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Absolute Errors in Forecasting Growth and the
Discount Rate
A fundamental difference between these two variables and cash ¬‚ow is
that value is nonlinear in r and g, whereas it is linear in cash ¬‚ow. We
will develop a formula to quantify the valuation error for any absolute


11. Strictly speaking, the error is really k, not k%. However, the description ¬‚ows better using the
percent sign after the k.
12. Again, this formula is the same with the midyear Gordon model, as the square root term
cancels out.




Part 4 Putting It All Together
394
error in calculating the discount rate or the growth rate, assuming cash
¬‚ow is forecast correctly.

De¬nitions
First we begin with some de¬nitions. Let:
V1 the correct value
V2 the erroneous value
r1 the correct discount rate
r2 the erroneous discount rate
g1 the correct growth rate
the erroneous growth rate13
g2
CF cash ¬‚ow, which we will assume to be correct in this section
the change in any value, which in our context means the error
We will consider a positive error to be when the erroneous value, discount
rate, or growth rate is higher than the correct value. For example, if g1
5% and g2 6%, then g g2 g1 1%; if g1 6% and g2 5%, then
g 1%.

The Mathematics
The correct valuation, according to the end-of-year Gordon model, is:
CF
V1 the correct value (11-9)
r1 g1
The erroneous value is:
CF
V2 the erroneous value (11-10)
r2 g2
The error, V V2 V1, equals:
CF CF 1 1
V CF (11-11)
r2 g2 r1 g1 r2 g2 r1 g1
In order to have a common denominator, we multiply the ¬rst term in
round brackets by (r1 g1)/(r1 g1) and we multiply the second term
in round brackets by (r2 g2)/(r2 g2).
(r1 g1) (r2 g2)
V CF (11-12)
(r1 g1)(r2 g2)
Rearranging the terms in the numerator, we get:
(r1 r2) (g1 g2)
V CF (11-13)
(r1 g1)(r2 g2)
Changing signs in the numerator:


13. Actually, only one of the two variables”r2 or g2”need be erroneous. The other one can be
correct, which would make it equal to its r1 or g1 counterpart.




CHAPTER 11 Measuring Valuation Uncertainty and Error 395
(r2 r1) (g2 g1)
V CF (11-14)
(r1 g1)(r2 g2)
which simpli¬es to:
r g
V CF (11-15)
(r1 g1)(r2 g2)
absolute effect of absolute error in r or g14

Example Using the Error Formula
Let™s use an example to demonstrate the error formula. Suppose cash ¬‚ow
is forecast next year at $100,000 and that the correct discount and growth
rate are 20% and 5%, respectively. The Gordon model multiple is 1/(0.25
0.05) 5, which leads to a valuation before discounts of $500,000.
Instead, the appraiser makes an error and uses a zero growth rate. His
erroneous Gordon model multiple will be 1/(0.25 0) 4, leading to a
$400,000 valuation. The appraiser™s error is an undervaluation of $400,000
$500,000 $100,000.
Using equation (11-15),
0 0.05 0.05
V $100,000 100,000
(0.25 0.05)(0.25 0) 0.2 0.25
0.05
100,000 $100,000
0.05

Relative Effects of Absolute Error in r and g
The relative valuation error, as before, is the valuation error in dollars
divided by the correct valuation, or:
CF( r g)/(r1 g1)(r2 g2)
V
% Error (11-16)
V CF/(r1 g1)
r g
V
% Error (11-17)
V r2 g2
relative effects of absolute error in r and g15


14. When r 0, then the formula using the midyear Gordon model is identical to equation
(11-15), with the addition of the term 1 r after the CF, but before the square brackets.
When there is an error in the discount rate, the error formula using the midyear Gordon
model is
(r1 g1) 1 r2 (r2 g2) 1 r1
CF
(r1 g1)(r2 g2)
The partial derivative for g is similar to the discrete equation for change:
V CF
g)2
g (r
Since it is a partial derivative, we hold r constant, which means r 0, and instead of
having r2 g2, we double up on r1 g1, which we can simplify to r g. Again, these
formulas are correct only when CF is forecast correctly.
15. This formula would be identical using the midyear Gordon model, as the 1 r would
appear in both numerators in equation (11-16) and cancel out.




Part 4 Putting It All Together
396
Example of Relative Valuation Error
From the previous example, the relative valuation error is
$400,000
1 20%
$500,000
a 20% undervaluation. Using equation (11-17), the relative error is
0 0.05 0.05
20%
0.25 0 0.25
which agrees with the previous calculation and demonstrates the accu-
racy of equation (11-17). It is important to be precise with the deltas, as
it is easy to confuse the sign. In equation (11-17) the numerator is r
g. It is easy to think that since there is a plus sign in front of g, we
should use a positive 0.05 instead of 0.05. This is incorrect, as we are
assuming that the appraiser™s error in the growth rate itself is negative,
i.e., the erroneous growth rate minus the correct growth rate, (V2 V1)
0 0.05 0.05.

Valuation Effects on Large Versus Small Firms
Next we look at the question of whether large or small ¬rms are more
affected by identical errors in absolute terms in the discount or growth
rate. The numerator of equation (11-17) will be the same regardless of
size. The denominator, however, will vary with size. Holding g2 constant,
r2 will be smaller for large ¬rms, as will r2 g2. Thus, the relative error,
as quanti¬ed in equation (11-17), will be larger for large ¬rms than small
¬rms, assuming equal growth rates.16
Table 11-3 demonstrates the above conclusion. Columns B through D
show valuation calculations for the huge ¬rm, as in Table 11-1. Historical
cash ¬‚ow was $300 million (B6), and we assume a constant 8% (B7)
growth rate as being correct, which leads to forecast cash ¬‚ow of $324
million (B8). Using the log size model, we get a discount rate of 15% (B9),
as shown in cells B14“B17. In B10, we calculate an end-of-year Gordon
model multiple of 14.2857, which differs from Table 11-1, where we were
using a midyear multiple. Multiplying row 8 by row 10 produces a value
of $4.63 billion (B11).
Column C contains the erroneous valuation, where the appraiser uses
a 9% growth rate (C7) instead of the correct 8% growth rate in B7. That
leads to a valuation of $5.45 billion (C11). The valuation error is $821.4
million (D11), which is C11 B11. Dividing the $821.4 million error by
the correct valuation of $4.63 million, the valuation error is 17.7% (D12).
We repeat the identical procedure with the small ¬rm in columns E“G
using the same growth and discount rate as the huge ¬rm, and the val-
uation error is 6.9% (G12). This demonstrates the accuracy of our conclu-
sion from equation (11-17) that equal absolute errors in the growth rate


16. As before, this is theoretically not true in CAPM, which should be independent of size.
However, in reality, is correlated to size.




CHAPTER 11 Measuring Valuation Uncertainty and Error 397
T A B L E 11-3

Absolute Errors in Forecasting Growth Rates


A B C D E F G

4 Huge Firm Small Firm

5 Correct Erroneous Error Correct Erroneous Error

6 Cash ¬‚ow-CFt 1 300,000,000 300,000,000 100,000 100,000
7 g growth rate 8% 9% 8% 9%
8 Cash ¬‚owt 324,000,000 327,000,000 108,000 109,000
9 Discount rate 15.0% 15.0% 26.0% 26.0%
10 Gordon multiple-end year 14.2857 16.6667 5.5556 5.8824
11 FMV 4,628,571,429 5,450,000,000 821,428,571 600,000 641,176 41,176

12 Percentage error 17.7% 6.9%
13 Verify discount rate

14 0.01204 * ln(FMV) 26.80% 26.99% 16.02% 16.10%
15 Add constant 41.72% 41.72% 41.72% 41.72%
16 Discount rate 14.92% 14.73% 25.70% 25.62%
17 Rounded 15% 15% 26% 26%




or discount rate cause larger relative valuation errors for large ¬rms than
small ¬rms.
Let™s now compare the magnitude of the effects of an error in cal-
culating cash ¬‚ow versus discount or growth rates. From equation (11-8),
a 1% relative error in forecasting cash ¬‚ows leads to a 1% valuation error.
From equation (11-17), a 1% absolute error in forecasting growth leads to
a valuation error of 0.01/(r2 g2). Using typical values for the denomi-
nator, the valuation error will most likely be in the range of 4“20% for
each 1% error in forecasting growth (or error in the discount rate). This
means we need to pay relatively more attention to forecasting growth rates and
discount rates than we do to producing the ¬rst year™s forecast of cash ¬‚ows,
and the larger the ¬rm, the more care we should be taking in the analysis.
Also, it is clear from (11-15) and (11-17) that it is the net error in both
r and g that drives the valuation error, not the error in either one indi-
vidually. Using the end-of-year Gordon model, equal errors in r and g
cancel each other out. With the more accurate midyear formula, errors in
g have slightly more impact on the value than errors in r, as an error in
r has opposite effects in the numerator and denominator.

Relative Effect of Relative Error in Forecasting Growth and
Discount Rates
We can investigate the impact of a k% relative error in estimating g by
restating the Gordon model in equation (11-18) below with the altered
growth rate (1 k)g. We denote the correct value as V1 and the incorrect
value as V2.
CF
V2 (11-18)
r (1 k)g
The ratio of the incorrect to the correct value is V2/V1, or:

Part 4 Putting It All Together
398
r g
V2
(11-19)
V1 r (1 k)g

The relative error in value resulting from a relative error in forecasting
growth will be (V2/V1) 1, or:

r g
% Error 1 (11-20)
r (1 k)g

relative error in value from relative error in growth

Thus, if both a large and small ¬rm have the same growth rate, then
the lower discount rate of the large ¬rm will lead to larger relative val-
uation errors in the large ¬rm than the small ¬rm. Note that for k 0,
(11-20) 0, as it should. When k is negative, which means we forecast
growth too low, the result is the same”the under-valuation is greater for
large ¬rms than small ¬rms.
A relative error in forecasting the discount rate shifts the (1 k) in
front of the r in (11-20) instead of being in front of the g. The formula is:

r g

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