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CHAPTER 4 Discount Rates as a Function of Log Size 155

APPENDIX A

Automating Iteration using Newton™s Method

This appendix is optional. It is mathematically dif¬cult and is more an-

alytically interesting than practical.

In this section we present a numerical method for automatically it-

erating to the correct log size discount rate. Isaac Newton invented an

iterative procedure using calculus to provide numerical solutions to equa-

tions with no analytic solution. Most calculus texts will have a section on

his method (for example, see Thomas 1972). His procedure involves mak-

ing an initial guess of the solution, then subtracting the equation itself

divided by its own ¬rst derivative to provide a second guess. We repeat

the process until we converge to a single answer.

The bene¬t of Newton™s method is that it will enable us to simply

enter assumptions for the cash ¬‚ow base and the perpetual growth, and

the spreadsheet will automatically calculate the value of the ¬rm without

our having to manually go through the iterations as we did in Tables

4-4A, B, and C. Remember, some iteration process is necessary when

using log size discount rates because the discount rate is not independent

of size, as it is using other discount rate models.

To use Newton™s procedure, we rewrite equation (4-13) as:

CF

Let f(V) V 0 (A4-1)

(a b ln V g)

bCF

f (V) 1 (A4-2)

g)2

V(a b ln V

Assuming our initial guess of value is V0, the formula that de¬nes our

next iteration of value, V1, is:

CF

V0

(a b ln V0 g)

V1 V0 (A4-3)

bCF

1

g)2

V0(a b ln V0

Table A4-5 shows Newton™s iterative process for the simplest valu-

ation. In B22“B26 we enter our initial guess of value of an arbitrary $2

trillion (B22), our forecast cash ¬‚ow base of $100,000 (B23), perpetual

growth g 7% (B24), and our regression coef¬cients a and b (B25 and

B26, which come from Table 4-1, E42 and E48, respectively).

In B7 we see our initial guess of $2 trillion. The iteration #2 value

of $280,530 (B8) is the result of the formula in the note immediately below

Table A4-5, which is equation (A4-3).33 B9 to B12 are simply the formula

in B8 copied to the remaining spreadsheet cells.

Once we have the formula, we can value any ¬rm with constant

growth in its cash ¬‚ows by simply changing the parameters in B23 to

B24.

33. Cell B7, our initial guess, is V0 in equation (A4-3).

PART 2 Calculating Discount Rates

156

T A B L E A4-5

Gordon Model Valuation Using Newton™s Iterative Process

A B

5 Iteration Value

6 t V(t)

0 2,000,000,000,000

8 1 280,530

9 2 612,879

10 3 599,634

11 4 599,625

12 5 599,625

14 Proof of Calculation:

16 Discount rate 23.68%

17 Gordon multiple 5.9963

18 CF FMV $599,625

30 Parameters

22 V(0) 2,000,000,000,000

23 CF 100,000

24 g 7%

25 a 37.50%

26 b 1.039%

29 Model Sensitivity

30 FMV Initial Guess V(0)

31 Explodes 3,000,000,000,000

32 599,625 2,000,000,000,000

33 599,625 27,000

34 Explodes 26,000

Formula in Cell B8:

B7 ((B7 (CF/(A B * LN(B7) G)))/(1 (B * CF)/(B7 * (A B * LN(B7) G) 2)))

Note: The above formula assumes an End-Year Gordon Model. Newton™s Method converges for the midyear Gordon Model, but too

slowly to be of practical use.

B31 to B34 show the sensitivity of the model to the initial guess. If

we guess poorly enough, the model will explode instead of converging

to the right answer. For this particular set of assumptions, an initial guess

of anywhere between $27,000 and $2 trillion will converge to the right

answer. Assumptions above $3 trillion or below $26,000 explode the

model.

Unfortunately, the midyear Gordon model, which is more accurate,

has a much more complex formula. The iterative process does converge,

but much too slowly to be of any practical use. One can use the end-of-

year Gordon model and multiply the result by the square root of (1 r).

CHAPTER 4 Discount Rates as a Function of Log Size 157

APPENDIX B

Mathematical Appendix

This appendix provides the mathematics behind the log size model, as

well as some philosophical analysis of the mathematics”speci¬cally on

the nature of exponential decay function and how that relates to phenom-

ena in physics as well as our log size model. This is intended more as

intellectual observation than as required information.

We will begin with two de¬nitions:

r return of a portfolio

S standard deviation of returns of the portfolio

Equation (B4-1) states that the return on a portfolio of securities (each

decile is a portfolio) varies positively with the risk of the portfolio, or:

r a1 b1S (B4-1)

This is a generalization of equation (4-1) in the chapter. This rela-

tionship is not directly observable for privately held ¬rms. Therefore, we

use the next equation, which is a generalization of equation (4-2) from

the chapter, to calculate expected return.

The parameter a1 is the regression estimate of the risk-free rate,34

while the parameter b1 is the regression estimate of the slope, which is

the return for each unit increase of risk undertaken, i.e., the standard

deviation of returns. Thus, b1 is the regression estimate of the price of or

the reward for taking on risk.

r a2 b 2 ln FMV, b 2 0 (B4-2)

Equation (B4-2) states that return decreases in a linear fashion with

the natural logarithm of ¬rm value. The parameter a2 is the regression

estimate of the return for a $1 ¬rm35 ”the valueless ¬rm”while the pa-

rameter b 2 is the regression estimate of the slope, which is the return for

each increase in ln FMV. Thus, b 2 is the regression estimate of the reduc-

tion in return investors accept for investing in smaller ¬rms. The terms

a1, a2, b1, and b 2 are all parameters determined in regression equations

(4-1) and (4-2).

Using all 73 years of stock market data, our regression estimate of

a1 6.56% (Table 4-1, D26), which compares well with the 73-year mean

Treasury Bond yield of 5.28%. It would initially appear that the log size

regression does a reasonable job of also providing an estimate of the risk-

free return. Unfortunately, it is not all that simple, as the log size estimate

using 60 years of data fares worse. The log size 60-year estimate of a1 is

8.90% (Table 4-1, E26), which is a long way off from the 60-year mean

treasury bond yield of 5.70% (Table 4-1, E27). Thus, eliminating the ¬rst

34. A zero risk asset would have no standard deviation of returns. Thus, S 0 and r a1.

35. A ¬rm worth $1 would have ln FMV ln $1 0. Thus in equation (B4-2), for FMV $1,

r a2.

PART 2 Calculating Discount Rates

158

13 years of data had the effect of shifting the regression line upwards and

¬‚attening it slightly.

We already knew from our analysis of Table 4-2 in the chapter that

using 60 years of data was the overall best choice because of its superi-

ority in the log size equation estimates, but it was not the best choice for

estimating equation (4-1). Its R 2 is lower and standard error is higher

than the 73-year results.

Focusing now on equation (B5-2), the log size equation, the 60-year

regression estimate of b1 1.0309% (Table 4-1, E48), which is signi¬-

cantly lower in absolute value than the 73-year estimate of 1.284%

(D48). The parameter b 2 is the reduction in return that comes about from

each unit increase in company value (in natural logarithms). The param-

eter a2 is the y-intercept. It is the return (discount rate) for a valueless

¬rm”more speci¬cally, a $1 ¬rm in value”as ln($1) 0.

Equating the right-hand sides of equation (B4-1) and (B4-2) and solv-

ing for S, we see how we are implicitly using the size of the ¬rm as a

proxy for risk.

a2 a1 b2

S ln FMV (B4-3)

b1 b1

Since a2 is the rate of return for the valueless ¬rm and a1 is the re-

gression estimate of the risk-free rate”¬‚awed as it is”the difference be-

tween them, a2 a1 is the equity premium for a $1.00 ¬rm, i.e., the val-

ueless ¬rm. Dividing by b1, the price of risk (or reward) for each

increment of standard deviation, we get (a2 a1)/b1, the standard devi-

ation of a $1 ¬rm. We then reduce our estimate of the standard deviation

by the ratio of the relative prices of risk in size divided by the price of

risk in standard deviation, and multiply that ratio by the log of the size

of the ¬rm. In other words, we start with the maximum risk, a $1 ¬rm,

and reduce the standard deviation by the appropriate price times the log

of the value of the ¬rm in order to calculate the standard deviation of the

¬rm.

Rearranging equation (B4-3), we get

(a1 a2) b1S

ln FMV (B4-4)

b2

Raising both sides of the equation as powers of e, the natural exponent,

we get:

(a1 a2) b1S (a1 a2) b1S

FMV e e e b , or (B4-5)

b2 b2 2

b1

(a1 a2)

kS

FMV Ae , where A e ,k 0 (B4-6)

b2

b2

Here we see that the value of the ¬rm or portfolio declines exponentially

with risk, i.e., the standard deviation.

Unfortunately, the standard deviation of most private ¬rms is un-

observable since there are no reliable market prices. Therefore, we must

CHAPTER 4 Discount Rates as a Function of Log Size 159

solve for the value of a private ¬rm another way. Restating equation

(B4-2),

r a2 b 2 ln(FMV) (B4-7)

Rearranging the equation, we get:

(r a2)

ln FMV (B4-8)

b2

Raising both sides by e, i.e., taking the antilog, we get:

(r a2)

FMV e (B4-9)

b2

or (B4-10)

a2

1

mr

FMV Ce , where C e and m

b2

b2

This shows the FMV of a ¬rm or portfolio declines exponentially

with the discount rate. This is reminiscent of a continuous time present

value formula; in this case, though, instead of traveling through time we

are traveling though expected rates of return. The same is true of equation

(B4-6), where we are traveling through degree of risk.

What Does the Exponential Relationship Mean?

Let™s try to get an intuitive feel for what an exponential relationship

means and why that makes intuitive sense. Equation (B4-6) shows that

the fair market value of the ¬rm is an exponentially declining function

of risk, as measured by the standard deviation of returns. Repeating equa-

tion (B4-6), FMV Ae k S, k 0. Because we ¬nd that risk itself is primarily

related to the size of the ¬rm, we come to a similar equation for size.

Cemr, m

Repeating equation (B4-10), we see that FMV 0.

In physics, radioactive minerals such as uranium decay exponen-

tially. That means that a constant proportion of uranium decays at every

moment. As the remaining portion of uranium is constantly less over time

due to the radioactive decay, the amount of decay at any moment in time

or during any ¬nite time period is always less than the previous period.

A graph of the amount of uranium remaining over time would be a

downward sloping curve, steep at ¬rst and increasing shallow over time.

Figure 4-3 shows an exponential decay curve.

It appears the same is true of the value of ¬rms. Instead of decaying

over time, their value decays over risk. Because it turns out that risk is

so closely related to size and the rate of return is so closely related to

size, the value also decays exponentially with the market rate of return,

i.e., the discount rate. The graph of exponential decay in value over risk

has the same general shape as the uranium decay curve.

Imagine the largest ship in the world sailing on a moderately stormy

ocean. You as a passenger hardly feel the effects of the storm. If instead

you sailed on a slightly smaller ship, you would feel the storm a bit more.

As we keep switching to increasingly smaller ships, the storm feels in-

PART 2 Calculating Discount Rates

160

creasingly powerful. The smallest ship on the NYSE might be akin to a

35-foot cabin cruiser, while appraisers often have to value little paddle-

boats, the passengers of which would be in danger of their lives while

the passengers of the General Electric boat would hardly feel the turbu-

lence.

That is my understanding of the principle underlying the size effect.

Size offers diversi¬cation of product and service. Size reduces transaction

costs in proportion to the entity, e.g., the proceeds of ¬‚oating a $1 million

stock issue after ¬‚otation costs are far less in percentage terms than ¬‚oat-

ing a $100 million stock issue. Large ¬rms have greater depth and breadth

of management, and greater staying power. Even the chances of beating

a bankruptcy exist for the largest businesses. Remember Chrysler? If it

were not a very big business, the government would never have jumped

in to rescue it. The same is true of the S&Ls. For these and other reasons,

the returns of big businesses ¬‚uctuate less than small businesses, which

means that the smaller the business, the greater the risk, the greater the

return.

The FMV of a ¬rm or portfolio declining exponentially with the dis-

count rate/risk is reminiscent of a continuous time present value formula,

e r t; in this case, though, instead of

where Present Value Principal

traveling through time we are traveling though expected rates of return/

risk.

CHAPTER 4 Discount Rates as a Function of Log Size 161

APPENDIX C

Abbreviated Review and Use

This abbreviated version of the chapter is intended for those who simply

wish to learn the model without the bene¬t of additional background and

explanation, or wish to use it as a quick reference for review.

INTRODUCTION

Historically, small companies have shown higher rates of return than

large ones, as evidenced by New York Stock Exchange (NYSE) data over

the past 73 years (Ibbotson Associates 1999). Further investigation into

this phenomenon has led to the discovery that return (the discount rate)

strongly correlates with the natural logarithm of the value of the ¬rm

(¬rm size), which has the following implications:

— The discount rate is a linear function negatively related to the

natural logarithm of the value of the ¬rm.

— The value of the ¬rm is an exponential decay function, decaying

with the investment rate of return (the discount rate).

Consequently, the value also decays in the same fashion with the

standard deviation of returns.

As we have already described regression analysis in Chapter 3, we

now apply these techniques to examine the statistical relationship be-

tween market returns, risk (measured by the standard deviation of re-

turns) and company size.

REGRESSION #1: RETURN VERSUS STANDARD

DEVIATION OF RETURNS

Columns A“F in Table 4-1 contain the input data from the Stocks, Bonds,

Bills and In¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the

regression analyses as well as the regression results. We use 73-year av-

erage returns in both regressions. For simplicity, we have collapsed 730

data points (73 years 10 deciles) into 73 data points by using averages.

Thus, the regressions are cross-sectional rather than time series. In Col-

umn A we list Ibbotson Associates™ (1999) division of the entire NYSE

into 10 different divisions”known as deciles”based on size, with the

largest ¬rms in decile 1 and the smallest in decile 10.36 Columns B through

F contain market data for each decile which is described below.

Note that the 73-year average market return in Column B rises with

each decile, as does the standard deviation of returns (Column C). Col-

umn D shows the 1998 market capitalization of each decile, which is the

price per share times the number of shares. It is also the fair market value

(FMV).

Dividing Column D (FMV) by Column F (the number of ¬rms in the

decile), we obtain Column G, the average capitalization, or the average

36. All of the underlying decile data in Ibbotson originate with the University of Chicago™s Center

for Research in Security Prices (CRSP), which also determines the composition of the deciles.

PART 2 Calculating Discount Rates