Svein-Arne Persson

Pricing of Unit-linked Life Insurance Policies

Abstract The key feature of unit-linked or equity-linked life insurance policies is the uncertain value of the future insurance benefit. By issuing unit-linked insurances that guarantees the policy-holder a minimum benefit, the insurance company is exposed to financial risk. The value of the insurance benefit is assumed to be a function of a particular stochastic process. We use the financial theory of arbitrage pricing and martingale theory to derive single premiums for different policies. We derive risk-minimizing trading strategies describing how the issuing company can reduce financial risk. We derive a partial differential equation for the market value of the premium reserve which we compare to Thieles equation of the actuarial sciences. Our equation contains some new terms stemming from our economic model. The interpretation of the principle of equivalence may be revisited in this framework; the principle still holds but under a new risk adjusted probability measure, equivalent to—but different fro...

Guaranteed Investment Contracts: Distributed and Undistributed Excess Return

Annual minimum rate of return guarantees are analyzed together with rules for distribution of positive excess return, i.e. investment returns in excess of the guaranteed minimum return. Together with the level of the annual minimum rate of return guarantee both the customers and the insurers fractions of the positive excess return are determined so that the market value of the insurers capital inflow (determined by the fraction of the positive excess return) equals the market value of the insurers capital outflow (determined by the minimum rate of return guarantee) at the inception of the contract. The analysis is undertaken both with and without a surplus distribution mechanism. The surplus distribution mechanism works through a bonus account that serves as a buffer in the following sense: in (‘bad’) years when the investment returns are lower than the minimum rate of return guarantee, funds are transferred from the bonus account to the customers account. In (‘good’) years when the investment returns are above the minimum rate of return guarantee, a part of the positive excess return is credited to the bonus account. In addition to characterizations of fair combinations of the level of the annual minimum rate of return guarantee and the sharing rules of the positive excess return, our analysis indicates that the presence of a surplus distribution mechanism allows the insurer to offer a much wider menu of contracts to the customer than without a surplus distribution mechanism.

Pricing rate of return guarantees in a Heath-Jarrow-Morton framework

Abstract Rate of return guarantees, included in many financial products, exist in two fundamentally different types. Maturity guarantees which are binding only at the expiration of the contract, and therefore, similar to financial options and multi-period guarantees which have the time to expiration divided into several subperiods with a binding guarantee for each subperiod. Relevant real-life examples are life insurance contracts and guaranteed investment contracts. We consider rate of return guarantees where the underlying rate of return is either (i) the rate of return on a stock investment or (ii) the short-term interest rates. Various types of these rate of return guarantees are priced in a general no-arbitrage Heath–Jarrow–Morton framework. We show that despite fundamental differences in the underlying rate of return processes ((i) or (ii)), the resulting pricing formulas for the guarantees are remarkably similar for maturity guarantees. For multi-period guarantees the presence of stochastic interest rates leads to intertemporal dependencies which complicates the valuation formulaes compared both to the case of maturity guarantees and the case of deterministic interest rates. Finally, we show how the term structure models of Vasicek (Vasicek, O., 1977. Journal of Financial Economics 5, 177–188) and Cox et al. (Cox, J.C., Ingersoll, Jr., J.E., Ross, S.A., 1985. Econometrica 53(2), 385–407) occur as special cases in our more general framework based on the model of Heath et al. (Heath, D., Jarrow, R.A., Morton, A.J., 1992. Econometrica 60 (1), 77–105).

Exotic Unit-Linked Life Insurance Contracts

This article integrates aspects of traditional insurance with advances in financial economics, yielding proper valuation and premium assessments of insurance benefits linked to various financial assets. Several new types of unit-linked life insurance contracts are discussed, with substantial potential for real-life applications. Compared to usual unit-linked products, these contracts offer added flexibility and/or altered exposure to financial risk for the insured and/or the insurer. The single premiums of these policies are calculated as expectations under a risk-adjusted probability measure (equivalent martingale measure), satisfying no-arbitrage conditions in financial markets.

Design and Pricing of Equity-Linked Life Insurance Under Stochastic Interest Rates

A valuation model for equity-linked life insurance contracts incorporating stochastic interest rates is presented. Our model generalizes some previous pricing results based on deterministic interest rates. Moreover, a design of a new equity-linked product with some appealing features is proposed and compared with the periodical premium contract of Brennan and Schwartz (1976). Our new product is very simple to price and may easily be hedged either by long positions in the mutual fund of linkage or by European call options on the same fund.

Stochastic interest rate in life insurance: The principle of equivalence revisited

Abstract Based on financial theory, a valuation model—including stochastic interest rates—for traditional life insurance contracts is derived. The interpretation of the principle of equivalence may be revisited in this framework; single premiums are found as expected present values under a risk adjusted probability measure. Using a specific model of the term structure some new formulas for the market value of various life insurance contracts are derived. A partial differential equation for the market values of the assurances, is deduced, corresponding to the traditional Thieles differential equation of classical actuarial sciences. This equation contains some interesting new terms.

Valuation of a multistate life insurance contract with random benefits

We present a model where the value of the life insurance benefit is random. The policy is at each point in time assumed to be in one of a finite number of states and the evolution of the policy through time is modelled by a time-continuous, non-homogeneous Markov chain. The insurance period of a life insurance contract is long compared to the contract period of a typical financial contingent claim. The value of the insurance benefit is assumed to follow a geometric Gaussian process which has certain appealing properties when dealing with such long contract periods. We use the martingale arbitrage pricing theory to derive the market value of a quite general life insurance policy and deduce the corresponding Thieles differential equation.

Credit Spreads and Incomplete Information

A new model is presented which produces credit spreads that do not converge to zero for short maturities. Our set-up includes incomplete, i.e., delayed and asymmetric information. When the financial market observes the companys earnings with a delay, the effect on both default policy and credit spreads is negligible, compared to the Leland (1994) model. When information is asymmetrically distributed between the management of the company and the financial market, short credit spreads do not converge to zero. This is result is similar to the Duffie and Lando (2001) model, although our simpler model improves some limitations in their set-up. Short interest rates from our model are used to illustrate effects similar to the dry-up in the interbank market experienced after the summer of 2007.

On the Pricing of Performance Sensitive Debt

Performance sensitive debt (PSD) contracts link a loans interest rate to a measure of the borrowers credit relevant performance, e.g., if the borrower becomes less credit worthy, the interest rate increases according to a predetermined schedule. We derive and empirically test a pricing model for PSD contracts and find that interest increasing contracts are priced reflecting a substantial risk of shocks to borrower credit quality. Borrowers using such contracts are of an overall higher credit quality compared to borrowers using interest decreasing contracts. These contracts are priced as if no risk of shocks to borrower credit quality is present.

Continuous Monitoring: Look Before You Leap

We present a model for pricing credit risk protection for a limited liability non-life insurance company. The protection is typically provided by a guaranty fund. In the case of continuous monitoring, i.e., where the market values of the companys assets and liabilities are continuously observable, and where the market values of assets and liabilities follow continuous processes, the regulators can liquidate the insurance company at the instant the market value of its assets equals the market value of its liabilities, implying that the credit protection is worthless. When jumps are included in the claims process, the protection provided by the guaranty fund has a strictly positive market value. We argue that the ability to continuously monitor the equity value of a company can be a new explanation for why jump processes may be important in models of credit risk.