Hi Michelle! As you know, we don't know the future with certainty and can only guess how an asset may grow or decline in value over time. There could be many different outcomes, and your question suggests that there are 5 growth rates that seem most
likely to occur: 6.75%, 8.25%, 9.5%, 7%, and 4%. Perhaps the 9.5% is achieved due to a bull market, and the 4% occurs because of a recession. Since the future is unknown, expectations must be developed as an "educated guess" on what might happen, and in
this case there are 5 possible scenarios.
Associated with each return is a probability that it might occur. For example, if we say that the 4% return is due to a recession, what's the chance that a recession will happen? In your question, the probability of the 4% return is 6%. Likewise, there's
a 19% chance that the asset will earn a 9.5% return over time.
The expected return combines the 5 possible scenarios into a single number that represents our best guess for what will happen with the asset. This is done by calculating the weighted average of the scenarios. In other words, you need to multiply the
return % by the probability % for each outcome, then add up the 5 results:
E[R] = R1P1 + R2P2 + R3P3 + R4P4 + R5P5
For example, scenario 1 is 6 3/4% times 0.39, which is 0.0675 x 0.39 = 0.026325. Repeat for the other four possible outcomes, then add up all five results, and you'll have your answer!
Hope this helps.