This paper investigates market-consistent valuation of insurance liabilities
in the context of, for instance, Solvency II and to some extent IFRS 4. We
propose an explicit and consistent framework for the valuation of insurance
liabilities which incorporates the Solvency II approach as a special case.

The proposed framework is based on dynamic replication over multiple
(one-year) time periods by a portfolio of assets with reliable market prices,
allowing for "limited liability" in the sense that the replication can in
general not always be continued. The asset portfolio consist of two parts: (1)
assets whose market price defines the value of the insurance liabilities, and
(2) capital funds used to cover risk which cannot be replicated. The capital
funds give rise to capital costs; the main exogenous input of the framework is
the condition on when the investment of the capital funds is acceptable.

We investigate existence of the value and show that the exact calculation of
the value has to be done recursively backwards in time, starting at the end of
the lifetime of the insurance liabilities. The main question only partially
considered in this paper is the uniqueness of the value. We derive upper bounds
on the value and, for the special case of replication by risk-free one-year
zero-coupon bonds, explicit recursive formulas for calculating the value.

Valuation in Solvency II and IFRS 4 is based on representing the value as a
sum of a "best estimate" and a "risk margin". In our framework, it turns out
that this split is not natural. Nonetheless, we show that a split can be
constructed as a simplification, and that it provides an upper bound on the
value under suitable conditions. We illustrate the general results by
explicitly calculating the value for a simple example.