Question

you invest $3,000 in a stock that has a 10% chance of a 2% return, a 70% chance of a 6% return and a 20% chance of a 15% return. what is your expected return after one year? 6.0% 8.8% 5.6% 7.4%

Explanation:

Step1: Recall Expected Value Formula

The expected return (in percentage) is calculated by multiplying each possible return by its corresponding probability and then summing these products. The formula for the expected value \( E(X) \) of a discrete random variable is \( E(X)=\sum_{i = 1}^{n}x_{i}P(x_{i}) \), where \( x_{i} \) is the return in percentage and \( P(x_{i}) \) is the probability of that return.

Step2: Identify Each Return and Probability

• For the first case: \( x_{1}=2\% = 0.02 \) and \( P(x_{1}) = 10\%=0.1 \)
• For the second case: \( x_{2}=6\% = 0.06 \) and \( P(x_{2}) = 70\%=0.7 \)
• For the third case: \( x_{3}=15\% = 0.15 \) and \( P(x_{3}) = 20\%=0.2 \)

Step3: Calculate Each Product

• First product: \( x_{1}P(x_{1})=0.02\times0.1 = 0.002 \)
• Second product: \( x_{2}P(x_{2})=0.06\times0.7 = 0.042 \)
• Third product: \( x_{3}P(x_{3})=0.15\times0.2 = 0.03 \)

Step4: Sum the Products

Sum these three products: \( E(X)=0.002 + 0.042+0.03=0.074 \)

Step5: Convert to Percentage

To convert the expected value (which is in decimal form) to a percentage, multiply by 100: \( 0.074\times100 = 7.4\% \)

Answer:

7.4% (corresponding to the option with 7.4%)