A confidence interval is not a range for the data

Question

You report a 95% confidence interval for a strategy's mean daily return: $[0.2\text{bp}, 1.8\text{bp}]$ . A colleague says: "Great, so 95% of our daily returns fall between 0.2bp and 1.8bp."

Is that right? Explain what the interval is actually about, and what object would answer your colleague's question.

Accepted Answer

The statement is wrong, and it confuses two very different quantities. What the CI describes. A confidence interval for the mean is about the average daily return \mu, and its width is driven by the standard error of the mean, s/\sqrt{n}. With many days of data, s/\sqrt{n} is tiny, so the interval [0.2, 1.8]bp is narrow, that narrowness reflects how precisely you have pinned down the mean, not how tightly individual days cluster. Why individual days are far wider. A single day's return has spread governed by s itself, not s/\sqrt{n}. Daily returns routinely swing by tens or hundreds of basis …

A confidence interval is not a range for the data

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