Category Archives: Risk Measures

What is a non-law-invariant risk measure?

Someone asked me recently, what is a risk-measure which is not law-invariant? Admittedly, the law invariance property seems so natural that any meaningful risk measure should have this property. Of course we can struggle to construct risk measures which are not law invariant, but are there any naturally occurring risk measures which do not have this property? The answer is yes, but let us take things from the start. Continue reading →

Zero risk

What does it mean for a random variable to exhibit zero risk? It of course depends on the risk measure we are using to quantify it. Continue reading →

Average value-at-risk of simple quadratic form

The average value-at-risk of a quadratic form y'y, where y\sim \mathcal{N}(0_n,I_n) is given by a particularly complex closed-loop formula which I’ll describe below.

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Strict monotonicity of expected shortfall

The expected shortfall, also known as average value-at-risk or conditional value-at-risk, is a coherent risk measure defined as

\mathrm{AV@R}_{\alpha}[Z]=\inf_{t\in\mathbb{R}} \{t+\alpha^{-1}\mathbb{E}[Z-t]_+\}

for Z\in\mathcal{Z}:=\mathcal{L}_p(\Omega,\mathcal{F},\mathrm{P}) for some p\in[1,+\infty].

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mathbabe

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Mathematix is the mathematician of the village of Asterix