Affordable Health Insurance Plans
Affordable Health Insurance Plans
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Finding Affordable Health Coverage
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What is the deductible? The deductible is the dollar amount of your medical bills you’ll have to pay before your coverage begins. If you don’t think you’ll need more than routine care, choosing a higher deductible means more affordable monthly premiums.
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Fund risk
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Important information
Risk factors you should consider before investing:
•Past performanceis not a guide to future performance.
•The value of shares and the income from them can
go down as well as up and you may get back less
than the amount invested.
•Movements in exchange rates can impact on both the level of income received and the capital value of your investment. If the currency of your country of residence strengthens against the currency in which the underlying investments of the Fund are made, the value of your investment will reduce and vice versa.
• The Fund invests in emerging markets which tend to be more volatile than mature markets and the value of your investment could move sharply up or down. In some circumstances, the underlying investments may become illiquid which may constrain the ability to realise some or all of the portfolio.
The registration and settlement arrangements in emerging markets may be less developed than in more
mature markets so the operational risks of investing are higher. Political risks and adverse economic
circumstances are more likely to arise, putting the value of your investment at risk.
Other important information:
The Fund is a unit-linked life fund issued by Aberdeen Asset Management Life and Pensions Limited.
Nothing herein constitutes investment, legal, tax or other advice and is not to be relied upon in making
an investment or other decision. No recommendation is made, positive or otherwise, regarding individual
securities mentioned.
Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets
Efficient Hedging and Pricing of Equity-Linked
Life Insurance Contracts on Several Risky Assets
no website reference, taken from PDF.
by Alexander Melnikov and Yuliya Romanyuk
Equity-linked insurance contracts have been studied since the middle of the 1970s. The payoff of such policies depends on two factors: the value of some underlying financial instrument(s) (hence the term equity-linked ), and some insurance-type event in the life of survival to a certain date, etc.). As such, the payoff contains both financial and insurance risk elements, which have to be priced so that the resulting premium is fair to both the seller and the buyer of the contract.
The famous results of Black and Scholes (1973) and Merton (1973) tell us that, in an idealized market setting, as long as the seller receives a price equal to the expectation under the risk-neutral probability measure of the discounted payoff, the seller can hedge this payoff perfectly – with probability of successful hedging equal to 1. Perfect hedging relies on the ability to trade the financial asset(s) underlying the payoff and the option itself so as to offset any movement in the values of the underlying asset(s) and the option. However, mortality risk cannot be offset in the same manner, since mortality is not (yet) traded, which makes the insurance market incomplete and renders perfect hedging of equity-linked life insurance contracts impossible.
The goal of this paper is to illustrate how to apply efficient hedging to minimize the shortfall risk when dealing with equity-linked life insurance contracts written on several risky assets. The shortfall risk is defined as the expectation of the potential loss from the imperfect hedging strategy, weighted by some loss function reflecting the hedger’s risk preferences. In this paper, the terms ‘risk preference’ and ‘loss function’ refer to the attitude of the hedger (the insurance firm underwriting the policy) toward financial market risk.
We do not discuss general risk preferences of market participants in the context of economic equilibrium.
Rather, we consider whether the company selling an equity-linked policy is risk-averse, risk-indifferent, or possibly a risk-taker. Such an approach is taken in Foellmer and Leukert (2000), who develop the efficient hedging methodology in a mathematical finance-type context of optimal hedging under budget constraints.
Building on the work of Kirch and Melnikov (2005), we extend the application of efficient hedging in the insurance context, showing that this method is a powerful tool that allows for many quantitative risk management possibilities, while at the same time being computationally practical, understandable, and justifiable not only to academics, but also to practitioners in the insurance industry.
We consider a single-premium equity-linked life insurance contract that enables its holder to receive the greater of the values of several risky assets (such as stocks) at the maturity of the contract, provided the policyholder survives to this date. We prove an interesting probabilistic result, which we refer to as the multi-asset theorem, that allows us to derive explicit pricing formulas for payoffs involving n risky assets.
We show how to apply the theorem in the context of the efficient hedging of payoffs on two risky assets by calculating formulas for optimal prices and maximal expected losses resulting from imperfect hedging for each of the risk-preference cases. In this, we extend the work of Melnikov and Romaniuk (2006)1 from a budget-constrainted investor utilizing quantile hedging for a contract with a single risky asset to a multi- asset Black- Scholes- Merton-type setting. We use historical values of the Dow Jones Industrial Average and Russell 2000 stock indexes to show how a company that sells an equity-linked contract on these indexes can assess, quantify, and hedge the resulting financial and insurance risk components based on a given risk preference. The significance of risk preferences on optimal hedging strategies is also illustrated numerically.
The paper is structured as follows motivates the question of optimal pricing of equity- linked insurance in general, the particular payoff under consideration, and the choice of hedging methodology.
Then we review briefly the existing literature on the pricing of equity-linked insurance
and hedging of payoffs on multiple risky assets, and explain our paper’s contribution. In sections 4 we discuss the financial and the insurance market settings. In section 6 we present the formal set-up of the problem in the context of pricing of equity-linked life insurance contracts via efficient hedging.
We present the multi-asset theorem and the formulas for optimal prices and shortfall risk amounts for each of the risk-preference cases of the hedger. We illustrate theoretical results on the use of efficient hedging in general, when the investor is unable or unwilling to put up the entire amount needed for a perfect hedge. We expand this topic by looking at maximal expected losses that the investor could bear based on the investor’s risk preference theoretically, and provide a numerical example.
We examine risk management strategies that measure and balance financial and insurance
risk elements when efficient hedging is applied to price equity-linked life insurance contracts.
2. Motivation
The insurance industry has grown at a tremendous pace in the past decade, especially with the development of new markets in Europe and Asia. Equity-linked (also known as ‘index-linked’) contracts, and contracts paying one unit of some risky asset (‘unit-linked’), have been especially successful. For example, in discussing world insurance growth in 1997, Swiss Re reported high growth in the life insurance business in Europe, North America, and the emerging markets in Western Europe, noting that the high growth
rates were spurred in particular by dynamic business in unit-linked and index-linked insurance products
(Swiss Re 1997). The National Association for Variable Annuities cited that, in the United States, the total industry sales of equity-indexed annuities grew from 0.2 to 12.6 billion dollars between 1995 and 2003 (NAVA 2004). Additionally, the Spanish Institute for Foreign Trade reported that, in Spain in 2000, the greatest growth occurred in unit-linked insurance products: 81 per cent compared with 21 per cent in other types (Spanish Institute for Foreign Trade 2002). Winterthur Life achieved a strong and remarkable growth in unit- linked business in Hong Kong in 2003, and launched markets for unit-linked insurance in 2001 and 2002 in Japan and Taiwan, respectively (Winterthur Life 2004). The Canadian market for segregated funds2 has also been quite successful, raising around 60 billion dollars in assets in 1999 (Hardy 2003).
Equity-linked and unit-linked businesses incorporate a wide variety of products, including variable annuities (United States), unit-linked insurance contracts (United Kingdom), equity-indexed annuities (United States), and segregated funds (Canada). The payoff on the maximum of several risky assets is embedded in some form in most products mentioned above. For example, in its simplest form, the payoff on the maximum of the values of a risky asset and a deterministic guarantee is incorporated in segregated
funds and indexed universal life insurance, where premiums earn interest based on the performance of some risky fund but the insurance firm also guarantees a minimum rate of return. This payoff is also studied
widely in the literature, for example, as an ‘asset-value guarantee’ (Brennan and Schwartz 1976, 1979; Boyle and Schwartz 1977), or ‘minimum death or maturity guarantee’ (Bacinello and Ortu 1993; Aase and Persson 1994). The payoff on the maximum of several risky assets is embedded in variable universal life products and segregated funds, where the premiums can be invested in one of the available risky funds,
This is the name usually associated with equity-linked products in Canada and the investor can generally switch premiums to a better (more profitable) fund at certain dates. Since the payoffs considered in this paper are building blocks for a variety of equity-linked products, there is value to knowing how to price them properly. Due to the rapid growth of equity-linked business, it is important to address the question of ‘correct’ pricing of equity-linked products in general. From the perspective of the insurance industry, the effects of failing to adopt adequate pricing and risk management models can be devastating. If the company overestimates and overprices its risks, the consumer will bear the financial burden of excessive insurance premiums, which may lead to government inquiries and regulation. If, on the other hand, the company undervalues its risks, it may face substantial losses and lose the confidence of investors and shareholders.
In a stable and efficient economy, it is desirable to have all firms operating optimally in some sense; for example, without the risk of large losses or even bankruptcy. If a big insurance firm defaults, the negative effects of this event could be felt by the financial markets and the national economy, as well as individuals, shareholders, and clients of the firm. Clearly, some of the events causing large losses to insurance companies
cannot be predicted (natural disasters, terrorist activities, etc.). However, fluctuations in the prices of risky assets and mortality patterns can be analyzed quantitatively and qualitatively to help build proper pricing tools for insurance firms. Thus the question of finding hedging methodologies that can assess and value financial and insurance risks, and provide appropriate risk management strategies, is of great interest and significance from both theoretical and practical perspectives.
One of the impediments to risk pricing arises from the growing market demand for flexible and personalized insurance products. To respond to this demand and compete with financial institutions, insurance firms quickly develop and advertise, along with traditional life and health insurance, comparative products for investment and wealth management (variable annuities, segregated funds, etc.). The latter instruments are attractive to investors, since they tend to have shorter maturities and more exposure to financial market risk than traditional insurance contracts. Moreover, the ‘assurance’ component (a guarantee of some sort paid upon the death of the investor or upon surival to the contract’s maturity), together with generous tax benefits of equity-linked products, make them very successful alternatives to traditional investment instruments.
Before examining the pricing of equity-linked insurance products in more detail, we review how insurance firms hedge these contracts currently, and discuss whether these methods are appropriate. Several general practices of insurance companies have been pointed out in the literature. For example, Bacinello (2001) notes that mortality risk is generally not included explicitly in the valuation of insurance policies.
Instead, firms account for this risk by using ‘safety-loaded’ life tables where survival probabilities are ‘loaded’ to reflect mortality/survival risks. Dahl (2004) states that, traditionally, insurance firms calculate premiums and reserves based on deterministic mortality and interest rates, and that to compensate for this, firms overprice financial and insurance risks, which results in higher than necessary premiums as well as room for error in estimating proper mortality and interest rates.
To illustrate how such ‘actuarial judgment’ may fail, Hardy (2003) brings up the case of Equitable Life (a large mutual company in the United Kingdom). In the 1980s, U.K. interest rates were higher than 10 per cent, and Equitable Life issued many contracts with guaranteed annuity options, in which the guarantees would move into the money only if interest rates fell below 6.5 per cent. Relying on their personal judgment, actuaries at Equitable Life believed that rates would never fall below 6.5 per cent.
However, in the 1990s interest rates did fall below 6.5 per cent and the policies were cashed in, generating such large guarantee liabilities that Equitable Life was forced to close to new business.
In this setting, one may argue that life insurance firms could buy options to manage the risks they carry from reinsurance companies. In this way, life insurance firms could insure their own risks. While reinsurance companies have been selling options that could be used to manage risks inherent in equity-linked contracts, the prices of these options have been such that life insurance firms selling the original equity-linked policies would be losing substantial portions of their expected profits. On top of that, insurance firms would be undertaking another risk – the risk of default of the reinsurer. Moreover, in some markets
(such as segregated funds in Canada), reinsurance companies are becoming more and more reluctant to provide reinsurance at prices that are acceptable to insurance firms (Hardy 2003).
The International Accounting Standards Board (IASB) has highlighted the problem of improper pricing of financial and insurance risks (Biffis 2005). The IASB recognizes that mortality risk cannot be ignored in actuarial calculations and recommends that firms price both financial and insurance risks explicitly. The efficient hedging approach, discussed in this paper, allows the insurance firm to price the risks arising from volatility in the prices of risky assets and mortality fluctuations. Also, the firm is able to choose and control
its desired risk exposure for both financial and insurance risk elements. Efficient hedging is intuitive, since the minimization of potential losses is an obvious and natural criterion for risk management, and flexible, since it enables the hedger to incorporate risk preferences when pricing equity-linked policies. Finally, it allows users to derive explicit formulas for the premiums of contracts in consideration. While numerical solutions can be found to many problems arising in finance, it is still desirable to have closed-form solutions
to these problems.
3. Literature Overview
The topic of pricing of equity-linked insurance contracts became popular soon after the celebrated papers by Black and Scholes (1973) and Merton (1973) on the valuation of call options. Since equity-linked contracts incorporate both financial and insurance risk elements, perfect hedging in the sense of Black and Scholes (1973) and Merton (1973) does not work: the mortality risk of the option holder cannot be offset by trading in the insurance market, since mortality is not a traded asset.4 This section reviews some of
the research on the pricing of risks entailed in equity-linked insurance products.5
Early contributions to the pricing of equity-linked insurance include Brennan and Schwartz (1976, 1979), Boyle and Schwartz (1977), and Delbaen (1986); among later authors on this topic are Bacinello and Ortu (1993), Aase and Persson (1994), Ekern and Persson (1996), Boyle and Hardy (1997), and Bacinello (2001). Note that most of these papers do not price financial and insurance risks explicitly.
For example, Ekern and Persson (1996) calculate premiums for a large variety of equity-linked contracts, including those with payoffs where the contract owner chooses the larger of the values of two risky assets (and possibly a guaranteed amount) at the maturity of the contract, similar to the payoffs considered here.
But the authors disregard mortality risk, calling it “unsystematic risk,” for which “the insurer does not receive any compensation.” The justification provided is the traditional argument that mortality risk can As noted in Musiela and Zariphopoulou (2004), very few existing optimization-type approaches to optimal pricing of equity-linked insurance have produced explicit formulas that are intuitive and convey the idea behind the methodology.
However, it appears that a market for mortality is slowly developing. Special thanks to the anonymous referee of Melnikov and Romaniuk (2006) for this interesting and useful observation.
F
or a more detailed literature review, please see Romanyuk (2006). maybe eliminated by selling a large number of equity-linked contracts. Note, however, that pooling mechanisms
do not seem to work.
Another method for the valuation of equity-linked contracts stems from the utility-based indifference pricing approach, introduced by Hodges and Neuberger (1989) in the context of incomplete markets due to transaction costs. Here, the premium for the contract is calculated in such a way as to make the hedger (the insurance company, in our case) indifferent between including and not including a specified number of contracts in their portfolio. The method is extended to equity-linked insurance by Young and Zariphopoulou (2002a,b), who look at utility-based pricing when insurance risks are independent of the
underlying financial asset (as in the setting considered in this paper), and Young (2003), who considers the situation where death benefits payable to the policyholder depend on the value evolution of the underlying
financial asset.
Moeller (1998, 2001) incorporates financial and group mortality risk and determines the optimal hedging strategy as the one minimizing squared errors in future costs of the strategy. Our paper uses the efficient hedging methodology, whose goal is to minimize the expected losses of the hedger. This approach and the related concept of quantile hedging, where the probability of successful hedging is maximized, were developed by Foellmer and Leukert (1999, 2000) without the context of pricing and hedging of equity-linked contracts.
A number of papers adapt quantile and efficient hedging to the insurance setting to price equity- linked contracts (Krutchenko and Melnikov 2001; Melnikov 2004a,b; Melnikov and Skornyakova 2005; Kirch and Melnikov 2005; Melnikov, Romaniuk, and Skornyakova 2005; Melnikov and Romaniuk 2006). Our work extends the research in the above contributions, where payoffs include one or two risky assets, by studying the application of efficient hedging in a more general financial setting with n risky assets, whose returns are modelled by correlated Wiener processes. We also add to the existing results by examining quantitatively and qualitatively the maximal expected losses resulting from imperfect hedging based on the risk preference
of the hedger.
European-type contracts with payoffs involving several risky assets have been studied by Margrabe (1978), who calculates the perfect hedging price of an option to exchange one risky asset for another (see also Davis 2002), and Stulz (1982), who derives analytical formulas for prices of European call and put options on the minimum or maximum of two risky assets in the classical Black-Scholes-Merton-type setting. Johnson (1987) generalizes the latter result to payoffs with n risky assets using a change of numeraire technique, the
characteristics of call/put options, and the lognormal properties of the underlying assets. Boyle and Tse (1990) present a fast and accurate approximation algorithm to value options on the maximum or minimum
of several assets. Boyle and Lin (1997) obtain upper bounds for prices of call options on several assets without making any assumptions about the probability distribution of these assets. Laamanen (2000) further extends the result of Johnson (1987) to the payoffs on m best of n risky assets by utilizing a recursive approach in pricing calculations. We will derive a more general probabilistic-type result that allows us to value not only payoffs with several assets, but also to calculate directly expectations resulting from such payoffs being contingent upon other events; for example, when using efficient hedging to price
equity-linked life insurance contracts.
As indicated by the anonymous referee of Melnikov and Romaniuk (2006).
Related contributions include Spivak and Cvitanic (1999), Cvitanic and Karatzas (1999), Cvitanic (2000), Nakano (2004),
and Kirch and Runggaldier (2004).
Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets
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