Integrating a Piecewise VaR Curve for a Triangular Fuzzy Number

Summary

The document asks how to derive average value at risk, also called conditional value at risk, from a piecewise value-at-risk curve for a triangular fuzzy number in a credibilistic setting. It gives separate formulas for VaR below and above the midpoint probability, then defines AVaR as the average of VaR over the interval from zero to the selected confidence level.

The author reports an extra term in the upper-interval expression and asks how it arises. The post contains no answer or worked derivation, so it does not establish whether the displayed AVaR expression is correct. Its useful content is the setup: integrating a piecewise VaR function requires accounting for the interval boundary when the integration range crosses the midpoint. Any application would need to verify the convention for VaR and the algebra under the chosen credibilistic definitions.

Key ideas

• The example defines VaR piecewise for a triangular fuzzy number.
• AVaR is framed as the average of VaR over probabilities up to the selected level.
• When the integration range crosses the midpoint, the integral must be split across the two VaR branches.
• The document poses an algebra question but does not provide a derivation or confirmed result.

Tags

# how to find CVaR/AVaR for triangular fuzzy no


# how to find CVaR/AVaR for triangular fuzzy no












While going through different methods of risk measure i came across AVaR/CVaR, while i was calculating AVaR/CVaR in credibilistic environment using VaR, i got stuck in the calculations

eg. For triangular fuzzy numbers $A =(a1,a2,a3)$ $$VaR(\alpha)= 2*(a1-a2)*\alpha -a1;$$ $$\alpha \leq 0.5$$ $$ = 2*(a2-a3)*\alpha +a3-2*a2; $$$$\alpha >0.5$$

Now, $$ AVaR(\alpha) = (1/\alpha) \int_0^\alpha Var(\beta)d\beta$$

which somehow results in $$AVaR(\alpha)= (a1-a2)*\alpha -a1;$$$$\alpha \leq 0.5$$ $$= (a2-a3)*\alpha +a3-2*a2-1/4*\alpha(a1-2*a2+a3)$$$$\alpha >0.5$$

Plz help me wd d term $$-1/4*\alpha(a1-2*a2+a3).$$ How to get it?

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