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: