Suppose that a firm's value $V$ grows over one year at a simple rate $^{\circ}_{R}$ that has a mean of $0.10$ and a variance of $0.04$ .If the firm's current value is $10 billion and it has one-year zero-coupon debt of face value $7 billion,what is the probability that the firm's assets will not be sufficient to repay the debt at the end of the year? Assume the firm value at the end of the year is normally distributed.
A)1.15%
B)2.28%
C)3.39%
D)5.12%

Question

Quizplus · Accepted Answer

Given the firm's current value $V_0 = \$10$ billion, the mean growth rate $\mu = 0.10$, and the variance $\sigma^2 = 0.04$, the firm's value at the end of the year can be modeled as a normal distribution. The face value of the debt is \$7 billion, so we need to find the probability that the firm's value at the end of the year, $V_1$, is less than \$7 billion.First, calculate the expected value of the firm at the end of the year:$$E[V_1] = V_0(1 + \mu) = \$10 \times (1 + 0.10) = \$11 \text{ billion}$$The standard deviation of the growth rate is the square root of the variance:$$\sigma = \sqrt{\sigma^2} = \sqrt{0.04} = 0.2$$To find the probability that the firm's value is less than \$7 billion, we need to find the Z-score of \$7 billion under this distribution:$$Z = \frac{X - \mu}{\sigma} = \frac{7 - 11}{2} = \frac{-4}{2} = -2$$Looking up the Z-score of -2 in the standard normal distribution table gives us a probability of approximately 0.0228 or 2.28%, which corresponds to the probability that the firm's assets will not be sufficient to repay the debt at the end of the year.

Suppose That a Firm's Value VVV Grows Over One Year at a Simple Rate

Suppose That a Firm's Value $V$ Grows Over One Year at a Simple Rate