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Valuation and Hedging of Limited Price Indexed Liabilities

Published online by Cambridge University Press:  10 June 2011

H.-C. Huang  and
A. J. G. Cairns
H.-C. Huang
Affiliation:
Department of Risk Management and Insurance, National Chengchi University, Taipei, Taiwan 116, ROC.
A. J. G. Cairns
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, U.K.
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Abstract

This paper considers the market or economic valuation and the hedging of Limited Price Indexed (LPI) liabilities. This involves finding optimal static and dynamic hedging strategies which minimise the riskiness of the investment portfolio relative to the liability.

In this paper we do not aim to find the perfect hedge in a perfect world. Instead, it is assumed that optimisation is restricted to three commonly used asset classes in pension funds: cash; long-term (or irredeemable) fixed-interest bonds; and long-dated index-linked bonds. The economic value of the liability is then defined as the value of the best matching portfolio using a mean/variance type of loss function. Specifically, we adopt the risk minimising approach of Föllmer & Sondermann (1986) and Schweizer & Föllmer (1988). Even with such a simple loss function, establishing the theoretically optimal solution can be difficult. We propose that a practical solution close to the theoretical optimum can be found using two approximations. First, we approximate the ‘true’ stochastic economic model by a vector autoregressive model of order one. Second, we use a sequence of linearisations to approximate non-linear by straightforward quadratic minimisation problems.

The proposed approach is illustrated with various numerical examples, and we compare the results of the approximately optimal hedging strategy with static strategies.

Keywords

LPI LiabilityStatic HedgingRegular RebalancingDynamic HedgingRisk Minimising Hedging
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 10 , Issue 3 , 01 August 2004 , pp. 627 - 663
Copyright
Copyright © Institute and Faculty of Actuaries 2004

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References

Black, F. & Scholes, M.S. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.CrossRefGoogle Scholar
Cairns, A.J.G. (1999). An analysis of the level of security provided by the minimum funding requirement. British Actuarial Journal, 5, 585–610.CrossRefGoogle Scholar
Cairns, A.J.G. (2000). Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time. ASTIN Bulletin 30, 19–55.CrossRefGoogle Scholar
Cairns, A.J.G. (2001). From financial economics to fair valuation. Proceedings of the 11th International AFIR Colloquium, Toronto, September 2001, 1, 135–166.Google Scholar
Cairns, A.J.G. & Parker, G. (1997). Stochastic pension fund modelling. Insurance: Mathematics and Economics, 21, 43–79.Google Scholar
Clarkson, R.S. (1995). Asset-liability modelling and the downside approach to risk. Transactions of the 25th International Congress of Actuaries, Brussels, 3, 1125–1136.Google Scholar
Dufresne, D. (1988). Moments of pension contributions and fund levels when rates of return are random. Journal of the Institute of Actuaries, 115, 535–544.CrossRefGoogle Scholar
Dufresne, D. (1989). Stability of pension systems when rates of return are random. Insurance: Mathematics and Economics, 8, 71–76.Google Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scandinavian Actuarial Journal, 1990, 39–79.CrossRefGoogle Scholar
Follmer, H. & Leukert, P. (1999). Quantile hedging. Finance and Stochastics, 3, 251–273.CrossRefGoogle Scholar
Follmer, H. & Leukert, P. (2000). Efficient hedging: cost versus shortfall risk. Finance and Stochastics, 4, 117–146.Google Scholar
Follmer, H. & Sondermann, D. (1986). Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics in Honor of Gerard Debreu, 205–233 (Hildenbrand, W., Mas-Collel, A., eds.) North Holland, Amsterdam.Google Scholar
Haberman, S. & Sung, J.-H. (1994). Dynamic approaches to pension funding. Insurance: Mathematics and Economics, 15, 151–162.Google Scholar
Head, S.J., Adkins, D.R., Cairns, A.J.G., Corvesor, A.J., Cule, D.O., Exley, C.J., Johnson, I.S., Spain, J.G. & Wise, A.J. (2000). Pension fund valuations and market values. British Actuarial Journal, 6, 55–141.CrossRefGoogle Scholar
Huang, H.-C. (2000). Stochastic modelling and control of pension plans. PhD Thesis, Heriot-Watt University.Google Scholar
Keel, A. & Muller, H.H. (1995). Efficient portfolios in the asset liability context. ASTIN Bulletin, 25, 33–48.CrossRefGoogle Scholar
Musiela, M. & Rutkowski, M. (1997). Martingale methods in financial modelling. Springer Verlag, Berlin.CrossRefGoogle Scholar
Schweizer, M. & Follmer, H. (1988). Hedging by sequential regression: an introduction to the mathematics of option trading. ASTIN Bulletin, 18, 147–160.Google Scholar
Srivastava, M.S. & Carter, E.M. (1983). An introduction to applied multivariate statistics, North Holland, Amsterdam.Google Scholar
Wei, W.S. (1990). Time series analysis: univariate and multivariate methods. Addison Wesley, Redwood City.Google Scholar
Wilkie, A.D. (1985). Portfolio selection in the presence of fixed liabilities: a comment on ‘The matching of assets to liabilities’. Journal of the Institute of Actuaries, 112, 229–277.CrossRefGoogle Scholar
Wilkie, A.D. (1995). More on a stochastic asset model for actuarial use. British Actuarial Journal, 1, 777–964.CrossRefGoogle Scholar
Wise, A.J. (1984a). A theoretical analysis of the matching of assets to liabilities. Journal of the Institute of Actuaries, 111, 375–402.CrossRefGoogle Scholar
Wise, A.J. (1984b). The matching of assets to liabilities. Journal of the Institute of Actuaries, 111, 445–501.CrossRefGoogle Scholar
Wise, A.J. (1987a). Matching and portfolio selection: Part I. Journal of the Institute of Actuaries, 114, 113–133.CrossRefGoogle Scholar
Wise, A.J. (1987b). Matching and portfolio selection: Part II. Journal of the Institute of Actuaries, 114, 551–568.CrossRefGoogle Scholar
Wise, A.J. (1989). Matching. Journal of the Institute of Actuaries, 116, 529–535.CrossRefGoogle Scholar
Yakoubov, Y., Teeger, M. & Duval, D. (1999). A stochastic investment model for asset and liability management. Proceedings of the 9th International AFIR Colloquium, Tokyo, August, 1999, Joint ASTIN/AFIR volume, 237-266. (Also presented to the Staple Inn Actuarial Society, 16 November, 1999.)Google Scholar