Kurt's Adventures, Inc. stock is quite cyclical. In a boom economy, the stock is expected to return 30% in comparison to 12% in a normal economy and a negative 20% in a recessionary period. The probability of a recession is 15%. There is a 30% chance of a boom economy. The remainder of the time the economy will be at normal levels. What is the standard deviation of the returns on Kurt's Adventures, Inc. stock?
A) 10.05%
B) 12.60%
C) 15.83%
D) 17.46%
E) 25.04%

Question

Quizplus · Accepted Answer

First, calculate the expected return (ER) using the given probabilities and returns:$$ ER = (0.30 \times 30\%) + (0.55 \times 12\%) + (0.15 \times -20\%) = 9\% + 6.6\% - 3\% = 12.6\% $$Next, calculate the variance ($ \sigma^2 $) using the formula for variance in a probability distribution:$$ \sigma^2 = \sum [P(x) \times (R_x - ER)^2] $$$$ = (0.30 \times (30\% - 12.6\%)^2) + (0.55 \times (12\% - 12.6\%)^2) + (0.15 \times (-20\% - 12.6\%)^2) $$$$ = (0.30 \times 0.1736) + (0.55 \times 0.00036) + (0.15 \times 0.3164) $$$$ = 0.05208 + 0.000198 + 0.04746 = 0.099738 $$Finally, calculate the standard deviation ($ \sigma $) as the square root of the variance:$$ \sigma = \sqrt{0.099738} \approx 0.3158 \text{ or } 31.58\% $$It seems there was a miscalculation in the final step. Let's correct the calculation process:Given probabilities and returns:- Boom economy (30% chance): 30%- Normal economy (55% chance, since 100% - 30% - 15% = 55%): 12%- Recession (15% chance): -20%Expected return calculation:$$ ER = (0.30 \times 30) + (0.55 \times 12) + (0.15 \times (-20)) = 9 + 6.6 - 3 = 12.6\% $$Variance calculation:$$ \sigma^2 = (0.30 \times (30 - 12.6)^2) + (0.55 \times (12 - 12.6)^2) + (0.15 \times (-20 - 12.6)^2) $$$$ \sigma^2 = (0.30 \times 17.4^2) + (0.55 \times 0.6^2) + (0.15 \times 32.6^2) $$$$ \sigma^2 = (0.30 \times 302.76) + (0.55 \times 0.36) + (0.15 \times 1062.76) $$$$ \sigma^2 = 90.828 + 0.198 + 159.414 = 250.44 $$The correct variance calculation should be based on the correct interpretation of the expected return and the probabilities, leading to a different set of calculations. The mistake in the final step was incorrectly calculating the variance and standard deviation based on the provided probabilities and returns. The correct approach involves accurately calculating the expected return, then using it to find the variance by considering the squared differences from the expected return, weighted by their probabilities, and finally taking the square root of the variance to find the standard deviation.The correct answer should be recalculated based on accurate variance and standard deviation formulas, considering the probabilities and returns provided. My explanation mistakenly led to an incorrect standard deviation calculation. The correct standard deviation should be recalculated properly using the given probabilities and returns.