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

% Error 1 (11-21)

(1 k)r g

relative error in value from relative error in r

Tables 11-4 through 11-4B: Examples Showing Effects on

Large Versus Small Firms

Table 11-4 shows the calculations for k 10% (B38) relative error in fore-

casting growth. Rows 5“6 contain the discount rate and growth rate for

a huge ¬rm in column B and a small ¬rm in column C, respectively. The

end-of-year Gordon model multiples are 50 (B7) and 5.5556 (C7) for the

huge and the small ¬rm, respectively. Multiplying the Gordon model

multiples by the forecast cash ¬‚ows in row 8 results in the correct values,

V1, in row 9 of $15 billion and $555,556, respectively.

Now let™s see what happens if we forecast growth too high by 10%

for each ¬rm. Row 10 shows the erroneously high growth rate of 9.9%.

Row 11 contains the new Gordon model multiples, and row 12 shows V2,

the incorrect values we obtain with the high growth rates. Row 13 shows

the ratio of the incorrect to the correct valuation, i.e., V2/V1, and Row 14

shows the relative error, (V2/V1) 1 81.82% for the huge ¬rm and

5.26% for the small ¬rm.

Rows 20“36 are a sensitivity analysis that show the relative valuation

errors for various combinations of r and g using equation (11-20), with

k 10%. Note that the bolded cells in F20 and F36 match the results in

row 14, con¬rming the accuracy of the error formula. This veri¬es our

observation from analysis of equation (11-20) that equal relative errors in

forecasting growth will create much larger relative valuation errors for

large ¬rms than small ¬rms, holding growth constant. All we need do is

notice that the relative errors in the sensitivity analysis decline as we

move down each column, and as small ¬rms have higher discount rates,

the lower cells represent the smaller ¬rms.

CHAPTER 11 Measuring Valuation Uncertainty and Error 399

T A B L E 11-4

Percent Valuation Error for 10% Relative Error in Growth

A B C D E F G

4 Description Huge Firm Small Firm

5 r 11% 27%

6 g 9% 9%

7 Gordon model 50.0000 5.5556

8 Cash ¬‚ow 300,000,000 100,000

9 V1 15,000,000,000 555,556

10 (1 PctError)*g 9.90% 9.90%

11 Gordon model 2 90.9091 5.8480

12 V2 27,272,727,273 584,795

13 V2/ V1 1.8182 1.0526

14 (V2/ V1) 1 81.82% 5.26%

16 Sensitivity Analysis: Valuation Error for Combinations of r and g

18 Growth rate g

19 Discount Rate r 5% 6% 7% 8% 9% 10%

20 11% 9.09% 13.64% 21.21% 36.36% 81.82% NA

21 12% 7.69% 11.11% 16.28% 25.00% 42.86% 100.00%

22 13% 6.67% 9.38% 13.21% 19.05% 29.03% 50.00%

23 14% 5.88% 8.11% 11.11% 15.38% 21.95% 33.33%

24 15% 5.26% 7.14% 9.59% 12.90% 17.65% 25.00%

25 16% 4.76% 6.38% 8.43% 11.11% 14.75% 20.00%

26 17% 4.35% 5.77% 7.53% 9.76% 12.68% 16.67%

27 18% 4.00% 5.26% 6.80% 8.70% 11.11% 14.29%

28 19% 3.70% 4.84% 6.19% 7.84% 9.89% 12.50%

29 20% 3.45% 4.48% 5.69% 7.14% 8.91% 11.11%

30 21% 3.23% 4.17% 5.26% 6.56% 8.11% 10.00%

31 22% 3.03% 3.90% 4.90% 6.06% 7.44% 9.09%

32 23% 2.86% 3.66% 4.58% 5.63% 6.87% 8.33%

33 24% 2.70% 3.45% 4.29% 5.26% 6.38% 7.69%

34 25% 2.56% 3.26% 4.05% 4.94% 5.96% 7.14%

35 26% 2.44% 3.09% 3.83% 4.65% 5.59% 6.67%

36 27% 2.33% 2.94% 3.63% 4.40% 5.26% 6.25%

38 Relative Error in g 10%

Formula in B20: (which copies to the other cells in the sensitivity analysis) (($A20 B$19)/($A20 ((1 $PctError)*B$19))) 1

Table 11-4A is identical to Table 11-4, with the one exception that the

growth rate is a negative 10% instead of a positive 10%. Table 11-4A

demonstrates that, assuming identical real growth rates, forecasting

growth too low also affects large ¬rms more than small ¬rms.

Table 11-4B is also identical to Table 11-4, except that it measures the

relative valuation error arising from relative errors in calculating the dis-

count rate. Table 11-4B uses equation (11-21) instead of equation (11-20)

to calculate the error. It demonstrates that relative errors in forecasting

the discount rate affect the valuation of large ¬rms more than the valu-

ation of small ¬rms, assuming identical real growth rates.

Table 11-5: Summary of Effects of Valuation Errors

Table 11-5 summarizes the effects of the valuation errors. Each cell in the

table contains three items:

Part 4 Putting It All Together

400

T A B L E 11-4A

Percent Valuation Error for 10% Relative Error in Growth

A B C D E F G

4 Description Huge Firm Small Firm

5 r 11% 27%

6 g 9% 9%

7 Gordon model 50.0000 5.5556

8 Cash Flow 300,000,000 100,000

9 V1 15,000,000,000 555,556

10 (1 PctError)*g 8.10% 8.10%

11 Gordon model 2 34.4828 5.2910

12 V2 10,344,827,586 529,101

13 V2/ V1 0.6897 0.9524

14 (V2/ V1) 1 31.03% 4.76%

16 Sensitivity Analysis: Valuation Error for Combinations of r and g

18 Growth rate g

19 Discount Rate r 5% 6% 7% 8% 9% 10%

20 11% 7.69% 10.71% 14.89% 21.05% 31.03% NA

21 12% 6.67% 9.09% 12.28% 16.67% 23.08% 33.33%

22 13% 5.88% 7.89% 10.45% 13.79% 18.37% 25.00%

23 14% 5.26% 6.98% 9.09% 11.76% 15.25% 20.00%

24 15% 4.76% 6.25% 8.05% 10.26% 13.04% 16.67%

25 16% 4.35% 5.66% 7.22% 9.09% 11.39% 14.29%

26 17% 4.00% 5.17% 6.54% 8.16% 10.11% 12.50%

27 18% 3.70% 4.76% 5.98% 7.41% 9.09% 11.11%

28 19% 3.45% 4.41% 5.51% 6.78% 8.26% 10.00%

29 20% 3.23% 4.11% 5.11% 6.25% 7.56% 9.09%

30 21% 3.03% 3.85% 4.76% 5.80% 6.98% 8.33%

31 22% 2.86% 3.61% 4.46% 5.41% 6.47% 7.69%

32 23% 2.70% 3.41% 4.19% 5.06% 6.04% 7.14%

33 24% 2.56% 3.23% 3.95% 4.76% 5.66% 6.67%

34 25% 2.44% 3.06% 3.74% 4.49% 5.33% 6.25%

35 26% 2.33% 2.91% 3.55% 4.26% 5.03% 5.88%

36 27% 2.22% 2.78% 3.38% 4.04% 4.76% 5.56%

38 Relative Error in g 10.0%