Über die Autorin bzw. den Autor

MARY HARDY is an Associate Professor and Associate Chair of Actuarial Science at the University of Waterloo, a Fellow of the Institute of Actuaries, and an Associate of the Society of Actuaries.

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PRAISE FOR INVESTMENT GUARANTEES

“In addition to being a valuable and innovative addition to the literature on risk management of equity-linked insurance, this book provides a uniquely clear demonstration of using different measures in a very practical context. A great way of showing actuaries how to ‘mind their Ps and Qs’!”

―Boris Brizeli
Principal and Chief Operations Officer
GE Insource Limited

“This is an extremely well written and comprehensive book which will be useful for both practitioners and graduate students. It provides a unique synthesis of techniques and tools in econometrics, financial engineering, and simulation applied to questions of fundamental importance in insurance.”

―Dr. Andrew Cairns
Reader in Financial and Actuarial Mathematics
Heriot-Watt University

“Investment Guarantees will become the reference book of choice for both actuaries and non-actuaries alike working in the field of guarantees associated with equity-linked products. The book provides an excellent balance between theory and practice in the course of contrasting actuarial modeling and option pricing theory as applied to guarantees on equity-linked products.”

―Larry M. Gorski
Life Actuary
Illinois Department of Insurance

“Professor Hardy strikes the right balance between theory development and practical demonstration, offering insights to both novices and experienced practitioners. Her comprehensive treatment of risk modeling of investment guarantees will benefit actuaries and financial engineers alike, allowing each to understand better the nature and management of equity exposure. Exceptionally readable, Investment Guarantees is an invaluable resource for insurance and financial professionals working with equity-linked insurance and annuity products.”

―Geoffrey H. Hancock
Principal Mercer Risk,
Finance & Insurance Consulting

Aus dem Klappentext

The object of life insurance is to provide financial security to policyholders and their families. But insurance markets around the world are changing, and policyholders want to enjoy the benefits of equity investments in conjunction with mortality protection. This idea has led to the development of equity-linked life insurance, and has probably left you searching for better ways to model these innovative investment guarantees as well as measure the risk associated with them.

Whether you?re involved with the product design, marketing, pricing and valuation, or risk management of equity-linked insurance, this book has something for you. Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance is a comprehensive guide that combines the econometric analysis of these investment models with their applications in pricing and risk management.

Designed with all equity-linked life insurance practitioners in mind, you?ll find both approaches to risk management of equity-linked insurance?the "actuarial" approach and the dynamic hedging approach?presented, discussed, and extensively illustrated with examples. Investment Guarantees opens with a discussion of various models, moves through modeling techniques, and addresses risk management in a straightforward, accessible manner. This unique resource will:

• Introduce the various types of investment guarantees commonly used in equity-linked insurance, including guaranteed death and living benefits
• Explore contracts which offer investment guarantees as part of the benefit package?variable annuities, segregated fund contracts, unit-linked contracts, equity-indexed annuities, and guaranteed annuity options
• Discuss stochastic models commonly used for equity returns, including ARCH and GARCH models, and regime switching models
• Illustrate parameter estimation using both Maximum Likelihood Estimation (MLE) and Markov Chain Monte Carlo Method (MCMC)?a Bayesian approach
• Explain how to model investment guarantees using the "actuarial" approach and the dynamic hedging approach to risk management
• Describe stochastic emerging cost modeling
• Examine the sources of forecast uncertainty

Investment Guarantees carefully pulls together all of the most recent models and methods that are useful in managing the risk associated with equity-linked insurance. Filled with professional insights and proven techniques, this book is a valuable one-stop reference that will allow you to better understand the theory and practice behind modeling and risk management for equity-linked life insurance.

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Investment Guarantees

John Wiley & Sons

Chapter One

INTRODUCTION

The objective of life insurance is to provide financial security to policyholders and their families. Traditionally, this security has been provided by means of a lump sum payable contingent on the death or survival of the insured life. The sum insured would be fixed and guaranteed. The policyholder would pay one or more premiums during the term of the contract for the right to the sum insured. Traditional actuarial techniques have focused on the assessment and management of life-contingent risks: mortality and morbidity. The investment side of insurance generally has not been regarded as a source of major risk. This was (and still is) a reasonable assumption, where guaranteed benefits can be broadly matched or immunized with fixed-interest instruments.

But insurance markets around the world are changing. The public has become more aware of investment opportunities outside the insurance sector, particularly in mutual fund type investment media. Policyholders want to enjoy the benefits of equity investment in conjunction with mortality protection, and insurers around the world have developed equity-linked contracts to meet this challenge. Although some contract types (such as universal life in North America) pass most of the asset risk to the policyholder and involve little or no investment risk for the insurer, it was natural for insurers to incorporate payment guarantees in these new contracts-this is consistent with the traditional insurance philosophy.

In the United Kingdom, unit-linked insurance rose in popularity in the late 1960s through to the late 1970s, typically combining a guaranteed minimum payment on death or maturity with a mutual fund type investment. These contracts also spread to areas such as Australia and South Africa, where U.K. insurance companies were influential. In the United States, variable annuities and equity-indexed annuities offer different forms of equity-linking guarantees. In Canada, segregated fund contracts became popular in the late 1990s, often incorporating complex guaranteed values on death or maturity. Germany recently introduced equity-linked endowment insurance. Similar contracts are also popular in many other jurisdictions. In this book the term equity-linked insurance is used to refer to any contract that incorporates guarantees dependent on the performance of a stock market indicator. We also use the term separate account insurance to refer to the group of products that includes variable annuities, segregated funds, and unit-linked insurance. For each of these products, some or all of the premium is invested in an equity fund that resembles a mutual fund. That fund is the separate account and forms the major part of the benefit to the policyholder. Separate account products are the source of some of the most important risk management challenges in modern insurance, and most of the examples in this book come from this class of insurance. The nature of the risk to the insurer tends to be low frequency in that the stock performance must be extremely poor for the investment guarantee to bite, and high severity in that, if the guarantee does bite, the potential liability is very large.

The assessment and management of financial risk is a very different proposition to the management of insurance risk. The management of insurance risk relies heavily on diversification. With many thousands of policies in force on lives that are largely independent, it is clear from the central limit theorem that there will be very little uncertainty about the total claims. Traditional actuarial techniques for pricing and reserving utilize deterministic methodology because the uncertainties involved are relatively minor. Deterministic techniques use "best estimate" values for interest rates, claim amounts, and (usually) claim numbers. Some allowance for uncertainty and random variation may be made implicitly, through an adjustment to the best estimate values. For example, we may use an interest rate that is 100 or 200 basis points less than the true best estimate. Using this rate will place a higher value on the liabilities than will using the best estimate as we assume lower investment income.

Investment guarantees require a different approach. There is generally only limited diversification amongst each cohort of policies. When a market indicator becomes unfavorable, it affects many policies at the same time. For the simplest contracts, either all policies in the cohort will generate claims or none will. We can no longer apply the central limit theorem. This kind of risk is referred to as systematic, systemic, or nondiversifiable risk. These terms are interchangeable.

Contrast a couple of simple examples:

* An insurer sells 10,000 term insurance contracts to independent lives, each having a probability of claim of 0.05 over the term of the contract. The expected number of claims is 500, and the standard deviation is 22 claims. The probability that more than, say, 600 claims arise is less than [10.sup.-5]. If the insurer wants to be very cautious not to underprice or underreserve, assuming a mortality rate of 6 percent for each life instead of the best estimate mortality rate of 5 percent for each life will absorb virtually all mortality risk.

* The insurer also sells 10,000 pure endowment equity-linked insurance contracts. The benefit under the insurance is related to an underlying stock price index. If the index value at the end of the term is greater than the starting value, then no benefit is payable. If the stock price index value at the end of the contract term is less than its starting value, then the insurer must pay a benefit. The probability that the stock price index has a value at the end of the term less than its starting value is 5 percent.

The expected number of claims under the equity-linked insurance is the same as that under the term insurance-that is 500 claims. However, the nature of the risk is that there is a 5 percent chance that all 10,000 contracts will generate claims, and a 95 percent chance that none of them will. It is not possible to capture this risk by adding a margin to the claim probability of 5 percent.

This simple equity-linked example illustrates that, for this kind of risk, the mean value for the number (or amount) of claims is not very useful. We can also see that no simple adjustment to the mean will capture the true risk. We cannot assume that a traditional deterministic valuation with some margin in the assumptions will be adequate. Instead we must utilize a more direct, stochastic approach to the assessment of the risk. This stochastic approach is the subject of this book.

The risks associated with many equity-linked benefits, such as variable-annuity death and maturity guarantees, are inherently associated with fairly extreme stock price movements-that is, we are interested in the tail of the stock price distribution. Traditional deterministic actuarial methodology does not deal with tail risk. We cannot rely on a few deterministic stock return scenarios generally accepted as "feasible." Our subjective assessment of feasibility is not scientific enough to be satisfactory, and experience-from the early 1970s or from October 1987, for example-shows us that those returns we might earlier have regarded as infeasible do, in fact, happen. A stochastic methodology...