2012/06/30 by Walter Farkas, Pablo Koch-Medina, Cosimo Munari · 50 citations
Computer Science · Decision Sciences · Economics, Econometrics and Finance · #Asset (computer security) #Capital asset pricing model #Cash #Collateral #Discounting #Mathematical finance #Optimization and Variational Analysis #Risk and Portfolio Optimization #Stochastic processes and financial applications #Subadditivity #Systematic risk #msc:06F30 #msc:46A55 #msc:46B40 #msc:46B42 #msc:91B30 #q-fin.RM
paper · pdf · doi:10.1007/s00780-013-0220-9
published in Finance and Stochastics 18(1), 145-173 (Springer Science and Business Media LLC)
openalex publication_date 2013/11/28 · crossref created 2013/11/28 · crossref issued 2013/11/29 · crossref published 2013/11/29 · crossref published-online 2013/11/29 · crossref published-print 2014/01/01 · arxiv created 2014/02/04 · arxiv updated 2014/02/05 · crossref deposited 2019/08/04 · openalex created_date 2020/11/23 · crossref indexed 2026/08/02 · openalex updated_date 2026/08/05
We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numeraire. However, discounting does not work in all financially relevant situations, typically when the eligible asset is a defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Tail Value-at-Risk on Lp spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.