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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