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CHAPTER 4 Discount Rates as a Function of Log Size 155
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:
Let f(V) V 0 (A4-1)
(a b ln V g)
f (V) 1 (A4-2)
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:
(a b ln V0 g)
V1 V0 (A4-3)
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

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

PART 2 Calculating Discount Rates
T A B L E A4-5

Gordon Model Valuation Using Newton™s Iterative Process


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
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
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
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
Rearranging equation (B4-3), we get
(a1 a2) b1S
ln FMV (B4-4)
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

(a1 a2)
FMV Ae , where A e ,k 0 (B4-6)
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
r a2 b 2 ln(FMV) (B4-7)
Rearranging the equation, we get:
(r a2)
ln FMV (B4-8)
Raising both sides by e, i.e., taking the antilog, we get:
(r a2)

FMV e (B4-9)

or (B4-10)
FMV Ce , where C e and m
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
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-
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
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/

CHAPTER 4 Discount Rates as a Function of Log Size 161
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.

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.

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
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

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