

PART 1 Forecasting Cash Flows
100
Letting n (the ¬nal payment period) 60, we want to calculate the PV
of payments 51“60, discounted to month 0. If we let F ¬nishing month
10, the formula n F 1 describes S, the starting month in C93
through C98. The formula n S 1 describes F, the ¬nishing month in
D93 through D98. For 1998, S 60 10 (D93) 1 51, and F 60
1 (C93) 1 60. Thus, our formulas give us the result that in calendar
1998 the amortization of principal is equal to the PV at t 0 of payments
51“60, which is correct.
For calendar 1999, S 60 22 (D94) 1 39, and F 60 11
(C94) 1 50. The amortization of principal in calendar 1999 is the PV
at t 0 of payments 39“50, which is also correct. Thus, the amortization
of principal in any year is equal to an ADF with no growth and end-of-
year cash ¬‚ows that run from n F 1 to n S 1. We begin the
calculation of this loan amortization ADF in equation (A3-6).
1 1 1
ADF ... (A3-6)
r)n F1
r)n F2
r)n S1
(1 (1 (1
Multiplying equation (A3-6) by 1/(1 r), we get:
1 1 1
ADF
r)n F2
r)n F3
1 r (1 (1
1 1
... (A3-7)
r)n S1
r)n S2
(1 (1
Subtracting equation (A3-7) from equation (A3-6), we get:
1 1 1
1 ADF (A3-8)
r)n F1
r)n S2
1 r (1 (1
The left-hand side of equation (A3-8) simpli¬es to r/(1 r) ADF. Mul-
tiplying both sides of equation (A3-8) by (1 r)/r, we come to:
1 r 1 1
ADF (A3-9)
r)n F1
r)n S2
r (1 (1
Canceling out the 1 r in the numerator and denominator, we arrive at
our ¬nal solution:
1 1 1
ADF
r)n F
r)n S1
r (1 (1
ADF formula for loan amortization (A3-10)
The spreadsheet formulas begin in column F of Rows 93 through 98.
Note that we multiply the ADF in equation (A3-10) by the monthly pay-
ment in F93 through F98 to calculate the PV of the loan. I is the monthly
interest rate 10%/12 months 0.833%, which is equivalent to r in
equation (A3-10).
The amortization in 1998 is $134,089 (E93), which equals:
1 1 1
ADF (A3-10a)
1.00833360 10
1.00833360 11
0.008333
The amortization in 1999 is $176,309, as per E94, which equals:

CHAPTER 3 Annuity Discount Factors and the Gordon Model 101
1 1 1
ADF (A3-10b)
1.00833360 22
1.00833360 11 1
0.008333
The principal amortization in cells E93 through E98 is equal to that
in column H of Section 1, which demonstrates the accuracy of equation
(A3-10).


The After-Tax Cost of a Loan
In our discussion in Table A3-3, Sections 2 and 3, we came to the insight
that principal amortizes in mirror image, and we used that understanding
to develop equation (A3-10) to calculate the principal amortization over
any given block of time. Now it is appropriate to present month-by-
month amortization of principal, as it will enable us to develop formulas
to calculate the PV of principal and interest of a loan. The primary prac-
tical application is to calculate the after-tax cost of a loan.
We begin with a month-by-month amortization. In the ¬rst month,
amortization equals the PVF for the last month. In the second month,
amortization equals the PVF for the second-to-last month, and we con-
tinue in that fashion. Mathematically, amortization is thus equal to:
1 1 1 1
Amort ... Pymt
r)n r)n 1
r)n 2
(1 (1 (1 1 r
(A3-11)
Note that this expression is the exact reverse of a simple series of cash
¬‚ows that solves to an end-of-year ADF with no growth, i.e., equation
(3-6d) in the body of the chapter. Thus, the total amortization equals equa-
tion (3-6d) Loan Payment Principal of the Loan. This is a rearrange-
ment of equation (3-21). Note that one should use the nominal interest
rate in this calculation.
Next we take the PV of equation (A3-11) at the nominal rate of in-
terest (when valuing a loan at a discount rate other than the nominal rate
of interest, see that discussion at the end of this chapter).
1 1 1
r)n r)n 1 r)n 2
(1 (1 (1
PV (Amort)
r)2 r)3
1 r (1 (1

1
1 r
... Pymt (A3-12)
r)n
(1


We can move the second denominator into the ¬rst denominator, and
equation (A3-12) simpli¬es to:
1 1 1
PV (Amort)
r)n 1
r)n 1
r)n 1
(1 (1 (1
1
... Pymt [n terms] (A3-13)
r)n 1
(1

PART 1 Forecasting Cash Flows
102
T A B L E A3-3

Amortization of Principal with Irregular Starting Point


A B C D E F G H I J K L M N O

4 SECTION 1: LOAN AMORTIZATION SCHEDULE

5 Pmt NPV Annual Aft-Tax
6 # Date Pmt Int Prin Bal Int Prin PVF Pymt NPV Cost-
Loan
7 0 02/28/98 1,000,000 1.0000
8 1 03/31/98 21,247 8,333 12,914 987,086 0.9917 21,071 17,766 12807 4959
9 2 04/30/98 21,247 8,226 13,021 974,065 0.9835 20,897 17,661 12807 4854
10 3 05/31/98 21,247 8,117 13,130 960,935 0.9754 20,725 17,558 12807 4751
11 4 06/30/98 21,247 8,008 13,239 947,696 0.9673 20,553 17,455 12807 4648
12 5 07/31/98 21,247 7,897 13,350 934,346 0.9594 20,383 17,353 12807 4546
13 6 08/31/98 21,247 7,786 13,461 920,885 0.9514 20,215 17,252 12807 4445
14 7 09/30/98 21,247 7,674 13,573 907,312 0.9436 20,048 17,152 12807 4345
15 8 10/31/98 21,247 7,561 13,686 893,626 0.9358 19,882 17,052 12807 4245
16 9 11/30/98 21,247 7,447 13,800 879,826 0.9280 19,718 16,954 12807 4147
17 10 12/31/98 21,247 7,332 13,915 865,911 78,381 134,089 0.9204 19,555 203,048 16,856 12807 4049
18 11 01/31/99 21,247 7,216 14,031 851,880 0.9128 19,393 16,759 12807 3952
19 12 02/28/99 21,247 7,099 14,148 837,732 0.9052 19,233 16,663 12807 3856
20 13 03/31/99 21,247 6,981 14,266 823,466 0.8977 19,074 16,567 12807 3760
21 14 04/30/99 21,247 6,862 14,385 809,081 0.8903 18,917 16,473 12807 3666
22 15 05/31/99 21,247 6,742 14,505 794,576 0.8830 18,760 16,379 12807 3572
23 16 06/30/99 21,247 6,621 14,626 779,951 0.8757 18,605 16,286 12807 3479
24 17 07/31/99 21,247 6,500 14,747 765,203 0.8684 18,451 16,194 12807 3387
25 18 08/31/99 21,247 6,377 14,870 750,333 0.8612 18,299 16,102 12807 3295
26 19 09/30/99 21,247 6,253 14,994 735,339 0.8541 18,148 16,011 12807 3204
27 20 10/31/99 21,247 6,128 15,119 720,220 0.8471 17,998 15,921 12807 3114
28 21 11/30/99 21,247 6,002 15,245 704,974 0.8401 17,849 15,832 12807 3025
29 22 12/31/99 21,247 5,875 15,372 689,602 78,656 176,309 0.8331 17,701 222,428 15,744 12807 2937
30 23 01/31/00 21,247 5,747 15,500 674,102 0.8262 17,555 15,656 12807 2849
31 24 02/28/00 21,247 5,618 15,630 658,472 0.8194 17,410 15,569 12807 2762
32 25 03/31/00 21,247 5,487 15,760 642,712 0.8126 17,266 15,482 12807 2675
33 26 04/30/00 21,247 5,356 15,891 626,821 0.8059 17,123 15,397 12807 2590
34 27 05/31/00 21,247 5,224 16,024 610,798 0.7993 16,982 15,312 12807 2505
35 28 06/30/00 21,247 5,090 16,157 594,641 0.7927 16,842 15,228 12807 2421
36 29 07/31/00 21,247 4,955 16,292 578,349 0.7861 16,702 15,144 12807 2337
37 30 08/31/00 21,247 4,820 16,427 561,922 0.7796 16,564 15,061 12807 2254
103
104
T A B L E A3-3 (continued)

Amortization of Principal with Irregular Starting Point


A B C D E F G H I J K L M N O

4 SECTION 1: LOAN AMORTIZATION SCHEDULE

5 Pmt NPV Annual Aft-Tax
6 # Date Pmt Int Prin Bal Int Prin PVF Pymt NPV Cost-
Loan
38 31 09/30/00 21,247 4,683 16,564 545,357 0.7732 16,427 14,979 12807 2172
39 32 10/31/00 21,247 4,545 16,702 528,655 0.7668 16,292 14,898 12807 2091
40 33 11/30/00 21,247 4,405 16,842 511,813 0.7604 16,157 14,817 12807 2010
41 34 12/31/00 21,247 4,265 16,982 494,831 60,194 194,771 0.7542 16,024 201,345 14,737 12807 1930
42 35 01/31/01 21,247 4,124 17,123 477,708 0.7479 15,891 14,657 12807 1850
43 36 04/29/61 21,247 3,981 17,266 460,442 0.7417 15,760 14,579 12807 1772
44 37 03/31/01 21,247 3,837 17,410 443,032 0.7356 15,630 14,501 12807 1694
45 38 04/30/01 21,247 3,692 17,555 425,476 0.7295 15,500 14,423 12807 1616
46 39 05/31/01 21,247 3,546 17,701 407,775 0.7235 15,372 14,346 12807 1539
47 40 06/30/01 21,247 3,398 17,849 389,926 0.7175 15,245 14,270 12807 1463
48 41 07/31/01 21,247 3,249 17,998 371,928 0.7116 15,119 14,194 12807 1387
49 42 08/31/01 21,247 3,099 18,148 353,781 0.7057 14,994 14,119 12807 1312
50 43 09/30/01 21,247 2,948 18,299 335,482 0.6999 14,870 14,045 12807 1238
51 44 10/31/01 21,247 2,796 18,451 317,031 0.6941 14,747 13,971 12807 1164
52 45 11/30/01 21,247 2,642 18,605 298,425 0.6884 14,626 13,898 12807 1091
53 46 12/31/01 21,247 2,487 18,760 279,665 39,799 215,166 0.6827 14,505 182,260 13,826 12807 1019
54 47 01/31/02 21,247 2,331 18,917 260,749 0.6770 14,385 13,754 12807 947
55 48 02/28/02 21,247 2,173 19,074 241,675 0.6714 14,266 13,682 12807 875
56 49 03/31/02 21,247 2,014 19,233 222,442 0.6659 14,148 13,612 12807 805
57 50 04/30/02 21,247 1,854 19,393 203,048 0.6604 14,031 13,541 12807 734
58 51 05/31/02 21,247 1,692 19,555 183,493 0.6549 13,915 13,472 12807 665
59 52 06/30/02 21,247 1,529 19,718 163,775 0.6495 13,800 13,403 12807 596
60 60 07/31/02 21,247 1,365 19,882 143,893 0.6441 13,686 13,334 12807 527
61 54 08/31/02 21,247 1,199 20,048 123,845 0.6388 13,573 13,267 12807 460
62 55 09/30/02 21,247 1,032 20,215 103,630 0.6335 13,461 13,199 12807 392
63 56 10/31/02 21,247 864 20,383 83,247 0.6283 13,350 13,133 12807 326
64 57 11/30/02 21,247 694 20,553 62,693 0.6231 13,239 13,066 12807 259
65 58 12/31/02 21,247 522 20,725 41,969 17,268 237,697 0.6180 13,130 164,984 13,001 12807 194
66 59 01/31/03 21,247 350 20,897 21,071 0.6129 13,021 12,936 12807 129
67 60 02/28/03 21,247 176 21,071 0 525 41,969 0.6078 12,914 25,935 12,871 12807 64
68 Totals 1,274,823 274,823 1,000,000 274,823 1,000,000 1,000,000 1,000,000 907,368
T A B L E A3-3 (continued)

Amortization of Principal with Irregular Starting Point


A B C D E F G H I J K L M N O

73 SECTION 2: SCHEDULE OF PRESENT VALUES CALCULATED BY ADF EQUATION (A3-5)
75 As Seen From The Beginning of Year
76 1998 1999 2000 2001 2002 2003 2004 Total
77 NPV 1998 payments [1] 203,048
78 NPV 1999 payments 222,428 241,675
79 NPV 2000 payments 201,345 218,767 241,675
80 NPV 2001 payments 182,260 198,031 218,767 241,675
81 NPV 2002 payments 164,984 179,260 198,031 218,767 241,675
82 NPV 2003 payments 25,935 28,179 31,130 34,390 37,991 41,969
83 NPV 2004 payments 0
84 Sum NPVs-all pymts 1,000,000 865,911 689,602 494,831 279,665 41,969 0 0
85 Reduction in NPV 134,089 176,309 194,771 215,166 237,697 41,969 1,000,000
86 Valuation date v 10 22 34 46 58
0
88 SECTION 3: AMORTIZATION CALCULATED AS THE PYMT * THE ADF in (A3-16)
90 Formulas For Principal Amortization, where:
91 Starting Finishing Prin I Monthly Interest 0.833%, n 60 Months,
92 Month Month Amort Pymt $21,247/Month
93 Calendar 1998 1 10 134,089 PYMT*(1/r)*((1/(1 r) (N $D93) (1/(1 r) (N $C93 1))))
94 Calendar 1999 11 22 176,309 PYMT*(1/r)*((1/(1 r) (N $D94) (1/(1 r) (N $C94 1))))
95 Calendar 2000 23 34 194,771 PYMT*(1/r)*((1/(1 r) (N $D95) (1/(1 r) (N $C95 1))))
96 Calendar 2001 35 46 215,166 PYMT*(1/r)*((1/(1 r) (N $D96) (1/(1 r) (N $C96 1))))
97 Calendar 2002 47 58 237,697 PYMT*(1/r)*((1/(1 r) (N $D97) (1/(1 r) (N $C97 1))))
98 Calendar 2003 59 60 41,969 PYMT*(1/r)*((1/(1 r) (N $D98) (1/(1 r) (N $C98 1))))
99 Total 1,000,000
105
106




T A B L E A3-3 (continued)

Amortization of Principal with Irregular Starting Point


A B C D E F G H I J K L M N O

102 Assumptions: After-Tax Cost of the Loan
104 Prin 1,000,000 (1 t) * Prin 0.600000 600,000
105 Int 10.0000% t*n/(1 r) (n 0.307368 307,368
1)*PYMT
106 Int-Mo 0.8333% Total L68 0.907368 907,368
107 Years 5
108 Months n 60 H106: (1 t) [t*N/(1 r) (N 1)*PYMT/P] Equation (A3-24a)
109 Pymt 21,247 I106: (1 t)*P [t*N/(1 r) (N 1)*PYMT] Equation (A3-23a)
110 Form-Prin 1,000,000
111 Start month S 3
112 x (1 g)/(1 r) 0.9917
113 y 1/(1 r) 0.9917
114 GM 1/r 120

Notes:
[1] Formula for D77 according to (A3-5): GM*(1 x ($D93 $C93 1))*y ($C93 A$86 1)*PYMT
n # months of cash ¬‚ow $D93 $C93 1, which is the ending month - beginning month 1. The exponent of y is the ending month - the valuation date); thus it is the discounting period. This formula copies both down and across,
i.e., it is the formula for all cells from D77 to I82. D78 D77 because there are 10 payments in 1998 and 12 in 1999“2002.
All the bracketed terms in equation (A3-13) are identical. Thus, the
PV of the amortization of principal, which we denote below as PV(P), is
equal to n any one of these terms the loan payment.
n
PV (Amort) PV (P) Pymt
r)n 1
(1
PV of principal payments (A3-14)
Restating equation (3-21) as equation (A3-15),
P
Pymt , (A3-15)
ADF
where ADF is de¬ned by equation (3-6d). Substituting equation (A3-15)
into equation (A3-14), we get:
n P
PV(P) (A3-16)
r)n 1
(1 ADF
The next section, in which we develop equations (3-16a) and (3-16b),
is somewhat of a digression from the previous and the subsequent dis-
cussion. We do not use equations (A3-16a) and (A3-16b) in our subse-
quent work. However, these formulas can be useful alternative forms
of (A3-16). Substituting in the de¬nition of the ADF, dividing through
by the principal, and solving the equation,22 another form of equation
(A3-16) is:
PV(P) n
(A3-16a)
r)n
P [(1 1](1 r)
Table A3-4 veri¬es the accuracy of this formula, which is my own
formula, to the best of my knowledge. For a ¬ve-year (60-month) loan at
12% per year, or 1% per month (A5 and A4, respectively), the present


T A B L E A3-4

PV of Principal Amortization

A B

4 r 1%
5 n 60
6 PV(P)/Pmt 32.69997718
7 Pmt/P $0.0222444
8 PV(P)/P $0.7273929
9 PV(P)/P $0.7273929
11 Cell Formulas:
13 B6: n/(1 r) (n 1)
14 B7: PMT(.01,60, 1)
15 B9: B7*B8
16 B10: (n*r)/(((1 r) n 1)*(1 r))




22. We do not show the steps to the solution, as we are not using this equation in our subsequent
work.


CHAPTER 3 Annuity Discount Factors and the Gordon Model 107
value of the principal divided by the loan payment is 32.69997718 (B6).
The formula for that cell appears in cell A13, and that formula is equation
(A3-14) after dividing both sides of the equation by the payment. In B7
we show the monthly payment per dollar of loan principal, which we
calculate using a standard spreadsheet ¬nancial function for a $1 loan
with 60 monthly payments at 1% interest (see cell A14 for the formula).
In B8 we multiply B6 B7. In B9 we test equation (A3-16a), and it comes
to the same answer as B8, i.e., the present value of the principal is
$0.7273929 per $1 of principal. That the two answers are identical dem-
onstrates the accuracy of equation (A3-16a). Of course, the present value
of the interest on a pretax basis is one minus that, or approximately $0.273
per $1 of principal.
In algebraic terms, the present value of the interest portion of a loan
per dollar of principal on a pretax basis is one minus (A3-16a), or:
PV(Int) n
1 (A3-16b)
n
P [(1 r) 1](1 r)
Resuming our discussion after the digression in the last several par-
agraphs, the PV of the interest portion of the payments is simply the PV
of the loan payments”which is the principal”minus the PV of the prin-
cipal portion, or:
PV(Int) P PV(P) (A3-17)
Substituting equation (A3-16) into equation (A3-17), we get:
n P n 1
PV(Int) P P1 (A3-18)
r)n 1
r)n 1
(1 ADF (1 ADF
The PV of the after-tax cost of the interest portion is (1 t) * (A3-18),
where t is the tax rate, or:
n 1
PV(Int)After-Tax (1 t) P 1 (A3-19)
r)n 1
(1 ADF
Thus, the after-tax cost of the loan, L, is (A3-16) plus (A3-19), or:
n P n 1
L (1 t)P 1 (A3-20)
r)n 1
r)n 1
(1 ADF (1 ADF
Factoring terms, we get:
n P
L [1 (1 t)] (1 t)P (A3-21)
r)n 1
(1 ADF
which simpli¬es to:
n P
L t (1 t)P (A3-22)
r)n 1
(1 ADF
Switching terms, our ¬nal equation for the after-tax cost of a loan is:
n P
L (1 t)P t after-tax cost of a loan (A3-23)
r)n 1
(1 ADF
Alternatively, using

PART 1 Forecasting Cash Flows
108
P
Loan Payment
ADF
we can restate equation (A3-23) as:
n
L (1 t)P t Pymt
r)n 1
(1 (A3-23a)
alternative expression”after-tax cost of loan
Equation (A3-23) gives us the equation for the after-tax cost of a loan
in dollars. We can restate equation (A3-23) to give us the after-tax cost of
the loan for each $1.00 of loan principal by dividing through by P.
L n 1
(1 t) t
r)n 1
P (1 ADF
after-tax cost of loan per each $1.00 of principal (A3-24)
Analyzing equation (A3-24), we can see the after-tax cost of the loan
is made up of two parts:
1. The after-tax cost of the principal, as if the entire loan payment
was tax-deductible, plus
2. The tax rate times the PV of the principal payments on the loan.
In item 1 we temporarily assume that principal and interest are tax-
deductible. This is actually true for ESOP loans, and the PV of an ESOP
loan is item 1. To adjust item 1 upwards for the lack of tax shield on the
principal of ordinary loans, in item 2 we add back the tax shield included
in item 1 that we do not really get. Of course, we can substitute the exact
expression for ADF in equation (A3-24) to keep the solution strictly in
terms of the variables t, n, and r.
We can derive an alternative expression for equation (A3-24) by di-
viding equation (A3-23a) by P:
L n Pymt
(1 t) t
r)n 1
P (1 P
alternative expression”after-tax cost of loan/$1 of principal
(A3-24a)
We demonstrate the accuracy of equations (A3-23a) and (A3-24a) in
Table A3-3. In Section 1, Column L is the after-tax cost of each loan pay-
ment. It is equal to the sum of [Principal (Column E) (1 Tax Rate)
Interest (Column D)] Present Value Factor (Column I). We assume
a 40% tax rate in this table. Thus cell L8, the after-tax cost of the ¬rst
month™s loan payment, is equal to [$12,914 (E8) (1 40%) $8,333
(D8)] 0.9917 (I8) $17,766. The sum of the after-tax cost of the loan
payments is $907,368 (L68).
We now move to Section 3, F102 to J109. Here we use equation (A3-
24a) to test if we get the same answer as the brute force approach in L68.
In I104 we show the PV of the principal after tax, corresponding to item
1 above, as $600,000 (H104 is the same, but for each $1.00 of principal).
In I105 we show the tax shield on the principal that we do not get at

CHAPTER 3 Annuity Discount Factors and the Gordon Model 109
$307,368. The sum of the two is $907,368 (I106), which matches L68 and
thus proves equation (A3-24a). Note that I106, which we calculate ac-
cording to equation (A3-23a), equals $0.907368, which is the correct after-
tax cost of the loan per each dollar of principal. When we multiply that
by the $1 million principal, we get the correct after-tax cost of the loan
in dollars, as per cell I106 and equation (A3-23a).


T A B L E A3-5

Present Value of a Loan at Discount Rate Different than Nominal Rate


A B C D E F G

5 Pmt
6 # Pmt Int Prin Bal PVF (r1) PV(P)

7 0 1,000,000 1.0000
8 1 21,247 8,333 12,914 987,086 0.9901 12,786
9 2 21,247 8,226 13,021 974,065 0.9803 12,765
10 3 21,247 8,117 13,130 960,935 0.9706 12,744
11 4 21,247 8,008 13,239 947,696 0.9610 12,723
12 5 21,247 7,897 13,350 934,346 0.9515 12,702
13 6 21,247 7,786 13,461 920,885 0.9420 12,681
14 7 21,247 7,674 13,573 907,312 0.9327 12,660
15 8 21,247 7,561 13,686 893,626 0.9235 12,639
16 9 21,247 7,447 13,800 879,826 0.9143 12,618
17 10 21,247 7,332 13,915 865,911 0.9053 12,597
18 11 21,247 7,216 14,031 851,880 0.8963 12,576
19 12 21,247 7,099 14,148 837,732 0.8874 12,556
20 13 21,247 6,981 14,266 823,466 0.8787 12,535
21 14 21,247 6,862 14,385 809,081 0.8700 12,514
22 15 21,247 6,742 14,505 794,576 0.8613 12,494
23 16 21,247 6,621 14,626 779,951 0.8528 12,473
24 17 21,247 6,500 14,747 765,203 0.8444 12,452
25 18 21,247 6,377 14,870 750,333 0.8360 12,432
26 19 21,247 6,253 14,994 735,339 0.8277 12,411
27 20 21,247 6,128 15,119 720,220 0.8195 12,391
28 21 21,247 6,002 15,245 704,974 0.8114 12,370
29 22 21,247 5,875 15,372 689,602 0.8034 12,350
30 23 21,247 5,747 15,500 674,102 0.7954 12,330
31 24 21,247 5,618 15,630 658,472 0.7876 12,309
32 25 21,247 5,487 15,760 642,712 0.7798 12,289
33 26 21,247 5,356 15,891 626,821 0.7720 12,269
34 27 21,247 5,224 16,024 610,798 0.7644 12,248
35 28 21,247 5,090 16,157 594,641 0.7568 12,228
36 29 21,247 4,955 16,292 578,349 0.7493 12,208
37 30 21,247 4,820 16,427 561,922 0.7419 12,188
38 31 21,247 4,683 16,564 545,357 0.7346 12,168
39 32 21,247 4,545 16,702 528,655 0.7273 12,148
40 33 21,247 4,405 16,842 511,813 0.7201 12,128
41 34 21,247 4,265 16,982 494,831 0.7130 12,108
42 35 21,247 4,124 17,123 477,708 0.7059 12,088
43 36 21,247 3,981 17,266 460,442 0.6989 12,068
44 37 21,247 3,837 17,410 443,032 0.6920 12,048
45 38 21,247 3,692 17,555 425,476 0.6852 12,028
46 39 21,247 3,546 17,701 407,775 0.6784 12,008
47 40 21,247 3,398 17,849 389,926 0.6717 11,988
48 41 21,247 3,249 17,998 371,928 0.6650 11,968
49 42 21,247 3,099 18,148 353,781 0.6584 11,949




PART 1 Forecasting Cash Flows
110
T A B L E A3-5 (continued)

Present Value of a Loan at Discount Rate Different than Nominal Rate


A B C D E F G

5 Pmt
6 # Pmt Int Prin Bal PVF (r1) PV(P)

50 43 21,247 2,948 18,299 335,482 0.6519 11,929
51 44 21,247 2,796 18,451 317,031 0.6454 11,909
52 45 21,247 2,642 18,605 298,425 0.6391 11,890
53 46 21,247 2,487 18,760 279,665 0.6327 11,870
54 47 21,247 2,331 18,917 260,749 0.6265 11,850
55 48 21,247 2,173 19,074 241,675 0.6203 11,831
56 49 21,247 2,014 19,233 222,442 0.6141 11,811
57 50 21,247 1,854 19,393 203,048 0.6080 11,792
58 51 21,247 1,692 19,555 183,493 0.6020 11,772
59 52 21,247 1,529 19,718 163,775 0.5961 11,753
60 53 21,247 1,365 19,882 143,893 0.5902 11,734
61 54 21,247 1,199 20,048 123,845 0.5843 11,714
62 55 21,247 1,032 20,215 103,630 0.5785 11,695
63 56 21,247 864 20,383 83,247 0.5728 11,676
64 57 21,247 694 20,553 62,693 0.5671 11,656
65 58 21,247 522 20,725 41,969 0.5615 11,637
66 59 21,247 350 20,897 21,071 0.5560 11,618
67 60 21,247 176 21,071 0 0.5504 11,599

68 Total 1,274,823 274,823 1,000,000 730,970
70 Assumptions:
72 Prin 1,000,000
73 Int 10.0000%
74 Int Mo r 0.8333%
75 Int 12.0000%
76 Int Mo r1 1.0000%
77 Years 5
78 Months n 60
79 Pymt 21,247
80 Start month S 3
81 (1/(r1 r))*((1/(1 r) n) (1/(1 r1) n))*PYMT 730,970




Present Value of the Principal when the Discount Rate is
Different than the Nominal Rate
When valuing a loan at a discount rate, r1, that is different than the nom-
inal rate of interest, r, the present value of principal is as follows:
1 1 1
r)n r)n 1 r)n 2
(1 (1 (1
PV (Amort)
r1)2 r1)3
1 r1 (1 (1

1
1 r
... Pymt (A3-25)
r1)n
(1

We can move the second denominator into the ¬rst to simplify the equa-
tion:



CHAPTER 3 Annuity Discount Factors and the Gordon Model 111
1 1
PV (Amort)
r)n(1 r)n 1(1 r1)2
(1 r1) (1
(A3-26)
1
... Pymt
r1)n
(1 r)(1
Multiplying both sides by (1 r)/(1 r1), we get:
1 r 1 1
PV (Amort) n1 2 n2
r1)3
1 r1 (1 r) (1 r1) (1 r) (1

1
... Pymt (A3-27)
r1)n
(1 r)(1
Subtracting equation (A3-27) from equation (A3-26) and simplifying, we
get:
r1 r 1 1
PV (Amort) Pymt
r)n(1 r1)n
1 r1 (1 r1) (1 r)(1
(A3-28)
This simpli¬es to:
1 1 1
PV (Amort) Pymt (A3-29)
r)n r1)n
r1 r (1 (1
Table A3-5 is almost identical to Section 1 of Table A3-3. We use a
nominal interest rate of 10% per year (B73), which is 0.8333% per month
(B74), and a discount rate of 12% per year (B75), or 1% per month (B76).
We discount the principal amortization at r1, the discount rate of 1%,
in Column F, so that Column G gives us the present value of the principal,
which totals $730,970 (G68). The Excel formula equivalent for equation
(A3-29) appears in cell A81, and the result of that formula appears in
G81, which matches the brute force calculation in G68, thus demonstrat-
ing the accuracy of the formula.


CONCLUSION
In this mathematical appendix to the ADF chapter, we have presented:
— ADFs with stub periods (partial years) for both midyear and
end-of-year.
— Tables to demonstrate their accuracy.
— ADFs to calculate the amortization of principal on a loan.
— A formula for the after-tax PV of a loan.




PART 1 Forecasting Cash Flows
112
PART TWO


Calculating Discount Rates




Part 2 of this book, Chapters 4, 5, and 6, deals with calculating discount
rates; discounting cash ¬‚ows is the second of the four steps in business
valuation.
Chapter 4 is a long chapter, with a signi¬cant amount of empirical
analysis of stock market returns. Our primary ¬nding is that returns are
negatively related to the logarithm of the size of the ¬rm. The most suc-
cessful measure of size in explaining returns of publicly held stocks is
market capitalization, though research by Grabowski and King shows
that many other measures of size also do a fairly good job of explaining
stock market returns.
In their 1999 article, Grabowski and King found the relationship of
return to three underlying variables: operating margin, the logarithm of
the coef¬cient of variation of operating margin, and the logarithm of the
coef¬cient of variation of return on equity. This is a very important re-
search result, and it is very important that professionals read and under-
stand their article. Even so, their methodology is based on Compustat
data, which leaves out the ¬rst 37 years of the New York Stock Exchange
data. As a consequence, their standard errors are higher than my log size
model, and appraisers should be familiar with both.
In this chapter, we:

— Develop the mathematics of potential log size equations.
— Analyze the statistical error in the log size equation for different
time periods and determine that the last 60 years, i.e., 1939“1998,
is the optimal time frame.
— Present research by Harrison that shows that the distribution of
stock market returns in the 18th century is the same as it is in
the 20th century and discuss its implications for which 20th
century data we should use.
— Give practical examples of using the log size equation.
— Compare log size to the capital asset pricing model (CAPM) for
accuracy.
— Discuss industry effects.



113




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
— Discuss industry effects.
— Present a claim that, with rare exceptions, valuations of small
and medium-sized privately held businesses do not require a
public guideline companies method (developing PE and other
types of multiples), as the log size model satis¬es the intent
behind the Revenue Ruling 59-60 requirement to use that
approach when it is relevant.
The last bullet point is very important; in my opinion, it frees ap-
praisers from wasting countless hours on an approach that is worse than
useless for valuing small ¬rms.1 The log size model itself saves much
time compared to using CAPM. The former literally takes one minute,
while the latter often requires one to two days of research. Log size is
also much more accurate for smaller ¬rms than is either CAPM or the
buildup approach. Using 1939“1998 data, the log size standard error of
the valuation estimate is only 41% as large as CAPM standard error. This
means that the CAPM 95% con¬dence intervals are approximately two
and one half times larger than the log size con¬dence intervals.2
Summarizing, log size has two advantages:
— It saves much time and money for the appraiser.
— It is far more accurate.
For those who prefer not to read through the research that leads to
our conclusions and simply want to learn how to use the log size model,
Appendix C presents a much shorter version of Chapter 4. It also serves
as a useful refresher for those who read Chapter 4 in its entirety but
periodically wish to refresh their skills and understanding.
Chapter 5 discusses arithmetic versus geometric mean returns. There
have been many articles in the professional literature arguing whether
arithmetic or geometric mean returns are most appropriate. For valuing
small businesses, the two measures can easily make a 100% difference in
the valuation, as geometric returns are always lower than arithmetic re-
turns (as long as returns are not identical in every period, which, of
course, they are not). Most of the arguments have centered around Pro-
fessor Ibbotson™s famous two-period example.
The majority of Chapter 5 consists of empirical evidence that arith-
metic mean returns do a better job than geometric means of explaining
log size results. Additionally, we spend some time discussing a very
mathematical article by Indro and Lee that argues for using a time
horizon-weighted average of the arithmetic and geometric means.
For those who use CAPM, whether in a direct equity approach or in
an invested capital approach, there is a trap into which many appraisers
fall, which is producing an answer that is internally inconsistent.
Common practice is to assume a degree of leverage”usually equal
to the subject company™s existing or industry average leverage”


1. When the subject company is close to the size of publicly traded ¬rms, say one half their size,
then the public guideline company approach is reasonable.
2. Using 1938“1997 data, the log size standard error was only 6% as large as CAPM™s standard
error. 1998 was a bad year for the log size model.




PART 2 Calculating Discount Rates
114
assuming book value for equity. This implies an equity for the ¬rm, which
is an ex-ante value of equity. The problem comes when the appraiser
stops after obtaining his or her valuation estimate. This is because the
calculated value of equity will almost always be inconsistent with the
value of equity that is implied in the leverage assumed in the calculation
of the CAPM discount rate.
In Chapter 6 we present an iterative method that solves the problem
by repeating the valuation calculations until the assumed and the calcu-
lated equity are equal.




PART 2 Calculating Discount Rates 115
CHAPTER 4


Discount Rates as a Function of
Log Size1




PRIOR RESEARCH
TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS
Regression #1: Return versus Standard Deviation of Returns
Regression #2: Return versus Log Size
Regression #3: Return versus Beta
Market Performance
Which Data to Choose?
Tables 4-2 and 4-2A: Regression Results for Different Time Periods
18th Century Stock Market Returns
Conclusion on Data Set
Recalculation of the Log Size Model Based on 60 Years
APPLICATION OF THE LOG SIZE MODEL
Discount Rates Based on the Log Size Model
Need for Annual Updating
Computation of Discount Rate Is an Iterative Process
Practical Illustration of the Log Size Model: Discounted Cash Flow
Valuations
The Second Iteration: Table 4-4B
Consistency in Levels of Value
Adding Speci¬c Company Adjustments to the DCF Analysis: Table
4-4C
Total Return versus Equity Premium
Adjustments to the Discount Rate
Discounted Cash Flow or Net Income?
DISCUSSION OF MODELS AND SIZE EFFECTS
CAPM



1. Adapted and reprinted with permission from Valuation (August 1994): 8“24 and The Valuation
Examiner (February/March 1997): 19“21.




117




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Sum Beta
The Fama“French Cost of Equity Model
Log Size Models
Heteroscedasticity
INDUSTRY EFFECTS
SATISFYING REVENUE RULING 59-60 WITHOUT A GUIDELINE
PUBLIC COMPANY METHOD
SUMMARY AND CONCLUSIONS
APPENDIX A: AUTOMATING ITERATION USING
NEWTON™S METHOD
APPENDIX B: MATHEMATICAL APPENDIX
APPENDIX C: ABBREVIATED REVIEW AND USE




PART 2 Calculating Discount Rates
118
PRIOR RESEARCH
Historically, small companies have shown higher rates of return when
compared to large ones, as evidenced by data for the New York Stock
Exchange (NYSE) over the past 73 years of its existence (Ibbotson Asso-
ciates 1999). The relationship between ¬rm size and rate of return was
¬rst published by Rolf Banz in 1981 and is now universally recognized.
Accordingly, company size has been included as a variable in several
models used to determine stock market returns.
Jacobs and Levy (1988) examined small ¬rm size as one of 25 vari-
ables associated with anomalous rates of return on stocks. They found
that small size was statistically signi¬cant both in single-variable and
multivariate form, although size effects appear to change over time, i.e.,
they are nonstationary. They found that the natural logarithm (log) of
market capitalization was negatively related to the rate of return.
Fama and French (1993) found they could explain historical market
returns well with a three-factor multiple regression model using ¬rm size,
the ratio of book equity to market equity (BE/ME), and the overall market
factor Rm Rf , i.e., the equity premium. The latter factor explained overall
returns to stocks across the board, but it did not explain differences from
one stock to another, or more precisely, from one portfolio to another.2
The entire variation in portfolio returns was explained by the ¬rst
two factors. Fama and French found BE/ME to be the more signi¬cant
factor in explaining the cross-sectional difference in returns, with ¬rm size
next; however, they consider both factors as proxies for risk. Furthermore,
they state, ˜˜Without a theory that speci¬es the exact form of the state
variables or common factors in returns, the choice of any particular ver-
sion of the factors is somewhat arbitrary. Thus detailed stories for the
slopes and average premiums associated with particular versions of the
factors are suggestive, but never de¬nitive.™™
Abrams (1994) showed strong statistical evidence that returns are
linearly related to the natural logarithm of the value of the ¬rm, as mea-
sured by market capitalization. He used this relationship to determine the
appropriate discount rate for privately held ¬rms. In a follow-up article,
Abrams (1997) further simpli¬ed the calculations by relating the natural
log of size to total return without splitting the result into the risk-free
rate plus the equity premium.
Grabowski and King (1995) also described the logarithmic relation-
ship between ¬rm size and market return. They later (Grabowski and
King 1996) demonstrated that a similar, but weaker, logarithmic relation-
ship exists for other measures of ¬rm size, including the book value of
common equity, ¬ve-year average net income, market value of invested
capital, ¬ve-year average EBITDA, sales, and number of employees. Their
latest research (Grabowski and King 1999) demonstrates a negative log-
arithmic relationship between returns and operating margin and a posi-


2. The regression coef¬cient is essentially beta controlled for size and BE/ME. After controlling for
the other two systematic variables, this beta is very close to 1 and explains only the market
premium overall. It does not explain any differentials in premiums across ¬rms or
portfolios, as the variation was insigni¬cant.




CHAPTER 4 Discount Rates as a Function of Log Size 119
tive logarithmic relationship between returns and the coef¬cient of vari-
ation of operating margin and accounting return on equity.
The discovery that return (the discount rate) has a negative linear
relationship to the natural logarithm of the value of the ¬rm means that
the value of the ¬rm decays exponentially with increasing rates of return.
We will also show that ¬rm value decays exponentially with the standard
deviation of returns.


TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS
Columns A“F in Table 4-1 contain the input data from the Stocks, Bonds,
Bills and In¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the
regression analyses as well as the regression results. We use the 73-year
average arithmetic returns in both regressions, from 1926 to 1998. For
simplicity, we have collapsed 730 data points (73 years 10 deciles) into
73 data points by using averages. Thus, the regressions are cross-sectional
rather than time series. Column A lists the entire NYSE divided into dif-
ferent groups (known as deciles) based on market capitalization as a
proxy for size, with the largest ¬rms in decile #1 and the smallest in decile
10.3 Columns B through F contain market data for each decile which is
described below.
Note that the 73-year average market return in Column B rises with
each decile. The standard deviation of returns (Column C) also rises with
each decile. Column D shows the 1998 market capitalization of each dec-
ile, with decile #1 containing 189 ¬rms (Column F) with a market capi-
talization of $5.986 trillion (D8). Market capitalization is the price per
share times the number of shares. We use it as a proxy for the fair market
value (FMV).
Dividing Column D (FMV) by Column F (the number of ¬rms in the
decile), we obtain Column G, the average capitalization, or the average
fair market value of the ¬rms in each decile. For example, the average
company in decile #1 has an FMV of $31.670 billion (G8, rounded), while
the average ¬rm in decile #10 has an FMV of $56.654 million (G17,
rounded).
Column H shows the percentage difference between each successive
decile. For example, the average ¬rm size in decile #9 ($146.3 million;
G16) is 158.2% (H16) larger than the average ¬rm size in decile #10 ($56.7
million; G17). The average ¬rm size in decile #8 is 92.5% larger (H15)
than that of decile #9, and so on.
The largest gap in absolute dollars and in percentages is between
decile #1 and decile #2, a difference of $26.1 billion (G8“G9), or 468.9%
(H8). Deciles #9 and #10 have the second-largest difference between them
in percentage terms (158.2%, per H16). Most deciles are only 45% to 70%
larger than the next-smaller one.
The difference in return (Column B) between deciles #1 and #2 is
1.6% and between deciles #9 and #10 is 3.2%, while the difference between


3
All of the underlying decile data in Ibbotson originate with the University of Chicago™s Center for
Research in Security Prices (CRSP), which also determines the composition of the deciles.




PART 2 Calculating Discount Rates
120
T A B L E 4-1

NYSE Data by Decile and Statistical Analysis: 1926“1998


A B C D E F G H I

4 Note [1] Note [1] Note [2] Note [2] Note [2] D/F
5 Y X1 X2
6 Recent Mkt % Change
7 Decile Mean Arith Return Std Dev Capitalization % Cap # Co.s Avg Cap FMV in Avg FMV Ln(FMV)
8 1 12.11% 18.90% 5,985,553,146,000 72.60% 189 31,669,593,365 468.9% 24.1786
9 2 13.66% 22.17% 1,052,131,226,000 12.76% 189 5,566,831,884 121.8% 22.4401
10 3 14.11% 23.95% 476,920,534,000 5.78% 190 2,510,108,074 73.2% 21.6436
11 4 14.76% 26.40% 273,895,749,000 3.32% 189 1,449,183,857 60.3% 21.0943
12 5 15.52% 27.24% 170,846,605,000 2.07% 189 903,950,291 49.2% 20.6223
13 6 15.60% 28.23% 114,517,587,000 1.39% 189 605,913,159 46.5% 20.2222
14 7 15.99% 30.58% 78,601,405,000 0.95% 190 413,691,605 46.9% 19.8406
15 8 17.05% 34.36% 53,218,441,000 0.65% 189 281,579,053 92.5% 19.4559
16 9 17.85% 37.02% 27,647,937,000 0.34% 189 146,285,381 158.2% 18.8011
17 10 21.03% 45.84% 10,764,268,000 0.13% 190 56,654,042 N/A 17.8525
18 Std deviation 2.48% 1,893
19 Value wtd index 12.73% NA 8,244,096,898,000 100.00%

23 1st Regression: Return F(Std Dev. of Returns)

25 1926“1998 1939“1998

26 Constant 6.56% 8.90%
27 72/60 year mean T-bond yield [Note 3] 5.28% 5.70%
28 Std err of Y est 0.27% 0.42%
29 R squared 98.95% 95.84%
30 Adjusted R squared 98.82% 95.31%
31 No. of observations 10 10
32 Degrees of freedom 8 8
33 X coef¬cient(s) 31.24% 30.79%
34 Std err of coef. 1.14% 2.27%
35 T 27.4 13.6
36 P .01% .01%
121
122
T A B L E 4-1 (continued)

NYSE Data by Decile and Statistical Analysis: 1926“1998


A B C D E F G H I

39 2nd Regression: Return F[LN(Mkt Capitalization)]
41 1926“1998 1939“1998
42 Constant 42.24% 37.50%
43 Std err of Y est. 0.82% 0.34%
44 R squared 90.37% 97.29%
45 Adjusted R squared 89.17% 96.95%
46 No. of observations 10 10
47 Degrees of freedom 8 8
48 X coef¬cient(s) 1.284% 1.039%
49 Std err of coef. 0.148% 0.061%
50 T 8.7 16.9
51 P .01% .01%

53 3rd Regression: Return F[Decile Beta]
54 Note [4]
55 1926“1998 1939“1998
56 Constant 2.78% NA
57 Std err of Y est 0.57% NA
58 R squared 95.30% NA
59 Adjusted R squared 94.71% NA
60 No. of observations 10 NA
61 Degrees of freedom 8 NA
62 X coef¬cient(s) 15.75% NA
63 Std err of coef. 1.24% NA
64 T 12.7 NA
65 P .01% NA

68 Assumptions:
69 Long-term gov™t bonds arithmetic mean income 1926“1998 [1] 5.20%
return
70 Long horizon equity premium [2] 8.0%

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