Ruilin Tian

Mortality Portfolio Risk Management

We provide a new method, the “MV+CVaR approach,” for managing unexpected mortality changes underlying annuities and life insurance. The MV+CVaR approach optimizes the mean–variance trade‐off of an insurers mortality portfolio, subject to constraints on downside risk. We apply the method of moments and the maximum entropy method to analyze the efficiency of MV+CVaR mortality portfolios relative to traditional Markowitz mean–variance portfolios. Our numerical examples illustrate the superiority of the MV+CVaR approach in mortality risk management and shed new light on natural hedging effects of annuities and life insurance.

Managing Capital Market and Longevity Risks in a Defined Benefit Pension Plan

This paper proposes a model for a defined benefit pension plan to minimize total funding variation while controlling expected total pension cost and funding downside risk throughout the life of a pension cohort. With this setup, we first investigate the plan’s optimal contribution and asset allocation strategies, given the projection of stochastic asset returns and random mortality evolutions. To manage longevity risk, the plan can use either the ground-up hedging strategy or the excess-risk hedging strategy. Our numerical examples demonstrate that the plan transfers more unexpected longevity risk with the excess-risk strategy due to its lower total hedge cost and more attractive structure.

Downside Risk Management of a Defined Benefit Plan Considering Longevity Basis Risk

To control downside risk of a defined benefit pension plan arising from unexpected mortality improvements and severe market turbulence, this article proposes an optimization model by imposing two conditional value at risk constraints to control tail risks of pension funding status and total pension costs. With this setup, we further examine two longevity risk hedging strategies subject to basis risk. While the existing literature suggests that the excess-risk hedging strategy is more attractive than the ground-up hedging strategy as the latter is more capital intensive and expensive, our numerical examples show that the excess-risk hedging strategy is much more vulnerable to longevity basis risk, which limits its applications for pension longevity risk management. Hence, our findings provide important insight on the effect of basis risk on longevity hedging strategies.

Portfolio Risk Management with CVaR-Like Constraints

Abstract A current research stream in the portfolio allocation literature develops models that take into account the asymmetric nature of asset return distributions. Our paper contributes to this research stream by extending the Krokhmal, Palmquist, and Uryasev approach. We add CVaR-like constraints in the traditional portfolio optimization problem to reshape the tails of the portfolio return distribution while not significantly affecting its mean and variance. We illustrate how to apply this approach, called the “MV + CVaR approach,” to manage tail risk of an insurer’s asset-liability portfolio. Finally, we compare the MV + CVaR approach with the traditional Markowitz method and a method recently introduced by Boyle and Ding. Our numerical analysis provides empirical support for the effectiveness of the MV + CVaR approach in controlling downside risk. Moreover, we find that the MV + CVaR approach may improve skewness of mean-variance portfolios, especially for high-variance portfolios.

Pension Risk Management in the Enterprise Risk Management Framework

This paper presents an enterprise risk management (ERM) model for a firm that is composed of a portfolio of capital investment projects and a defined benefit (DB) plan for its workforce. The firm faces the project, operational and hazard risks from its investment projects as well as the financial and longevity risks from its DB plan. The firm maximizes its capital market value net of pension contributions subject to constraints that control project, operational, hazard, financial and longevity risks as well as an overall risk. The analysis illustrates the importance of integrating pension risk into the firm’s ERM program by comparing firm value with and without integrating pension risk with other risks in an ERM program. We also show how pension hedging strategies can impact the firm’s net value under the ERM framework. While the existing literature suggests that a longevity swap is less expensive than a pension buy-out because the latter is more capital intensive, this analysis shows that the buy-out is more effective in increasing firm value.

Moment Problem and Its Applications to Risk Assessment

ABSTRACT This article discusses how to assess risk by computing the best upper and lower bounds on the expected value E[φ(X)], subject to the constraints E[Xi] = μi for i = 0, 1, 2, …, n. φ(x) can take the form of the indicator function in which the bounds on are calculated and the form φ(x) = (ϕ(x) − K)+ in which the bounds on financial payments are found. We solve the moment bounds on through three methods: the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the “point” estimate provided by the Johnson system of distributions. In addition, we propose a simpler formulation for the unimodal bounds on compared to the existing formulations in the literature. For those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of X. The more volatile the random variable X is, the looser the numerical bounds are, compared to their closed-form solutions.

Bounds on Tail Probabilities and Value at Risk Given Moment Information

We solve a moment problem to compute the best upper and lower bounds on the expected value E[φ(X)], subject to constraints E[X^i] = μ_i for i = 1, 2,...,n. By setting φ(x)=I_(-\inf,t], the indicator function for the event X ≤ t, we calculate the bounds on Pr(X ≤ t) = E[I_(-\inf,t]]. The bounds can be narrowed if more information about distribution classes is added. Specifically, we show how to find the bounds on a variable with unimodal distribution. In addition, given a set of moments, we present a moment-constrained maximum entropy method that provides “point” estimates on tail probabilities. As robustness check, we investigate how the bounds can be narrowed if more moments are considered. In addition, to check the accuracy and reliability of our numerical bounds, we compare our method with De Schepper and Heijnen (2010) which provides explicit expressions for the bounds. The semiparametric bounds are useful in risk analysis where there is only incomplete information concerning the random variable X, such as an insurance loss or an asset return. We show how the inversion of these bounds leads to approximations to bounds on Value at Risk (VaR). Besides helping to construct a representative distribution with given moments, the moment-constrained maximum-entropy method can be used to define risk neutral probabilities for asset pricing. To illustrate this idea, we present a numerical example.