To determine the range of returns expected 68 percent and 95 percent of the time, we can use the empirical rule (or the 68-95-99.7 rule) which applies to a normal distribution.

Given:

• Average return ($\mu$) = 6.4%
• Standard deviation ($\sigma$) = 9.7%

68 Percent Range

The empirical rule states that 68 percent of the data lies within one standard deviation from the mean. Therefore, we calculate the range as: $\text{Range} = \mu \pm \sigma$

1. Lower Bound: $6.4\% - 9.7\% = -3.3\%$
2. Upper Bound: $6.4\% + 9.7\% = 16.1\%$

So, the expected range of returns 68 percent of the time is: $-3.30\% \text{ to } 16.10\%$

95 Percent Range

For 95 percent of the data, we use two standard deviations from the mean: $\text{Range} = \mu \pm 2\sigma$

1. Lower Bound: $6.4\% - 2 \times 9.7\% = 6.4\% - 19.4\% = -13.0\%$
2. Upper Bound: $6.4\% + 2 \times 9.7\% = 6.4\% + 19.4\% = 25.8\%$

Thus, the expected range of returns 95 percent of the time is: $-13.00\% \text{ to } 25.80\%$

Final Answer Summary:

• 68 percent range: $-3.30\%$ to $16.10\%$
• 95 percent range: $-13.00\%$ to $25.80\%$

Would you like more details or have any questions?

1. What does the empirical rule (68-95-99.7 rule) mean in probability terms?
2. How would the range change if the standard deviation increased?
3. Why is the standard deviation a critical factor in investment risk?
4. How can we interpret the negative return in practical investment terms?
5. What is the implication of a high standard deviation on the expected range?

Tip: Standard deviation is a measure of how spread out the returns are around the average—higher values indicate more volatility.