While it is common knowledge that portfolio separation in a continuous-time lognormal market is due to the basic properties of the Gaussian distribution, the usual textbook exposition relies on dynamic programming and thus Itô stochastic calculus and the appropriate regularity conditions. This...

We develop a utility and asset pricing theory that features a novel measure of tail risk. Our model determines investor demand for both left and right-tail risk premia from an indifference curve incorporating tolerance for variance and tail risk. We show that the systematic tail risk factors...

Operational risk is being considered as an important risk component for financial institutions as evinced by the large sums of capital that are allocated to mitigate this risk. Therefore, risl measurement is of paramount concern for the purposes of capital allocation, hedging, and new product...

We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow...

Using equations that arise in quantum mechanics, this paper describes a way to more accurately and efficiently represent non-Gaussian return distributions than the standard method of invoking skewness and kurtosis. Then, it provides a new single intuitive number, defined here as the “crash...

We investigate portfolio diversification strategies based on hierarchical clustering. These hierarchical risk parity strategies use graph theory and unsupervised machine learning to build diversified portfolios by acknowledging the hierarchical structure of the investment universe. In this...

This paper proposes a new method to introduce coherent risk measures for risks with infinite expectation, such as those characterized by some Pareto distributions. Extensions of the conditional value at risk, the weighted conditional value at risk and other examples are given. Actuarial...

Suppose that a group of agents having divergent expectations can share risks efficiently. We examine how this group should behave collectively to manage these risks. We show that the beliefs of the representative agent is in general a function of the group.s wealth level, or equivalently, that...

This paper analyzes optimal risk sharing among agents that are endowed with either expected utility preferences or with dual utility preferences. We find that Pareto optimal risk redistributions and the competitive equilibria can be obtained via bargaining with a hypothetical representative...

This report discusses evolving expectations for board oversight of management's risk appetite and tolerance and the challenges boards face in meeting them. It also recommends steps to implement a board-driven, objective-centric approach to risk governance