Stock A is expected to return 14 percent in a normal economy and lose 21 percent in a recession.Stock B is expected to return 11 percent in a normal economy and 5 percent in a recession.The probability of the economy being normal is 75 percent and being recessionary is 25 percent.What is the covariance of these two securities?
A).007006
B).006563
C).005180
D).007309
E).006274

Question

Quizplus · Accepted Answer

The covariance of two securities, $Cov(A, B)$, can be calculated using the formula: $$Cov(A, B) = E[(R_A - E[R_A])(R_B - E[R_B])]$$ where $R_A$ and $R_B$ are the returns of stock A and B, respectively, and $E[R_A]$ and $E[R_B]$ are the expected returns of stock A and B, respectively.First, calculate the expected returns of stock A and B:- Expected return of stock A, $E[R_A] = (14\% \times 75\%) + (-21\% \times 25\%) = 10.5\% - 5.25\% = 5.25\%$- Expected return of stock B, $E[R_B] = (11\% \times 75\%) + (5\% \times 25\%) = 8.25\% + 1.25\% = 9.5\%$Next, calculate the covariance:- In a normal economy, the deviation from the expected return for both stocks is $(14\% - 5.25\%)(11\% - 9.5\%) = 8.75\% \times 1.5\% = 0.13125\%$- In a recession, the deviation from the expected return for both stocks is $(-21\% - 5.25\%)(5\% - 9.5\%) = -26.25\% \times -4.5\% = 1.18125\%$The weighted sum of these products, considering the probabilities of each economic condition, gives the covariance:$$Cov(A, B) = (0.13125\% \times 75\%) + (1.18125\% \times 25\%) = 0.0984375\% + 0.2953125\% = 0.39375\%$$However, the calculation above seems to have been misunderstood in the process. The correct approach involves directly applying the probabilities to the deviations from expected returns for each state of the economy:$$Cov(A, B) = P(Normal) \times (R_{A,Normal} - E[R_A]) \times (R_{B,Normal} - E[R_B]) + P(Recession) \times (R_{A,Recession} - E[R_A]) \times (R_{B,Recession} - E[R_B])$$$$= 0.75 \times (14\% - 5.25\%) \times (11\% - 9.5\%) + 0.25 \times (-21\% - 5.25\%) \times (5\% - 9.5\%)$$$$= 0.75 \times 0.0875 \times 0.015 + 0.25 \times (-0.2625) \times (-0.045)$$$$= 0.75 \times 0.0013125 + 0.25 \times 0.0118125$$$$= 0.000984375 + 0.002953125$$$$= 0.0039375$$This calculation does not directly match any of the provided options, indicating a mistake in the calculation process. The correct calculation involves accurately applying the formula for covariance, considering the probabilities and the deviations of returns in both economic states. The correct answer should be derived from accurately calculating these deviations and their expected values, but the explanation provided here contains a miscalculation. The correct approach is to use the formula and the given probabilities and returns to find the covariance, which involves detailed calculations not accurately reflected in the explanation above. The correct answer, based on the options provided and understanding that there was a mistake in the detailed calculation, aligns with the option that closely matches the expected outcome of such a calculation process.

Stock a Is Expected to Return 14 Percent in a Normal