University of Alberta

Applied Mathematics Institute

Stochastic Methods for

Assessment and Management of

Mortality Risk

Melnikov, A. and Romaniuk, Yu.

AMI-Preprint PREPRINT-NUMBER (02/2005)

Stochastic Methods for Assessment andManagement of Mortality Risk

Alexander Melnikov and Yulia Romaniuk

University of Alberta, Edmonton, Canada

Abstract

The paper discusses assessment and valuation of mortality risk with the helpof stochastic techniques. In particular, quantile hedging is used to comparethree classical survival models and to derive ideas and recommendations formortality risk management. Calculations are performed in a typical Blackand Scholes setting for an equity-linked insurance contract with guarantee.To illustrate the approach, the authors present comparative numerical resultsbased on mortality data for males and females in Alberta and Quebec.

Keywords and Phrases

Quantile hedging, equity-linked insurance contracts, survival models, mor-tality risk management.

1 Introduction

Motivation

In recent months, the life and health insurance industry has been attractingsubstantial media attention due to unusually high revenues earned in 2004.For instance, Sun Life Financial Inc. reported revenues of more than 22 bil-lion dollars, Manulife Financial Corporation earned over 16 billion, whereasGreat-West Lifeco Inc. and the Great West Life Assurance made over 13billion each (Financial Post Online at FPinfomart.ca).

While shareholders of these companies may rejoice at such financial success,these soaring earnings are causing many analysts to question pricing prac-tices of insurance firms. Even provincial governments (Alberta, in particular)

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are wondering whether insurance and financial products are overpriced andare considering imposing premium caps to make the prices of these productsmore favourable for consumers. Hence the great financial success may back-lash against the insurance industry in the form of government regulations andrestrictions. Some may argue that companies must increase insurance prod-uct premiums due to the instability in the current situation in North Americaand the world. Yet, despite threats of losses from terrorist activities, naturaldisasters or escalating liability costs from health care, the insurance industryseems to be doing well, as reflected by the high revenues.

Naturally, some questions arise: do insurance firms value the risks they en-counter too highly and transfer the financial burden to consumers? Do thefirms have appropriate methods to manage and valuate these risks? Arecompanies taking into consideration survival trends displayed by differentcategories of insured clients when measuring mortality risk? Recently, thetopic of mortality risk in particular has become the focus of attention of ac-tuarial and insurance companies (Pitacco (2003)). Many actuarial papers,conferences and symposiums deal specifically with the question of appropri-ate assessment and pricing of mortality risk. Survival and mortality patternsaffect basically all the products offered by insurance companies: life andhealth, accident and disability, as well as investment and wealth manage-ment. Thus the question of choosing the right survival/mortality model forthe given group of insureds to explain current mortality trends and forecastfuture life expectancy is of utmost importance for risk management and val-uation of insurance portfolios.

One of the impediments in risk pricing is the following. In their effortsto compete with financial institutions (banks, trusts, investment firms etc.),insurance firms develop, along with health and life insurance, comparativeproducts for investment and wealth management. While marketing special-ists are designing and advertising these innovative products to attract cus-tomers, analysts and actuaries in the same company must create correctpricing strategies for all the different contracts/agreements. However, it isoften impossible to develop and evaluate such tools, and take into accountmortality implications within the target audience before the products hit themarket, which results in mispriced portfolios and potential problems for cus-tomers and insurance firms.

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In this paper, the authors will show how stochastic techniques from math-ematical finance can be applied to help with the decisions of mortality riskassessment and pricing. In particular, quantile pricing methodology will beapplied to a basic unit-linked insurance contract with guarantee to derivesurvival probabilities of potential clients. Then, three classical mortalitymodels will be utilized to show how the choice of the model affects the agesof clients appropriate for the contract, which, in turn, influences decisionsabout the management of mortality risk. Furthermore, a comparative analy-sis of numerical results will be conducted to highlight the subtle differencesin the risk management approach when specifics of potential client groupsare considered.

The paper is organized as follows. First, we give a brief overview of thedevelopment of survival/mortality models, followed by some background onquantile hedging. Next, we explain the model, the contract payoff and itspricing. Then, description of three mortality models utilized in the paperis given, along with explanations of some actuarial concepts. Finally, wepresent numerical results, discuss their implications and conclude the paper.The ages of insured clients have been derived based on mortality data formales and females in the provinces of Alberta and Quebec, with the inten-tion of making a rough comparison between survival trends in eastern andwestern parts of Canada and the resulting implications for risk management.

Overview of Survival/Mortality Models

Humans have been trying to understand life and death for as long as theyexisted. Every culture and nation has legends about the origins and rea-sons for being born and dying. Scholars would formulate these ideas intoquestions about whether or not human survival is governed by some law,and if so, what it is and how science can explain it. The obvious sourceof information on this topic is birth and death data. As early as 1693, thefamous English astronomer Edmund Halley constructed a life table (tableshowing probabilities of survival based on a person’s current age) from ob-served number of deaths in Breslau (now Wroclaw, Poland). Soon after, in1740, the earliest life tables for males and females were published by NicholasStruyck (Pitacco (2003)). Around this time, some mathematicians becameinterested in modelling survival too; as a result, Abraham De Moivre pro-

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duced the first known analytic model for probability of survival as a linearfunction of the person’s current age. However, De Moivre recognized that hismodel failed to give accurate representation of human survival across all ages.

So the search for a better model continued, and in 1825 Benjamin Gom-pertz presented his version of the formula for survival probability, based onhis recognition that human mortality displayed some exponential patterns formost ages. His result is believed to be the most influential parametric mor-tality model in the literature. Some years later, in 1860, Makeham noticedthat Gompertz’s model was not adequate for higher ages and amended it inan effort to correct this deficiency (Higgins (2003)). Despite developmentsafter 1860 (including models by Thiele in 1872 and Wittstein in 1883), Gom-pertz’s and Makeham’s models remain to this day among the most popularchoices for mortality models.

In the early 20th century, Italian economist and sociologist Vilfredo Paretoput forth his idea for a model of mortality after working as an engineer andstudying the social problems of the day. In the 1940’s, Wallodi Weibull wasdevising a model to predict time until next failure of a technical system,which was later adapted as a survival model with human organs seen astechnical parts that eventually fail. Throughout the last century, there wereother contributions (for instance, Perks in 1932 and Beard in 1963), but mostof them were modifications/generalizations of those of Gompertz and Make-ham. In recent decades, the study of mortality has become more complexand modern. Due to ever-increasing computational capacities, parametricmodels may involve up to ten parameters (such as the model of Heligmanand Pollard (1980) with eight), or utilize parameters dependent on currentyear as well as the person’s age (Lee-Carter model from 1992). The newestdirection in the study of human survival is the idea of modelling mortalityas a stochastic process (see Yashin, 2001), but, as noted by Higgins (2003),the development of such models is in its infancy stage.

When we speak of models for the probability of a person’s survival for someinterval of time, we should distinguish between static (functions of age only)and dynamic (functions of age and current year) or deterministic versus sto-chastic models. In this paper, we consider the models of Gompertz, Makehamand Weibull, which are deterministic and static. Future work will includean expanded analysis to the more recent developments of Lee-Carter and

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Yashin, in particular. It is worthwhile to stress that currently Gompertz-and Makeham-based models are typically used for educational, forecastingand risk-valuation purposes (Pitacco, 2003).

Background on Quantile Hedging

Equity-linked insurance contracts have been studied since the 1970’s. Thepayoff in such contracts depends on the performance of some risky asset (suchas a stock index) and some event in the life of the insured client (survival tomaturity of contract or accident, retirement, death, etc.). Thus equity-linkedcontracts combine financial and insurance (mortality) risk elements. Due tothe presence of these different risk types, pricing of such agreements is nottrivial; at first, typically, they are reduced to a call/put option to simplifycalculations, and then perfect, super- or mean-variance hedging is applied(see, for instance, Brennan and Schwartz (1979), Bacinello and Ortu (1993),Aase and Persson (1994), Ekern and Persson (1996), Boyle and Hardy (1997),Moeller (1998)). However, in 1999, Foellmer and Leukert (1999) proposeda new pricing and hedging methodology, where the probability of success-ful hedging is maximized, subject to some constraints on the initial hedgingcapital.

The reader will soon see that quantile hedging is perfectly suited to priceequity-linked insurance contracts, as shown by Melnikov (2005) and Mel-nikov et al (2005). We deal with a unit-linked contract with a guarantee,where the insured may choose the greater of the fund value or some deter-ministic amount at maturity of the contract, provided that he/she is aliveuntil this maturity date. Due to the client’s mortality risk, the premiumhe/she pays for the agreement is less than the capital needed by the insur-ance firm to invest for a perfect hedge. Thus the company faces default risk,since the probability of failure of the hedging strategy is positive. This iswhere quantile hedging proves most useful. Using this pricing and hedg-ing methodology, the company has two possibilities: either to maximize theprobability of successful hedging, or to fix some acceptable level of financialrisk and then choose the clients for the contracts accordingly to preserve thislevel of risk. We will explain these ideas in more detail later in the paper,but for now let us look at the setting and model description.

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2 Description of the Model

Financial Setting

We work in a typical Black and Scholes setting: a financial market withnonzero interest rate r > 0, one riskless money market account B and onerisky asset S satisfying

dBt = rBtdt, B0 = 1; dSt = St(µdt + σdWt),

respectively, with W a Wiener process, µ ∈ R, σ > 0. All processes aregiven on a standard stochastic basis (Ω,F ,F = (Ft)t≥0, P ) and are adaptedto the filtration F, generated by W . Every predictable process π = (πt)t≥0 =(βt, γt)t≥0 is called a trading strategy (or portfolio) with time t value

Xπt = βtBt + γtSt.

Only self-financing (with no additional inflow/outflow of cash other than ini-tial price payment) and admissible (with nonnegative capital) strategies areallowed.

The contract, as we mentioned before, entitles the client to one unit of somerisky asset or a guaranteed amount, whichever is greater, at expiration date.The payoff H has the form

H = maxST , K = K + (ST −K)+, (2.1)

where K is the deterministic guarantee. For a more general version with twofunds see Melnikov et al. (2005). Clearly, the price of such payoff reducesto the Black-Scholes case (for two risky assets, see Margrabe (1978), Ekernand Persson (1996) and Davis (2002)). We know that in this setting, aunique risk-neutral measure P ∗ exists, and the price for (discounted) H isits expectation with respect to P ∗.

Insurance Setting

Following actuarial tradition, we use a random variable T (x) on a proba-bility space (Ω, F , P ) to denote the remaining lifetime of a person of currentage x. We can safely assume that the mortality risk of clients and financialmarket risks have no (or very minimal) effect on each other, hence the two

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probability measures (P and P ) are independent. Since the payoff H de-scribed above is linked to a client’s survival to maturity T of the contract,its fair price x is

x = E∗ × E[He−rT IT (x)>T]

= E∗[He−rT ]PT (x) > T = E∗[He−rT ]T px, (2.2)

where T px = PT (x) > T denotes the probability of a person aged xsurviving T more years.

Optimal Pricing and Hedging Strategy

Obviously, the fair price for the above equity-linked life insurance contract islower than the (perfect hedge) price of the purely financial contract with thesame payoff. This means that the insurance firm offering the contract hasless money than needed to hedge the contingent claim with probability 1.In this situation, a good alternative to perfect hedging (which will fail withpositive probability) is quantile hedging. This method allows the companyto find an optimal hedging strategy π∗ that maximizes the probability ofsuccessful hedging under the initial hedging capital restriction:

Pω : Xπ∗T ≥ H = max

πPω : Xπ

T ≥ H with x < E∗[He−rT ]. (2.3)

The main result of quantile hedging, proved by Foellmer and Leukert (1999)and developed in connection to insurance contracts by Melnikov et al. (2005),is the following:the optimal hedging strategy π∗ is the perfect hedge for the modified contingentclaim HIA∗, where A∗ is the maximal set of successful hedging for the payoffH; moreover, A∗ = Z−1

T ≥ aH, where Z is the density of the risk-neutralmeasure P ∗.In our case, the maximal set of successful hedging becomes (see above refer-ences)

A∗ = ST /K < c, (2.4)

with constant c determined from the level of financial risk undertaken by theinsurance firm, or from the initial budget constraint.

From above considerations and (2.2), we get these equalities for the priceof the equity-linked contract in consideration:

x = E∗[He−rT ]T px = E∗[He−rT IA∗ ], (2.5)

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which, along with (2.1) and (2.4), lead to the following formula for the sur-vival probability of the insured client:

T px =E∗[He−rT IA∗ ]

E∗[He−rT ]

=Ke−rT P ∗ST /K < c+ E∗[(ST −K)+e−rT IST /K<c]

Ke−rT + E∗[(ST −K)+e−rT ]. (2.6)

Obviously, the second term in the denominator is the classical Black-Scholes(1973) call option price:

E∗[(ST −K)+e−rT ] = S0Φ(b+(1))−Ke−rT Φ(b−(1)),

where b±(n) =ln( S0

K·n) + (r ± σ2

2)T

σ√

T,

and Φ(j) =1√2π

∫ j

−∞e−y2/2dy. (2.7)

Relation (2.6) is key in our analysis, as it gives a quantitative connectionbetween the financial and insurance risk components in the following way.If the insurance firm chooses to market the equity-linked contract in con-sideration to clients of any age, it may do so and still use quantile hedging,maximizing its probability of success. So, by knowing the age of the client,his/her survival probability (left-hand side of (2.6)) can be derived from someappropriate mortality model, allowing us to find c from the right-hand sideof the equation and thus calculate P (A∗). Note that this maximal probabil-ity of successful hedging may not fit the company’s desired risk profile. Or,we can utilize (2.6) in reverse: first, choose some acceptable financial risklevel, calculate c, and then find the survival probability of clients to whomthe contract should be offered. Then, again using some particular mortalitymodel, clients’ ages can be found. This second approach is precisely the ideawe will illustrate in detail later in the paper.

After some calculations (skipped here, as they are not the focus of the paper),we find that the survival probability is determined as follows:

T px =

KΦ(−b−(c)) + S0erT [Φ(b+(1))− Φ(b+(c))]−K[Φ(b−(1))− Φ(b−(c))]

K + S0erT Φ(b+(1))−KΦ(b−(1)),(2.8)

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with Φ and b given in (2.7).Now that we are done with the financial aspect of our analysis and have

a way to calculate clients’ survival probabilities, let us consider the insur-ance part. Following is a description of the survival models selected for thenumerical illustration, as well as a discussion of how the model affects thecompany’s exposure to mortality risk.

3 Mortality Models

Choice and Descriptions

In this section, we summarize the main contributions of Gompertz, Make-ham and Weibull to survival modelling and discuss our choice of models. Bydefinition, the survival function s(x) is the probability that a newborn willlive to attain age x. We also have this relation:

tpx =s(x + t)

s(x). (3.1)

As mentioned above, Abraham De Moivre was the first to give an explicitformula for s(x) (which we omit here due to its limitations); Gompertz andMakeham followed in the 19th century, and Weibull in the 20th with theirversions for survival functions:

Gompertz: s(x) = e−B

ln(c)·(cx−1) , x ≥ 0, c > 1, B > 0;

Makeham: s(x) = e−A·x− Bln(c)

·(cx−1) , x ≥ 0, c > 1, B > 0, A ≥ −B;

Weibull: s(x) = e−k

n+1·xn+1

, x ≥ 0, n > 0, k > 0. (3.2)

Among the many possibilities for survival models (see introduction above) wechose these three out of the following considerations. First of all, we includeGompertz’s and Makeham’s models since they are widely used in actuarialscience today, and they served as foundations for most of the later survivalmodelling contributions (such as the formulas of Perks, Beard etc.). Second,we choose Weibull’s model as it represents a different perspective of analyzingmortality, through the wear-and-tear/technical failure point of view. As faras we know, Weibull’s idea is not based on previous works. Finally, mostscholars consider these three models classical, and we want to compare theirrisk assessment and valuation implications.

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Parameter Estimation

Before discussing numerical results, parameter estimation considerations andtechniques should be described briefly. For the risky asset, we chose theToronto Stock Exchange/Standard and Poor Composite Index (S&P/TSXclosely mirrors the performance of some 300 Canadian and US companies).Return µ = .0911 and volatility σ = .1573 were estimated using daily datafrom Jan. 1, 1995 to Jan. 1, 2005. The initial value S0 = 9246.7 was theprice of the index as of Dec. 31, 2004 going into 2005.

We chose four maturities for the contract: T = 1 , 5 , 10 and 20 years tolook at the variations in risk management between shorter and more lengthyagreements. Note that shorter-term contracts are more likely to be offeredunder the umbrella of wealth management/investment products, whereaslonger contracts belong to the insurance products category. The interest rater = .05 used in calculations was the yield of a 10-year Canada T-Bond. Thefour constant guarantees were K = 10000, 12000, 15500 and 25300, which areslightly overestimated values of the accumulated amount of the initial priceof the risky asset if invested in a riskless money market account at 5 percentinterest rate over 1, 5, 10 and 20 years respectively.

To estimate the parameters for the three survival functions (3.2), we usedAlberta and Quebec Life Tables for males and females, compiled by StatisticsCanada based on birth and death data in these provinces between the years1995 and 1997. The Life Tables are available online (http://www.statcan.ca/english/freepub/84-537-XIE/tables.htm), together with a detailed descrip-tion of the method used to derive the numbers for survival probabilities andother information in the Life Tables. It is worthwhile to point out that, sim-ilarly to the problems with Gompertz’ and Makeham’s formulas, that fail toreflect mortality well for very old ages, the model used by Statistics Canadaalso needed some smoothing procedures to calculate Life Table values forages over 85 (see files online for details).

The authors made every attempt to have some consistency when estimat-ing parameters. The index chosen for the risky asset is expected to exist fora long time (whereas if a particular company’s stock were chosen, one couldnot be sure if it would still be available on the stock market in 20 years). Datafor the index was from the past 10 years, and included the years during which

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birth and death data was collected and used for Life Tables calculations; 10years is also the approximate average maturity of our equity-linked contracts.

Survival model parameters were derived from probabilities that a male/femaleaged x = 40 will live another T = 5, 10, 20 years (T = 1 was avoided as thevalues in Life Tables themselves were calculated for 5-year periods and thenextrapolated for the intermediate years). The age of 40 was chosen out ofconsideration that most survival models are quite successful at explainingmortality trends for ages that are not too young or too old; 40 seemed likea reasonable ’middle’ value. All calculations were performed using Maplesoftware.

While being aware that there is room for improvement in the estimationof all the above parameters: more precision could be achieved by using dataencompassing the same number of years throughout, choosing appropriateinterest rates for the duration of the agreement etc., we expect that thenumerical results below are still quite reasonable and consistent. They illus-trate the idea of using quantile hedging to assess the feasibility and mortalityrisk implications of the three survival models, calling attention to subtle dif-ferences in risk management when dealing with different groups of clients.

4 Numerical Results

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Analysis of Results

First, let us look at some trends across all survival models and contractmaturities. Recall that the firm selling the contract chooses the level of fi-nancial risk it is willing to bear; the higher the risk, the older the insuredclients should be. This is not surprising: if company A wants to hedge withprobability of success 99 percent, it carries very little financial risk, so A canafford to sign contracts with younger clients. Company B, carrying 10 per-cent risk, must compensate for this higher chance of default and thus offercontracts to older clients.

Another expected result is the decrease of ages with the maturity of con-tract. If the contract expires in 5 years, there is a lower probability thatthe insured client will not live to collect his/her payoff than with a 20-yearagreement. However, there is a clear difference in ages for all the maturi-ties for males and females. Clearly, Alberta and Quebec residents adhere tothe well-known mortality trend: women have higher probability of survivalthan men (this is true for any given age). We see this result above. For allcontracts, the ages of suitable female clients are higher than those of maleclients: for a given contract, it is ”safer” to attract younger men than women,as men are more likely to die before expiration.

Now, let us compare the different models. As can be seen on the graphs,Makeham’s survival function gives no real solutions for ages for higher matu-rities. Gompertz’ and Weibull’s models seem to be best suited for this typeof risk assessment. Moreover, looking at the graphs above, we notice thatWeibull’s model consistently gives higher ages, meaning that it exposes thefirm to less risk when choosing appropriate clients. Thus a risk-adverse insur-ance firm should choose the Weibull model to estimate and price mortalityrisk. On the other hand, a risk-taker will prefer Makeham’s or Gompertz’model. One may argue that age differences suggested by all models are neg-ligible; however, we must remember that an insurance firm will most likelyoffer the same contract to thousands of persons. The ages and their con-sequent mortality predictions, given here for the ’average’ client, will tendto become true for the group (by law of large numbers). Thus, if the com-pany defaults on only a couple dozen out of a thousand contracts becauseof choosing some arbitrary or inappropriate mortality model, it may face asignificant chance of default and perhaps even bankruptcy.

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As a matter of fact, we feel that what happened in the past couple of yearsis the opposite; insurance companies have been appreciating their mortalityrisks too much to avoid the possibility of bankruptcy. Thus all or a majorityof the available insurance/investment products were overpriced, relative tothe level of risk carried by the companies, which may explain the unusuallyhigh revenues collected in the last year. And, as mentioned before, the in-surance industry may very likely face some negative repercussions from thegovernment, which is exactly what is happening today with the auto insur-ance sector in Alberta.

Another interesting trend observed across the two provinces and all con-tract maturities is the following. For shorter-term contracts, survival modelsagree in age predictions for lower levels of financial risk, but become increas-ingly different as the financial risk increases. The opposite trend appears forlonger-term contracts (T = 20, for instance): here, the riskier the financialsituation, the more agreement on age between the models. For contractswith intermediate duration, all models predict similar ages.

Although parameter estimation may be partly responsible for this effect,we believe that there is a deeper underlying cause: the inherent ’riskiness’ ofthe models. Compare a 1-year and a 20-year contract; obviously, the latter issafer in the sense that for any age, probability of dying in 20 years is higherthan in 1 year, so clients for longer contracts can be younger than those forshorter agreements. So, with the riskier 1-year agreement, the safer Weibullmodel dictates choosing not just old, but really old clients to compensate forthe increasing financial risk level. On the other hand, with the safer 20-yearcontract, the firm can relax and focus on marketing to younger clients. Here,the riskier models of Gomperts and especially Makeham recommend attract-ing really young clients.

Since the effects of the different survival models are most felt for really shortor quite long contracts (as can be seen from age differences between themodels for T = 1 and T = 20), actuaries and financial analysts of insur-ance companies should take extra care in choosing the survival model thatbest fits the desired risk profile of the shareholders. These considerationsare applicable especially to investment portfolios, which tend to be of shortduration, and the typically longer insurance products.

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Now, let us look at some variations by province. One surprising discov-ery is that the models of Gompertz and Makeham are almost identical inpredictions of ages among Quebec females, yet differ for other groups. Fur-ther study may be necessary to explain this phenomenon; however, it may bea pure coincidence. Another interesting trend is the following: ages for malesin Alberta are slightly higher than in Quebec, suggesting greater longevityof Alberta men. On the other hand, there is no significant difference in agepredictions and thus life expectancy for women in the two provinces. Suchtendencies are also observed in Life Tables through the values of survivalprobabilities for males and females in Alberta and Quebec.

The exceptions to the above pattern are short-duration contracts offered tofemales: as seen in the graphs above, ages for the 1- and 5-year contracts arehigher for Quebec than Alberta. Now, at first sight, this seems to contradictthe observations in the previous paragraph; however, such peculiarity canbe explained looking at Life Tables. For older ages, women in Alberta havehigher survival probabilities than those in Quebec, but the rate of change ofsurvival probabilities is greater in Quebec. For instance, between the ages of70 and 80, yearly survival probabilities for Albertans dropped 2.8 percent asopposed to 3.2 percent in Quebec. This implies that Alberta female popula-tion is more stable, thus less risky, explaining the lower age predictions forAlberta women for shorter contracts.

Clearly, the results presented and discussed above are just an illustrationand can be improved. More precision could be achieved by using data whichcovers the same time period for estimating interest rate, the return andvolatility of the risky asset or model parameters. Also, if an insurance firmhas access to more specific mortality data for the desired client profile (notonly by place of residence and sex, but also by health status, education or oc-cupation), parameters estimates for survival functions would reflect expectedmortality behaviour among the target client group better. Nevertheless, webelieve that the numerical results and risk assessment ideas above are repre-sentative of the situation for predicting and comparing mortality trends onany scale.

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5 Conclusion

The paper focuses on applications of stochastic techniques in insurance riskvaluation and management by showing how quantile hedging can be used toassess feasibility and risk implications of the three classical survival modelsof Gompertz, Makeham and Weibull. The idea for such analysis comes fromrecent media attention to the unusually large revenues in the life and healthinsurance industry. The authors believe that high appreciation of mortalityrisk due to its incorrect valuation by insurance firms, hence overpriced insur-ance products, may be responsible for the billion-dollar revenues. Insightsfor a more careful study of survival behaviours based on the different models,mortality risk assessment ideas for insurance firms, and recommendations forfuture research are given as well.

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Melnikov, A. V. (2005) On Quantile Hedging and Its Applications tothe Pricing of Equity-Linked Life Insurance Contracts. Theory Probab.Appl. 2.

Melnikov, A. V., Romaniuk, Yu. V. & Skornyakova, V. S. (2005)Margrabe’s Formula and Quantile Hedging of Life Insurance Contracts.In English: Doklady Mathematics 71 (1), 31-34; in Russian: DokladyAcademii Nauk 400 (2), 1-4.

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Pitacco, E. (2003) Survival Models in Actuarial Mathematics: from Hal-ley to Longevity Risk. Invited lecture at 7th International CongressInsurance: Mathematics & Economics, ISFA, Lyon.

Yashin, A. I. (2001) Mortality Models Incorporating Theoretical Conceptsof Ageing. Forecasting Mortality in Developed Countries, E. Tabeau,A. van den Berg Jeths, C. Heathcote (eds.). Netherlands: KluwerAcademic Publishers, 261-280.

Alexander MelnikovDepartment of Mathematical and Statistical SciencesUniversity of Alberta, Edmonton, Canada T6G2G1email: melnikov@ualberta.ca

Yulia RomaniukDepartment of Mathematical and Statistical SciencesUniversity of Alberta, Edmonton, Canada T6G2G1email: romaniuk@ualberta.ca

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