Question

You are trying to develop a strategy for investing in two different stocks. The anticipated annual return for a​ $1,000 investment in each stock under four different economic conditions has the probability distribution shown to the right. Complete parts​ (a) through​ (c) below. Probability   Economic_condition   Stock_X   Stock_Y 0.1   Recession -90   -170 0.3   Slow_growth 20    50 0.4   Moderate_growth 110 150 0.2   Fast_growth 160 190 A. Compute the expected return for stock X and for stock Y. The expected return for stock X is ​(Type an integer or a decimal. Do not​ round.) The expected return for stock Y is ​(Type an integer or a decimal. Do not​ round.) B. Compute the standard deviation for stock X and for stock Y. The standard deviation for stock X is ​(Round to two decimal places as​ needed.) The standard deviation for stock Y is ​(Round to two decimal places as​ needed.) C. Would you invest in stock X or stock​ Y? Explain. Choose the correct answer below. A.Since the expected values are approximately the​ same, either stock can be invested in.​ However, stockX has a larger standard​ deviation, which results in a higher risk. Due to the higher risk of stockX​, stockY should be invested in. B.Since the expected values are approximately the​ same, either stock can be invested in.​ However, stockY has a larger standard​ deviation, which results in a higher risk. Due to the higher risk of stock Y​,stockX should be invested in.Your answer is not correct. C.Based on the expected ​value, stockY should be chosen. ​However, stockY has a larger standard ​deviation, resulting in a higher ​risk, which should be taken into consideration. D.Based on the expected ​value, stockX should be chosen. ​However, stockX has a larger standard ​deviation, resulting in a higher ​risk, which should be taken into consideration.

Answer 1

Answer A. First, let's just recreate the table to look like this:

Probability	Economic Condition	Stock_X	Stock_Y
0.1	Recession	-90	-170
0.3	Slow_growth	20	50
0.4	Moderate_growth	110	150
0.2	Fast_growth	160	190

Now, let's try to understand the table and then I'll explain the terms given in the question. There are 4 economic periods and their respective probabilities of happening are known. Then, the respective returns that each of stock X and y would give in these respective economic conditions is also known. Annual returns are usually expressed in a percentage form. Thus, we change them to a decimal format for easy calculations.

Probability	Economic Condition	Stock_X	Stock_Y
0.1	Recession	-0.9	-1.7
0.3	Slow_growth	0.2	0.5
0.4	Moderate_growth	1.1	1.5
0.2	Fast_growth	1.6	1.9

Now, we need to calculate the expected return on Stock X. This is the return that an individual expects a stock to earn over the next period. Of course, because this is only an expectation, the actual return may be either higher or lower. An individual’s expectation may simply be the average return per period a security has earned in the past.

Thus, we calculate the weighted average of the returns of Stock X in each of the conditions

Estimated return on Stock X = (0.1*(-0.9)) + (0.2*0.3) + (1.1*0.4) + (1.6*0.2)/4 = 0.1825 = 18.25%

Estimated return on Stock Y = (0.1*(-1.7)) + (0.3*0.5) + (0.4*1.5) + (0.2*1.9)/4 = 0.24 = 24%

B. Standard Deviation

For each stock, calculate the deviation of each possible return from the stock’s expected return given previously.

The deviations we have calculated are indications of the dispersion of returns. However, because some are positive and some are negative, it is difficult to work with them in this form. For example, if we were to simply add up all the deviations for a single company, we would get zero as the sum. To make the deviations more meaningful, we multiply each one by itself. Now all the numbers are positive, implying that their sum must be positive as well.

For each stock, calculate the average squared deviation, which is the variance:

Economic Condition	Rate of Return	Deviation from Expected return	Squared value of Deviation
Recession	-0.9	-1.0825	1.17180625
Slow_growth	0.2	0.0175	0.00030625
Moderate_growth	1.1	0.9175	0.84180625
Fast_growth	1.6	1.4175	2.00930625
		Average of Squared Deviations	1.00580625

Expected rate of return

0.1825

Thus, the variance of the returns on Stock X is 1.0058

Now, we know that standard deviation is calculated as the square root of the variance = = 1.003 approx, which is 100% approx.

Now, for stock Y, we do the same calculations

Economic Condition	Rate of Return	Deviation from Expected return	Squared value of Deviation
Recession	-1.7	-1.94	3.7636
Slow_growth	0.5	0.26	0.0676
Moderate_growth	1.5	1.26	1.5876
Fast_growth	1.9	1.66	2.7556
		Average of Squared Deviations	2.0436

Expected rate of return

0.24

Standard Deviation for Stock Y is given as = 143% approx

C. The expected values for both Stock X and Stock Y are not the same. Thus, options A and B are ruled out automatically. Based on the exptected returns, Stock Y gives a higher return i.e. 24%. However, it has a higher standard deviation that Stock X, which means higher risk. Thus, Option C is the correct answer.