Monetary utility over coherent risk ratios
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Johannes Leitner
SUMMARY
For a monetary utility functional U and a coherent risk measure ρ, both with compact scenario sets in Lq, we optimize the ratio α(V): = U(V)/ρ(V) over an (arbitrage-free) linear sub-space V⊆Lp, 1 ≤ p ≤ ∞, of attainable returns in an incomplete market model such that ρ > 0 on V \ {0}. If a solution Vˆ ∈ V with α(Vˆ) = α¯ V: = sup V∈Vα(V)∈[0,∞) exists, then the first order optimality condition allows to construct an absolutely continuous martingale measure for V as a convex combination Q¯+α¯VQ/1+α¯V of two probability measures Q¯, Q from the respective scenario sets defining U and ρ. Conversely, if α¯V ∈ [0,∞), then α¯V equals the smallest a∈[0,∞) such that Q¯+aQ/1+a is an absolutely continuous martingale measure for V for some probability measures Q¯, Q from the scenario sets defining U, ρ, and α¯V = ∞ holds iff such a convex combination does not exist.
© R. Oldenbourg Verlag, München
Articles in the same Issue
- Editorial preface
- Risk measurement with equivalent utility principles
- Dilatation monotone and comonotonic additive risk measures represented as Choquet integrals
- On distortion functionals
- Convex risk measures and the dynamics of their penalty functions
- Law invariant convex risk measures for portfolio vectors
- Robust utility maximization in a stochastic factor model
- Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints
- On the optimal risk allocation problem
- Monetary utility over coherent risk ratios
- Mean-risk optimization for index tracking
Articles in the same Issue
- Editorial preface
- Risk measurement with equivalent utility principles
- Dilatation monotone and comonotonic additive risk measures represented as Choquet integrals
- On distortion functionals
- Convex risk measures and the dynamics of their penalty functions
- Law invariant convex risk measures for portfolio vectors
- Robust utility maximization in a stochastic factor model
- Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints
- On the optimal risk allocation problem
- Monetary utility over coherent risk ratios
- Mean-risk optimization for index tracking
