To determine the range of returns we would expect for this asset over a given period, we’ll use the concept of the normal distribution. For a normal distribution:

1. 68% of observations fall within one standard deviation of the mean.
2. 95% of observations fall within two standard deviations of the mean.

Given:

• Average return (mean), $\mu = 11.7\%$
• Standard deviation, $\sigma = 19.5\%$

68% Range

For 68% of observations, the range would be: $\text{Range} = \mu \pm \sigma$

Calculating the lower bound: $\text{Lower bound} = 11.7\% - 19.5\% = -7.8\%$

Calculating the upper bound: $\text{Upper bound} = 11.7\% + 19.5\% = 31.2\%$

So, for 68% of the time, the expected range of returns is: $[-7.80\%, 31.20\%]$

95% Range

For 95% of observations, the range would be: $\text{Range} = \mu \pm 2\sigma$

Calculating the lower bound: $\text{Lower bound} = 11.7\% - (2 \times 19.5\%) = 11.7\% - 39.0\% = -27.3\%$

Calculating the upper bound: $\text{Upper bound} = 11.7\% + (2 \times 19.5\%) = 11.7\% + 39.0\% = 50.7\%$

So, for 95% of the time, the expected range of returns is: $[-27.30\%, 50.70\%]$

Final Answer:

• 68% Range: $[-7.80\%, 31.20\%]$
• 95% Range: $[-27.30\%, 50.70\%]$

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. What would the range be for 99% of the time?
2. How would the results change if the standard deviation were higher?
3. What implications does a high standard deviation have on investment risk?
4. How can understanding these ranges help in making investment decisions?
5. What is the probability of a return below -7.8%?

Tip: When dealing with financial returns, understanding the standard deviation can help gauge potential risk, as a higher standard deviation indicates more variability around the mean.