Mean Semivariance and PMPT Downside Risk

Summary

The document asks whether mean semivariance optimization and Post-Modern Portfolio Theory (PMPT) downside risk are equivalent when they use the same benchmark. It presents semivariance as the average squared shortfall of observed returns below a threshold, and PMPT downside risk as the square root of an integral over squared shortfalls weighted by the return distribution.

The comparison highlights a key distinction: semivariance is expressed as a second lower partial moment, while PMPT’s stated downside-risk measure takes the square root, making it a downside deviation. Equivalence therefore depends on whether the comparison is between semivariance and squared downside deviation, and on matching the benchmark and normalization conventions. The document poses the question but provides no proof, data, or resolution, so it serves as a conceptual prompt rather than a worked optimization method.

Key ideas

• Semivariance measures squared returns shortfalls below a chosen benchmark.
• The PMPT expression integrates squared shortfalls across the return distribution.
• The PMPT formula includes a square root, distinguishing downside deviation from semivariance itself.
• A valid comparison requires the same benchmark and compatible scaling conventions.

Tags

# Mean Semivariance Optimization VS PMPT


# Mean Semivariance Optimization VS PMPT












Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as:

$(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$

where $B$ is the benchmark rate, $R_{i}$ is the asset returns for asset $i$, and $T$ is the number of observations.

Post Modern Portfolio theory however, (https://en.wikipedia.org/wiki/Post-modern_portfolio_theory), defines downside risk (which according to my understanding should be equal to semivariance), as :

$ \sqrt {{\int_{-\infty}^{t}} ({t-r})^2 f(r)dr}$

where $t$ is the benchmark here, $r$ is the random variable representing the return for the distribution of annual returns $f(r)$, and $f(r)$ is the distribution for the annual returns.

Assuming $t$ is equal to $B$, would these two be equal, and can anyone prove so?

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