Assume the following:
What will be an ideal response?
benchmark index = Salomon Smith Barney BIG Bond Index
expected return for benchmark index = 7%
forward-looking tracking error relative to Lehman Aggregate Bond Index = 200 basis points
Assuming that returns are normally distributed, complete the following table:
Number of Standard Deviations Range for Portfolio Active Return Corresponding Range for Portfolio Return Probability
1
2
3
With an expected return of 7% and a standard deviation of 200 basis points or 2%, then a normal distribution implies there is about a 67% probability that values will be found between one standard deviation of either side of the mean. Thus, for a standard deviation of 1, the range on either side of the mean for portfolio active return is 1 standard deviation times 2% = 2%. The 2% deviation will be on both the left and right side of the 7% mean value. Thus, with a portfolio mean return of 7%, the corresponding range for portfolio return will be from 5% (7% – 2% = 5%) left of the mean value to 9% (7% + 2% = 9%) right of the mean value
Similarly, for a standard deviation of 2, the range on either side of the mean is 2 standard deviation times 2% = 4%. With a portfolio mean active return of 7%, the corresponding range for portfolio return will be from 7% – 4% = 3% to 7% + 4% = 11%. A normal distribution implies there a 96% probability that values will be found between two standard deviation of either side the mean.
Likewise, for a standard deviation of three, the range on either side of the mean is 3 standard deviation times 2% = 6%. With a portfolio mean active return of 7%, the corresponding range for portfolio return will be 7% – 6% = 1% and 7% + 6% = 13%. A normal distribution implies there a 99% probability that values will be found between two standard deviation of either side the mean.
The above values can all be found in the below table.
Number of Standard Deviations Range for Portfolio Active Return Corresponding Range for Portfolio Return Probability
1 2% 5%–9% 67%
2 4% 3%–11% 95%
3 6% 1%–13% 99%
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