Martha B's has total assets of €1,750.These assets are expected to increase in value to either €1,800 or €2,400 by next year.The company has a pure discount bond outstanding with a face
Value of €2,000.This bond matures in one year.Currently, Treasury bills are yielding 6%.What is the
Value of the equity in this ﬁrm?

Question

Quizplus · Accepted Answer

To calculate the value of the equity in the firm, we first need to determine the expected value of the assets and then subtract the present value of the bond's face value. The expected value of the assets is the average of the two possible outcomes: (€1,800 + €2,400) / 2 = €2,100. The present value of the bond's face value (€2,000) due in one year at a 6% discount rate is €2,000 / (1 + 0.06) = €1,886.79. The equity value is the expected value of the assets minus the present value of the debt, so €2,100 - €1,886.79 = €213.21. However, this calculation does not directly lead to any of the provided options because the correct approach involves considering the bond's payoff in each state and discounting those payoffs.The correct approach is to calculate the equity value considering the two scenarios:1. If assets increase to €1,800, the bondholders get paid the full €2,000 (since the bond has priority), and equity holders get nothing because the assets are worth less than the bond's face value.2. If assets increase to €2,400, the bondholders get paid €2,000, and equity holders get the remainder, which is €400.The probability of each scenario is not given, so we cannot calculate an expected value directly. However, we can calculate the present value of the equity in the second scenario and recognize that the first scenario results in zero equity value.The present value of the equity in the second scenario (€400) needs to be discounted back at the risk-free rate (6%). However, since we're not given probabilities, we focus on the scenario where equity holders receive something:PV of equity = €400 / (1 + 0.06) = €377.36This calculation also does not match the provided options, indicating a misunderstanding in the approach based on the information given. The correct method to value the equity involves considering the bond's repayment in each scenario and the cost of debt:- When assets are €1,800, equity is €0 because the bond (€2,000) is worth more than the assets.- When assets are €2,400, equity is €400 (€2,400 - €2,000).The value of equity is thus contingent on the asset's final state and the discount rate applied to the bond rather than the equity directly. The correct answer involves recognizing that the equity's value is derived from the option-like payoff structure of the firm's assets relative to its debt obligations. Given the misunderstanding in the initial explanation, the correct approach involves option pricing methods or a more detailed analysis of the bond's and equity's payoffs under each scenario, which isn't fully explored with the given information. Therefore, the selection of choice B was based on an incorrect calculation, and the correct valuation requires a more nuanced approach considering the value of the firm's assets relative to its debt at maturity and the potential outcomes for equity holders.

Martha B's Has Total Assets of €1,750