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The generalized PPF formulas are:

r)b

(1

PPF generalized PPFвЂ”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 PPFвЂ”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 п¬‚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 п¬‚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 п¬‚ows are

3-7 the cash п¬‚ows are equal to (1

g)t j b because the cash п¬‚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 п¬‚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 п¬Ѓrst cash п¬‚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-YearвЂ”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

5 6 1.00000 0.33490 0.33490 74% 74%

6 16 1.62889 0.05409 0.08810 19% 93%

7 26 2.65330 0.00874 0.02318 5% 98%

8 36 4.32194 0.00141 0.00610 1% 100%

9 46 7.03999 0.00023 0.00160 0% 100%

10 56 11.46740 0.00004 0.00042 0% 100%

11 66 18.67919 0.00001 0.00011 0% 100%

12 76 30.42643 0.00000 0.00003 0% 100%

13 86 49.56144 0.00000 0.00001 0% 100%

14 96 80.73037 0.00000 0.00000 0% 100%

15 Totals 0.45445 100%

17 Calculation of PPF by formula:

19 PPF

20 0.45445

22 Assumptions:

24 j Number of years between moves [1] 10

25 r Discount rate 20.0%

26 g Growth rate in net inc/cash п¬‚ow 5.0%

27 b Number of years from last cash п¬‚ow 4

29 Spreadsheet formulas:

31 A20: (1 r) b/((1 r) j (1 g) j) Equation (3-18a)

[1] As j decreases, the PV Factors and the PV increase. It is possible that you will have to add additional rows above Row 15 to

capture all the PV of the cash п¬‚ows. Otherwise, the PV in C20 will appear to be higher than the total of the cash п¬‚ows in D15.

PPFs in New versus Used Equipment Decisions

Another important use of PPFs is in new versus used equipment deci-

sions and in valuing used income-producing equipment. LetвЂ™s use a taxi-

cab as an example. The cab company can buy a new car or a used car.

Suppose a new car would last six years. It costs $20,000 to buy a new

one today, and we can model the cash п¬‚ows for its six-year expected life.

The cash п¬‚ows will consist of the purchase of the cab, income, gas-

oline, maintenance, insurance, etc. Each expense category has its own

pattern. Gas consumption is a variable expense that increases in dollars

over time with the rate of increase in gas prices. Maintenance is probably

low for the п¬Ѓrst two years and then begins increasing rapidly in Year 3

or 4.

We can then take the NPV of the cash п¬‚ows, and that represents the

NPV of operating a new cab for six years. It would be nice to compare

that with the NPV of operating a one-year-old cab for п¬Ѓve years (or any

other term desired). The problem is that these are different time periods.

We could use the lowest common multiple of 30 years (6 years 5 years)

and run the new cab cash п¬‚ows п¬Ѓve times and the used cab cycle six

times, but that is a lot of work. It is a far more elegant solution to use a

PPF for the new and the used equipment. The result of those computa-

CHAPTER 3 Annuity Discount Factors and the Gordon Model 83

tions will be the present value of keeping one new cab and one used cab

in service forever. We can then choose the one with the superior NPV.

Even though the cash п¬‚ows are contiguous, which is not true in the

periodic expense example, the cycle and the NPV of the cash п¬‚ows are

periodic. Every six years the operator buys a new cab. We can measure

the NPV of the п¬Ѓrst cab as of t 0. The operator buys the second cab

and uses it from Years 7вЂ“12. Its NPV as of the end of Year 6 (t 6) should

be the same as the NPV at t 0 of the п¬Ѓrst six yearsвЂ™ cash п¬‚ows, with a

growth rate for the rise in prices. If there are substantial difference in the

growth rates of income versus expenses or of the different categories of

expenses, then we can break the expenses into two or more subcategories

and apply a PPF to each subcategory, then add the NPVs together. Buying

a new cab every six years would then generate a series of NPVs with

constant growth at t 0, 6, 12, . . . . That repeating pattern is what enables

us to use a PPF to value the cash п¬‚ows.

We could perform this procedure for each different vintage of used

equipment, e.g., buying one-year-old cabs, two-year old cabs, etc. Our

п¬Ѓnal comparison would be the NPV of buying and operating a single cab

of each age (a new cab, one year old, two years old, etc.) forever. We then

simply choose the cab life with the highest NPV.

If equipment is not income producing, we can still the PPF to value

the periodic costs in perpetuity. Then the NPV would be negative.

ADFs IN LOAN MATHEMATICS

There are four related topics that should ideally all be together dealing

with the use of ADFs in loan mathematics to create formulas to calculate:

loan payments, principal amortization, the after-tax cost of a loan, and

the PV of a loan when the nominal and market rates differ. We will deal

with the п¬Ѓrst and the last topics in this section. Calculating the amorti-

zation of principal is mathematically very complex. To maintain read-

ability, it will be explained, along with the related problem of calculating

the after-tax cost of a loan, in the Appendix.

Calculating Loan Payments

We can use our earlier ADF results to easily create a formula to calculate

loan payments. We know that in the case of a п¬Ѓxed rate amortizing loan,

the principal must be equal to the PV of the payments when discounted

by the nominal rate of the loan. We can calculate the PV of the payments

using equation (3-6d) and the following deп¬Ѓnitions:

ADFNominal ADF at the nominal interest rate of the loan

ADFMkt ADF at the market interest rate of the loan

The nominal ADF is simply an end-of-year ADF with no growth.

Repeating equation (3-6d), the ADF is:

1

1

r)n

(1

ADFNominal

r

where r in this case is the nominal interest of the loan. If we use the

PART 1 Forecasting Cash Flows

84

market interest rate instead of the nominal rate, we get ADFMkt. We know

that the loan payment multiplied by the nominal ADF equals the prin-

cipal of the loan. Stating that as an equation:

Loan Payment ADFNominal Principal (3-20)

Dividing both sides of the equation by ADFNominal, we get:

Principal 1

Loan Payment Principal (3-21)

ADFNominal ADFNominal

Present Value of a Loan

The PV of a loan is the loan payment multiplied by the market rate ADF,

or:

PV Loan Payment ADFMkt (3-22)

From equation (3-21), the loan payment is the principal divided by the

nominal ADF. Substituting this into equation (3-22) gives us:

ADFMkt

PV of Loan Principal (3-23)

ADFNominal

The intuition behind this is the Principal 1/ADFNominal is the amount

of the loan payment. When we then multiply that by the ADFMkt, this

gives us the PV of the loan.

Table 3-10: Example of Equation (3-23)

Table 3-10 is an example of calculating the present value of a loan. The

assumptions appear in Table 3-10 in E77 to E82. We assume a $1 million

principal on a п¬Ѓve-year loan. The loan payment, calculated using ExcelвЂ™s

spreadsheet function, is $20,276.39 (E78) for 60 months. The annual loan

rate is 8% (E79), and the monthly rate is 0.667% (E80 E79/12). The

annual market rate of interest (the discount rate) on this loan is assumed

at 14% (I81), and the monthly market interest rate is 1.167% (I82

I81/12).

Column A shows the 60 months of payments. Column B shows the

monthly payment of $20,276.39 for 60 months. Columns C and D show

the PV factor and the PV of each monthвЂ™s payment at the nominal 8%

annual interest rate (0.667% monthly rate), while Columns E and F show

the same calculations at the market rate of 14% (1.167% monthly rate).

The present value factors in C6 to C65 total 49.31843, and present

value factors in E6 to E65 total 42.97702. Note also that the PV of the loan

at the nominal interest rate adds to the $1 million principal (D66), as it

should.

E70 is the ADF at 8% according to equation (3-6d). We show the

spreadsheet formula for E70 in A86. E71 is 1/ADFNominal $0.02027639,

the amount of loan payment for each $1 of principal. We multiply that

by the $1 million principal to obtain the loan payment of $20,276.39 in

F71, which matches E78, as it should. In E72 we calculate the ADF at the

market rate of interest, the formula for which is also equation (3-6d),

merely using the 1.167% monthly interest rate in the formula, which we

show in A88. In E73 we calculate the ratio of the market ADF to the

CHAPTER 3 Annuity Discount Factors and the Gordon Model 85

T A B L E 3-10

PV of Loan with Market Rate Nominal Rate: ADF, End-of-Year

A B C D E F

4 r 8% r 14%

5 Month Cash Flow PV Factor Present Value PV Factor Present Value

6 1 $20,276.39 0.99338 $ 20,142 0.98847 $ 20,043

7 2 $20,276.39 0.98680 $ 20,009 0.97707 $ 19,811

8 3 $20,276.39 0.98026 $ 19,876 0.96580 $ 19,583

9 4 $20,276.39 0.97377 $ 19,745 0.95466 $ 19,357

10 5 $20,276.39 0.96732 $ 19,614 0.94365 $ 19,134

11 6 $20,276.39 0.96092 $ 19,484 0.93277 $ 18,913

12 7 $20,276.39 0.95455 $ 19,355 0.92201 $ 18,695

13 8 $20,276.39 0.94823 $ 19,227 0.91138 $ 18,480

14 9 $20,276.39 0.94195 $ 19,099 0.90087 $ 18,266

15 10 $20,276.39 0.93571 $ 18,973 0.89048 $ 18,056

16 11 $20,276.39 0.92952 $ 18,847 0.88021 $ 17,848

17 12 $20,276.39 0.92336 $ 18,722 0.87006 $ 17,642

18 13 $20,276.39 0.91725 $ 18,598 0.86003 $ 17,438

19 14 $20,276.39 0.91117 $ 18,475 0.85011 $ 17,237

20 15 $20,276.39 0.90514 $ 18,353 0.84031 $ 17,038

21 16 $20,276.39 0.89914 $ 18,231 0.83062 $ 16,842

22 17 $20,276.39 0.89319 $ 18,111 0.82104 $ 16,648

23 18 $20,276.39 0.88727 $ 17,991 0.81157 $ 16,456

24 19 $20,276.39 0.88140 $ 17,872 0.80221 $ 16,266

25 20 $20,276.39 0.87556 $ 17,753 0.79296 $ 16,078

26 21 $20,276.39 0.86976 $ 17,636 0.78382 $ 15,893

27 22 $20,276.39 0.86400 $ 17,519 0.77478 $ 15,710

28 23 $20,276.39 0.85828 $ 17,403 0.76584 $ 15,529

29 24 $20,276.39 0.85260 $ 17,288 0.75701 $ 15,349

30 25 $20,276.39 0.84695 $ 17,173 0.74828 $ 15,172

31 26 $20,276.39 0.84134 $ 17,059 0.73965 $ 14,997

32 27 $20,276.39 0.83577 $ 16,946 0.73112 $ 14,824

33 28 $20,276.39 0.83023 $ 16,834 0.72269 $ 14,654

34 29 $20,276.39 0.82474 $ 16,723 0.71436 $ 14,485

35 30 $20,276.39 0.81927 $ 16,612 0.70612 $ 14,318

36 31 $20,276.39 0.81385 $ 16,502 0.69797 $ 14,152

37 32 $20,276.39 0.80846 $ 16,393 0.68993 $ 13,989

38 33 $20,276.39 0.80310 $ 16,284 0.68197 $ 13,828

39 34 $20,276.39 0.79779 $ 16,176 0.67410 $ 13,668

40 35 $20,276.39 0.79250 $ 16,069 0.66633 $ 13,511

41 36 $20,276.39 0.78725 $ 15,963 0.65865 $ 13,355

42 37 $20,276.39 0.78204 $ 15,857 0.65105 $ 13,201

43 38 $20,276.39 0.77686 $ 15,752 0.64354 $ 13,049

44 39 $20,276.39 0.77172 $ 15,648 0.63612 $ 12,898

45 40 $20,276.39 0.76661 $ 15,544 0.62879 $ 12,749

46 41 $20,276.39 0.76153 $ 15,441 0.62153 $ 12,602

47 42 $20,276.39 0.75649 $ 15,339 0.61437 $ 12,457

48 43 $20,276.39 0.75148 $ 15,237 0.60728 $ 12,313

49 44 $20,276.39 0.74650 $ 15,136 0.60028 $ 12,171

50 45 $20,276.39 0.74156 $ 15,036 0.59336 $ 12,031

51 46 $20,276.39 0.73665 $ 14,937 0.58651 $ 11,892

52 47 $20,276.39 0.73177 $ 14,838 0.57975 $ 11,755

53 48 $20,276.39 0.72692 $ 14,739 0.57306 $ 11,620

54 49 $20,276.39 0.72211 $ 14,642 0.56645 $ 11,486

55 50 $20,276.39 0.71732 $ 14,545 0.55992 $ 11,353

56 51 $20,276.39 0.71257 $ 14,448 0.55347 $ 11,222

57 52 $20,276.39 0.70785 $ 14,353 0.54708 $ 11,093

58 53 $20,276.39 0.70317 $ 14,258 0.54077 $ 10,965

59 54 $20,276.39 0.69851 $ 14,163 0.53454 $ 10,838

60 55 $20,276.39 0.69388 $ 14,069 0.52837 $ 10,714

61 56 $20,276.39 0.68929 $ 13,976 0.52228 $ 10,590

PART 1 Forecasting Cash Flows

86

T A B L E 3-10 (continued)

PV of Loan with Market Rate Nominal Rate: ADF, End-of-Year

A B C D E F

4 r 8% r 14%

5 Month Cash Flow PV Factor Present Value PV Factor Present Value

62 57 $20,276.39 0.68472 $ 13,884 0.51626 $ 10,468

63 58 $20,276.39 0.68019 $ 13,792 0.51030 $ 10,347

64 59 $20,276.39 0.67569 $ 13,700 0.50442 $ 10,228

65 60 $20,276.39 0.67121 $ 13,610 0.49860 $ 10,110

66 Totals $1,216,584 49.31843 $1,000,000 42.97702 $871,419

68 X Principal

69 Per $1 of $1 Million

70 ADF @ 8% C66 49.318433

71 Formula for payment 1/ADF 0.02027639 $20,276.39

72 ADF @ 14% E66 42.977016

73 ADF @ 14%/ADF @ 8% F66 0.871419 $871,419

75 Assumptions:

77 Principal $1,000,000

78 Loan payment $20,276.39

79 r Nominal discount rate-annual 8.0%

80 r1 Nominal discount rate-monthly 0.667%

81 r2 Market discount rate 14.0%

82 r3 Market discount rate 1.167%

84 Spreadsheet formulas:

86 E70: (1 1/(1 E80) 60)/E80

87 E71: 1/E70

88 E72: (1 1/(1 E82) 60)/E82

89 E73: E72/E70

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 mistakesвЂ”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 deп¬Ѓnitions that will aid in developing the mathematics. All other

deп¬Ѓnitions retain their same meaning as in the rest of the chapter.

Deп¬Ѓnitions

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 п¬‚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 п¬Ѓrm is its fair market value, and that is equal to its PE ratio times

the previous yearвЂ™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 п¬‚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 п¬‚ow (see Chapter 1). Now we have an ex-

pression for the FMV of the п¬Ѓrm15 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

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