Abstract
Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well‐defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black‐Scholes continuous time setting, with time dependent coefficients.
Highlights
The choice of risk measure has a significant effect on portfolio investment decisions
Value at risk VaR is probably the most famous among these measures, having featured heavily in various regulatory frameworks. It can be defined for a random variable X and a confidence level α by VaR X E X − qα, where qα is the α-quantile of X see e.g., 1, Equation 1.2 Another common definition is that the VaR of a loss distribution L is the smallest number xα such that P L > xα α
In this work we investigated continuous time portfolio selection under the notion of conditional capital at risk, within the Black-Scholes asset pricing paradigm, with time dependent coefficients
Summary
The choice of risk measure has a significant effect on portfolio investment decisions. The article motivated a number of authors 5, 8–13 to propose and investigate different types of coherent risk measures, all of which are tail mean-based risk measures One such measure, that does not suffer from the critical shortcomings of VaR and CaR, is conditional capital at risk CCaR. That does not suffer from the critical shortcomings of VaR and CaR, is conditional capital at risk CCaR This is defined in 14 as the difference between the riskless investment and the conditional expected wealth, under the condition that the wealth is smaller than the corresponding quantile, for a given risk level.
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