Optimal solution for immunizing arbitrarily scheduled multiple liabilities
- Authors: Kurochkin S.V.1, Rodina V.A.1
-
Affiliations:
- HSE University
- Issue: Vol 59, No 2 (2023)
- Pages: 87-99
- Section: Articles
- URL: https://ogarev-online.ru/0424-7388/article/view/141418
- DOI: https://doi.org/10.31857/S042473880025861-6
- ID: 141418
Open Access
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Abstract
Immunization, a control tool for interest rate dependent changes in the value of an asset portfolio given a similar dependency for a target liability portfolio, is central to portfolio management. A vast body of academic literature describes various immunization models either for the case of a single liability payout or assuming a specific change in the yield curve, or both. This paper is the first to propose an immunization solution for the case of multiple liability payouts assuming arbitrary changes in the yield curve. For the case of multiple liability payouts, we generalize M-Absolute, which is a risk measure proposed by Nawalkha и Chambers (1996), and estimate the proximity of payment streams with EMD (the Wasserstein distance) which is a well-known tool in machine learning. In line with Fong and Vasicek (1984), it is shown that portfolio’s interest rate risk is constrained to a product of two factors with one factor, EMD between asset and liability streams, being only dependent on the portfolio structure and the other factor, the sup-norm of the function of interest rate shocks, being solely determined by changes in the yield curve. We also show the unimprovability of the estimate and obtain, in an explicit form, a computational procedure for the optimal immunizing portfolio. The results are practically applicable as exemplified by the immunization of an annuity-type security with a portfolio of government bonds.
About the authors
Sergey Vladimirovich Kurochkin
HSE UniversityMoscow, Russia
Victoria Alekseevna Rodina
HSE UniversityRussia
References
- Бешенов С.В, Лапшин В.А. (2019). Параметрическая иммунизация процентного риска на основе моделей срочной структуры процентных ставок // Экономический журнал ВШЭ. Т. 23 (1). С. 9–31.
- Богачев В.И., Колесников А.В. (2012). Задача Монжа–Канторовича: достижения, связи и перспективы // Успехи математических наук. Т. 67. Вып. 5 (407). С. 3–110.
- Валландер С.С. (1973). Вычисление расстояния по Вассерштейну между распределениями вероятностей на прямой // Теория вероятностей и ее применения. Т. 18. Вып. 4. С. 824–827.
- Balbas A., Ibanez A. (1998). When can you immunize a bond portfolio? Journal of Banking and Finance, 22, 1571–1595.
- Balbas A., Ibanez A., Lopez S. (2002). Dispersion measures as immunization risk measures. Journal of Banking and Finance, 26 (6), 1229–1244.
- Bayliss C., Serra M., Nieto A., Juan A. (2020). Combining a matheuristic with simulation for risk management of stochastic assets and liabilities. Risks 8 (4), 131.
- Bierwag G. (1977). Immunization, duration, and the term structure of interest rates. Journal of Financial and Quantitative Analysis, 12 (5), 725–742.
- Bierwag G., Fooladi I., Roberts G. (1993). Designing an immunized portfolio: Is M-squared the key? Journal of Banking and Finance, 17, 1147–1170.
- Bierwag G., Kaufman G., Toevs A. (1983). Immunization strategies for funding multiple liabilities. Journal of Financial and Quantitative Analysis, 18 (1), 113–123.
- Chizat L. (2018). Unbalanced optimal transport: Dynamic and Kantorovich formulations. Journal of Functional Analysis, 274 (11), 3090–3123.
- De La Peña J.I., Iturricastillo I., Moreno R., Roman F., & Trigo E. (2021). Towards an immu-nization perfect model? International Journal of Finance & Economics, 26 (1), 1181–1196.
- Dutta G., Rao H., Basu S., Tiwari M. (2019). Asset liability management model with decision support system for life insurance companies: Computational results. Computers & Industrial Engineering, 128, 985–98.
- Fabozzi F.J., Fong H.G. (1985). Fixed income portfolio management. Appendix E: Derivation of risk immunization measures. Homewood Illinois: Dow Jones-Irwin.
- Fisher L., Weil R. (1971). Coping with the risk of interest rate fluctuations: Returns to bond-holders from naïve and optimal strategies. Journal of Business, 44 (4), 408–431.
- Fong G., Vasicek O. (1984). A risk minimizing strategy for portfolio immunization. Journal of Finance, 39 (5), 1541–1546.
- Ford P. (1991). Some Further Investigations into Cashflow Matching. AFIR Colloquium, Rome, Italy, 539–551.
- Ford P.E.B. (1991). Cashflow matching using modified linear programming. AFIR Colloquium, Brighton, United Kingdom, 3, 301–322.
- Gangbo W., Li W., Osher S., Puthawala M. (2019). Unnormalized Optimal transport. Journal of Computational Physics, 399, 108940.
- Hürlimann W. (2002). On immunization, stop-loss order and the maximum shiu measure. Insur-ance: Mathematics and Economics, 31, 315–325.
- Ingersoll J.Jr., Skelton J., Weil W. (1978). Duration forty years later. Journal of Financial and Quantitative Analysis, 13 (4), 627–650.
- Khang C. (1979). Bond immunization when short-term interest rates fluctuate more than long-term rates. Journal of Financial and Quantitative Analysis, 14 (5), 1085–1090.
- Kopa M., Rusý T. (2021). A decision-dependent randomness stochastic program for asset-liability management model with a pricing decision. Annals of Operations Research, 299, 241–271.
- Leibowitz M. (1986). The dedicated bond portfolio in pension funds – Part I: Motivations and ba-sics. Financial Analysts Journal, 42 (1), 68–75.
- Monge G. (1781). Mémoire sur la théorie des déblais et des remblais. Paris : De l'Imprimerie Royale.
- Montrucchio M., Peccati L. (1991). A note on shiu-fisher-weil immunization theorem. Insurance: Mathematics and Economics, 10, 125–131.
- Nawalkha S., Chambers D. (1996). An improved immunization strategy: M-absolute. Financial Analysts Journal, 52 (5), 69–76.
- Nawalkha S., Chambers D. (1997). The M-vector model: Derivation and testing of extensions to M-square. Journal of Portfolio Management, 23 (2), 92–98.
- Nawalkha S., Soto G., Zhang J. (2003). Generalized M-vector models for hedging interest rate risk. Journal of Banking and Finance, 27 (8), 1581–1604.
- Panaretos V., Zemel Y. (2019). Statistical aspects of wasserstein distances. Annual Review of Statistics and Its Application, 6, 405–431.
- Redington F. (1952). Review of the principles of life-office valuations. Journal of the Institute of Actuaries, 78 (3), 286–340.
- Rosenbloom E., Shiu E. (1990). The matching of assets with liabilities by goal programming. Managerial Finance, 16 (1), 23–26.
- Shiu E. (1987). On the Fisher–Weil immunization theorem. Insurance: Mathematics and Economics, 6, 259–266.
- Shiu E. (1990). On Redington’s theory of immunization. Insurance: Mathematics and Economics, 9, 171–175.
- Theobald M., Yallup P. (2009). Liability-driven investment: Multiple Liabilities and the question of the number of moments. European Journal of Finance, 16 (5), 413–435.
- Torres L., Pereira L., Amini H. (2021). A survey on optimal transport for machine learning: Theory and applications. arXiv: 2106.01963. doi: 10.48550/arXiv.2106.01963
- Van der Meer R., Smink M. (1993). Strategies and techniques for asset-liability management: An overview. Geneva Papers on Risk and Insurance, s and Practice, 18 (67), 144–157.
- Vanderhoof I. (1972). The interest rate assumption and the maturity structure of the assets of a life insurance company. Transactions of Society of Actuaries, 24 (69), 157–192.
- Weil R. (1973). Macaulay's duration: An appreciation. Journal of Business, 46 (4), 589–592.
