Advanced Mathematics Applied to Insurance: Risk Control for Long-Maturity Structured Products

In the last decade, insurance companies have widely developed long-maturity structured products to offer more attractive long-term investment supports to their clients. Also known as market-linked products, these pre-packaged investments can be roughly defined as savings or investment products where the return is linked to one or more underlying assets such as stocks and bonds. Risk management for these products is a challenging issue for the financial and insurance industry. Two characteristics in particular render classical approaches ill-adapted: their long-maturity feature and the huge notionals (total value of a trade) in assets involved, — a large notional induces an important market impact. “As a result, risk control for these products yields to questioning the procedures used for standard short-term product with moderate notional”, explains Bruno Bouchard, a professor of Mathematics and Finance at Université Paris Dauphine (France). “For the structured products, the classical linear pricing and hedging rules that are usually used are obsolete for a proper risk management”. In view of this fact, Bruno Bouchard and a team of collaborators, also working at the Probability, Statistics and Modeling Lab (LPSM) of Université de Paris, have undertaken a collaborative project with practitioners from AXA Risk Management aimed to “develop and improve models that can better take into account the major market imperfections” at play with structured products.  

“With such products, insurance companies are highly exposed, explains Jean-François Chassagneux. On one hand, their maturity is very long. The product’s lifespan can reach 30 years, and a lot can happen on the market over such long periods. It’s very hard to predict, and therefore to calibrate the model. For short-term products, we would use already matured products as benchmark, but there are no such long-term equivalents. On the other hand, these structured products have been sold to many individuals, so the notional values are huge. There is a lot of money involved”. Because they have an important risk exposure to market factors (especially equity, interest rates and volatility) in the long-term, insurance companies hold huge notional in assets to cover these liabilities and can represent important volumes of specific markets. This fact makes them strongly sensitive to market frictions and changes in market’s parameters. The present JRI aims at taking all the uncertainty into consideration and create mathematical models that can accommodate it. “The math is going to be very complicated, but our goal is to make it as simple as possible”, says Prof. Chassagneux. “Solutions to such models are usually represented via mathematical tools which are highly non-linear. This means that the risk you are taking when you manage a thousand of these products is not at all equivalent to a thousand times the risk you take when you manage one”.  

In addition to developing the non-linear mathematical models that will more accurately predict the risks associated with long-maturity structured products, the academic and practitioner teams will also put emphasis on putting them into practice. “We are not just solving the math here, we are also developing numerical methods to accommodate these complicated theories, and to make them applicable in the real world”. In both respect, collaboration between the academic and industry teams will be a major asset, as prof. Bouchard points out: “On the market modeling aspect, the AXA team has a precise knowledge of the current methodologies, the challenging issues as well as the market constraints (like the Solvency 2 rules — a set of regulatory requirements in European law that codifies the insurance regulation). In addition, their experience will permit to identify the major market imperfection factors and the way to quantify them. On the numerical aspect, their implementation experience will help to identify the main difficulties and provide intuition as well as new ideas in order to design efficient numerical algorithms.” Once the development stage is over, AXA Risk Management teams will use the output methodologies and apply them in the context of the Variable Annuity business, a type of long-maturity structured product that pays out a fixed stream of payments to an individual.  “They have access to market data, real VA Liabilities and also historical profit and loss statement associated to the strategies put in place to minimize the risk of adverse movements in the valued of the assets (hedging strategy). This will also be crucial to implement and test the new developed models”. 

In essence, the idea behind the present JRI is to build on advanced mathematic methods, namely non-linear models, to solve a very concrete financial issue: that of better managing the risks posed by structured products. The 2008 financial crisis exposed weaknesses in the risk management practices of such highly complex financial investments. By pooling academic and practitioner efforts, the project will undoubtedly make considerable progress towards better risk management of these products.