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

The firm's value at the end of the year can be calculated as $V(1 + ^\circ R)$. Given $V = 10$ billion and $^\circ R$ has a mean of $0.10$ and a variance of $0.04$, the mean value of the firm at the end of the year is $10 \times (1 + 0.10) = 11$ billion. The standard deviation of the growth rate $^\circ R$ is the square root of the variance, which is $\sqrt{0.04} = 0.2$ or 20%. Therefore, the standard deviation of the firm's value at the end of the year is $10 \times 0.2 = 2$ billion.To find the probability that the firm's assets will not be sufficient to repay the $7 billion debt, we need to find the probability that the firm's value at the end of the year is less than $7 billion. This requires calculating the Z-score, which is given by $(X - \mu) / \sigma$, where $X$ is the value of interest ($7 billion), $\mu$ is the mean value of the firm at the end of the year ($11 billion), and $\sigma$ is the standard deviation of the firm's value at the end of the year ($2 billion).The Z-score is $(7 - 11) / 2 = -2$. Looking up this Z-score in the standard normal distribution table gives a probability of approximately 2.28%, which corresponds to the probability that the firm's value will be less than $7 billion at the end of the year, indicating the firm will not be able to repay its debt.

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