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.

Special Cases for Midyear Cash Flows: No Growth, g 0

Letting g 0 in the equation above, we obtain the following ADF for

midyear cash ¬‚ows with no growth:

PART 1 Forecasting Cash Flows

68

T A B L E 3-2

ADF: Midyear 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 0.93250 0.93250

6 2 1.05100 0.05100 1.05100 0.81087 0.85223

7 3 1.10460 0.05360 1.10460 0.70511 0.77886

8 4 1.16094 0.05633 1.16094 0.61314 0.71181

9 5 1.22014 0.05921 1.22014 0.53316 0.65053

10 6 1.28237 0.06223 1.28237 0.46362 0.59453

11 7 1.34777 0.06540 1.34777 0.40315 0.54335

12 8 1.41651 0.06874 1.41651 0.35056 0.49658

13 9 1.48875 0.07224 1.48875 0.30484 0.45383

14 10 1.56468 0.07593 1.56468 0.26508 0.41476

15 Totals 6.42899

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 10.83213 4.40314 6.42899 6.42899

22 Assumptions:

24 n Number of years of cash ¬‚ows 10

25 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 SQRT(1 r)/(r g) 10.83213

30 Spreadsheet formulas:

32 B20: GM SQRT(1 r)/(r G)

33 C20: GM*x n

34 D20 B20 C20

35 E20 GM * (1 x n) This is equation (3-10b)

1 r 1 r

1

ADF midyear ADF, no growth (3-10c)

r)n

r (1 r

This follows the same type of logic as equation (3-6), with modi¬-

cation for growth being zero. The ¬rst and third terms on the RHS of

equation (3-10c) are midyear Gordon models for a constant $1 cash ¬‚ow.

g)n

Since there is no growth of cash ¬‚ows in this special case, the (1

in equation (3-10) simpli¬es to 1 and drops out of the equation. The

r)n discounts the second Gordon model term from t

1/(1 n back to

t 0, i.e., it reduces the PV of the perpetuity to time zero. Again, the

ADF is the difference of two perpetuities: the ¬rst one with cash ¬‚ows

from 1 to in¬nity, less the second one with cash ¬‚ows from n 1 to

in¬nity, the difference being cash ¬‚ows from 1 to n.

We can rewrite equation (3-10c) as equation (3-10d) by factoring out

the 1 r/r.

1 r 1

ADF 1 alternate expression for (3-10c),

r)n

r (1

midyear, no growth (3-10d)

CHAPTER 3 Annuity Discount Factors and the Gordon Model 69

Gordon Model

Letting n ’ in equation (3-10) leads us to the Gordon model.

1 r

PV CF Gordon model”midyear (3-10e)

(r g)