Valuation of contingent claims with mortality and interest rate risks

Mathematical and Computer Modelling 49 (2009) 1893–1904

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Valuation of contingent claims with mortality and interest rate risks Luka Jalen a , Rogemar Mamon b,∗ a

Department of Mathematical Sciences, Brunel University, Uxbridge, United Kingdom

b

Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada

article

info

Article history: Received 24 June 2008 Received in revised form 21 October 2008 Accepted 27 October 2008 Keywords: Stochastic models Mortality Bayes’ theorem Forward measure Death and survival benefits

a b s t r a c t We consider the pricing of life insurance contracts under stochastic mortality and interest rates assumed not independent of each other. Employing the method of change of measure together with the Bayes’ rule for conditional expectations, solution expressions for the value of common contracts are obtained. A demonstration of how to apply our proposed stochastic modelling approach to value survival and death benefits is provided. Using the Human Mortality Database and UK interest rates, we illustrate that the dependence between interest rate and mortality dynamics has considerable impact in the value of even a simple survival benefit. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Human mortality has improved significantly over the last few decades as documented in many actuarial and scientific publications (see for example Macdonald et al. [1], Currie, Durban and Eilers [2], and Renshaw and Haberman [3]). Although this is a positive development it brought considerable stress in pension and health care support for the elderly. Furthermore, contrary to traditional and deterministic approaches to mortality modelling, mortality is now widely accepted to be evolving in a stochastic fashion. As mortality is by its very nature a primary source of risk for a large number of products in life insurance, pensions and some other recently issued financial instruments it is imperative to understand better its dynamics. In particular, recent mortality trends have proved particularly challenging for the pricing and reserving of longterm mortality-linked contracts, such as contracts providing living benefits. Mortality will certainly continue to improve in the future with the advances made in the health sciences and medicine. This realisation, however, was not incorporated in mortality modelling even in the late 1970s. As noted in Bolton et al. [4], Boyle and Hardy [5], and Hardy [6] actuaries still use out-of-date mortality tables without explicit allowance for future mortality improvements when pricing and reserving for mortality-based contracts. Additionally, it is well known that insurance companies are also exposed to financial risks, and since their investments are predominantly on fixed income investments it means they are also heavily exposed to interest rate risk. However, there is still a considerable gap in the tools available for modelling these two types of risk. As pointed out in Cairns, Blake and Dowd [7], stochastic modelling of interest rate is very well developed whereas the theory of stochastic mortality risk modelling is still at its infancy. It can be observed that there are important similarities between mortality and interest rate modelling. Specifically, if we assume that mortality process is driven by a force of mortality similar to the short rate in interest rate modelling, we can quickly deduce that they must be positive processes, have term structures and are fundamentally stochastic in nature. These

∗ Corresponding address: Department of Statistical and Actuarial Sciences, 2nd floor Western Science Centre, The University of Western Ontario, London, Ontario, Canada N6A 5B7. Tel.: +1 519 661 2111x83625; fax: +1 519 661 3813. E-mail addresses: [email protected] (L. Jalen), [email protected] (R. Mamon). 0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.10.014

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similarities were exploited by Milevsky and Promislow [8], Cairns, Blake and Dowd [7], Biffis [9], Dahl [10], and Schrager [11] to model force of mortality using tools and techniques developed in interest rate modelling. Although we have been drawing parallels between the pricing of financial and mortality-linked instruments in this exposition, there are some important differences and specific problems with mortality risk modelling. Even though it is an accepted fact that interest rates are mean-reverting, this is not the case for mortality rates. Specifically, long-term stochastic improvements in mortality rates should not be mean-reverting to some deterministic projection. Otherwise, the inclusion of mean reversion implies that if mortality improvements have been faster than what have been expected in the past, then the potential for further mortality improvements will be significantly reduced in the future (see Cairns, Blake and Dowd [7]). There are a number of recent studies that have sought to model mortality as a random process. The first milestone in stochastic mortality modelling was marked by the work of Lee and Carter [12] that introduced a model for central mortality rates involving both age and time-dependent terms. The model was applied to US population data where the time dependency was modelled using a univariate ARIMA time series. Their idea was later extended and improved by several authors including Renshaw and Haberman [3] and Brouhns, Denuit and Vermunt [13]. Another approach that also models mortality as a stochastic variable in discrete time was proposed by Lee [14]. Lee took a deterministic projection of spot mortality rates as given, and then apply an adjustment that evolves stochastically over time. Similar approach was later used in the work of Cairns, Blake and Dowd [15]. These models were developed in discrete time but certain models were proposed to describe the dynamics of the force of mortality in continuous time. One of these, inspired by a previous work of Carriere [16], is contained in the study of Milevsky and Promislow [8] and assumed that the force of mortality µ(x, t ) has a Gompertz form µ(x, t ) = ζ0 exp(ζ1 x + σ Yt ) for some positive constants ζ0 and ζ1 . In this Gompertz form x refers to a life aged x and Yt is an Ornstein–Uhlenbeck process satisfying the stochastic differential equation (SDE) dYt = −bYt dt + dWt .

(1)

In (1) Wt denotes a standard Wiener process. The process in (1) is expected to grow exponentially but exhibits a mean reversion. Dahl [10] further improved Milevsky and Promislow’s approach and sought to model mortality intensity by a fairly general process, which includes the mean-reverting Brownian Gompertz model. In [10], a class of processes is developed by supposing that the force of mortality for every fixed x > 0 is governed by the SDE dµ(t , x + t ) = α (t , x, µ(t , x + t )) dt + σ µ (t , x, µ(t , x + t )) dWt . It is shown that if the drift α (t , x, µ(t , x + t )) and volatility σ µ (t , x, µ(t , x + t )) satisfy certain regularity conditions, the mortality model possesses an affine structure. That is, the survival probability p(t , T , x) from time t to T for a person of age x + t given the information up to time t can be expressed as



p(t , T , x) = E exp

 Z − t

T

  µ(u, x + u)du Mt

= exp (A(t , T , x) + B(t , T , x)µ(t , x + t )) , where the deterministic functions A(t , T , x) and B(t , T , x) for fixed x satisfy a system of ordinary differential equations (ODEs) involving α and σ µ . The filtration Mt is a sequence of non-decreasing sigma-fields generated by the process µ. Since the ODEs associated with (1) are generalised Riccati equations (see Biffis [9]) they can be solved using standard numerical methods when explicit solutions are not available. Recent work on affine mortality models was carried out by Luciano and Vigna [17]. These authors fitted different affine models to observed mortality data and compared their performance. The models considered in [17] are of the affine form, where the force of mortality follows either a CIR-like process or an Ornstein–Uhlenbeck (OU) process. It is established in [17] that the best fit is achieved when the force of mortality follows a non-mean reverting OU process. The pricing of mortality-linked products involved the modelling of both mortality and interest rates. Biffis and Millossovich [18], for instance, considered the pricing of guaranteed annuity options; however, they assumed that the interest and mortality factors are independent. In this paper we use the change of measure technique and application of Bayes’ theorem for conditional expectations to derive pricing expressions for common life insurance contracts when the dynamics of the financial and demographic factors are not assumed independent. Furthermore, we demonstrate the applications of the pricing equations we derive when mortality and interest rates are governed by affine diffusion processes. In that case we are able to fully exploit analytical tractability of affine processes to derive closed-form expressions for the most common life insurance contracts. We wish to emphasise that the main focus of this exposition is on pricing. Certainly, hedging of mortality-dependent contingent claims is an important consideration, and whilst it must be tackled, this is outside the scope of this paper. Readers who are interested on this issue are referred to Föllmer and Sondermann [19] who pioneered a risk minimisation hedging strategy under the presence of mortality rate uncertainty; such strategy was intended to analyse incomplete financial markets. Møller [20,21] then tailored this risk minimisation technique for insurance contracts. Pelsser [22] also developed a hedging methodology wherein swaptions are used to hedge guaranteed annuity options. In these previous works, however, mortality risk is assumed to be diversifiable. Hence, the actual insurance claims, intrinsically dependent on both sources of uncertainty, are simply replaced by similar claims which only have the financial risk. There is definitely a need to conduct further studies to advance and improve this area.

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This paper is organised as follows. In Section 2, we give the modelling framework for the evolution of both mortality and interest rate processes. Section 3 provides pricing formulae for a large class of life insurance contracts where both the development of mortality and interest rate dynamics are modelled stochastically. In Section 4, we consider an affine type specification for both interest and mortality rates and examine the corresponding implications in valuation. Section 5 presents a numerical implementation of our approach and Section 6 concludes. 2. Modelling framework 2.1. Mortality model In this section the basic components of a stochastic mortality model and notation are introduced. Consider the force of mortality µ(t , x) for an individual aged x at time t. Traditional mortality models implicitly assume the force of mortality is independent of age (see for example Bowers et al. [23]), that is µ(t , x) = µ(x) for all x and t. However, it is now widely recognised that over time mortality evolves in a stochastic manner. This is our motivation to model the force of mortality as a stochastic process in order to capture its time dependency and uncertainty of future developments. Write t

 Z

S (t , x) := exp −

µ(u, x + u)du

 (2)

0

for the survival function of a life aged x. Note, that if the force of mortality µ(t , x) is deterministic, then the survival function S (t , x) is simply the probability that an individual aged x at time zero will survive until a later time t. Furthermore, if we assume that the force of mortality is time independent, i.e., µ(t , x + t ) = µ(x + t ), expression (2) and results of any further analysis will simply reduce to those given in Bowers et al. [23]. For example, under the assumption of deterministic and time-independent force of mortality, formula (2) becomes

 Z

S (t , x) = exp −

x +t

 µ(y)dy .

x

However, we intend to make the force of mortality stochastic in this current discussion. Certainly, the survival function S (t , x) is a random variable. Note that S (t , x) is a survival probability whose value can only be observed at time t rather than at current time 0. In general, under a stochastic framework, survival probabilities can be obtained by taking the expected value of the random variable S (t , x). Let (Ω , M , P) be a probability space equipped with the filtration Mt generated by the evolution of mortality (µ(t , x)) up to time t. In other words, Mt provides a full information of development of mortality up to and including time t, but no information about how mortality rates will progress after time t. On (Ω , M , P) define the real-world or true survival probabilities measured at time t as follows. Let p(t , T , x) be the realworld probability for an individual aged x at time 0 who is still alive at the current time t and survives until later time T . Then p(t , T , x) = E



S (T , x)

 Mt . S ( t , x)

(3)

We note that p(t , T , x) corresponds to T −t px+t in standard actuarial notation as defined for instance in Bowers et al. [23]. Let τ (x) be a random residual lifetime of an individual aged x. In other words, τ (x) represents a future lifetime of an individual aged x. Similar to standard actuarial results (see Bowers et al. [23], Chapter 3)

P (τ (x) > T ) = E [S (T , x)] .

(4)

Moreover, the Mt -conditional density ft (.) of a random residual lifetime τ (x) of an individual aged x at time 0 on the set {τ (x) > t }, is given by



s

 Z

ft (s, x) = E µ(s, x + s) exp −

µ(u, x + u)du t

  Mt .

(5)

See Biffis [9] for further details of Eq. (5). In pricing various products in the financial markets, the risk-neutral valuation will be our starting point. In what follows we shall assume that there exists a risk-neutral measure Q that is absolutely continuous with respect to the real-world measure P when pricing contracts linked to mortality dynamics, such as endowments, insurance policies and mortalitylinked bonds. Consequently, the corresponding survival probabilities under the risk-neutral measure Q are pQ (t , T , x) = EQ



S (T , x)

 Mt . S ( t , x)

(6)

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2.2. Interest rate model We fix the stochastic R t basis (Ω , F , {Ft }, P) and take as given an adapted short rate process rt such that it satisfies the technical condition 0 r (s)ds < ∞ for all t > 0. The process rt represents the continuously compounded rate of interest of a riskless security. Consider a riskless money market   account Bt . The amount of money available at time t from investing

Rt

one unit at time 0 is given by Bt = exp

0

r (s)ds . We suppose that an equivalent martingale measure Q exists, under

which the gain from holding a risky security is a martingale after discounting by the money market account. From now on, we assume that the dynamics of all security processes are specified under a risk-neutral measure Q unless otherwise stated. We note that the zero-coupon bond price B(0, T ) is given by



T

 Z



r (s)ds

B(0, T ) = EQ exp −

.

(7)

0

3. Integrating the interest and force of mortality models In this section we introduce the general setup of a combined modelling framework that covers the development of both mortality and interest rate processes. We consider a filtered probability space (Ω , G, {Gt }, P), large enough to support a process r representing the evolution of interest rates and a process µ representing the evolution of mortality. The filtration {Gt } represents the information available up to time t as revealed by the processes r and µ. Write Mt ⊂ Gt for the filtration generated by the evolution of mortality up to time t as in Section 2. Similarly let Ft ⊂ Gt be the filtration generated by the evolution of short rate process r up to time t. Then we can express the filtration Gt as the smallest sigma-algebra generated by Mt and Ft . Formally, we write

Gt = M t ∨ F t , where Mt ∨ Ft is the filtration generated by σ (Mt ∪ Ft ).

3.1. Independent case Here, we develop the fair valuation of two basic actuarial benefits involved in standard insurance contracts. The payoffs are contingent on the survival or death of an individual over a pre-specified period of time, and in some instances, may be linked to other security prices. So far, no references are made to any specific model for interest rate or mortality dynamics. We simply suppose as well that the dynamics of interest rate model parameters are independent from those of the model for mortality development as assumed in previous investigations, such as Cairns et al. [7,15], Ballotta and Haberman [24], and Biffis [9]. The assumption that the short rate rt and the force of mortality µ(x, t ) are independent will significantly reduce the complexity of the pricing equations; it will allow the separate pricing of mortality risk from the pricing of financial risk. This is reasonable for a relatively short time horizon, although we are aware that in the long-run interest rates can be influenced by the relative size of population, which in turn, is influenced by mortality development (as well as fertility). Also in the short term, we recognise that a catastrophe event that seriously affects the size of population, such as major natural disasters or a nuclear war, can also affect interest rates. Survival benefit Let C be a bounded G-adapted process. The fair value BS (t , T , CT ) at time t of a survival benefit CT to be paid out at time T > t for an individual aged x at time 0 can be written as



T

 Z

BS (t , T , CT ) = E exp −

t





r (s)ds 1{τ >T } CT Gt T



 Z = 1{τ >t } E exp − t



  (r (s) + µ(s, x + s)) ds CT Gt ,

(8)

where τ = τ (x) is the residual lifetime of a life aged x as defined in (4). When the dynamics of the interest rates are independent from the dynamics of mortality, then



T

 Z

BS (t , T , CT ) = 1{τ >t } E exp −

t





 

T

 Z

r (s)ds CT Ft E exp −

µ(s, x + s)ds t

  Mt .

(9)

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Death benefit Assume again that C is a bounded G-adapted process. The fair value at time t of a death benefit BD (t , T , Cτ ) of amount Cτ payable at the time of death in case the insured aged x at time 0 dies before time T , with 0 ≤ t ≤ T can be expressed as τ



 Z BD (t , T , Cτ ) = E exp −



 r (s)ds 1{t <τ ≤T } Cτ Gt

t T

Z



u

 Z

E exp −

= 1{τ >t }

t

t

  (r (s) + µ(s, x + s)) ds µ(u, x + u)Cu Gt du.

(10)

The result in (10) is a direct consequence of expression (5). Again, if we assume independence of interest rate and mortality dynamics, we have BD (t , T , Cτ ) = 1{τ >t }

T

Z



 Z

u

E exp − t

t u



 Z × E exp − t





r (s)ds Cu Ft



  µ(s, x + s)ds µ(u, x + u) Mt du.

(11)

3.2. Dependent case In Section 3.1, we made the assumption that the dynamics of interest rates and mortality are independent. This assumption permits us to separate the evaluation of mortality risk from financial risk, thus enabling us to derive general pricing formulae for a generic class of life insurance contracts. We drop this assumption in this section. We consider stochastic processes rt and µ(x, t ) to be dependent. We use a change of measure technique to derive pricing equations for the survival benefit BS (t , T , CT ) and death benefit BD (t , T , Cτ ). Once we obtain BS (t , T , CT ) and BD (t , T , Cτ ), it is straightforward to derive expressions for the values of life insurance contracts, such as endowments annuities and various types of insurance. We shall be working under the forward measure PT defined on a filtration GT by setting the Radon–Nikodym derivative of PT with respect to the risk-neutral measure Q as dPT

dQ

 R T

= Λ0,T =

exp −

0

r (s)ds

 (12)

B(0, T )

GT

where B(0, T ) is defined as in (7). Let E T denote the expectation under the forward measure PT . From Bayes’ rule E T [H | Gt ] =

E [Λ0,T H | Gt ] E [Λ0,T | Gt ]

,

(13)

for a contingent claim H. Eq. (13) together with Eq. (12) implies that

h

E T [H | Gt ] =

 R T

E exp −

t

 i

r (s)ds H Gt

B(t , T )

or T



 Z E exp − t

  r (s)ds H Gt = B(t , T )E T [H | Gt ].

(14)

 R T

Eq. (14) can provide an alternative formula for both BS (t , T , CT ) and BD (t , T , Cτ ). Taking H = exp −

t

 µ(s, x + s)ds CT

in Eq. (14) and plugging this into Eq. (8), we obtain BS (t , T , CT ) = 1{τ >t } B(t , T )E

T



T

 Z exp −



 µ(s, x + s)ds CT Gt .

(15)

t

The goal is to separate the calculation of expectations involving interest rate dynamics, mortality and the benefit process under the forward measure. ˜ T that is absolutely continuous with respect to the forward To explicitly solve (15), we define an auxiliary measure P measure PT via the Radon–Nikodym derivative

dPT ˜T dP

GT

 R T

˜ 0,T = =Λ

exp −

0

µ(s, x + s)ds

p˜ (0, T , x)

 ,

(16)

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where p˜ (0, T , x) = E T

h

 R i T exp − 0 µ(s, x + s)ds .

˜ T . Then invoking Eqs. (14) and (15) the survival benefit value Let E˜ T denote the expectation under the auxiliary measure P BS (t , T , CT ) can be written as BS (t , T , CT ) = 1{τ >t } B(t , T )˜p(t , T , x)E˜ T CT Gt .

 

(17)

The main advantage of working with the forward measure is that the valuation is simpler when dealing with stochastic interest rates and force of mortality even without the independence assumption. In particular, we succeeded in splitting the expectation of a product (Eq. (8)) into a product of expectations, where the expectation of the last term must be taken under the auxiliary measure. Since we started with the assumption that both the short rate process and force of mortality follow affine processes, an explicit expression can be derived for the dynamics of the benefit process C under an auxiliary ˜ T given its dynamics under risk-neutral measure. This is made possible with the aid of Theorem A.1 discussed in measure P Appendix A. In a similar manner, using a forward measure PT we can also derive valuation expression for the death benefit value BD (t , T , Cτ ). Under the measure PT , the expression for the value of death benefit in (10) is BD (t , T , Cτ ) = 1{τ >t }

T

Z



u

 Z

B(t , u)E T exp −

t

t

  µ(s, x + s)ds µ(u, x + u)Cu Gt du.

In order to evaluate separately expectation involving the dynamics of mortality from that involving the dynamics of the u benefit process, we define another auxiliary measure P via the Radon–Nikodym derivative u



dP dPT

= Λ0,u =

µ(u, x + u) exp −

Ru 0

µ(s, x + s)ds

f t ( u)

GT


,

(18)

where t ≤ u ≤ T and f t (u) is the Gt -conditional density of a random residual lifetime τ (x) (see expression (5)) taken under u the forward measure PT . Applying Bayes’ rule (13) and changing measure to the auxiliary measure P the value of death benefit is given by BD (t , T , Cτ ) = 1{τ >t }

T

Z

B(t , u)f t (u)E t

u

u

h



i

Cu Gt du,

(19) u

where E denotes the expectation taken under the new auxiliary measure P . The difficulty in evaluating the values of the respective survival and death benefits BS (t , T , CT ) and BD (t , T , Cτ ) without ˜ T in Eq. (17) and the the independence assumption lies in the calculation of the expectation taken under the measure P u expectation in Eq. (18) taken under the measure P . In general, there are no analytical expressions for dynamics of stochastic process under such a measure. However, one can resort to analytically tractable models for the interest rate and force of mortality dynamics for which it is possible to express the dynamics under an auxiliary measure. One such class of stochastic models where this is feasible is the class of affine processes. When rt and µ(t , x) have affine specification, one can find analytical expression for their dynamics under the auxiliary measures using Theorem A.1 specified in Appendix A. An example is provided in the next section. 4. Example and illustration Suppose that both the short rate rt and the force of mortality µ(t ) follow stochastic processes of an affine type in its drift and volatility specification. Assume, for example, that the short rate follows a Vasicek model with constant parameters, i.e., its dynamics is given by the SDE drt = (ar − br rt )dt + c r dWtr .

(20)

Assume further that the force of mortality follows a relatively simple diffusion given by the equation µ dµt = aµ µt dt + c µ dWt ,

(21)

as put forward in Luciano and Vigna [17]. Note that the model in (21) allows negative values for the force of mortality. However, in our implementation in the following section, we demonstrate that using the optimal parameters estimated from the observed data, the probability of force of mortality hitting zero within the 35 time steps calculated using Eq. (45) is only about 10−10 . See Appendix B for the underlying theory. This shows that the force of mortality crossing zero and becoming negative is a remote possibility. Thus, this empirical evidence along with the results in Luciano and Vigna [17] sufficiently justifies our choice of the affine dynamics for force of mortality given in (21). We emphasise that in contrast to other developments in mortality modelling using affine processes and other attempts to price mortality-linked contracts, we do not assume the dynamics of the short rate and mortality development to be

L. Jalen, R. Mamon / Mathematical and Computer Modelling 49 (2009) 1893–1904

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independent. Instead, we use the change of measure approach developed in Section 3.2 to find explicit solutions for valuing basic mortality-linked instruments. As shown in Section 3.2 the price of survival benefit can be expressed as (Eq. (17)) BS (t , T , CT ) = 1τ >t B(t , T )˜p(t , T , x)E˜ T CT Gt .

 

Assuming further that the policy holder has survived to current time t, (τ > t), it simplifies to BS (t , T , CT ) = B(t , T )˜p(t , T , x)E˜ T CT Gt .

 

(22)

If rt has Vasicek dynamics then B(t , T ) has the explicit solution B(t , T ) = eAr (t ,T )rt +Br (t ,T ) ,

(23)

where Ar (t , T ) and Br (t , T ) are deterministic functions. The second factor p˜ (t , T , x) in (22) is not easy to evaluate. The difficulty comes from the fact that the expectation in



p˜ (t , T , x) = E T

T

 Z exp − t

  µ(s, x + s)ds Gt

(24)

is taken under the T -forward measure PT (see Eqs. (12)–(15)). In order to analytically derive the expression for p˜ (t , T , x) we need the dynamics of the force of mortality under the measure PT . By Theorem A.1 the evolution of µt under the T -forward measure follows the SDE dµt = (c µ c r Ar (t , T ) + aµ µt )dt + c µ dWtP , T

(25)

T

where WtP is a standard Brownian motion under a measure PT . Since the function Ar (t , T ) is deterministic, the dynamics of the force of mortality under a T -forward measure PT admits an affine form, therefore p˜ (t , T , x) can be expressed as p˜ (t , T , x) = eAµ (t ,T )µt +Bµ (t ,T ) .

(26)

Plugging (23) and (26) into Eq. (22), the value of survival benefit can be written as BS (t , T , CT ) = eAr (t ,T )rt +Br (t ,T )+Aµ (t ,T )µt +Bµ (t ,T ) E˜ T CT Gt .

 

(27)

From Theorem A.1, given the dynamics of the benefit process Ct under a risk-neutral measure we can also derive the ˜ T . Consequently, given the dynamics of the actual benefit Ct , the survival dynamics of the Ct under the auxiliary measure P benefit can be priced in an analytically tractable way without assuming the independence of the dynamics of the short rate from the dynamics of mortality development. Suppose the benefit consists of a fixed unit amount plus a variable amount equal to a percentage λ of the level of the short rate at the policy date. In other words Ct = 1 + λ rt .

(28) T

˜ . However, we know that under In Eq. (27) we need to derive the dynamics of the short rate rt under an auxiliary measure P a risk-neutral measure Q, the dynamics of the short rate is given by Eq. (20). Furthermore, it follows from Theorem A.1 that T

Q

t

Z

Ar (u, T )c r du

WtP = Wt −

(29)

0

is a standard Brownian motion under the T -forward measure PT . A similar line of reasoning shows that ˜T

T

t

Z

WtP = WtP −

Aµ (u, T )c µ du

(30)

0

˜ T . Combining (29) and (30) we can write down the dynamics is a standard Brownian motion under the auxiliary measure P T ˜ as of the short rate process rt under the auxiliary measure P drt = (ar + (c r )2 Ar (t , T ) + c µ c r Aµ (t , T ) − br rt )dt + c r dWtP . ˜T

Given the dynamics of rt under an auxiliary measure it is straightforward to calculate the expectation in Eq. (27). Therefore the value of survival benefit can be fully determined given a chosen functional form for the benefit process as in (28). In a similar manner, one can also use the change of measure technique to derive an explicit expression for the price of death benefit in (19). With the same assumptions regarding the dynamics of the short rate and the force of mortality processes we get similar expression for the price of the death benefit BD (t , T , Cτ ). The only factor in the product under the integral in Eq. (19) we need to consider is f t (u). Employing Theorem A.1 again and the fact that the force of mortality is an

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L. Jalen, R. Mamon / Mathematical and Computer Modelling 49 (2009) 1893–1904

Table 1 Actuarial fair prices for survival benefit with different times to maturity. BS (0, T , 1) value

T (years)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Independent case

Dependent case

0.92211574682795 0.84576852805949 0.77149720230694 0.69977599391098 0.63101430226575 0.56555716433548 0.50368626618530 0.44562142240926 0.39152246550173 0.34149150935876 0.29557557108280 0.25376955195146 0.21601959065656 0.18222680862647 0.15225146744143 0.12591755130357 0.10301777287750 0.08331897876781 0.06656790234859 0.05249717835692

0.93479234983310 0.87365541925090 0.81630686014195 0.76249382821873 0.71198836827754 0.66458360836166 0.62009062043372 0.57833583110336 0.53915888672416 0.50241089391967 0.46795297019147 0.43565505036976 0.40539490379685 0.37705732468225 0.35053346433776 0.32572027923266 0.30252007318847 0.28084011571057 0.26059232154637 0.24169297916488

affine process, the Mt -conditional density under the T -forward measure of a random residual lifetime in (5), can be written as f t (T ) = eAf (t ,T )µ(t )+Bf (t ,T ) Df (t , T )µ(t ) + Gf (t , T )




(31)

following Duffie et al. [25] and where Df (t , T ) and Gf (t , T ) are some deterministic functions. Therefore the death benefit in (19) has a fair value of BD (t , T , Cτ ) =

T

Z

eAr (t ,u)rt +Br (t ,u) eAf (t ,u)µ(t )+Bf (t ,u) Df (t , u)µ(t ) + Gf (t , u) E




u

Cu Gt du.







(32)

t

5. Numerical implementation In order to calculate the prices of basic mortality-linked instruments we need to calibrate the mortality and interest rate models first. The parameters estimated from fitting the models to observed data can then be readily used to value instruments in both dependent and independent case. We choose a well-known Vasicek model (20) for the stochastic representation of interest rates. For the mortality model an Ornstein–Uhlenbeck (OU) process (21) is used. This is motivated by the fact that the OU process was found to generate the best fit to observed data amongst all tested affine processes without jumps in the study of Luciano and Vigna [17]. The mortality table selected for the calibration is the observed UK generation tables for males born 1900. The observed mortality table is taken from the Human Mortality Database compiled by the University of California, Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). The data is also available at www.mortality.org or www.humanmortality.de. The mortality data set used in this paper was downloaded on 10 September 2006. In fitting the data, we have adopted the least squares method, considering the spreads between different observed and model survival probabilities. The initial value for µ(65, 0) is set to − ln (p(65)) in the calibration of the mortality model, as well as in all subsequent calculations. For the estimation of interest rate model parameters we use a different technique. A model was fitted to 1-month UK inter-bank loan data for the year 2003. The method selected in calibrating the interest rate model to LIBOR data is the maximum likelihood technique (see James and Webber [26]). We note that the data used for calibration of mortality and interest rate model are not consistent. However, as mentioned before our intention here is to show how the fitted parameters can be used to calculate the insurance product prices. After we calibrated the parameters of the interest rate and mortality models we could proceed to calculate the values of the contracts. For the data considered in this paper, the parameters for the interest rate model with specification in Eq. (20) are ar = 0.004089, br = 0.045398 and c r = 0.003789, whilst the parameters for the mortality model with specification (21) are aµ = 0.078282 and c µ = 0.002271. Table 1 displays the calculated prices of survival benefit with time to maturity between 1 and 20 years which pays 1 contingent on the survival to time T of an individual aged 65 (at current time 0). In the second column we present the actuarial fair values of survival benefit calculated under the assumption of independence between financial and demographic factors (Eq. (9)). The prices calculated without the assumption of independence between interest and mortality rates are depicted in the third column. In order to calculate values of the survival benefit

L. Jalen, R. Mamon / Mathematical and Computer Modelling 49 (2009) 1893–1904

Fig. 1. Relative difference

BS (0,T ,1)dependent BS (0,T ,1)independent

1901

with respect to maturity.

BS (0, T , 1) the expression in (27) needs to be implemented. In our fairly simple case we assumed the survival benefit to be deterministic, which simplifies the calculation of expected value under the auxiliary measure in (27). However, working with random payoffs is straightforward, provided we can express their dynamics under a risk-neutral measure, as explained in Section 4. From Table 1 it is apparent that the calculated survival benefit values differ considerably between the dependent and independent case. It can also be seen that the value of the survival benefit is noticeably higher when the independent assumption is dropped. This is somewhat expected since the independence assumption is relatively strong and disregards any correlation between demographic and financial factors. Furthermore, it is quite clear from Table 1 that the relative difference between the prices calculated with and without an independence assumption is increasing. Furthermore, the relative difference seems to increase exponentially with time to maturity and this is evident from Fig. 1. The assumption of independence between mortality development and financial factors leads to a considerable lower prices of mortality-dependent contracts compared to the prices generated under the dependence case. Therefore, based on the evidence suggested by our empirical work, one cannot ignore the dependence between interest and mortality risks in pricing instruments with long-term maturities. For the calculation of the values of death benefit in (32), the expected value under an auxiliary measure needs to be u

evaluated numerically. Nonetheless, since the Radon–Nikodym derivative ddPPT can readily be expressed as stated in (18), the calculation of the expected value of the benefit itself under an auxiliary measure is achievable, provided its dynamics under a risk-neutral measure Q is explicitly given. We stress that the analytical solutions for pricing survival and death benefits might not be available when more complicated mortality and interest rate models are assumed. However, as long as both interest and mortality rate models are of affine type, analytical tractability can still be achieved. Eqs. (27) and (31) are essentially linked to generalised Ricatti equations which can be solved using standard numerical methods as noted in Biffis [9].

6. Concluding remarks In this paper we contributed to the methodology of pricing insurance claims under the integrated framework of stochastic mortality and interest rates. Independence between financial and demographic risk factors is not assumed. By using change of probability measures technique and employing affine diffusion processes in the description of both the evolution of mortality and interest rates, we showed that our approach could lead to analytical expressions in the pricing problem of a number of life insurance contracts, either traditional, unit-linked or indexed. Numerical examples provided demonstrate the applicability of our results and theoretical contributions. Furthermore, numerical results presented suggest that the dependence between mortality and financial factors should not be ignored when pricing and reserving for the instruments with long-term maturities. More perspectives and insights about our proposed approach can be learnt by comparing it to other traditional valuation techniques designed for insurance and annuity contracts. For this purpose specifically, a performance index may be employed to quantify the benefit gained in using the change of measure method. This suggested comparison is one research direction that may be explored as a continuation of this investigation.

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Appendix A. Affine processes Theorem A.1. Let (Ω , F , P) be a probability space equipped with a filtration Ft , large enough to support a standard ndimensional Brownian motion Wt and suppose α(t , x): [0, ∞] × R → R has an affine dependency on the second argument. In other words, let α(t , x) = γ (t ) + δ(t )x. Assume further that β(t ) is an F -previsible n-dimensional vector process which satisfies growth condition



T

 Z

EP exp

1 2

|β(u)|2 du



< ∞.

(33)

0

i. If xt is a stochastic process admitting the dynamics dxt = α(t , xt )dt + β(t )∗ · dWt ,

(34)

 R  i T it follows that X (t , T ) := E exp − t xu du Ft has the exponential affine form. That is, h

X (t , T ) = eA(t ,T )xt +B(t ,T ) ,

(35)

where A(t , T ) and B(t , T ) are deterministic functions and denotes the transpose of a vector. ii. If we define an equivalent measure Q on Ω via a Radon–Nikodym derivative ∗

dQ ΛT := dP

 R T

exp −

=

0

xu du

X (0, T )

FT

 ,

then Q

Wt = WtP −

t

Z

A(u, T )β(u)du 0

is a standard n-dimensional Brownian motion under measure Q. Proof. The first part of the theorem is a known fact, proof for which can be found in various sources. We refer to Biffis [9] and references contained therein. Moreover, the deterministic functions A(t , T ) and B(t , T ) satisfy the system of ODEs At (t , T ) = 1 − δ(t )A(t , T ) Bt (t , T ) = −γ (t )A(t , T ) −

(36) 1 2

β(t )∗ · β(t )A(t , T )2 ,

(37)

where A(T , T ) = B(T , T ) = 0. Another approach which solely relies on the method of stochastic flows and forward measure and does not involve the solution of the above Ricatti equation can be found in Elliott and van der Hoek [27]. The result in part (ii) is immediate from the work of Mamon [28, eqn 23].  Appendix B. First hitting time density of an OU process with constant parameters Let Wt be a standard Brownian motion. The associated Ornstein–Uhlenbeck (OU) process Ut , with parameters µ ∈ R and

σ ∈ R, is defined as the solution to the stochastic differential equation dUt = µUt dt + σ dWt ,

U0 ∈ R.

(38)

Furthermore, the process Ut is a strong Markov process with infinitesimal generator, denoted by A, given by Af (x) = µx

σ 2 ∂ 2f ∂f + , ∂x 2 ∂ x2

x ∈ R.

(39)

When integrated, the stochastic differential equation (38) yields the realisation Ut = eµt



U0 + σ

Z

t

e−µs dWs



0

˜ t , such that for t ≥ 0. By the Dambis–Dubins–Schwartz theorem there exists a Brownian motion W t

Z

˜ τ (t ) , e−µs dWs = W 0

(40)

L. Jalen, R. Mamon / Mathematical and Computer Modelling 49 (2009) 1893–1904

1903

for any t ≥ 0, where τ (t ) = (2µ)−1 e2µt − 1 . Therefore the representation





Ut = eµt U0 + σ Wτ (t ) which is also known as Doob’s transform holds. For a fixed real number a, define the stopping time

λa = inf {t > 0 | Ut = a} . µ

Its law px→a (t ) is absolutely continuous with respect to a Lebesgue measure. We start with the assumption that µ is negative so that Ut is a recurrent process and therefore λa is finite. Finding the density of a first barrier hitting time of an OU-process is generally a difficult problem. There are several approximation methods to calculate the density, however they are fairly complicated. See for example Alili et al. [29], Alili and Patie [30], Lo and Hui [31], amongst others.  √ µ ˜ t = a 1 − 2µt . For the special case a = 0 however, there is a simple expression for px→0 (t ). Set λ˜a = inf t > 0 | W

˜ a a.s. Therefore, we can deduce As was described in Breiman [32], Doob’s transform implies the identity λa = λ µ px→0 (t ) = τ 0 (t )p0x→0 (τ (t )) .

(41)

Furthermore, by letting µ → 0 we recover the Brownian motion with volatility σ , hence p0x→a (t ) =

  | a − x| (a − x)2 . exp − √ 2 σ 2t σ 2π t 3

(42)

It follows from Eqs. (41) and (42) that µ

px→0 (t ) =

|x| √

σ 2π



µ sinh(µt )

3/2

x 2 µe µ t

 exp −

2σ 2 sinh(µt )

−

µt 2



.

(43)

Recall that if µ is positive, the process Ut is transient and the formula (43) no longer applies. Nevertheless, one can still µ find fairly simple formulae for the density of the first hitting time. Denote by Px the law of Ut where x = U0 ∈ R. As 0 before, letting µ → 0 we retrieve the law Px of a σ Wt started at x. Due to Girsanov’s theorem (see for example [33]), the absolute-continuity relationship µ dPx|Ft

  Z t
µ2 µ 2 2 2 Ws ds dPx|Ft = exp − 2 Wt − x − t − 2σ 2σ 2 0

(44) µ

−µ

holds for every t ≥ 0. From the chain rule and (44) we can deduce as in [33] that dPx|Ft = exp µ(Wt2 − x2 − t ) dPx|Ft for t > 0. The latter combined with the optional stopping theorem yields pµ x→a (t ) = exp

µ
 −µ 2 2 a − x − t px→a (t ). σ2




(45)

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