Calculating Downside Beta from Selected Return Observations

Summary

The document explains how to calculate downside beta when the condition is that the benchmark return is below zero. The answer selects only the observations that meet the benchmark condition, then applies the ordinary beta calculation to those paired asset and benchmark returns. It explicitly rejects adding zero-valued returns for periods that fail the filter.

A worked example illustrates the procedure using a small set of asset and benchmark returns. The selected rows include cases where the benchmark return is negative, as well as positive benchmark returns below the sample’s average; the resulting covariance is divided by the benchmark variance for those selected observations. The example reports a downside beta of -4.82 and an upside beta of -0.75. The explanation is specific to the stated downside threshold and the example’s selection rule. It does not discuss alternative definitions, statistical uncertainty, sample-size requirements, or how results change under other thresholds, so those choices should be made explicit in practical analysis.

Key ideas

• Downside beta can be computed by filtering paired returns according to a benchmark condition.
• After selecting the qualifying observations, use the ordinary beta calculation on those rows.
• Do not insert zeros for observations that fail the selection condition.
• The benchmark variance belongs in the denominator of the beta calculation.
• The threshold and selection rule should be stated because they define the resulting measure.

Tags

# Should I include zeros in downside beta calculation?


# Should I include zeros in downside beta calculation?












Downside beta is the beta coefficient for an asset and a benchmark restricting benchmark returns to be less than a given value. Let’s assume zero for simplicity.

We have:

If we have returns in period 1, 2, and 3, such that:

Asset: 5%, 3%, 8%

Benchmark: 4%, 2%, -1%

Would I run the beta using:

a) covar(5%, 3%; 4%, 2%)/var(4%, 2%)

or,

b) covar(5%, 3%, 0%; 4%, 2%, 0%)/var(4%, 2%, 0%)

With downside deviation we typically include the zeros for points which failed the positive/negative test, so unsure here.

## Answer by Brian B (score 3, accepted)

https://quant.stackexchange.com/a/68040

The conditional notation is indeed a bit confusing for those who do not spend a lot of time with mathematics. Downside beta is computed simply by taking only those data rows for which we see underperformance, and then doing the regular beta calculation.

So, for example, if you have percent returns like

| Asset | Benchmark |
| 5 | 4 |
| -2 | 2 |
| 8 | -1 |
| 4 | 6 |
| 3 | 1 |
| 3 | 5 |

Then the average market benchmark return is 2.8, so the rows you include are

| Asset | Benchmark |
| -2 | 2 |
| 8 | -1 |
| 3 | 1 |

and the calculation is $\beta^- = \frac{\mathrm{Cov}(\{-2,8,3\};\{2,-1,1\})}{\mathrm{Var}(\{2,-1,1\})}$ or -4.82. One does not insert zeros.

(Note: with this data our "upside" beta is -0.75.)

As an aside, downside beta is a really excellent risk measure to consider, and I am glad to see it receiving your attention.

Edits (per comments by @Henry Walter):

(A) Fixed Variance in denominator to be benchmark variance

(B) Fixed row choices

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.