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)

This can be split into the following terms:

1 r

CF

(r g)

The ¬rst term is the forecast net income for the ¬rst year, and the second

term is the Gordon model multiple for a midyear cash ¬‚ow.

STARTING PERIODS OTHER THAN YEAR 1

When cash ¬‚ows begin in any year other than 1, it is necessary to use a

more general (and complicated) ADF formula. We will present formulas

for both the end-of-year and midyear cash ¬‚ows when this occurs.

End-of-Year Formulas

In the following equations, S is the starting year of the cash ¬‚ows. The

end-of-year ADF is:

nS1

1 g

1 1 1

ADF

r)S 1

r g 1 r r g (1

generalized end-of-year ADF (3-11)

Note that when S 1, n S 1 n, and equation (3-11) reduces to

equation (3-6).

The intuition behind this formula is that if we are standing at point

t S 1 looking at the cash ¬‚ows that begin at S and end at n, they

would appear the same as if we were at t 0 looking at a normal series

of cash ¬‚ows that begin at t 1. The only difference is that there are n

cash ¬‚ows in the latter case and n (S 1) n S 1 cash ¬‚ows in

the former case.

Therefore, the term in square brackets, which is the PV of the cash

¬‚ows at t S 1, is the usual ADF formula, except that the exponent

of the second term in square brackets changes from n in equation (3-6)

to n S 1 in equation (3-11). If the cash ¬‚ows begin in a year later

than Year 1, S 1 and there are fewer years of cash ¬‚ows from S to n

than there are from 1 to n.7 From the end of Year S 1 to the end of

Year n, there are n (S 1) n S 1 years.

In order to calculate the PV as of t 0, it is necessary to discount

r)S 1. Note that at S

the cash ¬‚ows S 1 years using the term 1/(1

1, the term at the right”outside the brackets”becomes 1 and effectively

7. The converse is true for cash ¬‚ows beginning in the past, where S is less than 1.

PART 1 Forecasting Cash Flows

70

drops out of the equation. The exponent within the square brackets, n

S 1, simpli¬es to n, and (3-11) simpli¬es to (3-6).

An alternative form of (3-11) with the Gordon model speci¬cally fac-

tored out is:

nS1

1 g

1 1

ADF 1

r)S 1

r g 1 t (1

generalized end-of-year ADF”alternative form (3-11a)

Valuation Date 0

If the valuation date is different than t 0, then we do not discount by

the entire S 1 years. Letting the valuation date v, then we discount

back to t S v 1, the reason being that normally we discount S

1 years, but in this case we will discount only to v, not to zero. Therefore,

we discount S 1 v years, which we restate as S v 1. For example,

if we want to value cash ¬‚ows from t 23 months to 34 months as of t

8

10 months, then we discount 23 10 1 12 months, or 1 year.

This formula is important in calculating the reduction in principal for an

amortizing loan. The formula is:

nS1

1 g

1 1 1

ADF generalized ADF:

r)S v1

r g 1 r r g (1

(3-11b)

end-of-year

where v valuation date. We will demonstrate the accuracy of this for-

mula in Sections 2 and 3 of Table A3-3 in the Appendix.

Table 3-3: Example of Equation (3-11)

In Table 3-3, we begin with $1 of cash ¬‚ows (C7) at t 3.25 years, i.e.,

S 3.25 (G40). The discount rate is 15% (G42), and cash ¬‚ows grow at

5.1% (G43). In Year 4.25, cash ¬‚ow grows 5.1% $1.00 $0.051 (B8),

which is equal to the prior year cash ¬‚ow of $1.00 in C7 plus the growth

in the current year, for a total of $1.051 in C8. We continue in the same

fashion to calculate growth in cash ¬‚ows and the actual cash ¬‚ows

through the last year n 22.25.

g)t S, which

In Column D, we use the formula Cash Flow (1

duplicates the results in Column C. Thus, the formula in Column D is a

general formula for cash ¬‚ow in any period.9

Next, we discount the cash ¬‚ows to present value. In this table we

show both a two-step and a single-step discounting process.

8. We actually do this in Table A3-3 in the Appendix. In the context of loan payments, cash ¬‚ows

are ¬xed, which means g 0. Also, with loan payments we generally deal with time

measured in months, not years. To remain consistent, the discount rates must also be

monthly, not annual.

g)t S g)t 1, which is the formula that

9. Note that when cash ¬‚ows begin at t 1, then (1 (1

g)t S is truly a

describes the cash ¬‚ows in Column D in Tables 3-1 and 3-2. Thus, (1

general formula for the cash ¬‚ow.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 71

T A B L E 3-3

ADF with Cash Flows Starting in Year 3.25: End-of-Year Formula

A B C D E F G H

5 Cash Flow t S 1 t 0

g)t S

6 t (Yrs) Growth Cash Flow (1 PV Factor PV PV Factor PV

7 3.25 NA 1.00000 1.00000 0.86957 0.86957 0.63494 0.63494

8 4.25 0.05100 1.05100 1.05100 0.75614 0.79471 0.55212 0.58028

9 5.25 0.05360 1.10460 1.10460 0.65752 0.72629 0.48011 0.53032

10 6.25 0.05633 1.16094 1.16094 0.57175 0.66377 0.41748 0.48467

11 7.25 0.05921 1.22014 1.22014 0.49718 0.60663 0.36303 0.44295

12 8.25 0.06223 1.28237 1.28237 0.43233 0.55440 0.31568 0.40481

13 9.25 0.06540 1.34777 1.34777 0.37594 0.50668 0.27450 0.36997

14 10.25 0.06874 1.41651 1.41651 0.32690 0.46306 0.23870 0.33812

15 11.25 0.07224 1.48875 1.48875 0.28426 0.42320 0.20756 0.30901

16 12.25 0.07593 1.56468 1.56468 0.24718 0.38676 0.18049 0.28241

17 13.25 0.07980 1.64447 1.64447 0.21494 0.35347 0.15695 0.25810

18 14.25 0.08387 1.72834 1.72834 0.18691 0.32304 0.13648 0.23588

19 15.25 0.08815 1.81649 1.81649 0.16253 0.29523 0.11867 0.21557

20 16.25 0.09264 1.90913 1.90913 0.14133 0.26981 0.10320 0.19701

21 17.25 0.09737 2.00649 2.00649 0.12289 0.24659 0.08974 0.18005

22 18.25 0.10233 2.10883 2.10883 0.10686 0.22536 0.07803 0.16455

23 19.25 0.10755 2.21638 2.21638 0.09293 0.20596 0.06785 0.15039

24 20.25 0.11304 2.32941 2.32941 0.08081 0.18823 0.05900 0.13744

25 21.25 0.11880 2.44821 2.44821 0.07027 0.17202 0.05131 0.12561

26 22.25 0.12486 2.57307 2.57307 0.06110 0.15722 0.04461 0.11480

27 Pres. value (t 2.25 for column F, t 0 for column H) 8.43199 6.15687

28 Pres. value factor-discount from S 1 (t 2.25) to 0 0.73018

29 Present value (t 0) 6.15687

31 Calculation of PV by formulas:

32 Grand

33 Time S to In¬nity (n 1) to In¬nity S to n Total

34 t S 1 10.10101 1.66902 8.43199 8.43199

35 PV Factor 0.73018 0.73018 0.73018 0.73018

36 t 0 7.37555 1.21869 6.15687 6.15687

38 Assumptions:

40 S Beginning year of cash ¬‚ows (valuation at t 2.25) 3.25

41 n Ending year of cash ¬‚ows 22.25

42 r Discount rate 15.0%

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

44 x (1 g)/(1 r) 0.913913

45 Gordon model multiple GM [1/(r g)] 10.101010

47 Spreadsheet formulas:

49 B34: GM Gordon model for years 3.25 to in¬nity as of t 2.25

50 C34: GM*(x (n S 1)) Gordon model for years 23.25 to in¬nity as of t 2.25

51 D34: B34 C34

52 E34: GM*(1 x (n S 1)) grand total as of t S 1 2.25 years

53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0

54 Row 36: Row 34 * Row 35

PART 1 Forecasting Cash Flows

72

First, we demonstrate two-step discounting in Columns E and F. Col-

umn E contains the present value (PV) factors to discount the cash ¬‚ows

to t S 1, the formula for which is 1/(1 r)t S 1. Column F is the PV

as of t 2.25 Years. The present value of the cash ¬‚ows total $8.43199

(F27). F28 is the PV factor, 0.73018, to discount that result back to t 0

by multiplying it by F27, or $8.43199 0.73018 $6.15687 (F29).

In Columns G and H, we perform the same procedures, the only

difference being that Column G contains the PV factors to discount back

to t 0. Column H is the PV of the cash ¬‚ows, which totals the same

$6.15687 (H27), which is the same result as F29. This demonstrates that

the two-step and the one-step present value calculation lead to the same

results, as long as they are done properly.

Cell B34 contains the Gordon model multiple 10.10101 for cash ¬‚ows

from t S (3.25) to in¬nity, which we can see calculated in G45. C34 is

the Gordon model multiple for t n 1 to in¬nity, discounted to t

S 1. Subtracting C34 from B34, we get the cash ¬‚ows from S to n in

D34, or $8.43199, which also equals F27. Row 35 is the PV factor 0.73018,

and Row 34 Row 35 Row 36, the PV as of t 0. The total for cash

¬‚ows from S 3.25 to n appears in D36 as $6.15687.

In E34 we show the grand total cash ¬‚ows, as per equation (3-11).

The spreadsheet formula for E34 is in A52, where GM is the Gordon

model multiple. The $8.43199 is the total of the cash ¬‚ows from 3.25 to

22.25 as of t 2.25 and corresponds to the term in equation (3-11) in

square brackets. The PV factor 0.73018 is the term in equation (3-11) to

the right of the square brackets, and the one multiplied by the other is

the entirety of equation (3-11). Note that E36 D36 F29 H27, which

demonstrates the validity of equation (3-11).

Tables 3-4 through 3-6: Variations of Table 3-3 with S 0,

Negative Growth, and r g

Tables 3-4 through 3-6 are identical to Table 3-3. The only difference is

that Tables 3-4 through 3-6 have cash ¬‚ows that begin in Year 2, (S

2.00 in G40). Additionally, in Table 3-5 growth is a negative 5.1% (G43),

instead of the usual positive 5.1% in the other tables.

In Table 3-6, r g, so the discount rate is less than the growth rate,

which is impossible for a perpetuity but acceptable for a ¬nite annuity.

Note that the Gordon model multiple is 20 (B34 and G45), which by

itself would be a nonsense result. Nevertheless, it still works for a ¬nite

annuity, as the term for the cash ¬‚ows from n 1 to in¬nity is positive

and greater than the negative Gordon model multiple.10

In all cases, equation (3-11) performs perfectly, with D36 E36

F29 H27.

r)]n

10. This is so because [(1 g)/(1 1, so when we multiply that term by the GM”which is

negative”the resulting term is negative and of greater magnitude than the GM itself. Since

we are subtracting a larger negative from the negative GM, the overall result is a positive

number.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 73

T A B L E 3-4

ADF with Cash Flows Starting in Year 2.00: End-of-Year Formula

A B C D E F G H

5 Cash Flow t S 1 t 0

g)t S

6 t (Yrs) Growth Cash Flow (1 PV Factor PV PV Factor PV

7 2.00 NA 1.00000 1.00000 0.86957 0.86957 1.32250 1.32250

8 1.00 0.05100 1.05100 1.05100 0.75614 0.79471 0.15000 1.20865

9 0.00 0.05360 1.10460 1.10460 0.65752 0.72629 1.00000 1.10460

10 1.00 0.05633 1.16094 1.16094 0.57175 0.66377 0.86957 1.00951

11 2.00 0.05921 1.22014 1.22014 0.49718 0.60663 0.75614 0.92260

12 3.00 0.06223 1.28237 1.28237 0.43233 0.55440 0.65752 0.84318

13 4.00 0.06540 1.34777 1.34777 0.37594 0.50668 0.57175 0.77059

14 5.00 0.06874 1.41651 1.41651 0.32690 0.46306 0.49718 0.70425

15 6.00 0.07224 1.48875 1.48875 0.28426 0.42320 0.43233 0.64363

16 7.00 0.07593 1.56468 1.56468 0.24718 0.38676 0.37594 0.58822

17 8.00 0.07980 1.64447 1.64447 0.21494 0.35347 0.32690 0.53758

18 9.00 0.08387 1.72834 1.72834 0.18691 0.32304 0.28426 0.49130

19 10.00 0.08815 1.81649 1.81649 0.16253 0.29523 0.24718 0.44901

20 11.00 0.09264 1.90913 1.90913 0.14133 0.26981 0.21494 0.41035

21 12.00 0.09737 2.00649 2.00649 0.12289 0.24659 0.18691 0.37503

22 13.00 0.10233 2.10883 2.10883 0.10686 0.22536 0.16253 0.34274

23 14.00 0.10755 2.21638 2.21638 0.09293 0.20596 0.14133 0.31324

24 15.00 0.11304 2.32941 2.32941 0.08081 0.18823 0.12289 0.28627

25 16.00 0.11880 2.44821 2.44821 0.07027 0.17202 0.10686 0.26163

26 17.00 0.12486 2.57307 2.57307 0.06110 0.15722 0.09293 0.23910

27 Pres. value (t 2.25 for column F, t 0 for column H) 8.43199 12.8240

28 Pres. value factor-from S 1 (t 3.00) to 0 1.52088

29 Present value (t 0) 12.82400

31 Calculation of PV by formulas:

32 Grand

33 Time S to In¬nity (n 1) to In¬nity S to n Total

34 t S 1 10.10101 1.66902 8.43199 8.43199

35 PV factor 1.52088 1.52088 1.52088 0.73018

36 t 0 15.36237 2.53838 12.82400 12.82400

38 Assumptions:

40 S Beginning year of cash ¬‚ows (valuation at t 3.00) 2.00

41 n Ending year of cash ¬‚ows 17.00

42 r Discount rate 15.0%

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

44 x (1 g)/(1 r) 0.913913

45 Gordon model multiple GM [1/(r g)] 10.101010

47 Spreadsheet formulas:

49 B34: GM Gordon model for years 2.00 to in¬nity as of t 3.00

50 C34: GM*(x (n S 1)) Gordon model for years 18.00 to in¬nity as of t 3.00

51 D34: B34 C34

52 E34: GM*(1 x (n S 1)) grand total as of t S 1 3.00 years

53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0

54 Row 36: Row 34 * Row 35

PART 1 Forecasting Cash Flows

74

T A B L E 3-5

ADF with Cash Flows Starting in Year 2.00 with Negative Growth: End-of-Year Formula

A B C D E F G H

5 Cash Flow t S 1 t 0

g)t S

6 t (Yrs) Growth Cash Flow (1 PV Factor PV PV Factor PV

7 2.00 NA 1.00000 1.00000 0.86957 0.86957 1.32250 1.32250

8 1.00 0.05100 0.94900 0.94900 0.75614 0.71758 0.15000 1.09135

9 0.00 0.04840 0.90060 0.90060 0.65752 0.59216 1.00000 1.90060

10 1.00 0.04593 0.85467 0.85467 0.57175 0.48866 0.86957 0.74319

11 2.00 0.04359 0.81108 0.81108 0.49718 0.40325 0.75614 0.61329

12 3.00 0.04137 0.76972 0.76972 0.43233 0.33277 0.65752 0.50610

13 4.00 0.03926 0.73046 0.73046 0.37594 0.27461 0.57175 0.41764

14 5.00 0.03725 0.69321 0.69321 0.32690 0.22661 0.49718 0.34465

15 6.00 0.03535 0.65785 0.65785 0.28426 0.18700 0.43233 0.28441

16 7.00 0.03355 0.62430 0.62430 0.24718 0.15432 0.37594 0.23470

17 8.00 0.03184 0.59246 0.59246 0.21494 0.12735 0.32690 0.19368

18 9.00 0.03022 0.56225 0.56225 0.18691 0.10509 0.28426 0.15983

19 10.00 0.02867 0.53357 0.53357 0.16253 0.08672 0.24718 0.13189

20 11.00 0.02721 0.50636 0.50636 0.14133 0.07156 0.21494 0.10884

21 12.00 0.02582 0.48054 0.48054 0.12289 0.05906 0.18691 0.08982

22 13.00 0.02451 0.45603 0.45603 0.10686 0.04873 0.16253 0.07412

23 14.00 0.02326 0.43277 0.43277 0.09293 0.04022 0.14133 0.06116

24 15.00 0.02207 0.41070 0.41070 0.08081 0.03319 0.12289 0.05047

25 16.00 0.02095 0.38976 0.38976 0.07027 0.02739 0.10686 0.04165

26 17.00 0.01988 0.36988 0.36988 0.06110 0.02260 0.09293 0.03437

27 Pres. value (t 2.25 for column F, t 0 for column H) 4.86842 7.40426

28 Pres. value factor-from S 1 (t 3.00) to 0 1.52088

29 Present value (t 0) 7.40426

31 Calculation of PV by formulas:

32 Grand

33 Time S to In¬nity (n 1) to In¬nity S to n Total

34 t S 1 4.97512 0.10670 4.86842 4.86842

35 PV Factor 1.52088 1.52088 1.52088 1.52088

36 t 0 7.56654 0.16228 7.40426 7.40426

38 Assumptions:

40 S Beginning year of cash ¬‚ows (valuation at t 3.00) 2.00

41 n Ending year of cash ¬‚ows 17.00

42 r Discount rate 15.0%

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

44 x (1 g)/(1 r) 0.825217

45 Gordon model multiple GM [1/(r g)] 4.975124

47 Spreadsheet formulas:

49 B34: GM Gordon model for years 2.00 to in¬nity as of t 3.00

50 C34: GM*(x (n S 1)) Gordon model for years 18.00 to in¬nity as of t 3.00

51 D34: B34 C34

52 E34: GM*(1 x (n S 1)) grand total as of t S 1 3.00 years

53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0

54 Row 36: Row 34 * Row 35

CHAPTER 3 Annuity Discount Factors and the Gordon Model 75

T A B L E 3-6

ADF with Cash Flows Starting in Year 2.00 with g r: End-of-Year Formula

A B C D E F G H

5 Cash Flow t S 1 t 0

g)t S

6 t (Yrs) Growth Cash Flow (1 PV Factor PV PV Factor PV

7 2.00 NA 1.00000 1.00000 0.86957 0.86957 1.32250 1.32250

8 1.00 0.20000 1.20000 1.20000 0.75614 0.90737 0.15000 1.38000

9 0.00 0.24000 1.44000 1.44000 0.65752 0.94682 1.00000 1.44000

10 1.00 0.28800 1.72800 1.72800 0.57175 0.98799 0.86957 1.50261

11 2.00 0.34560 2.07360 2.07360 0.49718 1.03095 0.75614 1.56794

12 3.00 0.41472 2.48832 2.48832 0.43233 0.07577 0.65752 1.63611

13 4.00 0.49766 2.98598 2.98598 0.37594 1.12254 0.57175 1.70725

14 5.00 0.59720 3.58318 3.58318 0.32690 1.17135 0.49718 1.78147

15 6.00 0.71664 4.29982 4.29982 0.28426 1.22228 0.43233 1.85893

16 7.00 0.85996 5.15978 5.15978 0.24718 1.27542 0.37594 1.93975

17 8.00 1.03196 6.19174 6.19174 0.21494 1.33087 0.32690 2.02409

18 9.00 1.23835 7.43008 7.43008 0.18691 1.38874 0.28426 2.11209

19 10.00 1.48602 8.91610 8.91610 0.16253 1.44912 0.24718 2.20392

20 11.00 1.78322 10.69932 10.69932 0.14133 1.51212 0.21494 2.29974

21 12.00 2.13986 12.83918 12.83918 0.12289 1.57786 0.18691 2.39974

22 13.00 2.56784 15.40702 15.40702 0.10686 1.64647 0.16253 2.50407

23 14.00 3.08140 18.48843 18.48843 0.09293 1.71805 0.14133 2.61294

24 15.00 3.69769 22.18611 22.18611 0.08081 1.79275 0.12289 2.72655

25 16.00 4.43722 26.62333 26.62333 0.07027 1.87070 0.10686 2.84510

26 17.00 5.32467 31.94800 31.94800 0.06110 1.95203 0.09293 2.96880

27 Pres. value (t 3.00 for column F, t 0 for column H) 26.84876 40.83361

28 Pres. value factor-From S 1 (t 3.00) to 0 1.52088

29 Present Value (t 0) 40.83361

31 Calculation of PV by formulas:

32 Grand

33 Time S to In¬nity (n 1) to In¬nity S to n Total

34 t S 1 20.00000 46.84876 26.84876 26.84876

35 PV Factor 1.52088 1.52088 1.52088 1.52088

36 t 0 30.41750 71.25111 40.83361 40.83361

38 Assumptions:

40 S Beginning year of cash ¬‚ows (valuation at t 3.00) 2.00

41 n Ending year of cash ¬‚ows 17.00

42 r Discount rate 15.0%

43 g Growth rate in net inc/cash ¬‚ow 20.0%

44 x (1 g)/(1 r) 1.043478

45 Gordon model multiple GM [1/(r g)] 20.000000

47 Spreadsheet formulas:

49 B34: GM Gordon model for years 2.00 to in¬nity as of t 3.00

50 C34: GM*(x (n S 1)) Gordon model for years 18.00 to in¬nity as of t 3.00

51 D34: B34 C34

52 E34: GM*(1 x (n S 1)) grand total as of t S 1 3.00 years

53 Row 35: 1/(1 r) (S 1) present value factor from t S 1 back to t 0

54 Row 36: Row 34 * Row 35

PART 1 Forecasting Cash Flows

76

Special Case: No Growth, g 0

Setting g 0, equation (3-11) reduces to:

1 1 1 1

ADF

r)n S1

r)S 1

r (1 r (1

1 1 1

1 ADF: no growth (3-11c)

r)n S1

r)S 1

r (1 (1

This formula is useful in calculating loan amortization, as the reader can

see in the loan amortization section of the Appendix to this chapter.

Generalized Gordon Model

If we start with cash ¬‚ows at any year other than Year 1, then we have

to use a generalized Gordon model. Letting n ’ in equation (3-11), the

end-of-year formula is:

1 1

PV CF (3-11d)

r)S 1

(r g) (1

This is the formula for the PV of the reversion (the cash ¬‚ows from t

n 1 to in¬nity) that every appraiser uses in every discounted cash ¬‚ow

analysis. This is exactly what appraisers do in calculating the PV of the

reversion, i.e., the in¬nity of time that follows the discounted cash ¬‚ow

forecasts for the ¬rst n years. For example, suppose we do a ¬ve-year

forecast of cash ¬‚ows in a discounted cash ¬‚ow analysis and calculate its

PV. We must then calculate the PV of the reversion, which is the sixth-

year cash ¬‚ow multiplied by the Gordon model and then discounted ¬ve

years to t 0, or:

1 1

PV CF6 (3-11e)

r)5

r g (1

The reason we discount ¬ve years and not six is that after discount-

ing the ¬rst ¬ve years™ cash ¬‚ows to PV, we are standing at the end of

Year 5 looking at the in¬nity of cash ¬‚ows that we forecast to occur be-

ginning with Year 6. The Gordon model requires us to use the ¬rst fore-

cast year™s cash ¬‚ow, which is why we use CF6 and not CF5, but we still

must discount the cash ¬‚ows from the end of Year 5, or ¬ve years. The

¬rst two terms on the right-hand side of equation (3-11d) give us the

formula for the PV of the cash ¬‚ows from Years 6 to in¬nity as of

the end of Year 5, and the ¬nal term on the right discounts that back to

t 0.

Midyear Formula

When the starting period is not in Year 1, the midyear ADF formula is:

nS1

1 r 1 r

1 g 1

ADF

r)S 1

r g 1 r r g (1

nS1

1 r 1 g 1

1 (3-12)

r)S 1

r g 1 r (1

Note that at S 1, the term at the right”outside the brackets”becomes

CHAPTER 3 Annuity Discount Factors and the Gordon Model 77

1 and effectively drops out of the equation, which renders equation

(3-12) equivalent to equation (3-10). The midyear ADF in equation (3-12)

is identical to the end-of-year ADF in equation (3-11), except that we

replace the two Gordon model 1 r terms with the value 1 in the latter.

PERIODIC PERPETUITY FACTORS (PPFs): PERPETUITIES

FOR PERIODIC CASH FLOWS

Thus far, all ADFs and Gordon model perpetuities have been for contig-

uous cash ¬‚ows. In this section we develop perpetuities for periodic cash

¬‚ows that occur only at regular intervals or cycles. To my knowledge,

these formulas are my own creation, and I call them periodic perpetuity

factors (PPFs). PPFs are really Gordon model multiples for periodic (non-

contiguous) cash ¬‚ows and for contiguous cash ¬‚ows that have repeating

patterns.

The example we use here arose in Chapter 2 in dealing with moving

expenses. Every small to midsize company that is growing in real terms

moves periodically. We will assume a move occurs every 10 years, al-

though we will derive formulas that can handle any periodicity. To fur-

ther simplify the initial mathematics, we will assume the last move oc-

curred in the last historical year of analysis. Later we will relax that

assumption to handle different timing of the cash ¬‚ows.

Suppose our subject company moved last year, and the move cost

$20,000. We expect to move every 10 years, and moving costs increase at

g 5% per year. The PPFs are the present values of these periodic cash

¬‚ows for both midyear and end-of-year assumptions.

The Mathematical Formulas

For every $1.00 of forecast moving costs in Year 10, the PV of the lifetime

expected moving costs would be as follows in equation (3-13):

g)10

(1 (1 g)

1

PV (3-13)

r)10 r)20

(1 (1 (1 r)

The $1.00 grows at rate g for 10 years, and we discount it back to PV for

10 years. We follow the same pattern at 20 years, 30 years, etc. to in¬nity.

r)]10, we get:

Multiplying equation (3-13) by [(1 g)/(1

10

g)10 g)20

1 g (1 (1 (1 g)

PV (3-14)

r)20 r)30

1 r (1 (1 (1 r)

Subtracting equation (3-14) from equation (3-13), we get:

10

1 g 1

1 PV (3-15)

r)10

1 r (1

The left-hand side of equation (3-15) simpli¬es to

r)10 g)10

(1 (1

PV

r)10

(1

Multiplying both sides of equation (3-15) by the inverse,

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

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 ¬‚ow 5.0%

27 b Number of years from last cash ¬‚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 ¬‚ows. Otherwise, the PV in C20 will appear to be higher than the total of the cash ¬‚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. Let™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 ¬‚ows for its six-year expected life.

The cash ¬‚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 ¬rst two years and then begins increasing rapidly in Year 3

or 4.

We can then take the NPV of the cash ¬‚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 ¬ve 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 ¬‚ows ¬ve 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 ¬‚ows are contiguous, which is not true in the

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

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

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

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

be the same as the NPV at t 0 of the ¬rst six years™ cash ¬‚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 ¬‚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

¬nal 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 ¬rst 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 ¬xed 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 de¬nitions:

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 ¬ve-year loan. The loan payment, calculated using Excel™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 month™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 mistakes”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 de¬nitions that will aid in developing the mathematics. All other

de¬nitions retain their same meaning as in the rest of the chapter.

De¬nitions

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

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

pression for the FMV of the ¬rm15 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

growth rate times the midyear Gordon model multiple. In reality, inves-

14. I wish to thank Larry Kasper for pointing out the need for this.

15. Assuming the present value of the cash ¬‚ows of the ¬rm is its FMV. This ignores valuation

discounts, an acceptable simpli¬cation in this limited context.

PART 1 Forecasting Cash Flows

88

tors do not expect constant growth to perpetuity. They usually have ex-

pectations of uneven growth for a few years and a vague, long-run ex-

pectation of growth thereafter that they approximate as being constant.

Therefore, we should look at g, the perpetual growth rate in cash ¬‚ow,

as an average growth rate over the in¬nite period of time that we are

modeling.

We should be very clear that the earnings base in the PE multiple

and the Gordon model are different. The former is the immediate prior

year and the latter is the ¬rst forecast year. When an appraiser develops

PE multiples from guideline companies, whether publicly or privately

owned, he should multiply the PE multiple from the guideline companies

(after appropriate adjustments) by the subject company™s prior year earn-

ings. When using a discounted cash ¬‚ow approach, the appraiser should

multiply the Gordon model by the ¬rst forecast year™s earnings. Using the

wrong earnings will cause an error in the valuation by a factor of one

plus the forecast one-year growth rate.

CONCLUSIONS

We can see that there is a family of annuity discount factors (ADFs), from

the simplest case of an ordinary annuity to the most complicated case of

an annuity with stub periods (fractional years), as discussed in the Ap-

pendix. The elements that determine which formula to use are:

— Whether the cash ¬‚ows are midyear versus end-of-year.

— When the cash ¬‚ows begin (Year 1 versus any other time).

— If they occur every year or at regular, skipped intervals (or have

repeating cycles).

— Whether or not the constant growth is zero.

— Whether there is a stub period.

For cash ¬‚ows without a stub period, the ADF is the difference of

two Gordon model perpetuities. The ¬rst term is the perpetuity from S

to in¬nity, where S is the starting year of the cash ¬‚ow. The second term

is the perpetuity starting at n 1 (where n is the ¬nal cash ¬‚ow in the

annuity) going to in¬nity. For cash ¬‚ows with a stub period, the preced-

ing statement is true with the addition of a third term for the single cash

¬‚ow of the stub period itself, discounted to PV.

While this chapter contains some complex algebra, the focus has been

on the intuitive explanation of each ADF. The most dif¬cult mathematics

have been moved to the Appendix, which contains the formulas for ADF

with stub periods and some advanced material on the use of ADFs in

calculating loan amortization. ADFs are also used for practical applica-

tions in Chris Mercer™s quantitative marketability discount model (see

Chapter 7), periodic expenses such as moving costs and losses from law-

suits, ESOP valuation, in reducing a seller-subsidized loan to its cash

equivalent price in Chapter 10, and to calculate loan payments.

We have performed a rigorous derivation of the PE multiple and the

Gordon model. This derivation demonstrates that the PE multiple equals

one minus the earnings retention rate times one plus the one-year growth

CHAPTER 3 Annuity Discount Factors and the Gordon Model 89

T A B L E 3-11

ADF Equation Numbers

With Growth No Growth

Formulas in the Chapter End-of-Year Midyear End-of-Year Midyear

Ordinary ADF (3-6) to (3-6b) (3-10) to (3-10b) (3-6d) (3-10c) & (3-10d)

Gordon model (3-7) (3-10e)

Starting cash ¬‚ow not t 1 (3-11) & (3-11a) (3-12) (3-11c)

Valuation date v (3-11b)

Gordon model for starting CF not 1 (3-11d)

Periodic expenses (3-18) (3-19)

Periodic expenses-¬‚exible timing (3-18a) (3-19a)

Loan payment (3-21)

Relationship of Gordon model to PE (3-28)

Formulas in the Appendix

ADF with stub period (A3-3) (A3-4)

Amortization of loan principal (A3-10)

PV of loan after-tax (A3-24) & (A3-25)

rate times the midyear Gordon model multiple. Furthermore, we showed

how the former uses the prior year™s earnings, while the latter uses the

¬rst forecast year™s earnings. Many appraisers have found that confusing,

and hopefully this section of the chapter will do much to eliminate that

confusion.

Because there are so many ADFs for different purposes and assump-

tions, we include Table 3-11 to point the reader to the correct ADF equa-

tion.

BIBLIOGRAPHY

Gordon, M. J., and E. Shapiro. 1956. ˜˜Capital Equipment Analysis: The Required Rate of

Pro¬t,™™ Management Science 3: 102“110.

Gordon, M. J. 1962. The Investment, Financing, and Valuation of the Corporation, 2d ed.

Homewood, Ill.: R. D. Irwin.

Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing and Supporting

Marketability Discounts in the Appraisal of Closely Held Business Interests. Memphis,

Tenn.: Peabody.

Williams, J. B. The Theory of Investment Value. 1938. Cambridge, Mass.: Harvard University

Press.

APPENDIX

INTRODUCTION

This appendix is an extension of the material developed in the chapter.

The topics that we cover are:

— Developing ADFs for cash ¬‚ows that end on a fractional year

(stub period).

— Developing ADFs for loan mathematics, consisting of calculating

the amortization of principal in loans and the net after-tax cost of

a loan.

PART 1 Forecasting Cash Flows

90

This appendix is truly for the mathematically brave. The topics cov-

ered and formulas developed are esoteric and less practically useful than

the formulas in the chapter, though the formula for the after-tax cost of

a loan may be useful to some practitioners. The material in this appendix

is included primarily for reference. Nevertheless, even those not com-

pletely comfortable with the dif¬cult mathematics can bene¬t from fo-

cusing on the verbal explanations before the equations and the develop-

ment of the ¬rst one or two equations in the derivation of each of the

formulas. The rest is just the tedious math, which can be skipped.

THE ADF WITH STUB PERIODS (FRACTIONAL YEARS)

We will now develop a formula to handle annuities that have stub peri-

ods, constant growth in cash ¬‚ows, and cash ¬‚ows that start at any time.

To the best of my knowledge, I invented this formula. In this section we

will assume midyear cash ¬‚ows and later present the formula for end-

of-year cash ¬‚ows.

Let™s begin with constructing a timeline of the cash ¬‚ows in Figure

A3-1, using the following de¬nitions and assumptions:

De¬nitions

S time (in years) of the ¬rst cash ¬‚ow for end-of-year cash

¬‚ows. For midyear cash ¬‚ows, S end of the year in which the

¬rst cash ¬‚ow occurs 3.25 years in this example, which means

the cash ¬‚ow for that year begins at t 2.25 years and we assume

the cash ¬‚ow occurs in the middle of the year, or S 0.5

3.25 0.5 2.75 years.

n end of the last whole year™s cash ¬‚ows 12.25 years in this

example

z end of the stub period 12.60 years.

p proportion of a full year represented by the stub period

z n 12.60 12.25 0.35 years

g constant growth rate in cash ¬‚ows 5.1%

t point in time, measured in years

The Cash Flows

We assume the ¬rst cash ¬‚ow of $1.00 (Figure A3-1, cell C4) occurs during

year S (S is for starting cash ¬‚ow), where t 2.25 to t 3.25 years. For

F I G U R E A3-1

Timeline of Cash Flows

Row \ Col. B C D E F G H

1 Year (numeric) 3.25 4.25 5.25 ¦ 12.25 12.60

2 Year (symbolic) S S+1 S+2 ¦ n z

g(1+g)n-S-1

3 Growth (in $) 0 g g(1+g) ¦ NA

(1+g)2 (1+g)n-S p(1+g)n-S+1

4 Cash Flow 1 1+g ¦

CHAPTER 3 Annuity Discount Factors and the Gordon Model 91

simplicity, we denote that the cash ¬‚ow is for the year ending at t 3.25

years (cell C1). Note that for Year 3.25, there is no growth in the cash

¬‚ow, i.e., cell B3 0.

The following year is 4.25 (cell D1), or S 1 (cell D2). The $1.00

grows at a rate of g (cell D3), so the ending cash ¬‚ow is 1 g (cell D4).

tS

g)4.25 3.25.

Note that the ending cash ¬‚ow is equal to (1 g) (1

For Year 5.25, or S 2 (cell E2), growth in cash ¬‚ows is g times the

prior year™s cash ¬‚ow of (1 g), or g (1 g) (cell E3), which leads to a

cash ¬‚ow equal to the prior year™s cash ¬‚ow plus this year™s growth, or

(1 g) g(1 g) (1 g) (1 g) (1 g)2 [cell E4]. Again, the cash

g)t S (1 g)5.25 3.25.

¬‚ow equals (1

For the year 6.25, or S 3, which is not shown in Figure A3-1, cash

g)2, so cash ¬‚ows are (1 g)2 g)2 g)2

¬‚ows grow g(1 g(1 (1

g)3 (1 g)t S g)6.25 3.25.

(1 g) (1 (1

We continue in this fashion through the last whole year of cash ¬‚ows,

which we call Year n (Column G). In our example, n 12.25 years (cell

nS

G1). The cash ¬‚ows during Year n are equal to (1 g) [cell G4].

Had we completed one more full year, the cash ¬‚ows would have

extended to Year 13.25, or Year n 1. If so, the cash ¬‚ow would have

nS1

been (1 g) . However, since the stub year™s cash ¬‚ow is only for a

partial year, the ending cash ¬‚ow is multiplied by p”the fractional por-

g)n S 1.

tion of the year”leading to an ending cash ¬‚ow of p(1

It is important to recognize that there may be other ways of speci-

fying how the partial year affects the cash ¬‚ows. For example, it is pos-

sible, but very unlikely, that the cash ¬‚ows can be based on a legal doc-

ument that speci¬es that only the growth rate itself will be fractional, but

the corpus of the cash ¬‚ow will not diminish for the partial year. We

could calculate a solution to this ADF, but we will not, as it is very un-

likely to be of any practical use and we have already demonstrated how

to model the most likely method of splitting the cash ¬‚ows in the frac-

tional year. The point is that modeling the fractional year cash ¬‚ows de-

pends on the agreement and/or the underlying scenario, and one should

not blindly charge off into the sunset applying a formula developed un-

der an assumption that does not apply in another case.

Discounting Periods

The ¬rst cash ¬‚ow occurs during the year that spans from

t 2.25 to t 3.25. We assume the cash ¬‚ows occur evenly throughout

the year, which is tantamount to assuming all cash ¬‚ows occur on average

halfway through the year, i.e., at Year 2.75. Therefore as of time zero,

de¬ned as t 0, the ¬rst $1 cash ¬‚ow has a present value of

1 1

r)2.75 r)S 0.5

(1 (1

We will be discounting the cash ¬‚ows in two stages because that will

later enable us to provide a more intuitive explanation of our results. Our

¬rst discounting of cash ¬‚ows will be to t S 1, the beginning of

the ¬rst year of cash ¬‚ows. The ¬rst year™s cash ¬‚ow then receives a dis-

PART 1 Forecasting Cash Flows

92

r)0.5, the second year™s cash ¬‚ows receive a discount

count of 1/(1

r)1.5, etc. Thus, the denominators here are identical to those

of 1/(1

for cash ¬‚ows that would begin in Year 1 instead of S.

The Equations

The PV of our series of cash ¬‚ows as of t S 1 is:

(1 g)

1

PV

r)0.5 r)1.5

(1 (1

g)n S

g)n S 1