A stock had returns of 8 percent,39 percent,11 percent,and −24 percent for the past four years.Which one of the following best describes the probability that this stock will not lose more than 43 percent in any one given year?
A)92.5 percent
B)95.0 percent
C)97.5 percent
D)84.0 percent
E)99.5 percent

Question

Quizplus · Accepted Answer

To calculate the probability that the stock will not lose more than 43 percent in any one given year, we need to find the minimum and maximum possible returns based on the given data. The minimum possible return is -24 percent, which means that the stock can lose up to 24 percent in a year. Therefore, the probability of not losing more than 43 percent is the probability of the stock returning more than -43 percent in any one year. To calculate this probability, we can use Chebyshev's inequality, which states that the probability of a random variable deviating from its mean by more than k standard deviations is at most 1/k^2. In this case, we can estimate the standard deviation of the stock returns as the square root of the variance, which is the average of the squared differences from the mean. Using the given data, we get: mean = (8+39+11-24)/4 = 8.5% variance = ((8-8.5)^2 + (39-8.5)^2 + (11-8.5)^2 + (-24-8.5)^2)/4 = 776.5% standard deviation = sqrt(variance) = 27.9% Now, we want to find the probability of the stock returning more than -43 percent, which is -43 - 8.5 = -51.5 percent deviation from the mean. So, using Chebyshev's inequality with k=2: P(return > -43%) >= 1 - (variance/k^2)/(-51.5% - 8.5%)^2 P(return > -43%) >= 1 - (776.5%/4)/(-0.6)^2 P(return > -43%) >= 0.975 P(return <= -43%) <= 0.025 Therefore, the probability that the stock will not lose more than 43 percent in any one given year is at least 97.5 percent (option C).

A Stock Had Returns of 8 Percent,39 Percent,11 Percent,and −24