PART 1 Forecasting Cash Flows

78

r)10

(1

r)10 g)10

(1 (1

we come to:

r)10

(1 1

PV (3-16)

r)10 g)10 (1 r)10

(1 (1

r)10 in the numerator and denominator, the so-

Canceling out (1

lution is:

1

PV (3-17)

r)10 g)10

(1 (1

We can generalize this formula to other periods of cash ¬‚ows by

letting cash ¬‚ows occur every j years. The PV of the cash ¬‚ows is the

same, except that we replace each 10 in equation (3-17) with a j in equa-

tion (3-18). Additionally, we rename the term PV as PPF, the periodic

perpetuity factor. Therefore, the PPF for $1 of payment, ¬rst occurring in

year j, is:

1

PPF PPF”end-of-year (3-18)

r) j g) j

(1 (1

The midyear PPF is again our familiar result of 1 r times the

end-of-year PPF, or:

1 r

PPF PPF”midyear (3-19)

r) j g) j

(1 (1

Note that for j 1, equations (3-18) and (3-19) reduce to the Gordon

model. As you will see further below, the above two formulas only work

if the last cash ¬‚ow occurred in the immediate prior year, i.e., t 1. In

the section on other starting years, we generalize these two formulas to

equations (3-18a) and (3-18b) to be able to handle different starting times.

Tables 3-7 and 3-8: Examples of Equations

(3-18) and (3-19)

We begin in Table 3-7 with $1.00 (B5) of moving expenses11 that we fore-

cast to occur in the next move, 10 years from now. The second move,

g)10

which we expect to occur in 20 years, should cost (1 $1.62889

(B6), assuming a 5% (D26) constant growth rate (g) in the cost. We dis-

count cash ¬‚ows at a 20% discount rate (D25).

Column A shows time in 10-year increments going up to 100 years.

Cells B5 to B14 contain the forecast cash ¬‚ows and are equal to (1 g)t j,

where t 10, 20, 30, . . . , 100 years and j 10. Actually, time should

continue to t , but at a 20% discount rate and 5% growth rate, the

11. Another common periodic expense that is less predictable than moving expenses is losses from

lawsuits. Rather than use the actual loss from the last lawsuit, one should use a base-level,

long-run average loss, which will grow at a rate of g.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 79

T A B L E 3-7

Periodic Perpetuity Factor (PPF): End-of-Year Formula

A B C D E F

Cash Flow PV Factor

g)t j r)t

4 t(Yrs) (1 1/(1 PV % PV Cum % PV

5 10 1.00000 0.16151 0.16151 74% 74%

6 20 1.62889 0.02608 0.04249 19% 93%

7 30 2.65330 0.00421 0.01118 5% 98%

8 40 4.32194 0.00068 0.00294 1% 100%

9 50 7.03999 0.00011 0.00077 0% 100%

10 60 11.46740 0.00002 0.00020 0% 100%

11 70 18.67919 0.00000 0.00005 0% 100%

12 80 30.42643 0.00000 0.00001 0% 100%

13 90 49.56144 0.00000 0.00000 0% 100%

14 100 80.73037 0.00000 0.00000 0% 100%

15 Totals 0.21916 100%

17 Calculation of PPF by formula:

19 PPF

20 0.21916

22 Assumptions:

24 j Number of years between moves 10

25 r Discount rate 20.0%

26 g Growth rate in moving costs 5.0%

28 Spreadsheet formulas:

30 A20: 1/((1 r) j (1 g) j) Equation (3-18)

T A B L E 3-8

Periodic Perpetuity Factor (PPF): Midyear Formula

A B C D E F

Cash Flow V Factor

g)t j r)t 0.5)

4 t (Yrs) (1 1/(1 PV % PV Cum % PV

5 10 1.00000 0.17692 0.17692 74% 74%

6 20 1.62889 0.02857 0.04654 19% 93%

7 30 2.65330 0.00461 0.01224 5% 98%

8 40 4.32194 0.00075 0.00322 1% 100%

9 50 7.03999 0.00012 0.00085 0% 100%

10 60 11.46740 0.00002 0.00022 0% 100%

11 70 18.67919 0.00000 0.00006 0% 100%

12 80 30.42643 0.00000 0.00002 0% 100%

13 90 49.56144 0.00000 0.00000 0% 100%

14 100 80.73037 0.00000 0.00000 0% 100%

15 Totals 0.24008 100%

17 Calculation of PPF by formula:

19 PPF

20 0.24008

22 Assumptions:

24 j Number of years between moves 10

25 r Discount rate 20.0%

26 g Growth rate in moving costs 5.0%

28 Spreadsheet formulas:

30 A20: SQRT(1 r)/((1 r) j (1 g) j) Equation (3-19)

PART 1 Forecasting Cash Flows

80

present value factors nullify all cash ¬‚ows after year 40.12 Column C con-

tains a standard present value factor, where

1

PV

r)t

(1

Column D, the present value of the cash ¬‚ows, equals Column B

Column C. Cell D15, the total PV, equals $0.21916 for every $1.00 of mov-

ing expenses in the next move. This is the ¬nal result using the ˜˜brute

force™™ method of scheduling all the cash ¬‚ows and discounting them to

PV. Cell A20 contains the formula for equation (3-18), and the result is

$0.21916, which demonstrates the accuracy of the formula. Note that the

formula for A20 appears at A30.

To calculate the PV of $20,000 of the previous year™s moving expense

growing at 5% per year and occurring every 10 years, we forecast the

cost of the next move by multiplying the $20,000 by 1.0510 $32,577.89.

We then multiply the cost of the next move by the PPF, i.e., $32,577.89

0.21916 (A20) $7,139.83 before corporate taxes. Assuming a 40% tax

rate, that rounds to $4,284 after tax. Since this is an expense, we must

remember to subtract it from”not add it to”the value we calculated

before moving expenses.13 For example, suppose we calculated a mar-

ketable minority interest FMV of $1,004,284 before moving expenses. The

¬nal marketable minority FMV would be $1 million.

Column E shows the percentage of the PV contributed by each move.

Seventy-four percent (E5) of the PV comes from the ¬rst move (Year 10),

and 19% from the second move (Year 20, at E6). Column F shows the

cumulative PV. The ¬rst two moves cumulatively account for 93% (F6) of

the entire PV generated by all moves, and the ¬rst three moves account

for 98% (F7) of the PV. Thus, in most circumstances we need not worry

about the argument that after attaining a certain size a company tends to

not move anymore. As long as it moves at least twice, the PPF will be

accurate.

Table 3-8 is identical to Table 3-7, except that it is testing equation

(3-19), the midyear formula, instead of the end-of-year formula, equation

(3-18). Again C20 D15, which veri¬es the formula.

Other Starting Years

Another question to address is what happens when the periodic expense

occurred before the prior year. Using our moving expense every 10 years

example, suppose the subject company last moved 4 years ago. It will be

another 6 years, not 10 years, to the next move. The easiest way to handle

this situation is ¬rst to value the cash ¬‚ows from a point in time where

12. Of course, at a higher growth rate and the same discount rate, it will take longer for the

present value factors to nullify the growth. The converse is also true.

13. We accomplish this by removing moving expenses from historical costs before developing our

forecast of expenses (see Chapter 2).

CHAPTER 3 Annuity Discount Factors and the Gordon Model 81

we can use the ADF equations in (3-18) and (3-19) and then adjust. Thus,

if we choose t 4 as our temporary valuation date, all cash ¬‚ows will

be spaced every 10 years, and the ADF formulas (3-18) and (3-19) apply.

We then roll forward to t 0 by multiplying the preliminary PPF by

b

(1 r) .

The generalized PPF formulas are:

r)b

(1

PPF generalized PPF”end-of-year (3-18a)

r) j g) j

(1 (1

The midyear generalized PPF is again our familiar result of 1 r

times the end-of-year PPF, or:

r)b

1 r (1

PPF generalized PPF”midyear (3-19a)

r) j g) j

(1 (1

Note that for j 1 and b 0, equations (3-18a) and (3-19a) reduce to the

Gordon model.

It is important to roll forward the cash ¬‚ow properly. With the

$20,000 move occurring 4 years ago, our forecast of the next move is still

1.0510

$20,000 $32,577.89. Whether the last move occurred 4 years

ago or yesterday, the forecast cost of the next move is the same 10 years

growth. The present value, and therefore the PPF, is different for the two

different moves, and that is captured in the numerator of the PPF, as we

have already discussed.

It is also important to recognize that the valuation date is at t 0,

which is the end of the prior year. Thus, if the valuation date is January

1, 1998, the end of the prior year is December 31, 1997. If the move oc-

curred, for example, in December 1995, then that is 2 years ago and b

2. We would use an end-of-year assumption, which means using the for-

mula in equation (3-18a). If the move occurred in June 1995, we use the

formula in equation (3-19a), and b still equals 2.

Table 3-9 is identical to Table 3-7, except that the expenses occur in

Years 6, 16, . . . instead of 10, 20, . . . . The nominal cash ¬‚ows are identical

to Table 3-7, but the formula that generates them is different. In Table

g)t j. In Table 3-9 the cash ¬‚ows are

3-7 the cash ¬‚ows are equal to (1

g)t j b because the cash ¬‚ows still grow at the rate g for 10

equal to (1

years from the last move, not just the 6 years to the next move. However,

the cash ¬‚ows in Table 3-9 are discounted 6 years instead of 10 years. The

PPF is $0.45445. The calculation by formula in A20 matches the brute

force calculation in D15, which demonstrates the validity of equation

(3-18a).

Modifying the moving expense example in Table 3-7, the PV of all

moving costs throughout time equals $20,000 1.62889 $0.45445

$14,805.14. Assuming a 40% tax rate, the after-tax present value of the

perpetuity of moving costs is $8,883, compared to the $4,284 we calcu-

lated in the discussion of Table 3-7. The present value of moving costs is

higher in this example, because the ¬rst cash ¬‚ow occurs in Year 6 instead

of Year 10.

PART 1 Forecasting Cash Flows

82

T A B L E 3-9

Periodic Perpetuity Factor (PPF): End-of-Year”Cash Flows Begin Year 6

A B C D E F

Cash Flow PV Factor

g)t j b r)t

4 t (Yrs) (1 1/(1 PV % PV Cum % PV