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Root number
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423901 |
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Semester
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FS2023 |
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Type of course
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Lecture |
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Allocation to subject
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Statistics |
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Type of exam
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not defined |
| Title |
Risk measures |
| Description |
- Simple examples of random variables and their risks. Why are is
the expectation and the variance not adequate risk measures?
- Value-at-Risk, quantiles, main properties (with
proofs). Subadditivity of VaR for normal random vectors (proof),
violation of the subadditivity property in general.
- General monetary risk measures, acceptance sets, recovering the
risk measure from the acceptance set (proof).
- Coherent risk measures. Acceptance sets of coherent risk
measures (proof).
- Properties of risk measures defined on a finite probability space.
- Lower semicontinuity property of risk measures
(formulation). Definition of sigma(Lp,Lq)-topology.
- Representation of sigma(Lp,Lq)-lower semicontinuous
coherent risk measures (with the proof based on the bipolar
theorem).
- Convex risk measures. Representation of convex risk measures
(without proof). Penalty function. Entropic risk measure and its
penalty function (proof).
- Average Value-at-Risk, its special cases for alpha=0 and
alpha=1. Its expression as an expectation (proof).
- Representation of AVaR using supremum of
E mu (-X) (proof). Relations of AVaR to WCE and ES (proof).
- Law invariance, representation of convex law invariant risk
measures on Lp using the quantile functions (with proof using the
Hardy-Littlewood inequality) and on L infinity
using measures on (0,1] (proof).
- The minimal property of AVaR (without proof).
- Construction of risk measures using moments and loss functions.
- Distorted probability and the corresponding risk measure
(without proof).
- Non-additive measure, 2-alternating probabilities,
concavity of the corresponding distortion functions (proof that the
concavity of psi implies 2-alterantion).
- The Choquet integral and its basic properties, sublinearity
(proof that sublinearity implies 2-alternation).
- Characterisation of convex law invariant risk measures using the
penalised Choquet integral with respect to the distorted probability
(formulation).
- Comonotonic random variables, characterisation.
- Comonotonic additive risk measures, their coherency property,
and characterisation using the Choquet integral and as an integral
of AVaR. |
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ILIAS-Link (Learning resource for course)
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Registrations are transmitted from CTS to (no admission in ILIAS possible).
ILIAS
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Link to another web site
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| Lecturers |
PD Dr.
Andrii Ilienko, Institute of Mathematical Statistics and Actuarial Science ✉
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ECTS
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4 |
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Recognition as optional course possible
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Yes |
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Grading
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1 to 6 |
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| Dates |
Friday 10:15-13:00 Weekly
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Friday 30/6/2023 10:00-11:30
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| Rooms |
Hörraum B078, Exakte Wissenschaften, ExWi
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| Students please consult the detailed view for complete information on dates, rooms and planned podcasts. |
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Öffentlicher Bereich
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