To solve this problem, we need to find the expected value of the annual rate of return. The expected value [tex]\( E(X) \)[/tex] is calculated by multiplying each possible return by its probability and then summing up those products.
The formula for expected value [tex]\( E(X) \)[/tex] is:
[tex]\[ E(X) = \sum (x_i \cdot p_i) \][/tex]
where:
- [tex]\( x_i \)[/tex] is the return,
- [tex]\( p_i \)[/tex] is the probability of that return.
Let's break this down step-by-step.
1. Identify the returns and their probabilities:
- The return of [tex]\( 20\% \)[/tex] has a probability of [tex]\( 0.5 \)[/tex].
- The return of [tex]\( 15\% \)[/tex] has a probability of [tex]\( 0.3 \)[/tex].
- The return of [tex]\( 10\% \)[/tex] has a probability of [tex]\( 0.2 \)[/tex].
2. Convert the percentages to decimals for calculation:
- [tex]\( 20\% \)[/tex] is [tex]\( 0.2 \)[/tex].
- [tex]\( 15\% \)[/tex] is [tex]\( 0.15 \)[/tex].
- [tex]\( 10\% \)[/tex] is [tex]\( 0.1 \)[/tex].
3. Calculate the expected value by multiplying each return by its probability, and then summing the results:
[tex]\[
E(X) = (0.2 \times 0.5) + (0.15 \times 0.3) + (0.1 \times 0.2)
\][/tex]
Perform the individual multiplications:
- [tex]\( 0.2 \times 0.5 = 0.1 \)[/tex]
- [tex]\( 0.15 \times 0.3 = 0.045 \)[/tex]
- [tex]\( 0.1 \times 0.2 = 0.02 \)[/tex]
4. Sum the products:
[tex]\[
E(X) = 0.1 + 0.045 + 0.02 = 0.165
\][/tex]
To express this expected value as a percentage, multiply by 100:
[tex]\[
0.165 \times 100 = 16.5\%
\][/tex]
Therefore, the expected value of the rate of return for the MNP Company, Inc. stock is:
[tex]\[
\boxed{16.5\%}
\][/tex]
Hence, the correct answer is [tex]\( \boxed{B} \)[/tex].
