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Herr Euler is so prolific that perhaps this has not been asked before. I read a paper - arxiv.org/pdf/0708.2542.pdf "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle Dirk Tasche" - showing a proof of the "EULER ALLOCATION METHOD" but after trying to follow it through it then referred to fuzzy logic game theory invented in the 50s by Shapley et al. On page 5 it refers: "Denault (2001) derived the Euler principle by game-theoretic considerations". Since Euler didnt have access to that, how did he come up with the concept. Its used to allocate VAR in risk. I think perhaps its just saying that is you have a lot of gradients making a continual surface then the average gradient is the natural allocation. Is there more to it than that? The article is by Dirk Tasche. (sic p5)

Theory

The Euler allocation principle may be applied to any risk measure that is homogeneous of degree 1 in the sense of Definition A.1 (see Appendix A for full or formula below) and differentiable in an appropriate sense. ...

(sic part of Appendix A.1)

Theorem A.1 (Euler’s theorem on homogeneous functions) Let U ⊂ Rn be an open set and f :U → $R^{n}$ be a continuously differentiable function. Then f is homogeneous of degree τ if and only if it satisfies the following equation:

$\tau f\left( u\right) = \sum ^{n}_{i=1}u_{i}\dfrac{\partial f\left(u\right)}{\partial u_{i}}$

$u=\left( u_{1},\ldots u_{n}\right) \in U, h>0$

rupert
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  • Euler was not around in the 1960s, unless maybe if you spent your time in HappyHippieHosen-land =) – mathreadler Sep 11 '17 at 08:16
  • Use of a search engine reveals that Euler wrote a paper General investigations on the mortality and the multiplication of the human race at the end of which he investigates life insurance strategies. Documents here – Gribouillis Sep 11 '17 at 08:25
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    Erm... I'm sorry to inconvenience you, but would you mind telling us what you are talking about? Thank you very much, indeed! –  Sep 11 '17 at 08:26
  • Apologies. This is one of Dirk's articles on this allocation principle: https://arxiv.org/pdf/0708.2542.pdf "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle Dirk Tasche" On page 5 it refers: "Denault (2001) derived the Euler principle by game-theoretic considerations". I admit Euler has so many principles that I need to be specific. – rupert Sep 11 '17 at 11:34
  • 2 Theory The Euler allocation principle may be applied to any risk measure that is homogeneous of degree 1 in the sense of Definition A.1 (see Appendix A) and differentiable in an appropriate sense. ... Theorem A.1 (Euler’s theorem on homogeneous functions) Let U ⊂ Rn be an open set and f :U → R be a continuously differentiable function. Then f is homogeneous of degree τ if and only if it satisfies the following equation: see Appendix A1 to Dirk's paper – rupert Sep 11 '17 at 11:46
  • Thanks Gribouillis for the link. I was hoping for a interpretation and ideally proof though but have looked at the archive and will hope to return to it. – rupert Sep 11 '17 at 11:49

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