Why Expected Shortfall Is Positively Homogeneous
Summary
The document explains why expected shortfall scales linearly when portfolio returns are multiplied by a positive constant. It writes expected shortfall as the average return in the lower tail up to a quantile, then considers a scaled random variable. Scaling changes the quantile and density in corresponding ways; substituting these into the integral and changing variables gives the original expected shortfall multiplied by the same constant.
The argument applies to distributions on the real line when the stated density-based formula is valid and the relevant integrals exist. The response also points to the broader measure-theoretic view, where positive homogeneity follows from the integral’s behavior under positive scaling. The post does not address other coherence properties, such as subadditivity, or practical estimation of expected shortfall from finite data.
Key ideas
- Expected shortfall can be expressed as an integral over the lower tail of a return distribution.
- Multiplying returns by a positive constant scales the corresponding quantile by that constant.
- A change of variables in the expected-shortfall integral shows that the measure scales linearly.
- The density-based derivation assumes the relevant distribution and integrals satisfy its regularity conditions.
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Full text
# Coherent risk measure
# Coherent risk measure
One of the characteristics of a Coherent risk measure is `Positive homogeneity` (ref, https://en.wikipedia.org/wiki/Coherent_risk_measure).
`Expected Shortfall` is considered to be a conhorent risk measure. Does it mean that it satisfy `Positive homogeneity` for all kinds of distribution of underlying risk factor?
I understand that if the statistical distribution is Location and scale invariant (e.g. `Normal distribution`), then `Expected shortfall` can be assumed to follow `Positive homogeneity`, but how can it be generalised for all kind of statistical distributions with support as Real line?
Thanks for any input.
## Answer by Rylan (score 1, accepted)
https://quant.stackexchange.com/a/76291
It should be. Consider a random variable $X$ with support $\mathbb{R}$ that we'll think of as returns in a portfolio, with cdf $F$ with inverse $F^{-1}$. (Let's assume also that it has a pdf $p$.)
Then, for a confidence level $\alpha$, for example $95\%$, the expected shortfall is
$$\frac{1}{1-\alpha} \int_{-\infty}^{F^{-1}(1-\alpha)}xp(x)dx$$
Let $c>0$ be a positive scalar, and consider the distribution of $Y=cX$. This random variable has cdf $F_c$ and pdf $p_c$. Similarly, it has expected shortfall given by
$$\frac{1}{1-\alpha} \int_{-\infty}^{F_c^{-1}(1-\alpha)}yp_c(y)dy$$
Considering the properties of $F_c$, we see that $F_c(x) = P( Y< x) = P(cX < x) = P(X < \frac{x}{c}) = F(\frac{x}{c})$, and similarly $F_c^{-1}(x) = cF^{-1}(x)$. and so $p_c(x) = \frac{p(\frac{x}{c})}{c}$.
This gives us:
$$\frac{1}{1-\alpha} \int_{-\infty}^{cF^{-1}(1-\alpha)}y\frac{p(\frac{y}{c})}{c}dy$$
A change of variables $y=cx, dy=cdx$ gives us
$$\frac{1}{1-\alpha} \int_{-\infty}^{F^{-1}(1-\alpha)}cxp(x)dx = \frac{c}{1-\alpha} \int_{-\infty}^{F^{-1}(1-\alpha)}xp(x)dx$$
which is exactly $c$ times our initial definition.
## Answer by Gabriele Bonomi (score 0)
https://quant.stackexchange.com/a/76290
As long as you're happy that PnL distribution is a positive $\Sigma$-measurable function (can it not be?) then, yes, ES measure is always positively homogeneous.
https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_is_Positive_HomogeneousShown 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.