which can be the case with annuities, royalties, etc. The former is com-

monly known as the midyear assumption, while the latter is known as the

end-of-year (or end year) assumption.

Another important concept related to time that can be confusing is

the valuation date, the point in time to which we discount the cash ¬‚ows.

The valuation date is rarely the same as the ¬rst cash ¬‚ow. The most

common valuation date in this chapter is as of time zero, i.e., t 0. The

cash ¬‚ows usually, but not always, either begin during Year 1 or occur at

the end of Year 1.

ADF WITH END-OF-YEAR CASH FLOWS

The ADF is the present value of a series of cash ¬‚ows over n years with

constant growth, beginning with $1 of cash ¬‚ow in Year 1. We multiply

by the ¬rst year™s forecast cash ¬‚ow by the ADF to arrive at the PV of

the cash ¬‚ow stream. For example, if the ADF is 9.367 and the ¬rst year™s

cash ¬‚ow is $10,000, then the PV of the annuity is 9.367 $10,000

$93,670.

We begin the calculation of the ADF by de¬ning the cash ¬‚ows and

discounting them to their present value. Initially, for simplicity, we as-

sume end-of-year cash ¬‚ows. The PV of an annuity of $1, paid at the end

of the year for each of n years, is:

g)n 1

$1 (1 g) $1 (1

$1

PV (3-1)

r)1 r)2 r)n

(1 (1 (1

Factoring out the $1:

g)n 1

(1 g) (1

1

PV $1 (3-1a)

r)1 r)2 r)n

(1 (1 (1

The ADF is the PV of the constant growth cash ¬‚ows per $1 of starting

year cash ¬‚ow. Dividing both sides of equation (3-1a) by $1, the left-hand

side becomes PV/$1, which equals the ADF. Thus, equation (3-1a) sim-

pli¬es to:

g)n 1

(1 g) (1

1

ADF (3-1b)

r)1 r)2 r)n

(1 (1 (1

The numerators in equation (3-1b) are the forecast cash ¬‚ows them-

CHAPTER 3 Annuity Discount Factors and the Gordon Model 61

selves, and the denominators are the present value factors for each cash

¬‚ow. As mentioned previously, the ¬rst year™s cash ¬‚ow in an ADF cal-

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