(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%

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

1. The formula for the valuation error.

2. The equation number containing the error formula.

3. Whether the error is larger for large ¬rms, small ¬rms, or there

is no difference.

The upper half of the table shows the valuation effects of absolute

errors in forecasting the variables (cash ¬‚ow, discount rate, and growth

rate), and the lower half of the table shows the valuation effects of relative

errors in forecasting the variables.

In 10 of the 12 cells in the table that contain error formulas, the

valuation errors are greater for large ¬rms than for small ¬rms. Only

equation (11-5), which is the relative valuation error resulting from a dol-

lar error in forecasting cash ¬‚ows, affects small ¬rms more than large

¬rms. Equation (11-8), the relative valuation error resulting from a relative

error in forecasting cash ¬‚ows, affects both small and large ¬rms alike. It

CHAPTER 11 Measuring Valuation Uncertainty and Error 401

T A B L E 11-4B

Percent Valuation Error for 10% Relative Error in Discount Rate

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 12.10% 29.70%

11 Gordon model 2 32.2581 4.8309

12 V2 9,677,419,355 483,092

13 V2/ V1 0.6452 0.8696

14 (V2/ V1) 1 35.48% 13.04%

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% 15.49% 18.03% 21.57% 26.83% 35.48% 52.38%

21 12% 14.63% 16.67% 19.35% 23.08% 28.57% 37.50%

22 13% 13.98% 15.66% 17.81% 20.63% 24.53% 30.23%

23 14% 13.46% 14.89% 16.67% 18.92% 21.88% 25.93%

24 15% 13.04% 14.29% 15.79% 17.65% 20.00% 23.08%

25 16% 12.70% 13.79% 15.09% 16.67% 18.60% 21.05%

26 17% 12.41% 13.39% 14.53% 15.89% 17.53% 19.54%

27 18% 12.16% 13.04% 14.06% 15.25% 16.67% 18.37%

28 19% 11.95% 12.75% 13.67% 14.73% 15.97% 17.43%

29 20% 11.76% 12.50% 13.33% 14.29% 15.38% 16.67%

30 21% 11.60% 12.28% 13.04% 13.91% 14.89% 16.03%

31 22% 11.46% 12.09% 12.79% 13.58% 14.47% 15.49%

32 23% 11.33% 11.92% 12.57% 13.29% 14.11% 15.03%

33 24% 11.21% 11.76% 12.37% 13.04% 13.79% 14.63%

34 25% 11.11% 11.63% 12.20% 12.82% 13.51% 14.29%

35 26% 11.02% 11.50% 12.04% 12.62% 13.27% 13.98%

36 27% 10.93% 11.39% 11.89% 12.44% 13.04% 13.71%

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

is not surprising that the only two exceptions to the greater impact of

valuation errors being on large ¬rms comes from cash ¬‚ows, as value is

linear in cash ¬‚ows. The nonlinear relationship of value to discount rate

and growth rate causes errors in those two variables to impact the val-

uation of large ¬rms far more than small ¬rms and to impact the value

of both more than errors in cash ¬‚ow.

Errors in forecasting growth have the greatest impact on value. Value

is positively related to forecast growth. Errors in forecasting discount

rates are a close second in effect,17 though opposite in sign. Value is neg-

atively related to discount rate. Errors in forecasting the ¬rst year™s cash

¬‚ow by far have the least impact on value.

17. Again, this result comes from using the midyear Gordon model, not the end-of-year formula.

Part 4 Putting It All Together

402

T A B L E 11-5

Summary of Effects of Valuation Errors

Valuation Effects of Absolute Errors in the Variables [1]

Valuation Error Cash Flow Discount Rate r Growth Rate g

Absolute ($) 1 r g r g

V CF V CF V CF

(r g) (r1 g1)(r2 g2) (r1 g1)(r2 g2)

(11-3) (11-15) (11-15)

Large ¬rms Large ¬rms Large ¬rms

Relative (%) V CF r g r g

V V

V CF V (r2 g2) V (r2 g2)

(11-5) (11-17) Note [3] (11-17) Note [3]

Small ¬rms Large ¬rms Large ¬rms

Valuation Effects of Relative Errors in the Variables [1]

Valuation Error Cash Flow Discount Rate r Growth Rate g

Absolute ($) V kV1 Note [4] Note [4]

Note [2] NA NA

Large ¬rms Large ¬rms Large ¬rms

Relative (%) V2 r g r g

1 k %Error 1 %Error 1

V1 r (1 k)g

(1 k)r g

(11-8) (11-21) (11-20)

No difference Large ¬rms Large ¬rms

[1] Each cell shows the formula for the valuation error, the equation number in the chapter for the formula, and whether the valuation error is larger for large ¬rms, small ¬rms, or there is

no difference.

[2] This formula is not explicitly calculated in the chapter. We can calculate it as: V2 V1 [(1 k)V1 V1] kV1.

[3] While there is no difference in the magnitude of valuation errors arising from an error in r or g when we measure value by the end-of-year Gordon model, when we use the midyear

Gordon model, errors in g have slightly more impact than errors in r (and much more impact than errors in cash ¬‚ow).

[4] Omitted because these expressions are complex and add little to understanding the topic.

Another issue in valuation error in using the log size model is that