A Benchmark-Relative Downside Risk Measure

Summary

The document proposes a risk indicator for evaluating portfolio returns against a benchmark. Its numerator is the expected portfolio return minus the expected benchmark return. Its denominator aggregates squared shortfalls in cases where the portfolio return falls below the benchmark, using their joint distribution. This adapts the target-relative logic of the Sortino ratio by making the target vary with benchmark returns rather than setting it to a fixed scalar.

The proposal is presented as a question, not a validated measure: no derivation, sample calculation, comparison with established metrics, or empirical evidence is supplied. Its usefulness would depend on choices such as whether to normalize the downside term, how to estimate the joint distribution, and how to interpret the measure when the benchmark-relative shortfall is small or absent. The exchange provides a framework to evaluate, but does not resolve whether the formula is formally sound or preferable to existing benchmark-relative risk measures.

Key ideas

• The proposed numerator is expected portfolio return minus expected benchmark return.
• The denominator aggregates squared portfolio shortfalls relative to the benchmark.
• The benchmark return acts as a varying target, extending the target-relative idea behind the Sortino ratio.
• The proposal contains no empirical validation or comparison with established performance measures.

Tags

# Benchmarking risk


# Benchmarking risk












Given the portfolio return $R$ and the benchmark return $B$, I want to define a risk indicator, measuring the ability to beat the benchmark ($R>B$), given the downside risk taken; the latter not intended as an absolute loss, but as the risk of falling below the benchmark.

This measure could be set as:

$$ \frac{\mathbb{E}R-\mathbb{E}B} {\int_{r\leq b}\, (r-b)^2 f_{R,B}(r,b)\mathrm{d}r\mathrm{d}b} $$

where $f_{R,B}$ is the joint density for $R$ and $B$.

The result is Sortino-like, but the target here is not a scalar, but the return of the benchmark.

Do you find any conceptual or formal error in this approach? or can you suggest a better way to implement the idea?

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.