Pension funds and insurers face difficulties in hedging their longevity risk, which is the uncertainty of how long their clients will live. A possible solution could be using longevity-linked securities to transfer some of this risk to other parties. However, these securities may not match the actual mortality rates of the insurer’s clients, resulting in a potential loss due to basis risk. In this paper, we measure this basis risk through the pricing of a longevity derivative under Solvency II. We also compare this method with other common pricing methods in finance. We explore and evaluate different hedging strategies for insurers, using a multi-population model derived from a two-dimensional Hull and White model that captures the dynamics of mortality over time.

The life annuity business is heavily exposed to longevity risk. Risk transfer solutions are not yet fully developed, and when available they are expensive. A significant part of the risk must therefore be retained by the life insurer. So far, most of the research work on longevity risk has been mainly concerned with capital requirements and specific risk transfer solutions. However, the impact of longevity risk on shareholder value also deserves attention. While it is commonly accepted that a market-consistent valuation should be performed in this respect, the definition of a fair shareholder value for a life insurance business is not trivial. In this paper, we develop a multi-period market-consistent shareholder value model for a life annuity business. The model allows for systematic and idiosyncratic longevity risk and includes the most significant variables affecting shareholder value: the cost of capital (which in a market-consistent setting must be quantified in terms of frictional and agency costs, net of the value of the limited liability put option), policyholder demand elasticity and the cost of alternative longevity risk management solutions, namely indemnity-based and index-based solutions. We show how the model can be used for assessing the impact of different longevity risk management strategies on life insurer shareholder value and solvency.

An optimal reinsurance problem from the perspective of an insurer is studied in this paper, where an upper limit is imposed on a reinsurer's expected loss over a prescribed level. In order to reduce the moral hazard, we assume that both the insurer and the reinsurer are obligated to pay more as the amount of loss increases in a typical reinsurance treaty. We further assume that the optimization criterion preserves the convex order. Such a criterion is very general as most of the criteria for optimal reinsurance problems in the literature preserve the convex order. When the reinsurance premium is calculated as a function of the actuarial value of coverage, we show via a stochastic dominance approach that any admissible reinsurance policy is dominated by a stop-loss reinsurance or a two-layer reinsurance, depending upon the amount of the reinsurance premium. Moreover, we obtain a similar result to Mossin's Theorem and find that it is optimal for the insurer to cede a loss as much as possible under the net premium principle. To further examine the reinsurance premium for the optimal piecewise linear reinsurance policy, we assume the expected value premium principle and derive the optimal reinsurance explicitly under (1) the criterion of minimizing the variance of the insurer's risk exposure, and (2) the criterion of minimizing the risk-adjusted value of the insurer's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

This Working Paper should not be reported as representing the views of the IMF.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate.

In today's financial system, complex financial institutions are connected through an opaque network of financial exposures. These connections contribute to financial deepening and greater savings allocation efficiency, but are also unstable channels of contagion. Basel III and Solvency II should improve the stability of these connections, but could have unintended consequences for cost of capital, funding patterns, interconnectedness, and risk migration.

In this paper, we investigate the problem of purchasing a reinsurance policy that minimizes the risk-adjusted value of an insurer's liability, where the valuation is carried out using a cost-of-capital approach. In order to exclude the moral hazard, we assume that both the insurer and reinsurer are obligated to pay more for larger loss in a typical reinsurance treaty. Moreover, the reinsurance premium principle is assumed to satisfy three axioms: law invariance, risk loading and preserving convex order. The proposed class of premium principles is quite general in the sense that it contains all the widely used premium principles except Esscher principle listed in Young (2004). When capital at risk is measured by the value at risk (VaR) or conditional value at risk (CVaR), we find it is optimal for the insurer to cede two separate layers over the prescribed premium principles. By imposing an additional weak constraint on the premium principle, we further get that the reinsurance in the form of a layer is optimal. Finally, to illustrate the applicability of our results, we derive explicitly the optimal one-layer reinsurance for expected value principle and Wang's premium principle, and show that two-layer reinsurance may be optimal for Dutch premium principle.

This paper investigates market-consistent valuation of insurance liabilities in the context of Solvency II among others 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 replication over multiple (one-year) time periods by a periodically updated 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 consists 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 in 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. 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. In the paper, we only partially consider the question of the uniqueness of 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.

A major outcome of ERM activities in insurance companies has been the bringing together of all of the key risks in the company, to be managed collectively in a holistic fashion. The authors of this paper believe that an ERM framework also needs to look beyond the company, and have regard to the risk management needs of investors, from the point of view of the contribution of the insurance company to the overall risk and reward of their total investment portfolios. To meet these needs, the ERM framework needs to provide sufficient information on topics such as systematic risk, potential correlations of earnings from future new business with macroeconomic trends, other risks to franchise value, and sources of model risk within the company. The paper does not provide solutions for the issues described above; but limits itself to describing and discussing the direction for some important new initiatives in ERM activities.

Most businesses have assets financed by capital providers. The cost of capital is a measure of the returns required by those capital providers. Its main use is to set a target for the profits, which must be achieved on the firm's assets in order to satisfy equity and bond holders.

This paper describes the classical theory of the cost of capital, and then applies it to the special case of banking and insurance firms. We develop implications for product pricing, performance measurement and capital structure optimisation.

Capital and cost of capital form a bridge between the insurance firm and the financial markets. The term capital is used in various ways. In current parlance, economic capital is frequently used to mean capital calculated using a risk-based measure which is independent of the regulatory requirements. In this paper we discuss the concept of target capital, where the firm takes account of three different approaches to risk appetite: regulatory capital plus a buffer; rating agency views; and the views of shareholders, where they make commitments to customers and wish to protect franchise value. We describe how, when blending these views, the firm needs to understand the trade-offs between too high and too low amounts of capital, with reference to the double taxation burden, insurance gearing (leverage of premiums to capital ratio), and the impact of the firm's credit rating on maximising franchise value. We then discuss the main drivers of the cost of capital, which we define as the required total return on the market value of the firm, as determined by reference to the opportunity cost of alternative investments of equivalent risk. We explain that, because the stock market value of the firm is not the same as the capital held inside the firm, the cost of capital derived from external studies cannot be directly applied to internal measures of target return such as return on equity (ROE); it is necessary to translate between the two measures. We separate the risk of the firm between the investment risk and the insurance risk. We describe the frictional costs of investing in an insurance firm, and explain the role of parameter and model risk arising from the uncertainty of the future claim costs of the firm. We describe the findings of two studies of the actual historical stock market returns of United States P&C companies. One of them suggests that applying the Fama-French model produces higher and more accurate cost of capital estimates than the capital asset pricing model (CAPM) method. This is explained by linking the price to book ratio to the costs of financial distress, which are particularly important for general insurance firms, given the influence of insurance strength ratings from the rating agencies. Finally, we attempt to estimate the risk load required in premiums to compensate investors for the elements of cost of capital which we have described, in a way that combines the financial economic approaches to insurance target returns with the traditional actuarial approaches to assessing the risks in the insurance business.