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 numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing 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 L p 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.
We model a Systemically Important Financial Institution (SIFI) that is too big (or too interconnected) to fail. Without credible regulation and strong supervision, the shareholders of this institution might deliberately let its managers take excessive risk. We propose a solution to this problem, showing how insurance against systemic shocks can be provided without generating moral hazard. The solution involves levying a systemic tax needed to cover the costs of future crises and more importantly establishing a Systemic Risk Authority endowed with special resolution powers, including the control of bankers' compensation packages during crisis periods.
The pricing kernel is an important link between economics and finance. In standard models of financial economics, it is proportional to the aggregate marginal utility in the economy. We first show that none of the three standard assumptions (completeness, risk aversion, and correct beliefs) is needed for the pricing kernel to be generally decreasing. If at least one of the three assumptions is violated, the pricing kernel can have increasing parts. We explain the economic principles that lead to an increasing part in the pricing kernel and compare the resulting pricing kernels with the empirical pricing kernel estimated in Jackwerth (2000, Rev. Financ. Stud., 13, 433–451).
