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We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSM), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSM admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined. Funding: M. Ghossoub and D. Saunders acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada in the form of Discovery Grants [Grants 2018-03961 and 2017-04220, respectively].

In this paper, we discuss optimal design for stylized Intergenerational Risk Sharing (IRS) pension plans. We study an IRS plan under which both contributions and pension benefits are adjusted based on the level of pension assets. Our optimization focuses on the stability of members’ lifetime consumption, both in the contribution and benefit phases, through formulating the optimization as an ergodic control problem. We illustrate the drawbacks of unconstrained optimization and demonstrate the importance of including regulatory requirements for the sake of fairness across generations. In this formulation, the employers are not included; implicitly, it is assumed that employer contributions would be paid as salary to the workers if not required for the plan, so the employer risk is eliminated.

This paper provides an in-depth investigation of Epstein-Zin preferences under an uncertain lifetime. These models have been gaining popularity as normative preference models in life-cycle applications. Though extensively used, it is unclear if the preference models are well-understood by the users. To this end, this paper studies the decision-making under Epstein-Zin preferences. Specifically, analytical solutions for a simple consumption and savings problem are derived, isolating the impact of relative risk aversion (RRA), elasticity of intertemporal substitution (EIS), time discounting, and risks stemming from mortality, investment, and inflation. We investigate three Epstein-Zin preference models that differ in their treatment of mortality risk, and find that some lead to normatively implausible solutions. Importantly, we find that the EIS is not always monotone in its effect on consumption volatility over time, meaning that its interpretation can be ambiguous when considering an uncertain future lifetime. This has been misinterpreted in the literature to date. We also show that one particular Epstein-Zin specification is not necessarily a generalization of expected discounted utility maximization in the case of constant relative risk aversion, as many works wrongly claim.

In this paper we consider three types of embedded options in pension benefit design. The first is the Florida second election (FSE) option, which has been offered to public employees in the state of Florida since 2002. The state runs both defined contribution (DC) and defined benefit (DB) pension plans. Employees who initially join the DC plan have the option to convert to the (DB) plan at a time of their choosing. The cost of the switch is assessed in terms of the ABO (Accrued Benefit Obligation), which is the expected present value of the accrued DB pension at the time of the switch. If the ABO is greater than the DC account, the employee is required to fund the difference. The second is the DB Underpin option, also known as a floor offset plan, under which the employee participates in a DC plan, but with a guaranteed minimum benefit based on a traditional DB formula. The third option can be considered a variation on each of the first two. We remove the requirement from the FSE option for employees to fund any shortfall at the switching date. The resulting plan is similar to the DB underpin, but with the possibility of early exercise. We adopt an arbitrage-free pricing methodology to value each option. We analyse and value the optimal switching strategy for the employee by constructing an exercise frontier, and we illustrate numerically the difference between the FSE, DB Underpin and Early Exercise DB Underpin options.

The Wilkie economic scenario generator has had a significant influence on economic scenario generators (ESGs) since the first formal publication in Wilkie (1986). In this paper we update the model parameters using U.S.data to 2014, and review the model performance. In particular, we consider stationarity assumptions, parameter stability, and structural breaks.

Scenarios are the language of Risk. While scenario analysis and stress testing have been an explicit part of risk management methodologies and systems for over two decades, the typical scenario and stress testing tools haven’t evolved much and are still generally quite static and largely subjective. In this paper, we present a simple and powerful approach to create meaningful stress scenarios for risk management and investment analysis of multi-asset portfolios, which effectively combines economic forecasts and “expert” views with portfolio simulation methods.Expert scenarios are typically described in terms of a small number of key economic variables or factors. However, when applied to a portfolio, they are incomplete – they generally do not describe what occurs to all relevant market risk factors that affect the portfolio. We need to understand how these market risk factors behave, conditional on the outcome of the economic factors. The key insight to our approach is that the conditional expectation, or more generally the full conditional distribution of all the factors, and of the portfolio P&L, can be estimated directly from a pre-computed simulation using Least Squares Regressions (LSR). All the conditional scenario analytics can be derived from the regression results. We refer to this approach as Least Squares Stress Testing (LSST). LSST is a simulation-based conditional scenario generation method that offers many advantages over more traditional analytical methods. Simulation techniques are simple, flexible, and provide very transparent results, which are auditable and easy to explain. LSST can be applied to a large number of risk factors, which can follow completely general joint stochastic processes, with fat-tails, non-parametric and general codependence structures, autocorrelation, etc. LSST further produces explicit risk factor P&L contributions.We illustrate the application of the methodology through the analysis of the performance of a real-life portfolio under recent economic research reports.