The Infinitum company's value of outstanding equity is USD 400 million, and you have estimated its beta to be 1.2. Infinitum has four-year zero-coupon debt outstanding with a face value of USD 100 million that currently trades for USD 80 million. The firm pays no dividends and reinvests all of its earnings. The risk-free rate of interest is currently 4.00% and is continuously compounded, while the standard deviation of the return on firm's assets is equal to 0.6. From the standard normal distribution table, please use the 4-digit value. Using the Black-Scholes formula, the estimated unlevered beta of the firm is closest to:单项选择题
A
A. 1.80
B
B. 1.02
C
C. 1.41
D
D. 0.20

Question

The Infinitum company's value of outstanding equity is USD 400 million, and you have estimated its beta to be 1.2. Infinitum has four-year zero-coupon debt outstanding with a face value of USD 100 million that currently trades for USD 80 million. The firm pays no dividends and reinvests all of its earnings. The risk-free rate of interest is currently 4.00% and is continuously compounded, while the standard deviation of the return on firm's assets is equal to 0.6. From the standard normal distribution table, please use the 4-digit value. Using the Black-Scholes formula, the estimated unlevered beta of the firm is closest to:单项选择题

A

A. 1.80

B

B. 1.02

C

C. 1.41

D

D. 0.20

BlackTom AI · Accepted Answer

We begin by identifying the key inputs: the levered beta given for the firm is 1.2, and the market values provided are equity 400 million and debt 80 million (market value of debt). First, determine the debt-to-equity ratio that feeds the beta unlevering formula.

D/E = 80 / 400 = 0.20.

Under the classic Hamada/Modigliani-Miller framework (with taxes ignored or t ≈ 0 for this unlevering step), the relation between levered beta (βL) and unlevered beta (βU) is:
βL = βU × [1 + (1 − t) × (D/E)]. If we assume taxes are negligible (t ≈ 0), this simplifies to:
βL = βU × [1 + D/E] = βU × (1 + 0.20) = βU × 1.20.
Thus,
beta_unlevered = beta_levered / 1.20 = 1.2 / 1.20 = 1.0.

Now compare this computed unlevered beta to the available answer choices. The exact calculation yields 1.0, which is closest to 1.02 among the options.

Why the other options are not correct:
- A. 1.80: This would require a much larger unlevered beta, implying a far higher asset risk than implied by the given levered beta and debt level; it does not fit the standard levered/unlevered relationship with D/E = 0.20.
- C. 1.41: Similar issue as A; this would indicate substantially more asset risk than what the levered beta and modest debt load imply under the usual formula.
- D. 0.20: This is far too small for an unlevered beta given an equity beta of 1.2; it would suggest extremely low asset risk, inconsistent with the provided levered beta.