... (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

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— 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

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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%