culation is always de¬ned as $1. With constant growth in cash ¬‚ow, each

successive year is (1 g) times the previous year™s cash ¬‚ow, which

g)n 1. The cash ¬‚ow is not

means that the cash ¬‚ow in period n is (1

(1 g)n, because the ¬rst year™s cash ¬‚ow is $1.00, not 1 g. For example,

if g 10%, the ¬rst year™s cash ¬‚ow is, by de¬nition, $1.00. The second

year™s cash ¬‚ow is 1.1 $1.00 $1.10. The third year™s cash ¬‚ow is 1.1

2

$1.00 1.21. The fourth year™s cash ¬‚ow is 1.13 $1.00

$1.10 1.1

$1.331, etc. The denominators in equation (3-1b) discount the cash

¬‚ows in the numerator to their present value.

Next, we begin a series of algebraic manipulations which will ulti-

mately enable us to solve for the ADF and specify it in a formula. Mul-

tiplying equation (3-1b) by (1 g)/(1 r), we get:

g)n 1 g)n

(1 g) (1 g) (1 (1

ADF (3-2)

r)2 r)n 1

(1 r) (1 (1 r)n (1

Notice that most of the terms in equation (3-2) are identical to equation

(3-1b). We next subtract equation (3-2) from equation (3-1b). All of the

terms in the middle of the equation are identical and thus drop out. The

only terms that remain on the RHS after the subtraction are the ¬rst term

on the RHS of equation (3-1b) and the last term on the RHS of equation

(3-2).

g)n

1 g (1

1

ADF ADF (3-3)

r)n 1

1 r 1 r (1

Next, we wish to simplify only the left-hand side of equation (3-3):

1 g 1 g

ADF ADF ADF 1 (3-3a)

1 r 1 r

Multiplying the 1 in the square brackets on the RHS of the equation

by (1 r)/(1 r), we get:

1 g 1 g

1 r

ADF 1 ADF

1 r 1 r 1 r

(1 r) (1 g) r g

ADF ADF (3-3b)

1 r 1 r

Substituting the last expression of equation (3-3b) into the left-hand

side of equation (3-3), we get:

g)n

(r g) (1

1

ADF (3-4)

r)n 1

(1 r) (1 r) (1

Multiplying both sides of the equation by (1 r)/(r g), we obtain:

g)n

(1

(1 r) 1

ADF (3-5)

r)n 1

(r g) (1 r) (1

After canceling out the (1 r), this simpli¬es to:

PART 1 Forecasting Cash Flows

62

n

1 g

1 1

ADF (3-6)

r g 1 r r g

ADF with growth and end-of-year cash ¬‚ows

There are three alternative ways to regroup the terms in equation

(3-6) that will prove useful, which we label as equations (3-6a), (3-6b),

and (3-6c). In the ¬rst alternative expression for equation (3-6), we split

up the ¬rst term in the square brackets into two separate terms, placing

the denominator at the far right.

1 1 1

g)n

ADF (1

r)n

r g r g (1 (3-6a)

first alternative expression for (3-6)

We derive the second alternative expression by simply factoring out

the 1/(r g) from equation (3-6) and restate the equation as equation

(3-6b). It has the advantage of being more compact than equation (3-6).

n

1 g

1

ADF 1

r g 1 r (3-6b)

second alternative expression for (3-6)

After we develop some additional results, we will be able to explain

equations (3-6) through (3-6b) intuitively. In the meantime, we will make

some substitutions in equation (3-6b) that will greatly simplify its form

and eventually make the ADF much more intuitive.

Note that the ¬rst term on the right-hand-side of equation (3-6b) is

the classical Gordon model multiple, 1/(r g). Let™s denote it GM. The

next substitution that will simplify the expression is to let x (1 g)/

(1 r). Then we can restate equation (3-6b) as:

xn)

ADF GM (1 third alternative expression for (3-6) (3-6c)

Behavior of the ADF with Growth

The ADF is inversely related to r and directly related to g, i.e., an increase

in the discount rate decreases the ADF and vice-versa, while an increase

in the growth rate causes an increase in the ADF, and vice-versa.

Special Case of ADF when g 0: The Ordinary Annuity

When g 0, there is no growth in cash ¬‚ows, and equation (3-6) sim-

pli¬es to equation (3-6d), the formula for an ordinary annuity.

1

1

r)n

1 1 1 (1

ADF , or ADF (3-6d)

r)n r

r (1 r

1/r is the PV of a perpetuity that is constant in nominal dollars, or a

Gordon model with g 0.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 63

Special Case when n ’ and r g: The Gordon Model

The Gordon model is a ¬nancial formula that every business appraiser

knows”at least in the end-of-year form. It is the formula necessary to

calculate the present value of the perpetuity with constant growth in cash

¬‚ows in the terminal period (also known as the residual or reversion

period), i.e., from years n 1 to in¬nity (after discounting the ¬rst n

years of cash ¬‚ows or net income). To be valid, the growth rate must be

less than the discount rate.

What few practitioners know, however, is that the Gordon model is

merely a special case of the ADF. The Gordon model contains two ad-

ditional assumptions that the ADF in equation (3-6) does not have.

— The time horizon is in¬nite, which means that we assume cash

¬‚ows will grow at the constant rate of g forever. This means that

n, the terminal year of the cash ¬‚ows, equals in¬nity.

— The discount rate is greater than the growth rate, i.e., r g.

Since r g,

n

1 g

1 r

goes to zero as n goes to in¬nity. Therefore, the entire term in square

brackets in equation (3-6) goes to zero, which simpli¬es to:

1

ADF Gordon model multiple, end-of-year cash flows (3-7)

r g

Equation (3-7) is the end-of-year Gordon model multiple. In other

words, the Gordon model multiple is just a special case of the ADF when

n equals in¬nity. Using this multiple, we obtain the Gordon model, with

end-of-year cash ¬‚ows:

CF

PV (3-8)

(r g)

Another way of expressing equation (3-8) is rewriting it as:

1

PV CF (3-9)

(r g)

Thus, the present value of a perpetuity with growth contains two terms

conceptually:

— CF, the starting year™s forecast cash ¬‚ow.3

— 1/(r g), the Gordon model multiple, which when multiplied

by the ¬rst year™s forecast cash ¬‚ow gives us the present value of

the perpetuity.

3. Note that you do not use historical cash ¬‚ow (or earnings).

PART 1 Forecasting Cash Flows

64

Intuitively Understanding Equations (3-6) and (3-6a)

Now that we understand the Gordon model, we can gain deeper insight

into equation (3-6). The ADF is the difference of two perpetuities. The

¬rst term, 1/(r g), is the PV as of t 0 of a perpetuity with cash ¬‚ows

going from t 1 to in¬nity. The second term is the PV as of t 0 of a

perpetuity going from t n 1 to in¬nity, which is explained in the

next paragraph. The difference of the two is the PV as of t 0 of the

annuity from t 1 to n.

g)n

Let™s give an intuitive explanation of equation (3-6a). The (1

is the forecast cash ¬‚ow4 for Year (n 1), which we then multiply by

1/(r g), our familiar Gordon model multiple. The result is the PV as

of t n of the forecast cash ¬‚ows from n 1 to in¬nity. Dividing by

n

(1 r) transforms the PV as of t n to the PV as of t 0.

Relationship between the ADF and the Gordon Model

The relationship between the ADF and Gordon model is so intimate that

we can derive the Gordon model from the ADF and vice-versa. The ADF

is the difference of two Gordon models, as illustrated graphically below

in Figure 3-1.

In graphical terms, the top line represents the Gordon model with

cash ¬‚ows from t 1 to in¬nity (our valuation date is actually time zero,

which is not shown on the graph). The cash ¬‚ows in the second Gordon

model begin at t n 1 and continue to in¬nity. The difference between

these two Gordon models is simply the ADF from t 1 to n.

F I G U R E 3-1

Timeline of the ADF and Gordon Model

Gordon 1’∞

Minus

Gordon n+1’∞

Equals

ADF 1’n

∞

1 n n+1

Table 3-1: Proof of ADF Equations (3-6) through (3-6b)

Table 3-1 is the valuation of a 10-year annuity, with a discount rate of

15% and an annual growth rate of 5.1%. All assumptions appear in cells

4. The ¬rst year™s cash ¬‚ow is 1, or (1 + g)0. The second year™s cash ¬‚ow is (1 + g)1. In general,

cash ¬‚ow in Year t (1 + g)t 1.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 65

T A B L E 3-1

ADF: End-of-Year Formula

A B C D E F

g)t 1

4 t (Yrs) Cash Flow (CF) Growth in CF (1 PV Factor NPV

5 1 1.00000 0.00000 1.00000 0.86957 0.86957

6 2 1.05100 0.05100 1.05100 0.75614 0.79471

7 3 1.10460 0.05360 1.10460 0.65752 0.72629

8 4 1.16094 0.05633 1.16094 0.57175 0.66377

9 5 1.22014 0.05921 1.22014 0.49718 0.60663

10 6 1.28237 0.06223 1.28237 0.43233 0.55440

11 7 1.34777 0.06540 1.34777 0.37594 0.50668

12 8 1.41651 0.06874 1.41651 0.32690 0.46306

13 9 1.48875 0.07224 1.48875 0.28426 0.42320

14 10 1.56468 0.07593 1.56468 0.24718 0.38676

15 Totals 5.99506

17 Calculation of NPV by formulas:

18 Grand

19 Time 1 to In¬nity (n 1) to In¬nity 1 to n Total

20 NPV 10.10101 4.10595 5.99506 5.99506

22 Assumptions:

24 n Number of years of cash ¬‚ows 10

24 r Discount rate 15.0%

26 g Growth rate in net inc/cash ¬‚ow 5.1%

27 x (1 g)/(1 r) 0.9139

28 Gordon model multiple GM 1/(r g) 10.101010

30 Spreadsheet formulas:

32 B20: GM 1/(r g)

33 C20: GM*x n

34 D20 B20 C20

35 E20 GM * (1 x n) This is equation (3-6c)

0.9139 (F27).5 If

F24 to F28. Recall that we de¬ne x (1 g)/(1 r)

this were a perpetuity, the Gordon model multiple would be 10.101010

(F28).

We begin with a cash ¬‚ow of $1.00 at the end of Year 1 (B5). Column

C shows the annual growth in cash ¬‚ows at 5.1%.6 The cash ¬‚ow in

Column B is always equal to the previous cash ¬‚ow plus the growth in

the current period, where Cash Flowt Cash Flowt 1 Growtht. Column

D replicates the cash ¬‚ow in Column C using the formula Cash Flow

(1 g)t 1, which thus provides us with a general formula for the cash

¬‚ows. We multiply the cash ¬‚ows in Column C by the end-of-year present

value factor in Column E to arrive at the present value of the cash ¬‚ows

5. As mentioned in a previous footnote, we use i synonymously with r.

6. We can use the same formulas for other time periods, e.g., months instead of years. Then we

must use the monthly growth rate of 5.1%/12 0.4267% instead of the annual.

PART 1 Forecasting Cash Flows

66

in Column F. The sum of the present values of the 10 years of cash ¬‚ows

is 5.99506 in F15. This is the ˜˜brute force™™ method of calculating the an-

nuity.

As we will demonstrate, equation (3-6) is a more compact and ele-

gant solution. Cell B20 contains the end-of-year Gordon multiple results

of the ¬rst term in equation (3-6), which equals F28. This is the present

value of the perpetuity of $1.00 growing at a constant 5.1% from Year 1

to in¬nity. In C20 we subtract the present value of the perpetuity from

Year n 1 to in¬nity, which equals 4.10595 and is the term in equation

(3-6) in square brackets. The difference of the two perpetuities is 5.99506,

which equals F15, our brute force solution. Finally, E20 is the formula for

the entire equation, which equals the same 5.99506 calculated in D20 and

F15, proving the validity of equation (3-6), including its components. We

show the formulas for Row 20 at the bottom of Table 3-1. Note that the

formula in E20 is equation (3-6c).

A Brief Summary

To help you decide if you should read on, let™s take a look at what we

have covered so far, what we will cover in the remainder of the chapter,

and how dif¬cult the material will be. We have thus far derived the end-

of-year ADF, examined its special cases (the Gordon model and the no-

growth formula), explained the intimate relationship of the ADF and the

Gordon model, explained the intuition behind the components of the

ADF model, and proved the model with an example.

The reader now should understand the principles of ADFs and Gor-

don models. If you are having dif¬culty with the mathematics, you may

wish to skip to the sections on Periodic Perpetuity Factors (PPFs) and

Relationship of the Gordon Model to the Price/Earnings Ratio, which are

of practical signi¬cance to most readers. However, you now should un-

derstand almost everything you will need to easily comprehend the rest

of the chapter. The rest of the chapter is primarily simple variations of

the derivations we have done thus far.

In the remainder of the chapter, we will cover:

— The midyear version of the ADF (with the same special cases of

the Gordon model and g 0).

— Starting periods for the cash ¬‚ows that are different than Year 1,

which is of practical signi¬cance in discounted cash ¬‚ow analysis

in the calculation of the PV of the reversion.

— Calculating periodic perpetuity factors (PPFs), which are a

variation of the Gordon model for periodic expenses such as

moving expense and losses from lawsuits. Additionally, PPFs are

useful for decisions in buying new versus used income-

producing equipment (such as CAT scans, ships, or taxicabs) and

for calculating the value of used equipment.

— Calculating loan payments.

— Calculating the present value of loans.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 67

— The relationship of the Gordon model to the PE multiple, the

misunderstanding of which may well be the single most

common source of technical error in business valuation.

MIDYEAR CASH FLOWS

Most businesses have cash ¬‚ows that more or less occur evenly through-

out the year. In a present value sense, this is approximately equivalent to

having all cash ¬‚ows occur midway through the year. Thus, in valuing

most businesses, it is appropriate to use midyear cash ¬‚ows rather than

end-of-year cash ¬‚ows.

Midyear cash ¬‚ows occur six months (one half-year) earlier than end-

of-year cash ¬‚ows. We derive this formula in exactly the same fashion as

equation (3-6). We start with equation (3-1b); however, the denominators,

which are the time periods by which we discount the cash ¬‚ows, are one

half-year less than those in equation (3-1b). We adjust for this difference

by multiplying every numerator by 1 r, which has the same effect

as reducing the denominators by 0.5 years. We then factor the 1 r

out of the sequence, resulting in a the midyear ADF that equals 1 r

times the end-of-year ADF.

n

1 r 1 r

1 g

ADF midyear ADF (3-10)

r g 1 r r g

We interpret equation (3-10) in exactly the same fashion as equation

(3-6). We can factor out the Gordon model multiple as before and restate

equation (3-10) as equations (3-10a) and (3-10b) below. Note that equa-

tions (3-10a) and (3-10b) are identical to equations (3-6b) and (3-6c), re-

spectively, except that the Gordon model multiple is midyear instead of

end-of-year.

n

1 r 1 g

ADF 1 alternative expression for (3-10)

r g 1 r

(3-10a)

n

ADF GM (1 x ) second alternative expression for (3-10) (3-10b)

Table 3-2: Example of Equation (3-10) through (3-10b)

Table 3-2 is identical to Table 3-1, except that here we use the midyear

rather than end-of-year ADF. Note that the Gordon model multiple (GM)

in B20 and F28 is 10.83213 versus 10.101010 in Table 3-1. The GM in Table

3-2 is exactly 1 r times the GM in Table 3-1, i.e., 10.1010 1.15

10.83213. This demonstrates the validity of equations (3-10) through

(3-10b), the midyear ADF.