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Life Reinsurance Pricing – Reinsuring The “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts Copyright 2005 PartnerRe 96 Pitts Bay Road Pembroke HM 08, Bermuda Authors Frank Pinette, Head of Life PartnerRe, Paris [email protected] Paul-Antoine Darbellay Pricing Actuary PartnerRe, Zurich Photos and graphs PartnerRe Keystone Corbis Image For more copies of this publication or for permission to reprint, please contact: Celia Powell Corporate Communications 96 Pitts Bay Road Pembroke HM 08, Bermuda [email protected] Telephone (1 441) 292 0888 Telefax (1 441) 292 70 10 This publication is also available for download under www.partnerre.com Revised reprint May 2005 Original publication, December 2000 Introduction The investment risk can cause serious concern. Stock markets are volatile and annual fluctuations of around 40% are not unrealistic. This research paper discusses a number of aspects relating to the “guaranteed minimum death benefit” in unit-linked contracts. It deals briefly with the different types of guarantees found in the market, then examines the forms of reinsurance that are potentially suitable and assesses the risks associated with such covers. Finally, by using the Black & Scholes model, it analyzes both a pricing method and a method of reserving that focus on the important differences between the so-called “traditional” and “financial” risks in reinsurance. These fall into two key categories: random fluctuation and systematic risk (accumulation of risk). Therefore it is in the reinsurer’s best interest, when covering these guaranteed minimum death benefits, to carefully evaluate its maximum level of liability. Unit-linked products have become very popular in the French insurance marketplace over the last four years. The policyholder premiums are PartnerRe 2005 Reinsuring the “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts invested in investment funds with the policyholder assuming the investment risks. These products are usually sold with an attaching death benefit. The most common types of death benefit available in the market are: - The “guaranteed minimum death benefit” that protects the beneficiary, in the event of the insured’s death, against a possible drop in the value of the unit-linked fund and also guarantees a pre-determined minimum equal to the total value of the premium paid. - The indexed (interest-rate) guaranteed death benefit that provides the beneficiary, in the event of the insured’s death, with similar protection as above and, in addition, a pre-determined interest-rate minimum guarantee on the premium paid (for example 3.5%). 3 The Different Forms of Reinsurance Two forms of reinsurance exist: Individual Method This method consists of reinsuring, on a weekly or monthly basis, for each individual insured the difference between the amount guaranteed in the event of death and the surrender value, when the latter is lower than the guaranteed minimum death benefit. A weekly or monthly mortality table is applied to this difference. This may be the ideal method but it is constraining for the ceding company and requires a sophisticated administrative backup system. Global Method Premium expressed as a percentage of the reserves. In this method, the reinsurer prices the death risk with a premium loading calculated on the annual mathematical reserve. The reinsurer’s approach is a global one and the price is determined according to average age and to underlying asset volatility. The obvious drawback of this method is that the ceding company pays a high premium when the sum at risk reduces (increase in reserves) and conversely. Premium expressed as a percentage of the original premium. As with the method above, this is a global method whereby a reinsurance premium is payable on the premium contributed by the insured. This method is the focus of this paper. Guaranteed Minimum Death Benefits: The Reinsurer’s Viewpoint Firstly, we examine the risks associated with this guarantee and then the pricing method covering the assumptions and related problems and finally the issue of reserving. The Risk Traditional Life Insurance Contracts In traditional life insurance contracts, the investment risk is held by the direct insurer. The direct insurer assumes both the mortality and the investment risk. The latter is generally small as the capital is usually invested in investment assets with moderately low volatili- 4 PartnerRe 2005 Reinsuring the “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts ty. The mortality risk can be controlled when reliable statistics are available for a relatively substantial insurance portfolio. When dealing with this type of portfolio, the capability to evaluate significant numbers of similar risks from many sources reduces the probability of wide fluctuations and it is possible to forecast the number of deaths fairly accurately i. e. the law of large number applies. Unit-Linked Contracts In unit-linked contracts, the investment risk is borne by the insured. The investment risk comes from stock market fluctuations in the value of the assets representing the insured’s assets. However, this no longer holds entirely true when a guaranteed minimum death benefit is included in the policy providing a death benefit that would never be less than a predetermined minimum. This means that if, at the time of the insured’s death, the asset value has fallen below the level of the guaranteed minimum sum, the insurer will pay the difference. Therefore there are two essential features to this type of contract: the mortality risk and the investment risk. As pointed out above, unlike the stock market fluctuations, the mortality risk can be controlled. The investment risk can cause serious concern. Stock markets are volatile and annual fluctuations of around 40% are not unrealistic. The mortality risk in this type of cover is in essence a minor feature compared with the investment risk. There is also an additional risk to be considered: the systemic risk. Stock market fluctuations impact on all contracts of the same type. There is no spreading of risks between different forms of contract; on the contrary, the danger posed by accumulation of risk is greater. Furthermore, wide market fluctuations may have serious impact on both the insurer and the reinsurer and reduce their capacity to assume their liabilities. Pricing Method Calculating The Premium A solution for pricing this type of product is to combine death and survival probabilities (mortality risk) with financial derivatives (investment risk). The Black & Scholes Model Using the Black & Scholes model, we can calculate the price of a European put. The basic option-pricing formula is as follows: Specifically, r T X ST = The force of interest corresponding to the risk-free rate of return = Time until expiration of the option contract = The exercise price (guaranteed minimum death benefit) = Price of the underlying stock at T (random variable) We can use the following formula, where: SP = Single reinsurance premium qx = Probability of death between ages x and x +1 p = Probability for a policyholder aged x to i x be alive at age x + i Pi = Present value of a European put option exercised at time i i = Number of years covered The first part, multiplying the survival and death probabilities, is the equivalent of a term assurance without any discount factor (already included in the put price). This step enables us to determine the number or percentage of a group of insureds whose death benefit may have to be increased if the said benefit were to be lower than a pre-determined minimum. As the price of this cover is given by the put, all that is necessary is to multiply our probabilities by the put price. Summing over the number of contract years, we obtain the single reinsurance premium. The price of a European put dictates for the buyer the right, not the obligation, to sell underlying stock at a pre-determined price, the exercise price, and at a pre-determined date, the expiry date of the option contract. We now examine how the put price is obtained more closely. PartnerRe 2005 Reinsuring the “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts The difference between the exercise price and the underlying stock price is covered by the reinsurer if the underlying stock value is lower than the guaranteed minimum death benefit (the exercise price). This is the reason why we apply the maximum of this difference and zero. We then determine the expected value of this maximum to obtain the mean reinsurance coverage that we discount at the risk-free interest rate. The Black & Scholes model is based on strict assumptions. It is worthwhile examining these assumptions that we apply implicitly when we use this model to evaluate an option or devise a cover strategy. The Assumptions The Black & Scholes model assumes that the financial markets are efficient, that the interest rate is constant, and that the option price is contingent upon the risk-free interest rate, the underlying security, its lifetime, its volatility and the exercise price. It also assumes that share price movements follow a continuous stochastic log-normal distribution. It is possible to set up a perfect “risk-free” cover by using this model (as it assumes that the gains and losses from buying securities and buying/selling options cannot diverge from one another for any length of time). The Black & Scholes model implies that this situation is, at any time and continuously, risk-adjustable, according to the movements in stock 5 prices, ensuring that the cover remains risk free until the expiry of the option. true for a call) and that, as a general rule (although this depends on the discount period), it is an increasing function of the time until expiration (the same as for a call). Furthermore it is a decreasing function of the risk-free rate (the opposite is true for a call) and finally, it is a decreasing function of the underlying stock price (the opposite applies to a call). This hypothesis suggests that markets are efficient. In an efficient market, prices reflect, instantaneously and without bias, all (price) relevant information relating to the evaluation of financial instruments, and the market recognizes all information and follows it correctly. Also, it implies that there are no arbitrage opportunities (arbitrage being the possibility of making a riskless profit by, for example, buying stock on one market and selling it immediately at a higher price on another market, to outperform the market). Pricing Problems How valid are the assumptions of the Black & Scholes model in the context of insurance? The duration of the cover and the volatility calculation are undoubtedly the most problematic assumptions when applying this model to insurance. The Parameters Pricing an option using the Black & Scholes model is derived from the five following parameters: The Black & Scholes model is normally used for short intervals of time (2, 3, 6 months) and not for periods covering several years as is the case for life insurance. As market conditions are constantly changing, predicting market behavior is usually a short-run description, not extending over a number of years. Thus the model’s parameters change over time presen- - Price of the underlying stock - Exercise price (guaranteed minimum death benefit) - Risk-free interest rate - Duration - Price volatility of the underlying stock 350 Stock Price 300 250 200 150 100 120 115 110 105 95 100 85 90 80 70 75 65 60 55 45 50 35 40 25 30 20 15 5 0 0 10 50 Time [Days] Simulation of the stock price using a risk-free interest rate of 3% and a volatility of 25 %. Thus the price of the option does not depend on the return from the underlying stock but on the risk-free rate. This is derived from the assumptions used in the Black & Scholes model as it is possible to obtain a perfect riskfree cover. The model does not depend on expected share returns and ignores any parameter representing market participant preferences. It works whatever the attitude investors may have towards risks. It is worth noting that the put is an increasing function of volatility (the same as for a call) and also of the strike price (the opposite is 6 PartnerRe 2005 Reinsuring the “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts ting a completely different price option forecast. Present market conditions indicate an upward movement in asset volatility. How should volatility be measured then over relatively long intervals of time? Note again that in the European-style options model the underlying stock does not pay dividends so market capitalization is implied. However, the insurer generally deducts costs so the total value of the dividends is not reinvested. This creates a bias. Bearing these problems in mind, it is worthwhile comparing results obtained from the Black & Scholes model with more pragmatic methods such as the maximum probable loss or average mean probable loss or, quite simply, with the results obtained from using one’s own market fluctuation predictions. Reserving Here we need only revert to the traditional mathematical reserving formula. The following is a possible formula. The present value of future benefits represents the current value of benefits due to the client in the event of death when the value of the fund falls below the guaranteed minimum capital. Mathematical reserve = Present value of future benefits – Present value of future premiums If we use the single reinsurance premium method described above, the formula can be written as follows: Mathematical reserve at time interval t = Present value at time interval t of future benefits = This means recalculating the new current value of future benefits every year. This requires applying new market measurements and recalculating the single premium. We now need to analyze the parameters that may change from one year to the next more closely. The price of the underlying security will obviously vary each year as will the risk-free rate and the duration. The volatility is readjustable yearly in correlation with empirical data. Only the exercise price remains constant. It is quickly apparent that these changes in the Black & Scholes model parameters may result PartnerRe 2005 Reinsuring the “Guaranteed Minimum Death Benefit” Attached to Unit-Linked Contracts in substantial fluctuations in the evaluation of the annual mathematical reserves. Conclusion We have briefly examined how to reinsure and reserve guaranteed minimum death benefits attached to unit-linked contracts. We have noted that these methods derive from strict assumptions and that deviations in these assumptions impact widely on the results obtained. Our underwriting methodology is based on the Black & Scholes model and consequently on financial derivatives. This means that the reinsurer sells a put anticipating a bullish or flat stock market whereas the cedant buys a put to protect against a bearish market, each party positioning itself accordingly with its market expectations. This type of contract requires that we evaluate the parameters we use as accurately as possible. However, as we have seen, we have to confront the problem of long-term coverage in life insurance. This implies that when evaluating the criteria used in pricing this type of risk we cannot allow for the actual evolution of market behavior, which will always be different from what we anticipate today. Furthermore, there is an important gearing effect in this type of contract. For example, the total risk exposure could be 40 times the premium. The reinsurance cover contains an important financial component. As we have seen, the reinsurer’s liability is triggered by mortality. The policyholder needs to die to trigger the reinsurer’s liability but, as we have pointed out, this mortality risk can be controlled: we can easily predict the mortality rate of a reasonably sized portfolio. However, an even greater trigger is the performance of the stock market at the time of death. It is obvious that it is the stock market that governs the reinsurer’s liability. Furthermore, we have to remember that there is no spreading of risks between different forms of contract; on the contrary, the risk is systemic. As the risk is systematic, it is vital for the reinsurer to establish the level of potential loss limits not to be exceeded (accumulation control). By calculating the maximum amount to be covered (the total sum at risk, all contracts combined), the reinsurer will be able to meet all potential liabilities at all times. 7 www.partnerre.com