Maintaining liquid asset portfolios involves a high carry cost and is mandatory by law for most financial institutions. Taking this into account a financial institution's aim is to manage a liquid asset portfolio in an “optimal” way, such that it keeps the minimum required liquid assets to comply with regulations. In this paper we propose a multi-stage dynamic stochastic programming model for liquid asset portfolio management. The model allows for portfolio rebalancing decisions over a multi-period...

Let ${𝔽}_{}^{}$F be a filtration andτbe a random time. Let ${𝔾}_{}^{}$G be the progressive enlargement of ${𝔽}_{}^{}$F withτ. We study the following formula, called the optional splitting formula: For any ${𝔾}_{}^{}$G-optional processY, there exists an ${𝔽}_{}^{}$F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being $ℬ[0,\infty ]\otimes 𝒪\left({𝔽}_{}^{}\right)$ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y=Y^{\text{'}}{1}_{[0,\tau )}^{}+Y^{\text{'}\text{'}}\left(\tau \right){\mathbb{1}}_{[\tau ,\infty )}^{}.$Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,τk). We are interested in this formula because of its fundamental role in many...

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are...