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,