Final answer:
A portfolio's variance equals the weighted average of variances of its assets when returns are perfectly positively correlated. Expected return is the weighted sum of all possible returns. The safest investment has the lowest variance, and the riskiest has the highest, whereas the investment with the highest expected return has the best average performance.
Step-by-step explanation:
The portfolio's variance would be equal to the weighted average of the variances of the assets in the portfolio only when the portfolio's assets' returns are perfectly positively correlated. This occurs because under perfect positive correlation, all assets in the portfolio would move in tandem and therefore, the diversification benefits which typically reduce the portfolio's variance, do not apply. In most practical scenarios, assets do not have a perfect positive correlation, and the portfolio's variance is reduced through diversification, supported by the principle that assets' returns are usually less than perfectly correlated.
Expected return for each investment is the weighted sum of all possible returns, multiplied by their respective probabilities. To determine the safest investment, one would typically look for the investment with the least variance or standard deviation, since it implies less fluctuation in returns. Conversely, the riskiest investment would be the one with the highest variance, indicating greater uncertainty in returns. The investment with the highest expected return could be different from the safest and riskiest, as it is a measure of average performance over time rather than variability.