14. I wish to thank Larry Kasper for pointing out the need for this.

15. Assuming the present value of the cash ¬‚ows of the ¬rm is its FMV. This ignores valuation

discounts, an acceptable simpli¬cation in this limited context.

PART 1 Forecasting Cash Flows

88

tors do not expect constant growth to perpetuity. They usually have ex-

pectations of uneven growth for a few years and a vague, long-run ex-

pectation of growth thereafter that they approximate as being constant.

Therefore, we should look at g, the perpetual growth rate in cash ¬‚ow,

as an average growth rate over the in¬nite period of time that we are

modeling.

We should be very clear that the earnings base in the PE multiple

and the Gordon model are different. The former is the immediate prior

year and the latter is the ¬rst forecast year. When an appraiser develops

PE multiples from guideline companies, whether publicly or privately

owned, he should multiply the PE multiple from the guideline companies

(after appropriate adjustments) by the subject company™s prior year earn-

ings. When using a discounted cash ¬‚ow approach, the appraiser should

multiply the Gordon model by the ¬rst forecast year™s earnings. Using the

wrong earnings will cause an error in the valuation by a factor of one

plus the forecast one-year growth rate.

CONCLUSIONS

We can see that there is a family of annuity discount factors (ADFs), from

the simplest case of an ordinary annuity to the most complicated case of

an annuity with stub periods (fractional years), as discussed in the Ap-

pendix. The elements that determine which formula to use are:

— Whether the cash ¬‚ows are midyear versus end-of-year.

— When the cash ¬‚ows begin (Year 1 versus any other time).

— If they occur every year or at regular, skipped intervals (or have

repeating cycles).

— Whether or not the constant growth is zero.

— Whether there is a stub period.

For cash ¬‚ows without a stub period, the ADF is the difference of

two Gordon model perpetuities. The ¬rst term is the perpetuity from S

to in¬nity, where S is the starting year of the cash ¬‚ow. The second term

is the perpetuity starting at n 1 (where n is the ¬nal cash ¬‚ow in the

annuity) going to in¬nity. For cash ¬‚ows with a stub period, the preced-

ing statement is true with the addition of a third term for the single cash

¬‚ow of the stub period itself, discounted to PV.

While this chapter contains some complex algebra, the focus has been

on the intuitive explanation of each ADF. The most dif¬cult mathematics

have been moved to the Appendix, which contains the formulas for ADF

with stub periods and some advanced material on the use of ADFs in

calculating loan amortization. ADFs are also used for practical applica-

tions in Chris Mercer™s quantitative marketability discount model (see

Chapter 7), periodic expenses such as moving costs and losses from law-

suits, ESOP valuation, in reducing a seller-subsidized loan to its cash

equivalent price in Chapter 10, and to calculate loan payments.

We have performed a rigorous derivation of the PE multiple and the

Gordon model. This derivation demonstrates that the PE multiple equals

one minus the earnings retention rate times one plus the one-year growth

CHAPTER 3 Annuity Discount Factors and the Gordon Model 89

T A B L E 3-11

ADF Equation Numbers

With Growth No Growth

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

Ordinary ADF (3-6) to (3-6b) (3-10) to (3-10b) (3-6d) (3-10c) & (3-10d)

Gordon model (3-7) (3-10e)

Starting cash ¬‚ow not t 1 (3-11) & (3-11a) (3-12) (3-11c)

Valuation date v (3-11b)

Gordon model for starting CF not 1 (3-11d)

Periodic expenses (3-18) (3-19)

Periodic expenses-¬‚exible timing (3-18a) (3-19a)

Loan payment (3-21)

Relationship of Gordon model to PE (3-28)

Formulas in the Appendix

ADF with stub period (A3-3) (A3-4)

Amortization of loan principal (A3-10)

PV of loan after-tax (A3-24) & (A3-25)

rate times the midyear Gordon model multiple. Furthermore, we showed

how the former uses the prior year™s earnings, while the latter uses the

¬rst forecast year™s earnings. Many appraisers have found that confusing,

and hopefully this section of the chapter will do much to eliminate that

confusion.

Because there are so many ADFs for different purposes and assump-

tions, we include Table 3-11 to point the reader to the correct ADF equa-

tion.

BIBLIOGRAPHY

Gordon, M. J., and E. Shapiro. 1956. ˜˜Capital Equipment Analysis: The Required Rate of

Pro¬t,™™ Management Science 3: 102“110.

Gordon, M. J. 1962. The Investment, Financing, and Valuation of the Corporation, 2d ed.

Homewood, Ill.: R. D. Irwin.

Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing and Supporting

Marketability Discounts in the Appraisal of Closely Held Business Interests. Memphis,

Tenn.: Peabody.

Williams, J. B. The Theory of Investment Value. 1938. Cambridge, Mass.: Harvard University

Press.

APPENDIX

INTRODUCTION

This appendix is an extension of the material developed in the chapter.

The topics that we cover are:

— Developing ADFs for cash ¬‚ows that end on a fractional year

(stub period).

— Developing ADFs for loan mathematics, consisting of calculating

the amortization of principal in loans and the net after-tax cost of

a loan.

PART 1 Forecasting Cash Flows

90

This appendix is truly for the mathematically brave. The topics cov-

ered and formulas developed are esoteric and less practically useful than

the formulas in the chapter, though the formula for the after-tax cost of

a loan may be useful to some practitioners. The material in this appendix

is included primarily for reference. Nevertheless, even those not com-

pletely comfortable with the dif¬cult mathematics can bene¬t from fo-

cusing on the verbal explanations before the equations and the develop-

ment of the ¬rst one or two equations in the derivation of each of the

formulas. The rest is just the tedious math, which can be skipped.

THE ADF WITH STUB PERIODS (FRACTIONAL YEARS)

We will now develop a formula to handle annuities that have stub peri-

ods, constant growth in cash ¬‚ows, and cash ¬‚ows that start at any time.

To the best of my knowledge, I invented this formula. In this section we

will assume midyear cash ¬‚ows and later present the formula for end-

of-year cash ¬‚ows.

Let™s begin with constructing a timeline of the cash ¬‚ows in Figure

A3-1, using the following de¬nitions and assumptions:

De¬nitions

S time (in years) of the ¬rst cash ¬‚ow for end-of-year cash

¬‚ows. For midyear cash ¬‚ows, S end of the year in which the

¬rst cash ¬‚ow occurs 3.25 years in this example, which means

the cash ¬‚ow for that year begins at t 2.25 years and we assume

the cash ¬‚ow occurs in the middle of the year, or S 0.5

3.25 0.5 2.75 years.

n end of the last whole year™s cash ¬‚ows 12.25 years in this

example

z end of the stub period 12.60 years.

p proportion of a full year represented by the stub period

z n 12.60 12.25 0.35 years

g constant growth rate in cash ¬‚ows 5.1%

t point in time, measured in years

The Cash Flows

We assume the ¬rst cash ¬‚ow of $1.00 (Figure A3-1, cell C4) occurs during

year S (S is for starting cash ¬‚ow), where t 2.25 to t 3.25 years. For

F I G U R E A3-1

Timeline of Cash Flows

Row \ Col. B C D E F G H

1 Year (numeric) 3.25 4.25 5.25 ¦ 12.25 12.60

2 Year (symbolic) S S+1 S+2 ¦ n z

g(1+g)n-S-1

3 Growth (in $) 0 g g(1+g) ¦ NA

(1+g)2 (1+g)n-S p(1+g)n-S+1

4 Cash Flow 1 1+g ¦

CHAPTER 3 Annuity Discount Factors and the Gordon Model 91

simplicity, we denote that the cash ¬‚ow is for the year ending at t 3.25

years (cell C1). Note that for Year 3.25, there is no growth in the cash

¬‚ow, i.e., cell B3 0.

The following year is 4.25 (cell D1), or S 1 (cell D2). The $1.00

grows at a rate of g (cell D3), so the ending cash ¬‚ow is 1 g (cell D4).

tS

g)4.25 3.25.

Note that the ending cash ¬‚ow is equal to (1 g) (1

For Year 5.25, or S 2 (cell E2), growth in cash ¬‚ows is g times the

prior year™s cash ¬‚ow of (1 g), or g (1 g) (cell E3), which leads to a

cash ¬‚ow equal to the prior year™s cash ¬‚ow plus this year™s growth, or

(1 g) g(1 g) (1 g) (1 g) (1 g)2 [cell E4]. Again, the cash

g)t S (1 g)5.25 3.25.

¬‚ow equals (1

For the year 6.25, or S 3, which is not shown in Figure A3-1, cash

g)2, so cash ¬‚ows are (1 g)2 g)2 g)2

¬‚ows grow g(1 g(1 (1

g)3 (1 g)t S g)6.25 3.25.

(1 g) (1 (1

We continue in this fashion through the last whole year of cash ¬‚ows,

which we call Year n (Column G). In our example, n 12.25 years (cell

nS

G1). The cash ¬‚ows during Year n are equal to (1 g) [cell G4].

Had we completed one more full year, the cash ¬‚ows would have

extended to Year 13.25, or Year n 1. If so, the cash ¬‚ow would have

nS1

been (1 g) . However, since the stub year™s cash ¬‚ow is only for a

partial year, the ending cash ¬‚ow is multiplied by p”the fractional por-

g)n S 1.

tion of the year”leading to an ending cash ¬‚ow of p(1

It is important to recognize that there may be other ways of speci-

fying how the partial year affects the cash ¬‚ows. For example, it is pos-

sible, but very unlikely, that the cash ¬‚ows can be based on a legal doc-

ument that speci¬es that only the growth rate itself will be fractional, but

the corpus of the cash ¬‚ow will not diminish for the partial year. We

could calculate a solution to this ADF, but we will not, as it is very un-

likely to be of any practical use and we have already demonstrated how

to model the most likely method of splitting the cash ¬‚ows in the frac-

tional year. The point is that modeling the fractional year cash ¬‚ows de-

pends on the agreement and/or the underlying scenario, and one should

not blindly charge off into the sunset applying a formula developed un-

der an assumption that does not apply in another case.

Discounting Periods

The ¬rst cash ¬‚ow occurs during the year that spans from

t 2.25 to t 3.25. We assume the cash ¬‚ows occur evenly throughout

the year, which is tantamount to assuming all cash ¬‚ows occur on average

halfway through the year, i.e., at Year 2.75. Therefore as of time zero,

de¬ned as t 0, the ¬rst $1 cash ¬‚ow has a present value of

1 1

r)2.75 r)S 0.5

(1 (1

We will be discounting the cash ¬‚ows in two stages because that will

later enable us to provide a more intuitive explanation of our results. Our

¬rst discounting of cash ¬‚ows will be to t S 1, the beginning of

the ¬rst year of cash ¬‚ows. The ¬rst year™s cash ¬‚ow then receives a dis-

PART 1 Forecasting Cash Flows

92

r)0.5, the second year™s cash ¬‚ows receive a discount

count of 1/(1

r)1.5, etc. Thus, the denominators here are identical to those

of 1/(1

for cash ¬‚ows that would begin in Year 1 instead of S.

The Equations

The PV of our series of cash ¬‚ows as of t S 1 is:

(1 g)

1

PV

r)0.5 r)1.5

(1 (1

g)n S

g)n S 1

(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: