Maturity Guarantees Research Articles

It's RILA time: An introduction to registered index‐linked annuities

AbstractRegistered index‐linked annuities (RILAs) are increasingly popular equity‐based retirement savings products offered by US life insurance companies. They combine features of fixed‐index annuities and traditional variable annuities (TVAs), offering investors equity exposure with downside protection in a tax‐deferred setting. This article introduces RILAs to the academic literature by describing the products' key features, developing a general pricing model, and deriving the providers' hedging strategy by decomposing their liabilities into short‐term European options. Numerical illustrations show that RILAs offer investors similar risk profiles (in the long run) as TVAs with maturity guarantees, and that many products currently sold appear to be priced quite favorably for investors. For providers, RILAs may be a preferable alternative or complement to TVAs as they greatly simplify the management of the embedded equity risk and can naturally reduce the TVA capital requirements. These features position RILAs as a viable long‐term solution for this product space.

Multi-year analysis of solvency capital in life insurance

With the commencement of the Solvency II directive, insurers in the European Union need to provide a projection of their solvency figures into the future (as part of the Own Risk and Solvency Assessment, ORSA). This is a highly complex task since future solvency figures depend on the development of numerous (stochastic) risk factors. The required evaluations are numerically challenging, which in practice forces companies to limit their analyses to only a few selected deterministic scenarios. These deterministic scenarios clearly cannot describe the full probability distribution of a company’s future solvency situation. The focus of this paper is on financial guarantees in participating life insurance products. In particular, we study two major types of interest rate guarantees in life insurance, a maturity guarantee and a (path-dependent) cliquet-style guarantee. In order to derive entire probability distributions of future solvency ratios, we limit the model framework to two sources of risk (a Hull–White model for interest rates and a geometric Brownian motion for stocks). This partly leads to closed-form solutions of the market-consistent valuation of the liabilities, ensures higher accuracy in computations and less numerical effort. Furthermore, the model allows for a detailed analysis of the impact of the different types of interest rate guarantees on the future solvency situation. Our results suggest that the type of guarantee has a significant impact on the long-term solvency of the company.

It's RILA Time: An Introduction to Registered Index-Linked Annuities

Registered index-linked annuities (RILAs) are increasingly popular equity-based retirement savings products offered by U.S. life insurance companies. They combine features of fixed-index annuities and traditional variable annuities (TVAs), offering investors equity exposure with downside protection in a tax-deferred setting. This article introduces RILAs to the academic literature by describing the products' key features, developing a general pricing model, and deriving the providers' hedging strategy by decomposing their liabilities into short-term European options. Numerical illustrations show that RILAs offer investors similar risk profiles (in the long run) as TVAs with maturity guarantees, and that many products currently sold appear to be priced quite favorably for investors. For providers, RILAs may be a preferable alternative or complement to TVAs as they greatly simplify the management of the embedded equity risk and can naturally reduce the TVA capital requirements. These features position RILAs as a viable long-term solution for this product space.

Optimal asset allocation for participating contracts under the VaR and PI constraint

ABSTRACTParticipating contracts provide a maturity guarantee for the policyholder. However, the terminal payoff to the policyholder should be related to financial risks of participating insurance contracts. We investigate an optimal investment problem under a joint value-at-risk and portfolio insurance constraint faced by the insurer who offers participating contracts. The insurer aims to maximize the expected utility of the terminal payoff to the insurer. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the joint value-at-risk and the portfolio insurance constraint impacts the optimal terminal wealth.

Investment, agency conflicts, debt maturity, and loan guarantees by negotiation

We consider an irreversible investment, of which the sunk cost is financed by a finite-term debt after entering into an option-for-guarantee swap (OGS) with negotiation. The OGS is a three-party agreement among a lender (bank), an insurer, and a borrower (entrepreneur), where the bank lends at a given interest rate to the entrepreneur and if the borrower defaults on debt, the insurer must pay all the principal and remaining interests to the lender instead of the borrower. In return for the guarantee, the borrower must allocate a perpetual American call option to purchase a fraction (guarantee cost) of his equity at a given strike price. We find that the investment threshold decreases but the exercise threshold of the insurer’s option increases with the borrower’s bargaining power. Both the investment and exercise threshold increase with debt maturity, but there is a U-shaped relation between the guarantee cost and debt maturity. The borrower postpones investment once the funding gap or project risk increases. The swap may overcome the inefficiencies from asset substitution and debt overhang, strongly depending on the debt maturity and borrower’s bargaining power.

A Framework to Charge for Unit-Linked Contracts When Considering Guaranteed Risk

Risk management for investment guarantees with unit-linked contracts is very critical for the insurer. This paper proposes a framework to charge for guaranteed risk when the insurer reserves for the guaranteed risk. This framework can facilitate the calculation of risk reserves and charge for the investment guarantees. In this framework the charge is determined by two criteria of meeting the insurance company?s target internal rate of return and simultaneously the reserving standard. The framework is built on the stochastic cash-flow analysis. For illustrative purposes, we set up quantile reserves as the actuarial reserving standard in our framework. In this framework, the procedure to work out the charges is in reverse. In our numerical illustration, we investigate a unit-linked policy with maturity guarantees. Our framework would also apply to other types of contacts, guarantees and reserving standard.

Quantile hedging on equity-linked life insurance contracts with transaction costs

This paper analyzes the application of quantile hedging on equity-linked life insurance contracts in the presence of transaction costs. Following the time-based replication strategy, we present the explicit expressions for the present values of expected hedging errors and transaction costs. The results are derived by using the adjusted hedging volatility σ̄ proposed by Leland. Furthermore, the estimated values of expected hedging errors, transaction costs and total costs are obtained from a simulation approach for comparison. Finally, the costs of maturity guarantee for equity-linked life insurance contracts inclusive of transaction costs are discussed.

Pricing maturity guarantee under a refracted Brownian motion

The cost of maturity minimum guarantee of an investment fund is typically deducted upfront or at the beginning of each period, together with other costs such as sales commissions and management fees. This practice gives a significant negative impact on the short term returns of the investment. In this paper, we propose a way to compensate the cost of investment guarantee by continuous withdrawals whenever the fund value stays below or at a predetermined threshold. The corresponding dynamics of fund price follows a (geometric) refracted Brownian motion, where the two drifts are state dependent. We derive integral formulas for the prices of fund and maturity minimum guarantee. Given the cost deduction threshold and the maturity guarantee level, the fair deduction rate can be determined numerically.

Pricing maturity guarantee with dynamic withdrawal benefit

Motivated by the importance of withdrawal benefits for enhancing sales of variable annuities, we propose a new equity-linked product which provides a dynamic withdrawal benefit (DWB) during the contract period and a minimum guarantee at contract maturity. The term DWB is coined to reflect the duality between it and dynamic fund protection. Under the Black–Scholes framework and using results pertaining to reflected Brownian motion, we obtain explicit pricing formulas for the DWB payment stream and the maturity guarantee. These pricing formulas are also derived by means of Esscher transforms, which is another seminal contribution by Gerber to finance. In particular, we show that there are closed-form formulas for pricing European put and call options on a traded asset whose price can be modeled as the exponential of a reflected Brownian motion.

Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model

We tackle the problem of computing fair periodical premiums of an equity-linked policy with a maturity guarantee and an embedded surrender option. We consider the policy as a Bermudan-style contingent claim that can be exercised at the premium payment dates. The evaluation framework is based on a discretization of a bivariate model that considers the joint evolution of the equity value with stochastic interest rates. To deeply reduce the computational complexity of the pricing problem we use the singular points framework that allows us to compute accurate upper and lower estimates of the policy premiums.

Valuation of the interest rate guarantee embedded in defined contribution pension plans

In this research, we derive the valuation formulae for a defined contribution pension plan associated with the minimum rate of return guarantees. Different from the previous studies, we work on the rate of return guarantee which is linked to the δ -year spot rate. The payoffs of interest rate guarantees can be viewed as a function of the exchange option. By employing Margrabe’s [Margrabe, W., 1978. The value of an option to exchange one asset for another. Journal of Finance 33, 177–186] option pricing approach, we derive general pricing formulae under the assumptions that the interest rate dynamics follow a single-factor HJM (1992) [Heath. D. et al., 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105] interest rate model and the asset prices follow a geometric Brownian motion. The volatility of the forward rates is assumed to be exponentially decaying. The formula is explicit for valuing maturity guarantee (type-I guarantee). For multi-period guarantee (type-II guarantee), the analytical formula only exists when the guaranteed rate is the one-year spot rate. The accuracy of the valuation formulae is illustrated with numerical analysis. We also investigate the effect of mortality and the sensitivity of key parameters on the value of the guarantee. We find that type-II guarantee is much more costly than the type-I guarantee, especially with a long duration policy. The closed form solution provides the advantage in valuing pension guarantees.

A General Approach for Pricing Rollover Options

When an insurance company sells a mutual fund with death and maturity guarantees to its client, it may consider allowing the client to extend the guarantee for some more years. If the renewal only happens once, a so-called rollover option is implied in the contract. In this paper, we show how the generalized Bermudan option can be applied to the special case of the rollover option. By avoiding the heavy mathematical tools which are necessary to prove the existence of a hedging strategy, we will focus on the calculations that are common in the Black-Scholestype analysis. Contrary to Bilodeau (1997) who analyzed the one-time renewal, we can refer to the results on the (generalized) Bermudan option for which the existence of a hedging strategy was already proved. We will see that the strike price has to be adjusted if the contract is renewed in order to explicitly calculate the price of the contract.

The effect of management discretion on hedging and fair valuation of participating policies with maturity guarantees

In this paper we consider how an insurer should invest in order to hedge the maturity guarantees inherent in participating policies. Many papers have considered the case where the guarantee is increased each year according to the performance of an exogenously given reference portfolio subject to some guaranteed rate. However, in this paper we will consider the more realistic case whereby the reference portfolio is replaced by the insurer’s own investments which are controlled completely at the discretion of the insurer’s management. Hence in our case any change in the insurer’s investment strategy leads to a change in the underlying value process of the participating contract. We use a binomial tree model to show how this risk can be hedged, and hence calculate the fair value of the contract at the outset.

Defined Contribution Based Pension Plans

ABSTRACTIn this paper we address the problem of valuing (corporate) pension plans, and in particular defined contribution based pension plans. Several pension plans are proposed, both with and without rate of return guarantees. Both maturity and annual guarantees are considered. Emphasis is also given to the risk management of pension plans.To tie the analysis closer to real-world problems, we allow for both periodic premium and pension payments and mortality risk to be taken into account. Some new results on forward-start guarantees are also derived.

Mixture Gaussian Time Series Modeling of Long-Term Market Returns

Stochastic modeling of investment returns is an important topic for actuaries who deal with variable annuity and segregated fund investment guarantees. The traditional lognormal stock return model is simple, but it is generally less appealing for longer-term problems. In recent years, the use of regime-switching lognormal (RSLN) processes for modeling maturity guarantees has been gaining popularity. In this paper we introduce the class of mixture Gaussian time series processes for modeling long-term stock market returns. It offers an alternative class of models to actuaries who may be experimenting with the RSLN process. We use monthly data from the Toronto Stock Exchange 300 and the Standard and Poor-s 500 indices to illustrate the mixture time series modeling procedures, and we compare the fits of the mixture models to the lognormal and RSLN models. Finally, we give a numerical example comparing risk measures for a simple segregated fund contract under different stochastic return models.

Investment Guarantees in the South African Life Insurance Industry

This paper explores the risks faced by South African life insurance companies arising from the provision of investment guarantees in products sold. The current thinking and practice of the larger South African life insurance companies regarding investment guarantees is set out following their responses to a survey. The paper examines the forms of investment guarantee available and the business issues created by the writing of these guarantees. These include issues around the design and pricing of new business, as well as the risk management of in-force business. The paper also compares existing methods used internationally to value life insurance business with investment guarantees, focusing on the use of stochastic models. The different allowances for risk within each valuation method and the appropriateness of these allowances when valuing investment guarantees are considered. The stochastic models compared include both statistically based real-world models and market-consistent state-price-deflator or risk-neutral models. Practical issues around the building of such asset–liability stochastic models are briefly discussed. Finally, the authors put forward their own views of possible developments in the future within South Africa that may impact on life insurance business with investment guarantees, and the possible implications. Keywords: Investment guarantee; maturity guarantee; South Africa; non-profit guarantee; smoothed-bonus business; asset–liability matching; market risk premium; financial options; hedging; equity volatility; stochastic modelling; market-consistent valuation South African Actuarial Journal: 2003 3: 29-76

Reserving, Pricing and Hedging For Policies with Guaranteed Annuity Options

ABSTRACTIn this paper we consider reserving and pricing methodologies for a pensions-type contract with a simple form of guaranteed annuity option. We consider only unit-linked contracts, but our methodologies and, to some extent, our numerical results would apply also to with-profits contracts.The Report of the Annuity Guarantees Working Party (Bolton et al., 1997), presented the results of a very interesting survey, as at the end of 1996, of life assurance companies offering guaranteed annuity options. There was no consensus at that time among the companies on how to reserve for such options. The Report discussed several approaches to reserving, but concluded that it was unable to recommend a single approach. This paper is an attempt to fill that gap.We investigate two approaches to reserving and pricing. In the first sections of the paper we consider quantile, and conditional tail expectation, reserves. The methodology we adopt here is very close to that proposed by the Maturity Guarantees Working Party in its Report to the profession (Ford et al., 1980). We show how these policies could have been reserved for in 1985, and what would have been the outcome of using the proposed method.In a later section we consider the feasibility of using option pricing methodology to dynamically hedge a guaranteed annuity option. It is shown that this is possible within the context of the model we propose, but we submit that, in practical terms, dynamic hedging is not a complete solution to the problem since suitable tradeable assets do not in practice exist.Finally, we describe several enhancements to our models and methodology, which would make them even more realistic, though generally they would have the effect of increasing the required contingency reserves

An object-oriented framework for valuing shout options on high-performance computer architectures

A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. Complex versions of these options are embedded in financial products which offer various types of maturity guarantees such as segregated funds marketed by Canadian insurance companies. The value of these options can be determined by solving a collection of coupled partial differential equations. In this work we develop an object-oriented framework for valuing these contracts which is capable of exploiting modern high-performance computer architectures. We use this framework to study practical aspects of valuing and hedging these contracts.

Understanding the Behavior and Hedging of Segregated Funds Offering the Reset Feature

Segregated funds have become an extremely popular Canadian investment vehicle. These instruments provide long-term maturity guarantees and often include complex option features. One controversial aspect is the reset feature, which provides the ability to lock in market gains. Recently, regulators have announced that firms offering these products will be subject to new capital requirements. This paper discusses the effects of volatility, interest rates, investor optimality, and product design on the cost of providing a segregated fund guarantee. For each scenario, the authors provide the appropriate management expense ratio (MER) that should be charged and demonstrate the current liability using a given fixed MER. The paper also investigates intuitive reasons that cause the reset feature to require such a dramatic increase in the hedging costs. Finally, an approximate method for handling the reset feature is presented that can be computed very efficiently, provided the correct proportional fee is charged.

The reset decision for segregated fund maturity guarantees

A segregated fund is a mutual fund whose value is protected by a maturity guarantee. An investor in such a fund may have the option of periodically resetting this guarantee to extend its life and increase the guaranteed amount. We model this guarantee reset decision using a discrete time Markov chain and analyse a decision strategy based upon a return threshold. We characterise the optimal value for this threshold and examine its properties. We show that the intuitive strategy of resetting the guarantee whenever the fund’s value has increased is not optimal but seems to provide good results.