Standardizing Active Returns Against a Cash Benchmark

Summary

This question considers how to express a fund’s return as a distance from an expected mean in standard deviation units. For a standalone fund, it gives the familiar return-over-volatility calculation, with annualization by the square root of time when relevant. For a fund measured against a benchmark, it substitutes active return and tracking error to describe relative performance.

The unresolved case is a cash benchmark, which has a return but is treated as having no volatility. The question asks whether tracking error in that case reduces to the fund’s own volatility. It provides a numerical illustration for the standalone calculation, but no response or conclusion for the cash-benchmark case. Interpretation also depends on the assumed return distribution, measurement horizon, and whether the benchmark return is constant over the period.

Key ideas

• A return z-score expresses performance relative to a mean in units of volatility.
• Active performance can be standardized using fund-minus-benchmark return and tracking error.
• The question asks whether a cash benchmark makes tracking error equal to fund volatility.
• The document leaves that case unanswered and does not establish distributional assumptions.

Tags

# z-score of an active return with a no-volatility benchmark


# z-score of an active return with a no-volatility benchmark












I don't know how to approach the problem I am having. Basically, the statement I am trying to make is: the fund's return is X standard distribution away from the mean.

Normally, for a single fund, you should just take its

```
(return - 0)/volatility*sqrt(t)
```

if the return was 37% and the volatility was 40%, it would be (0.37 - 0)/.40 = 0.925 standard deviations away from the mean. The probability of this or lower happening is ~17.88%.

For an active return, you would do the same thing, but do active return/tracking error

```
(return_fund - return_bench)/tracking_error*sqrt(t)
```

Now here is where I am stuck. What if the benchmark is cash. This has a return, but no risk, so would we still approach this the same way? Meaning the tracking_error used is just the fund's volatility?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.