Firm Valuation With Uncertain Cash Flows

Problem:

You are considering acquiring a firm that you believe can generate expected cash flows of $10,000 a year forever. However, you recognize that those cash flows are uncertain. a. Suppose you believe that the beta of the firm is .4. How much is the firm worth if the risk free rate is 4% and the expected rate of return on the market portfolio is 11%? b. By how much will you overvalue the firm if its beta is actually .6?

Solution:

To value the firm using the Capital Asset Pricing Model (CAPM) and a perpetuity formula, we follow these steps:

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Given:

• Expected cash flows = $10,000 per year forever

• $R_f = 4\% = 0.04$
  

• $R_m = 11\% = 0.11$
  

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Step 1: Use CAPM to Find the Required Return $r$

$$r = R_f + \beta (R_m - R_f)$$

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a. Valuation if β = 0.4

$$r = 0.04 + 0.4(0.11 - 0.04) = 0.04 + 0.4(0.07) = 0.04 + 0.028 = 0.068 = 6.8\%$$

Now value the firm using the perpetuity formula:

$$\text{Value} = \frac{\text{Cash Flow}}{r} = \frac{10,000}{0.068} \approx \boxed{\$147,059}$$

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b. If β = 0.6, what is the actual value?

$$r = 0.04 + 0.6(0.07) = 0.04 + 0.042 = 0.082 = 8.2\%$$
$$\text{True Value} = \frac{10,000}{0.082} \approx \boxed{\$121,951}$$

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Overvaluation:

$$\text{Overvaluation} = 147,059 - 121,951 = \boxed{\$25,108}$$

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✅ Final Answers:

• a. Firm value with β = 0.4: $147,059

• b. Overvaluation if true β = 0.6: $25,108