1. C. Bernard, C. Y. Liu, N. MacGillivray and J. Zhang, "Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence." 2013, Dependence Modelling, 1, 37-53.
During the spring term of 2012, I worked closely with Dr. Bernard to expand previous work done by Nelson (2004) and Tankov (2011) on finding bounds for bivariate distribution functions with constraints on given quartiles. In particular, we were able to provide weaker conditions (partial information on dependence) to ensure both the Upper and Lower Bounds of these functions are Copulas.
We then developed an application of our theorems to industry, which has a direct impact in reinsurance, bivariate derivative pricing, and multivariate risk modelling. For example, reinsurers who carry life and automobile business typically have accurate descriptions of the marginal distributions of both businesses but have limited information on their correlation (i.e. the joint distribution). This is especially true under regime-switching conditions when we migrate from a stable market (where the copula and dependence are accurately quantified) to catastrophic market conditions (e.g., depressions, recessions) when the dependence and copula are unknown. We provide both theoretical developments as well as numerical analysis for this application.
2. N. MacGillivray, "Valuation of variable annuities with target volatility." Research note with Dr. David Saunders and Dr. Mary Hardy.
Most recently, I worked with Dr. Mary Hardy and Dr. Saunders on evaluating variable annuities with target volatility. Although there has been considerable research done already on variable annuities, little has been done academically to assess the impact of target volatility. We worked to assess whether insurers are charging a fair price for these contracts in light of the benefits they are providing; in particular, we focused on annuities with a Guaranteed Lifetime Withdrawal feature that allows policyholders to withdraw a fixed percentage of their account value at retirement, valid for life. With the added benefit of target volatility rebalancing, the insurer charges a sliding fee based on the level of observed volatility. We made several simplifying assumptions in our model for ease of computation: rebalancing between a risky and riskless asset, fixed target portfolio volatilities, and a constant interest rate.
The principle purpose of this research was to quantify the impact of rebalancing based on volatility. To model this impact, we chose to represent the risky asset with a Heston model making both the asset and volatility stochastic. We developed a Matlab program to calibrate the Heston model to current market conditions and then wrote code to model and price the annuity using two-staged Monte Carlo simulations. First, we simulated our model from the valuation date until retirement of the policyholder and then computed 500 simulations from retirement until death and repeated this entire process 5,000 times. We completed these simulations across 80 different withdrawal guarantees (ranging from 0% to 8% per annum) and also provided 95% Confidence Interval bounds. The conclusion of our investigation was that the "fair" guarantee withdrawal rate was between 6.01% and 6.84%, compared to observed industry rates offered between 4% and 5%, meaning that insurers were overcharging for contracts with a target volatility feature.