Assembling Individual ILS into an Optimal Portfolio

Abstract

Much has been written about individual insurance-linked securities (ILS), a.k.a. Cat Bonds, in the 20 years of their existence. This includes pricing, investor returns, liquidity and their advantages/disadvantages to investors—they provide good returns and excellent diversification. Much less has been written about how individual ILS should be assembled into an optimal portfolio. We seek to address that question—how do you underwrite (i.e., assemble) a portfolio of ILS that maximizes return and does not take excessive risk. Traditional reinsurance underwriters used to contain the excessive risk question by a “silo-ed” approach to selecting different risks. Nowadays, more sophisticated approaches are available. In the capital market modern portfolio theory (MPT) or mean-variance optimization (MVO) is used to select high return assets that are not highly correlated so as to have a portfolio of low variation. This requires knowledge of correlation and covariance of assets as well as an assumption of log normality of return. In insurance and ILS in particular, such covariance measure are not forthcoming and, more important, returns are not normal—but long tailed. Instead, in this chapter, we use a scenario approach and stochastic optimization in which the co-movement of outcomes is captured implicitly via each scenario, rather than explicitly via correlation coefficients. The portfolio is optimized over all scenarios, subject to pre-specified risk constraints.

All models are wrong; some are useful.

George E. P. Box.

Appendices

Appendix 1: The Price Indication Sheets

Appendix 2: A (Incomplete) List of Model Assumptions

• Initial portfolio is assumed to be cash of $500 million.

• It is assumed that there are no initial holdings. Subsequent runs of the portfolio could start with an initial portfolio, then allowing transaction costs to be taken into account.

• It is assumed that the AIR risk profile for each bond as of its issue or remodelling is still relevant as of evaluation date. Sixteen bonds are eliminated from the outstandings that do not satisfy that criteria.

• Prices are mid-market prices from a dealer Price Indication sheet as of 9/30/2014 (as augmented by LFC when prices on that sheet were not available).

• Risk profiles and scenario sets are all from the AIR remodelled data.

• The AIR data are for the Standard Scenario set often referred to as the SSST set, not the WSST.

• Deals are acquired or valued at secondary market prices.

• AIR losses of principal characterizing the effect of each scenario are applied to the “market value” of the deal, rather than the original issue principal, since that is the true value of loss.

• It is assumed that the horizon under examination is one year. Deals maturing in under one year are assumed to be replaced in kind; deals maturing in over one year are assumed to be consistent with current discount margins or yields less scenario loss.

• It is assumed the investor can acquire up to 10% of the market value of an issue. That parameter may change from deal to deal or institution to institution.

Appendix 3: Selected Related Publications (Reverse Chronological Order)

• Morton Lane, “Using Stochastic Optimization to Underwrite a Portfolio of Insurance Linked Securities”, available at http://www.lanefinacialllc.com, 2016

• Morton Lane and Jerome Kreuser, “ILS Market-Derived Metrics: Implications for Risk Adjustment Transforms and Capital Allocation”, Alternative (Re)insurance Strategies, (London: Risk Books, 2012), Chapter 21

• Adolfo Pena “Portfolio Optimization in a Dedicated Hedge Fund”, Alternative (Re)insurance Strategies, (London: Risk Books, 2012), Chapter 19

• Jerome Kreuser and Morton Lane, “Optimal Insurance and Reinsurance Portfolios, Implied Pricing, Allocating Retrocessional Costs and Capital Allocation”, available at http://www.lanefinacialllc.com, (presented at the “Adventures in Risk” conference in Christchurch, New Zealand, September 2007)

• Jerome Kreuser and Morton Lane, “An Introduction to the Benefits of Optimization Models for Underwriting Portfolio Selection”, available at http://www.lanefinancialllc.com, (presented at Proceedings of the 28th International Congress of Actuaries, Paris, June 2006)

• Alexander McNeil, Rüdiger Frey and Paul Embrechts, Quantitative Risk Management: Concepts, Techniques, and Tools, (Princeton, NJ and Oxford, UK: Princeton Series in Finance, 2005)

• Shaun S. Wang, “Cat Bond Pricing Using Probability Transforms”, (Geneva Papers: Etudes et Dossiers on “Insurance and the State of the Art in Cat Bond Pricing”, No. 278) 19–29, January 2004

• Shaun S. Wang, “A Universal Framework for Pricing Financial and Insurance Risks”, ASTIN Bulletin 32(2), 2002, 213–234

• Marco Avellaneda and Peter Laurence, Quantitative Modelling of Derivative Securities, (Boca Raton, FL: Chapman & Hall/CRC, 2000)

• Hans Bühlmann, Freddy Delbaen, Paul Embrechts and Albert N. Shiryaev, “On Esscher Transforms in Discrete Finance Models”, ASTIN Bulletin 28(2), 1998, 171–186

• Morton Lane and Jerome Kreuser, Designing Investment Strategies for Fixed Income Portfolios, The World Bank, eds. A.V. Fiacco and K.O Kortanek, (New York: Springer-Verlag, 1980) 98–134

• Morton Lane, “Interest Rate Futures and Management of Bank Portfolios”, Investment Manager’s Handbook, ed. S. N. Levine, (Homewood, IL: Dow Jones-Irwin, 1980) 777–788

• Morton Lane and P. Hutchinson, “A Model for Managing a Certificate of Deposit Portfolio under Uncertainty”, Stochastic Programming, ed. M.A.H. Dempster (London: Academic Press, 1980) 473–493

• Morton Lane and Steven Littlechild, “A Stochastic Programming Approach to Weather-Dependent Pricing for Water Resources”, Stochastic Programming, ed. M.A.H. Dempster, (London: Academic Press, 1980) 357–368

• Morton Lane and Steven Littlechild, “Weather Dependent Pricing for Water Resources in the Texas High Plains”, Water Resource Research 12, 1976, 599–605

• Morton Lane, “Short-Term Management for Bank Portfolios”, Journal of Bank Research 5, 1974, 102–119

• Morton Lane, “A Chance Constrained Model for Water Reservoir Control”, Water Resource Research 92, 1973, 937–938

• Kenneth Arrow and Gerard Debreu, 1954, “Existence of an Equilibrium for a Competitive Economy”, Econometrica 22(3), July 1954, 265–290.