Sitco has a total of $12 million in cash and marketable securities. Free cash flows during the coming year are expected to be $47 million with a standard deviation of $31 million. Assume that Sitco's free cash flows are approximately normally distributed. What is the probability that Sitco will run out of cash during the coming year?
A) 29.98%
B) 34.83%
C) 97.13%
D) 2.87%

Question

Quizplus · Accepted Answer

We need to use the formula for z-score: (X - μ) / σ, where X is the value we are interested in (in this case, the amount of free cash flows that would cause Sitco to run out of cash), μ is the mean of the free cash flows, and σ is the standard deviation of the free cash flows. Since we want to find the probability of running out of cash, we want to find the area under the curve to the left of the z-score corresponding to that value of X.

First, we need to find the z-score corresponding to the value of X that would cause Sitco to run out of cash. This can be calculated as (X - μ) / σ = (0 - 47) / 31 = -1.5161. Using a standard normal distribution table, we can find that the area to the left of this z-score is 0.0630.

However, we need to adjust this probability because we are interested in the probability of running out of cash, which means we want to find the probability that free cash flows will be less than or equal to zero. To make this adjustment, we need to subtract the probability of free cash flows being between zero and the mean (which we can find using the same table and the z-score corresponding to zero: (0 - 47) / 31 = -1.5161), and then divide by the probability of free cash flows being less than the mean. Doing this gives us:

P(X ≤ 0) = [P(X ≤ 0) - P(μ ≤ X ≤ 0)] / P(X < μ)
= [0.0630 - 0.4708] / 0.5
= 0.0944

Finally, we can convert this to a percentage by multiplying by 100, giving us 2.87%. Therefore, the answer is D.

Sitco Has a Total of $12 Million in Cash and Marketable