Abstract
The concept of best-estimate, prescribed by regulators to value insurance liabilities for accounting and solvency purposes, has recently been discussed extensively in the industry and related academic literature. To differentiate hedgeable and non-hedgeable risks in a general case, recent literature defines best-estimates using orthogonal projections of a claim on the space of replicable payoffs. In this paper, we apply this concept of best-estimate to long-maturity claims in a market with reinvestment risk, since in this case the total liability cannot easily be separated into hedgeable and non-hedgeable parts. We assume that a limited number of short-maturity bonds are traded, and derive the best-estimate price of bonds with longer maturities, thus obtaining a best-estimate yield curve. We therefore use the multifactor Vasiˇcek model and derive within this framework closed-form expressions for the best-estimate prices of long-term bonds.
Highlights
Life insurers are often faced with the problem of valuing liabilities with long maturities
We show that the number of bonds available and the correlation structure between the bond prices influence the best-estimate price of the long-maturity bond
We apply the definition of best-estimate given by [6] in terms of span-deflators to price long-term bonds in a market with reinvestment risk
Summary
Life insurers are often faced with the problem of valuing liabilities with long maturities. We attempt to calculate the best-estimate price of a long term zero-coupon bond in a market with reinvestment risk, where “best-estimate” is as prescribed by Solvency II regulations. We apply their definition of best-estimate reserves to value zero-coupon bonds with long maturities in financial markets with reinvestment risk, whose hedgeable part cannot be separated by a conditional expectation. Reinvestment risk stems from the fact that only a limited number of bonds are available on the market In this context, we value a zero-coupon bond with time to maturity longer than the longest traded bond. The best-estimate obtained takes into account the lack of liquidity on the long-maturity bond market To analyse this valuation method we use the multifactor Vasiček model since it provides a flexible modeling framework with tractable formulas for the valuation of long-term bonds.
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