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

(1 p(1

... (A3-1)

r)n S 0.5

r)n S 1 0.5p

(1 (1

Note that the exponent in the denominator of the last term (the frac-

tional year) is equal to the one before it (the last whole year) plus 1„2 year

to bring us to the end of Year n, plus 1„2 of the fractional year, thus main-

taining a midyear assumption.

We already have a solution to the PV of the whole years in the body

of the chapter”equation (3-10). Thus, the PV of the entire series of cash

¬‚ows as of t S 1 is equation (3-10) plus the ¬nal term in equation

(A3-1), or:

nS1

g)n S 1

1 r 1 r

1 g p(1

NPV (A3-2)

r)n S 1 0.5p

r g 1 r r g (1

The next step is to discount the PV from t S 1 to t 0. We do

S1

this by multiplying by 1/(1 r) . The result is our annuity discount

factor for midyear cash ¬‚ows with a stub period.

nS1

1 r 1 r

1 g

NPV

r g 1 r r g

g)n S 1

p(1 1

(A3-3)

r)n S 1 0.5p r)S 1

(1 (1

The ADF formula for end-of-year cash ¬‚ows with a stub period is:

nS1

1 g

1 1

ADF

r g 1 r r g

g)n S 1

p(1 1

(A3-4)

r)(z S 1) r)S 1

(1 (1

The individual terms in equation (A3-4) have the same meaning as

in the midyear cash ¬‚ows of equation (A3-3). To easily see the derivation

of the end-of-year (EOY) model from the midyear, note that an EOY

model in equation (A3-1) would require the exponent in each denomi-

nator to be 0.5 years larger, which changes the 1 r term in equation

(A3-3) to 1. 1/(r g) is the EOY Gordon model formula. The only other

difference is the discount factor in the rightmost term in the braces

of equations (A3-3) and (A3-4). In the former, we discount the stub pe-

CHAPTER 3 Annuity Discount Factors and the Gordon Model 93

r)n S 1 0.5p

riod cash ¬‚ow by (1 , while in the latter we discount by

r)(z S 1).

(1

Tables A3-1 and A3-2: Example of Equations [A3-3]

and [A3-4]

Table A3-1 is an example of the midyear ADF with a fractional year cash

¬‚ow, and Table A3-2 is an example using end-of-year cash ¬‚ows. Table

A3-2 has the identical structure and meaning as Table A3-1”merely us-

ing end-of-year formulas rather than midyear. Therefore, we will explain

only Table A3-1.

In the ¬rst part of Table A3-1, we will use a ˜˜brute force™™ method of

scheduling out the cash ¬‚ows, calculating their present values, and then

summing them. Later we will directly test the formulas and demonstrate

they produce the same result as the brute force method.

Brute Force Method of Calculating PV of Cash Flows

Rows 7 through 17 in Table A3-1 are a detailed listing of the cash ¬‚ows

and their present values each year. The ¬rst cash ¬‚ows begin in Row 7

at Year 2.25 and ¬nish at t 3.25, with Year 2.75 as the midpoint from

which we discount. We will refer to the years by the ending year, i.e., the

cash ¬‚ow in Row 7 is for the year ending at t 3.25. Assumptions of the

model begin in Row 33.

We begin with $1.00 of cash ¬‚ow for the year ending at t 3.25 (C7).

Column B shows the growth in cash ¬‚ows and is equal to g 5.1%

multiplied by the previous period™s cash ¬‚ow. In B8 the calculation is

$1.00 5.1% $0.051. The cash ¬‚ow in C8 is C7 B8, or $1.00 $.051

$1.051. We repeat this pattern through Row 16, the last whole year™s

cash ¬‚ow.

Column D replicates Column C using the formula cash ¬‚ow

g)t S for all cells except D17, which is the fractional year cash ¬‚ow.

(1

g)n S 1, where multiplying by p 0.35

The formula for that cell is p(1

years converts what would have been the cash ¬‚ow for the whole year

n 1 (and would have been $1.64447) into the fractional year cash ¬‚ow

of $0.57557.16 Note that in that formula, n 12.25 years, the last whole

year.

We show the present values of the cash ¬‚ows as of t S 1 in

Columns E and F and the present values as of t 0 in Columns G and

H. The discount rate is 15% (G36).

Column E contains the present value factors (PVFs), and its formula

17

is

1

PVF

r)t S 0.5

(1

Column F is Column C (or Column D, as the results are identical) times

16. See cell A45 for the formula in the spreadsheet.

17. The intuition behind the exponent is that we are discounting from t to S 1, which is equal to

1 years. Using a midyear convention, we always discount from 1„2

t (S 1) t S

year earlier than end-of-year, which reduces the exponent to t S 0.5. The 0.5 reverts to

1 in the end-of-year formula.

PART 1 Forecasting Cash Flows

94

T A B L E A3-1

ADF with Fractional Year: Midyear Formula

A B C D E F G H

5 Cash Flows t S 1 t 0

g)t S

r)t S 0.5

r)t 0.5

6 t (Yrs) Growth Cash Flow (1 PVF 1/(1 PV PVF 1/(1 PV

7 3.25 NA 1.00000 1.00000 0.93250 0.93250 0.68090 0.68090

8 4.25 0.05100 1.05100 1.05100 0.81087 0.85223 0.59208 0.62228

9 5.25 0.05360 1.10460 1.10460 0.70511 0.77886 0.51486 0.56871

10 6.25 0.05633 1.16094 1.16094 0.61314 0.71181 0.44770 0.51975

11 7.25 0.05921 1.22014 1.22014 0.53316 0.65053 0.38930 0.47501

12 8.25 0.06223 1.28237 1.28237 0.46362 0.59453 0.33853 0.43412

13 9.25 0.06540 1.34777 1.34777 0.40315 0.54335 0.29437 0.39674

14 10.25 0.06874 1.41651 1.41651 0.35056 0.49658 0.25597 0.36259

15 11.25 0.07224 1.48875 1.48875 0.30484 0.45383 0.22259 0.33138

16 12.25 0.07593 1.56468 1.56468 0.26508 0.41476 0.19355 0.30285

17 12.60 NA 0.57557 0.57557 0.24121 0.13883 0.17613 0.10137

18 Totals for whole years 3.25 12.25 6.42899 4.69432

19 Add fractional year 12.60 0.13833 0.10137

20 Grand total (t S 1 in Column G and t 0 in Column I) 6.56782 4.79469

21 Present value factor-discount from S 1 (t 2.25) to 0 0.73018

22 Grand total (t 0) 4.79569

24 Calculation of PV by formulas:

25 Grand

26 Whole Yrs Frac Yr Total Total

27 t S 1 6.42899 0.13883 6.56782

28 PV Factor 0.73018 0.73018

29 t 0 4.69432 0.10137 4.79469 4.79569

31 Assumptions:

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

34 n Ending year of cash ¬‚ows-whole year 12.25

35 z Ending year of cash ¬‚ows-stub year 12.60

36 r Discount rate 15.0%

37 g growth rate in cash ¬‚ow 5.1%

38 p proportion of year in the stub period 0.35

39 Midpoint n 0.5 p midpoint of the fractional year 12.425

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

41 Gordon model multiple GM Sqrt (1 r)/(r g) 10.832127

43 Spreadsheet Formulas:

45 C17, D17: p*(1 g) (n s 1) stub period cash ¬‚ow

46 E17: 1/(1 r) (n S 1 0.5*p) stub period present value factor at t 2.25

47 G17: 1/(1 r) (n 0.5*p) stub period present value factor for t 0

48 B27: GM*(1 x (n S 1)) ADF for years 3.25 to 32.25 at t 2.25

49 C27: p*(1 g) (n S 1)/(1 r) (n S 1 0.5*p) PV of stub period CF at t 2.25

50 B28, C28: 1/(1 r) (S 1) present value factor at t S 1 2.25

51 E29: (GM*(1 x (n S 1)) p*(1 G) (n S 1)/(1 r) (n S 1 0.5*p))*(1/(1 r) (S 1))

Note: E29 is the formula for the Grand Total

CHAPTER 3 Annuity Discount Factors and the Gordon Model 95

T A B L E A3-2

ADF with Fractional Year: Midyear Formula

A B C D E F G H

5 Cash Flows t S 1 t 0

g)t S

r)t S1

r)t

6 t (Yrs) Growth Cash Flow (1 PVF 1/(1 PV PVF 1/(1 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 12.60 NA 0.57557 0.57557 0.23538 0.13548 0.17187 0.09892

18 Totals for whole years 3.25 22.25 5.99506 4.37747

19 Add fractional year 22.60 0.13548 0.09892

20 Grand total (t S 1 in Column G and t 0 in Column H) 6.13054 4.47640

21 Present value factor-discount from S 1 (t 2.25) to 0 0.73018

22 Grand total (t 0) 4.47640