nominal ADF, which is E72 divided by E70 and equals 0.871419. In F73

we multiply E73 by the $1 million principal to obtain the present value

of the loan of $871,419. Note that this matches our brute force calculation

in F66, as it should.

RELATIONSHIP OF THE GORDON MODEL

TO THE PRICE/EARNINGS RATIO

In this section, we will mathematically derive the relationship between

the price/earnings (PE) ratio and the Gordon model. The confusion be-

tween the two leads to possibly more mistakes by appraisers than any

other single source of mistakes”I have seen numerous reports in which

the appraiser used the wrong earnings base. Understanding this section

should clear the potential confusion that exists. First, we will begin with

some de¬nitions that will aid in developing the mathematics. All other

de¬nitions retain their same meaning as in the rest of the chapter.

De¬nitions

Pt stock price at time t

Et historical earnings in the prior year (usually the prior 12

months)

CHAPTER 3 Annuity Discount Factors and the Gordon Model 87

Et 1 forecast earnings in the upcoming year

b earnings retention rate. Thus, cash ¬‚ow to shareholders equals

(1 b) earnings.

g1 one-year forecast growth rate in earnings, i.e., E t 1/E t 1

PE price/earnings ratio Pt/E t

Mathematical Derivation

We begin with the statement that the market capitalization of a publicly

held ¬rm is its fair market value, and that is equal to its PE ratio times

the previous year™s historical earnings:

Pt

FMV * Et (3-24)

Et

We repeat equation (3-10e) below as equation (3-25), with one

change. We will assume that forecast cash ¬‚ow to shareholders, CFt 1, is

E t 1, where b is the earnings retention rate.14 The

equal to (1 b)

earnings retention rate is the sum total of all the reconciling items be-

tween net income and cash ¬‚ow (see Chapter 1). Now we have an ex-

pression for the FMV of the ¬rm15 according to the midyear Gordon

model.

1 r

FMV (1 b) E t midyear Gordon model (3-25)

1

(r g)

Substituting Et E t (1 g1) into equation (3-25), we come to:

1

1 r

FMV (1 b) E t (1 g1) (3-26)

(r g)

The left-hand sides of equations (3-24) and (3-26) are the same. There-

fore, we can equate the right-hand sides of those equations.

1 r

Pt

* Et (1 b) E t (1 g1) (3-27)

Et (r g)

E t cancels out on both sides of the equation. Additionally, we use the

simpler notation PE for the price-earnings multiple. Thus, equation (3-27)

reduces to:

1 r

PE (1 b) (1 g1)

r g

relationship of PE to Gordon model multiple (3-28)

The left-hand term is the price-earnings multiple and the right-hand

term is one minus the earnings retention rate times one plus the one-year

growth rate times the midyear Gordon model multiple. In reality, inves-

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

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

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

PART 1 Forecasting Cash Flows

88

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

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

pectation of growth thereafter that they approximate as being constant.

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

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

modeling.

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

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

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

PE multiples from guideline companies, whether publicly or privately

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

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

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

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

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

plus the forecast one-year growth rate.

CONCLUSIONS

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

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

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

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

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

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

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

repeating cycles).

— Whether or not the constant growth is zero.

— Whether there is a stub period.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Gordon model. This derivation demonstrates that the PE multiple equals

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

CHAPTER 3 Annuity Discount Factors and the Gordon Model 89

T A B L E 3-11

ADF Equation Numbers

With Growth No Growth

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

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

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

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

Valuation date v (3-11b)

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

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

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

Loan payment (3-21)

Relationship of Gordon model to PE (3-28)

Formulas in the Appendix

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

Amortization of loan principal (A3-10)

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

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

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

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

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

confusion.

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

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

tion.

BIBLIOGRAPHY

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

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

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

Homewood, Ill.: R. D. Irwin.

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

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

Tenn.: Peabody.

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

Press.

APPENDIX

INTRODUCTION

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

The topics that we cover are:

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

(stub period).

— Developing ADFs for loan mathematics, consisting of calculating

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

a loan.

PART 1 Forecasting Cash Flows

90

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

ered and formulas developed are esoteric and less practically useful than

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

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

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

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

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

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

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

THE ADF WITH STUB PERIODS (FRACTIONAL YEARS)

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

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

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

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

of-year cash ¬‚ows.

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

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

De¬nitions

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

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

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

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

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

3.25 0.5 2.75 years.

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

example

z end of the stub period 12.60 years.

p proportion of a full year represented by the stub period

z n 12.60 12.25 0.35 years

g constant growth rate in cash ¬‚ows 5.1%

t point in time, measured in years

The Cash Flows

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

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

F I G U R E A3-1

Timeline of Cash Flows

Row \ Col. B C D E F G H

1 Year (numeric) 3.25 4.25 5.25 ¦ 12.25 12.60

2 Year (symbolic) S S+1 S+2 ¦ n z

g(1+g)n-S-1

3 Growth (in $) 0 g g(1+g) ¦ NA

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

4 Cash Flow 1 1+g ¦

CHAPTER 3 Annuity Discount Factors and the Gordon Model 91

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

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

¬‚ow, i.e., cell B3 0.

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

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

tS

g)4.25 3.25.

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

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

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

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

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

g)t S (1 g)5.25 3.25.

¬‚ow equals (1

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

g)2, so cash ¬‚ows are (1 g)2 g)2 g)2

¬‚ows grow g(1 g(1 (1

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