14 df SS MS F Signi¬cance F

15 Regression 7 0.6354 0.0908 11.9009 1.810E-08

16 Residual 45 0.3432 0.0076

17 Total 52 0.9786

19 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

20 Intercept 0.0673 0.1082 0.6221 0.5370 0.2854 0.1507

21 Rev2 4.629E-18 9.913E-19 4.6698 0.0000 6.626E-18 2.633E-18

22 Shares sold-$ 3.619E-09 1.199E-09 3.0169 0.0042 6.035E-09 1.203E-09

23 Mkt cap 4.789E-10 1.790E-10 2.6754 0.0104 1.184E-10 8.394E-10

24 Earn stab 0.1038 0.0402 2.5831 0.0131 0.1848 0.0229

25 Rev stab 0.1824 0.0531 3.4315 0.0013 0.2894 0.0753

26 AvgYrs2Sell 0.1722 0.0362 4.7569 0.0000 0.0993 0.2451

27 Price stab 0.0037 8.316E-04 4.3909 0.0001 0.0020 0.0053

Source: Management Planning, Inc. Princeton NJ (except for AvgYrs2Sell and Rev 2 , which we derived from their data)

239

240

T A B L E 7-5 (continued)

Abrams Regression of Management Planning Study Data

A B C D E F G

32 Regression #2 (Without Price Stability)

34 Regression Statistics

35 Multiple R 0.7064

36 R square 0.4990

37 Adjusted R square 0.4337

38 Standard error 0.1032

39 Observations 53

41 ANOVA

42 df SS MS F Signi¬cance F

43 Regression 6 0.4883 0.0814 7.6365 0.0000

44 Residual 46 0.4903 0.0107

45 Total 52 0.9786

47 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

48 Intercept 0.1292 0.1165 1.1089 0.2732 0.1053 0.3637

Rev2

49 5.39E-18 1.15E-18 4.6740 0.0000 7.71E-18 3.07E-18

50 Shares sold-$ 4.39E-09 1.40E-09 3.1287 0.0030 7.21E-09 1.57E-09

51 Mkt cap 6.10E-10 2.09E-10 2.9249 0.0053 1.90E-10 1.03E-09

52 Earn stab 0.1381 0.0466 2.9626 0.0048 0.2319 0.0443

53 Rev stab 0.1800 0.0628 2.8653 0.0063 0.3065 0.0536

54 AvgYrs2Sell 0.1368 0.0417 3.2790 0.0020 0.0528 0.2208

other 40.53% of variation in the discounts that remains unexplained is

due to two possible sources: other signi¬cant independent variables of

which I (and Management Planning, Inc.) do not know, and random var-

iation. The standard error of the y-estimate is 8.7% (B10 rounded). We

can form approximate 95% con¬dence intervals around the y-estimate by

adding and subtracting two standard errors, or 17.4%.

Cell B20 contains the regression estimate of the y-intercept, and B21

through B27 contain the regression coef¬cients for the independent var-

iables. The t-statistics are in D20 through D27. Only the y-intercept itself

is not signi¬cant at the 95% con¬dence level. The market capitalization

and earnings stability variables are signi¬cant at the 98% level,45 and all

the other variables are signi¬cant at the 99 % con¬dence level.

Note that several of the variables are similar to Grabowski and King™s

results (Grabowski and King 1999), discussed in Chapter 5. They found

that the coef¬cient of variations (in log form) of operating margin and

return on equity are statistically signi¬cant in explaining stock market

returns. Here we ¬nd that the stability of revenues and earnings (as well

as the coef¬cient of variation of stock market prices) explain restricted

stock discounts. Thus, these variables are signi¬cant in determining the

value of the underlying companies, assuming they are marketable, and

in determining restricted stock discounts when restrictions exist.

I obtained regression #2 in Table 7-5 by regressing all the indepen-

dent variables in the ¬rst regression except for price stability. The adjusted

R 2 has dropped to 43.37% (B37), indicating that regression #1 is superior

when price data are available, which generally it is for restricted stock

studies and is not for calculating DLOM for privately held businesses.

The second regression is not recommended for the calculation of re-

stricted stock discounts, but it will be useful in other contexts.

Using the Put Option Model to Calculate DLOM

of Restricted Stock

Chaffe (1993) wrote a brilliant article in which he reasoned that buying a

hypothetical put option on Section 144 restricted stock would ˜˜buy™™ mar-

ketability and that the cost of that put option is an excellent measure of

the discount for lack of marketability of the stock. For puts, the Black“

Scholes option pricing model has the following formula:

Rf t

P E N( d2)e S N( d1)

where:

S stock price

N( ) cumulative normal density function

E exercise price

Rf risk-free rate, i.e., treasury rate of the same term as the option

t time remaining to expiration of the option

t0.5]

d1 [ln(S/E) (Rf 0.5 variance) t]/[std dev

t0.5]

d2 d1 [std dev

We have suf¬cient daily price history on 13 of the stocks in Table

45. The statistical signi¬cance is one minus the P-value, which is in E20 through E27.

CHAPTER 7 Adjusting for Levels of Control and Marketability 241

7-5 to derive the proper annualized standard deviation (std dev) of con-

tinuously compounded returns to test Chaffe™s approach.

Annualized Standard Deviation of Continuously Compounded

Returns. Table 7-6 is a sample calculation of the annualized standard

deviation of continuously compounded returns for Chantal Pharmaceu-

tical, Inc. (CHTL), which is one of the 13 stocks. The purpose of this table

is to demonstrate how to calculate the standard deviation.

Column A shows the date, column B shows the closing price, and

columns C and D show the continuously compounded returns. The sam-

ple period is just over 6 months and ends the day prior to the transaction

date.

We calculate continuously compounded returns over 10-trading-day

intervals for CHTL stock.46 The reason for using 10-day intervals in our

T A B L E 7-6

Calculation of Continuously Compounded Standard Deviation

Chantal Pharmaceutical, Inc.”CHTL

A B C D

6 Date Close Interval Returns

7 1/31/95 $2.1650

8 2/7/95 $2.2500

9 2/14/95 $2.5660 0.169928

10 2/22/95 $2.8440 0.234281

11 3/1/95 $2.6250 0.022733

12 3/8/95 $2.9410 0.033538

13 3/15/95 $2.4480 0.069810

14 3/22/95 $2.5000 0.162459

15 3/29/95 $2.2500 0.084341

16 4/5/95 $2.0360 0.205304

17 4/12/95 $2.2220 0.012523

18 4/20/95 $2.1910 0.073371

19 4/27/95 $2.6950 0.192991

20 5/4/95 $2.6600 0.193968

21 5/11/95 $2.5660 0.049050

22 5/18/95 $2.5620 0.037538

23 5/25/95 $2.9740 0.147560

24 6/2/95 $3.3120 0.256764

25 6/9/95 $5.1250 0.544223

26 6/16/95 $6.0000 0.594207

27 6/23/95 $5.8135 0.126052

28 6/30/95 $6.4440 0.071390

29 7/10/95 $6.5680 0.122027

30 7/17/95 $6.6250 0.027701

31 7/24/95 $8.0000 0.197232

32 7/31/95 $7.1250 0.072759

33 8/7/95 $7.8120 0.023781 0.092051

34 Interval standard deviation”CHTL 0.16900 0.20175

35 Annualized 0.84901 1.03298

36 Average of standard deviations 0.94099

46. The only exception is the return from 7/31/95 to 8/7/95, which is in cell D33.

PART 3 Adjusting for Control and Marketability

242

calculation instead of daily intervals is that the bid“ask spread on the

stock may create apparent volatility that is not really present. This is

because the quoted closing prices are from the last trade. In Nasdaq trad-

ing, one sells to a dealer at the bid price and buys at the ask price. If on

successive days the last price of the day is switching randomly from a

bid to an ask price and back, this can cause us to measure a considerable

amount of apparent volatility that is not really there. By using 10-day

intervals, we minimize this measurement error caused by the spread.

We start with the 1/31/95 closing price in column C and the 2/7/

95 closing price in column D. For example, the 10-trading-day return from

1/31/95 (A7) to 2/14/95 (A9) is calculated as follows: return Ln(B9/

B7) Ln(2.5660/2.1650) 0.169928 (C9).

Using this methodology, we get two measures of standard deviation:

0.16900 (C34) and 0.20175 (D34). To convert to the annualized standard

deviation, we must multiply each interval standard deviation by the

square root of the number of intervals that would occur in a year. The

equation is as follows:

SQRT

annualized interval returns

# of interval returns in sample period

365 days per year

days in sample period

For example, the sample period in column C is the time period from

the close of trading on January 31, 1995, to the close of trading on August

7, 1995, or 188 days, and there are 13 calculated returns. Therefore the

annualized standard deviation of returns is:

0.1690 SQRT(13 365/188)

annualized

0.1690 SQRT(25.2394) 0.84901 (cell C35)

The 13 trading periods that span 188 days would become 25.2394 trading

periods in one year (25.2394 13 365/188). The square root of the

25.2394 trading periods is 5.0239. We multiply the sample standard de-

viation of 0.1690 by 5.0239 0.84901 to annualize the standard deviation.

Similarly, the annualized standard deviation of returns in column D is

1.03298 (D35), and the average of the two is 0.94099 (D36).

Calculation of the Discount. Table 7-7 is the Black“Scholes put op-

tion calculation of the restricted stock discount. We begin in cell B5 with

S, the stock price on the valuation date of August 8, 1995, of $8.875. We

then assume that E, the exercise price, is identical (B6).

B7 is the time in years from the valuation date to marketability. Ac-

cording to SEC Rule 144, the shares have a two-year period of restriction

before the ¬rst portion of the block can be sold. At 2.25 years the rest can

be sold. The weighted average time to sell is 2.125 years (B7, transferred

from Table 7-5, I17) for this particular block of Chantal.

B8 shows the two-year Treasury rate, which was 5.90% as of the

transaction date. B9 contains the annualized standard deviation of returns

CHAPTER 7 Adjusting for Levels of Control and Marketability 243

T A B L E 7-7

Black“Scholes Put Option”CHTL

A B

5 S Stk price on valuation date $8.875

6 E Exercise price $8.875

7 t time to expiration in yrs (Table 7-5, I17) 2.125

8 r risk-free rate [1] 5.90%

9 stdev standard deviation (Table 7-6, D36) 0.941

10 var variance 0.885

11 d1 1st Black-Scholes parameter [2] 0.777

12 d2 2nd Black-Scholes parameter 3] (0.594)

13 N( d1) cum normal density function 0.219

14 N( d2) cum normal density function 0.724

[E*N( d2)*e rt ] S*N( d1)

15 P $3.73

16 P/S 42.0%

Note: Values are for European options. The put option formula can be found in Options Futures and Other Derivatives, 3rd Ed. by

John C. Hull, Prentice Hall, 1997, pp. 241 and 242.

[1] 2 Year Treasury rate on transaction date, 8/8/95 (Source: Federal Reserve)

.5 * var) * t]/[stdev *t0.5], where variance and standard deviation are expressed in annual terms.

[2] d1 [ln (S/E) (r

[std dev * t0.5]

[3] d2 d1

for CHTL of 0.941, transferred from Table 7-6, cell D36, while B10 is var-

iance, merely the square of B9.

Cells B11 and B12 are the calculation of the two Black“Scholes par-

ameters, d1 and d 2. B13 and B14 are the cumulative normal density func-

tions for d1 and d 2. For example, look at cell B13, which is N( 0.777)

0.219. This requires some explanation. The cumulative normal table

from which the 0.219 came assumes the normal distribution has been

standardized to a mean of zero and standard deviation of 1.47 This means

that there is a 21.9% probability that our variable is less than or equal to