The premium and the risk of a life policy in the presence of interest rate fluctuations

Insurance: Mathematics and Economics 35 (2004) 537–551

The premium and the risk of a life policy inthe presence of interest rate fluctuations

Nan Wang∗, Russell Gerrard, Steven Haberman

Faculty of Actuarial Science & Statistics, Cass Business School, City University, 106 Bunhill Row, London, UK

Received November 2003; received in revised form July 2004; accepted 8 July 2004

Abstract

In this work, we consider the premium setting and the associated risk with regard to a life insurance policy when the interestrate process is a diffusion. Explicit formulas and numerical examples are given for the premiums determined by the equivalenceprinciple under some well-known interest rate models such as the Vasicek model and the CIR model. Two types of premiumdesign, single deposit premium and continuous premium, are considered. A martingale related to the discounted reserve processis introduced. With this martingale, an analysis of the risk of ruin is also carried out on a collective basis when the premium isthe natural premium plus an initial positive loading. A bound of exponential type is derived for the ruin probability.© 2004 Elsevier B.V. All rights reserved.

MSC:IM13; IM30; IB10; IE51

Keywords:Instantaneous interest rate; Equivalence principle; Natural premium; Feynman–Kac formula; Martingale; Marcinkiewicz–Zygmund

inequality

1. Introduction

This work considers the setting of the premium and the risk of ruin for a life insurance policy in the presence ofinterest rate fluctuations.

In traditional actuarial investigations, the interest rate is assumed to be deterministic and hence there isonly one source of uncertainty, the mortality uncertainty, to be considered. Concerns about the effects of in-cluding a stochastic interest rate in the model have been growing during the last decade. The literature hastended to focus on annuities and the model adopted to describe the interest rate uncertainty, in a continu-

∗ Corresponding author.E-mail address:[email protected] (N. Wang).

0167-6687/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2004.07.004

538 N. Wang et al. / Insurance: Mathematics and Economics 35 (2004) 537–551

ous framework, has usually involved the use of a Brownian motion. See for exampleBeekman and Fuelling(1990, 1991), Dufresne (1990), De Schepper et al. (1992), Parker (1994), Perry and Stadje (2001), Perry et al.(2003).

Let rt, t ≥ 0 be the instantaneous interest rate process (or, force of interest). The discount factor e−∫ t

0rs ds is

then the central concern. As stated inParker (1994), there can be two ways to model the randomness in actuar-ial studies, (i) directly modellingrt ; or (ii) modelling the accumulated processYt = ∫ t0 rs ds. Most of the aboveworks adopt the latter approach by modellingYt = θt +Xt , whereXt is a Brownian motion, a reflected Brownianmotion, or an Ornstein-Uhlenbeck process. In this work, we adopt the former and, as in most of the financial lit-erature (wherert is called the short rate), we assume thatrt is a diffusion described by the stochastic differentialequation

drt = µ(rt, t) dt + σ(rt, t) dWt, t ≥ 0, (1)

whereµ(r, t), σ(r, t) are properR× R+ → R functions andWt is a standard Brownian motion. Readers may refertoDuffie (1996), Bjork (1998), or to the original works such asVasicek (1977), Cox et al. (1985), Heath et al. (1992)for a discussion of the frequently used models.

We point out that, in the financial literature, models of the interest rate could be constructed under an ar-tificial probability measure called risk neutral measure (which is obtained from the real probability measurevia Girsanov’s theorem) for the purpose of arbitrage-free pricing of a derivative security. Since the work ofBrennan and Schwartz (1976), the idea of contingent claim pricing theory has been used to price life insur-ance products, expecially equity-linked products and, more recently, participating life insurance (see, for exam-ple, Grosen and Jørgensen (2000)) and annuity products with implicit guarantees (see, for example,Ballottaand Haberman (2003)). In this work, however, we will continue to use the traditional equivalence principle inorder to determine the premium. The probability measure in this work is therefore the real probability mea-sure.

The life insurance policy considered in this paper pays a fixed benefit to a policyholder (or his beneficiary)if he dies before the maturity time. The premium for this policy can be set up in two ways, a single deposit atthe time of inception of the policy or a continuous stream payable until maturity or the death of the policyholder,whichever comes first. This work considers premiums that satisfy the equivalence principle, i.e., the present value ofthe premium payments meets on average the present value of the benefit payoff. We present a computable formulafor the premium ifµ(r, t) andσ(r, t) in model (1) satisfy

µ(r, t) = α1(t)r + α2(t), σ2(r, t) = β1(t)r + β2(t) (2)

whereαi(t) andβi(t) are properR+ → R functions fori = 1,2. Numerical comparisons in respect of three models—a constant short rate, the Vasicek model and the Cox-Ingersoll-Ross model (CIR)—are carried out based on theformulae which are explicitly available for these three cases.

Of the two types of premium setting, single deposit premium and continuous premium, the latter is shown to besuperior in the sense that it may be less affected by interest rate fluctuations. In the case that the premium setting isthe natural premium (seeSection 3.2), the interest rate has no influence at all.

In addition to analysing the premium, we also study the risk of ruin on a collective basis (i.e., assuming that wehave a portfolio ofnpolicyholders, each with the same policy) for the natural premium.Frostig et al. (2003)similarlyconsider the ruin probability under a discrete framework, where recursive formulae and an exponential bound areprovided. In this paper, an exponential bound for the ruin probability is also provided using a martingale approach.A sufficient initial fund is shown to be necessary and the number of policyholders is also critical in reducing therisk regardless of the nature of the interest rate.

N. Wang et al. / Insurance: Mathematics and Economics 35 (2004) 537–551 539

The structure of the paper is the following.Section 2presents the dynamics of the reserve process.Section 3provides a discussion of equivalence premiums.Section 4introduces a martingale related to the reserve process,andSection 5presents a risk analysis by employing the martingale introduced inSection 4.

2. The reserve process

Let the time horizon be [0,L] and the benefit payoff beM. Consider two types of premiums, a single premiumpaymentD at the starting time 0 and a continuous premium payment with time ratev(t) until the maturity timeL orthe death of the policyholder. The policy is a temporary life insurance ifL is finite and is a whole life insurance ifL = ∞.

Denote byRt the policy reserve at timet and denote byNt the counting process with respect to the policyholder’ssurvival status, i.e.,Nt = 0 if the policyholder is alive at time pointt andNt = 1 if the policyholder is dead. In aninfinitesimal time period (t, t +t], the change of the reserve includes the interest increasertRtt, the premiumincomev(t)(1 −Nt)t and the claim amountM(Nt+t −Nt). The dynamics ofRt is then

dRt = rtRt dt + v(t)(1 −Nt) dt −MdNt, 0 ≤ t ≤ L

which, written in integral form, is

Rt = De

∫ t0ru du +

∫ t

0e∫ tsru du

v(s)(1 −Ns) ds−M

∫ t

0e∫ tsru du dNs.

For the sake of convenience, we denote below

zs,t = e−∫ tsru du

, 0 ≤ s < t.

Rt is then

Rt = Dz−10,t +

∫ t

0z−1s,t v(s)(1 −Ns) ds−M

∫ t

0z−1s,t dNs

= z−10,t

[D+

∫ t

0z0,sv(s)(1 −Ns) ds−M

∫ t

0z0,s dNs

].

(3)

The present value of the reserve will be denoted below asRt , i.e.,

Rt := D+∫ t

0z0,sv(s)(1 −Ns) ds−M

∫ t

0z0,s dNs. (4)

Herev(t) = 0 if a single premium is applied andD = 0 if a continuous premium is applied.

The present value of the payoff on death,M∫ t

0 z0,s dNs, can also be expressed in another way asMe−∫ T

0rs ds1T≤t

whereT is the death time of the policyholder and 1T≤t is the indicator function of eventT ≤ t. It is obvious thatP(T > t) = P(Nt = 0). From now on, we writeλ(t) as the hazard rate (force of mortality) of the policyholder, andhence have the expression

P(T > t) = P(Nt = 0) = e−∫ t

0λ(s) ds

for the survival probability. Depending on the context, it is sometimes convenient to use theT-expression andsometimes convenient to use theNt-expression.


3. Equivalence premiums

3.1. The discount factor

Finding the expectation of the discount factor follows the same procedure as finding the price of a zero couponbond. We write this out for completeness.

We have introduced two sources of uncertainties, the mortality uncertainty represented by the counting processNtand the market uncertainty represented by the Brownian motionWt . These two sources of uncertainties are assumed tobe independent throughout this work. Suppose (Ω1,F

(1)(F(1)t ), P1) is the probability space containing the Brownian

motionWt (F(1)t , t ≥ 0 is the augmented filtration of the Brownian motionWt) and (Ω2,F

(2)(F(2)t ), P2) is the

probability space containing the counting processNt (F(2)t , t ≥ 0 is the filtration generated by the counting process

Nt). The probability space we will be working on is then the product space (Ω = Ω1 ×Ω2,F = F(1) × F(2)(Ft =F(1)t × F(2)

t ), P = P1 × P2).Define

f (r; s, t) := Er;s(zs,t), s < t. (5)

whereEr;s(·) represents the conditional expectationE(· | rs = r). Clearly,Er;s(zs,t)

is the same thing asEr;s1

(zs,t)

in space (Ω1,F(1), P1). We will not mention this equivalence hereafter with respect either toWt or toNt (in space

(Ω2,F(2), P2)). Sincef (r; t, t) = 1 for any (r, t) ∈ R× R+, therefore, by the Feynman-Kac theorem (see page 366

of Karatzas and Shreve, 1991), f (r; s, t) must solve

Af (r; s, t) − rf (r; s, t) = 0, (r, s) ∈ R× [0, t)

f (r; t, t) = 1, r ∈ R

whereAf (r; s, t) = fs(r; s, t) + fr(r; s, t)µ(r, s) + 1

2σ2(r, s)frr(r; s, t).

If we choose a model as in (2),f (r; s, t) will admit an affine structure:

f (r; s, t) = eA(t−s)+B(t−s)r, (6)

whereA(s) andB(s) satisfyB′(s) = α1(t − s)B(s) + 1

2β1(t − s)B2(s) − 1; B(0) = 0

A′(s) = α2(t − s)B(s) + 12β2(t − s)B2(s); A(0) = 0.

(7)

The frequently used short rate models such as the Vasicek model and the CIR model have affine structure andf (r; s, t) is explicitly available for these two models (see (10), (11) inSection 4).

In what follows, the stochastic processrt is assumed to start from a particular point, sayr0 = r (r ∈ R+), whichimplies that we know the interest rate today when the policy is issued. The measure (expectation) corresponding tothis initial condition will be written asPr (Er).

3.2. The premiums

In order to find the premium satisfying the equivalence principle, we setErRL = 0. Noting the relation be-

tweenNt and the time of deathT mentioned inSection 2, and noting also thatT has densityλ(t)e−∫ t

0λ(s) ds, we


have

Er(∫ L

0z0,t dNt

)= Er

(e−∫ T

0rt dt1T≤L

)=∫ L

0f (r; 0, t)λ(t)e−

∫ t0λ(s) ds dt.

By a simple calculation one can verify that the equationErRL = 0 is equivalent to

D+∫ L

0f (r; 0, t)e−

∫ t0λ(s) ds[v(t) −Mλ(t)] dt = 0 (8)

and this is the equation that the premium should satisfy.For the single premium case we have

D = M

∫ L

0f (r; 0, t)λ(t)e−

∫ t0λ(s) ds dt.

For the continuous premium case with a constant time rate (in practice, the payments are usually made monthly),we have

v = M∫ L

0 f (r; 0, t)λ(t)e−∫ t

0λ(s) ds dt∫ L

0 f (r; 0, t)e−∫ t

0λ(s) ds dt

.

Supposeλ(t) is continuous in [0,L]. Then there is a pointξ ∈ [0, L] such that

v = Mλ(ξ).

In the case thatλ(t) is fairly flat in the interval [0,L], the interest rate will have little influence on the choice ofconstantv since it only affects the choice ofξ.

In addition to the two premium schemes above, we have another obvious and important choice

v(t) = Mλ(t), (9)

which is often referred to in the actuarial literature asthe natural premium(see for exampleHaberman and Pitacco,1999). A feature of this premium is that it is free of the interest rate.

The formula (8) can also be used to determine the premium of another type of life insurance, the annuity. It isnot difficult to see that the single premium determined by the equivalence principle for an annuity with payoff ratec (constant) is theD in (8) whileL,M andv(t) are set to be:L = ∞,M = 0 andv(t) ≡ −c.

Except for the natural premium, the premiums determined by (8) depend onr, the current interest rate. An insurercan eliminate this dependence by taking the expectation with respect to a proper distributionπ(r), i.e., replacingf (r; 0, t) with

∫f (r; 0, t) dπ(r). Under many choices ofµ(r, t) andσ(r, t), model (1) produces a stationary processrt .

Thus, the choice ofπ(r) can be the stationary distribution ofrt . For example, the Vasicek model (µ(r, t) = a(b− r),σ(r, t) = σ wherea, b andσ are constants) has a stationary normal distributionN(b,Σ2), whereΣ2 = σ2/2a. Thestationary distribution of the CIR model (µ(r, t) = a(b− r), σ(r, t) = σ

√r) is a gamma distribution with meanb

and varianceσ2b/2a.


3.3. Numerical examples for three interest rate models

In this section, we calculate the premiums with respect to different interest models. We will try three models:

rt = b, (constant)

drt = a(b− rt) dt + σdWt, (Vasicek)

drt = a(b− rt) dt + σ√rt dWt. (Cox–Ingersoll–Ross)

For the Vasicek model, the functionf (r; s, t) satisfies (see for example page 260 ofBjork, 1998)

ln f (r; s, t) = [1 − e−a(t−s) − a(t − s)](4a2b− 2σ2) − σ2[1 − e−a(t−s)]2

4a3 − [1 − e−a(t−s)]ra

. (10)

For the Cox–Ingersoll–Ross model (see, for example, page 261 ofBjork, 1998),

f (r; s, t) =[

2γe(a+γ)(t−s)/2

(γ + a)(eγ(t−s) − 1) + 2γ

]2ab/σ2

exp

[− 2(eγ(t−s) − 1)r

(γ + a)(eγ(t−s) − 1) + 2γ

](11)

whereγ = √a2 + 2σ2.

In order to make the comparison among the three models reasonable, we letr0 = b in the following numericalexamples.

SupposeL = 10 year andλ(t) ≡ λ = 0.04 (the remaining life time is 25 years on average) and the death payoff is aunit amount. Numerical calculations are made forr0 = b = 0.05,a = 0.1,0.5,1 andσ = 0.01,0.02, . . . ,0.09,0.1.We want to compare the single premiumD resulting from the Vasicek model corresponding to those parameterswith that resulting from the fixed interest ratert ≡ b in order to see if there are significant differences among them.Table 1gives the numerical results.

The single premium for the fixed interest rate casert ≡ 0.05 is 0.26276. Compared with the values in the table,the premium for the fixed interest rate case is lower and the differences become larger asadecreases andσ increases.Note thata → 0 leads to a Brownian motion model for the short rate. So, the differences are expected to be largerbetween a fixed short rate model and a Brownian motion short rate model.

With the same settings of the parametersa, b andσ, the CIR model is more stable than the Vasicek model sincethe interest ratert is usually less than 1 and henceσ > σ

√rt . Table 2gives the numerical results for the single

premium for the CIR model:The premiums inTable 2are also higher than the premium with the fixed interest ratert ≡ 0.05. But the differences

are much smaller than those observed for the Vasicek model with the same settings ofa, b andσ.The reason that the premium for the fixed interest rate case (equal to the mean) is lower can be seen from Jensen’s

inequality

f (r; 0, t) = Er(

e−∫ t

0rs ds)

≥ e−Er∫ t

0rs ds

.

Table 1Single premium (the Vasicek model):b = 0.05,r0 = 0.05,λ = 0.04,L = 10,M = 1

a σ

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.1 0.26325 0.26472 0.26724 0.27086 0.27570 0.28190 0.28967 0.29927 0.31106 0.325510.5 0.26287 0.26322 0.26379 0.26459 0.26564 0.26693 0.26847 0.27026 0.27233 0.274671.0 0.26280 0.26292 0.26311 0.26340 0.26376 0.26420 0.26473 0.26533 0.26602 0.26680


Table 2Single premium (the CIR model):b = 0.05,r0 = 0.05,λ = 0.04,L = 10,M = 1

a σ

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.1 0.26278 0.26286 0.26298 0.26314 0.26336 0.26362 0.26391 0.26425 0.26463 0.265040.5 0.26276 0.26278 0.26281 0.26285 0.26290 0.26296 0.26304 0.26312 0.26321 0.263321.0 0.26276 0.26277 0.26278 0.26279 0.26281 0.26283 0.26286 0.26289 0.26292 0.26296

Table 3Continuous level premium:a = 0.5, b = 0.05,r0 = 0.05,L = 10,M = 1, λ(t) = 0.0006(t + 50)

Model σ

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Vasicek 0.0325995 0.0326016 0.0326051 0.0326101 0.0326165 0.0326243 0.0326335 0.0326442 0.0326564 0.0326701CIR 0.0325988 0.0325989 0.0325991 0.0325994 0.0325997 0.0326000 0.0326005 0.0326010 0.0326016 0.0326022

However, we can not jump to the conclusion that the CIR model will give a result which is always lower than thatgiven by the Vasicek model, because comparing the results of the two models with samea, bandσ is not the way todraw conclusions. Recall the stationary distributions of the two models in the last subsection. For the same settingsof a, b andσ, the two stationary distributions do not agree on the variance. A comparison of the two models ismeaningful if the two stationary distributions have the same expectation and variance. For our numerical example,it would be more proper to compare the respective results of the Vasicek model withσ = 0.01 or 0.02 with thoseof the CIR model withσ = 0.05 or 0.09.

The hazard rate is constant in the above example, so, for whatever setting of the interest rate, the continuouslevel premium isv = 0.04 (which is also the natural premium). In order to measure the influence of interest ratefluctuations on the continuous premium, we need to introduce a hazard rate that depends on age. As an example,we takeλ(t) to be a linear function of time parametert, λ(t) = k(x+ t), wherek is a constant andx is the age of thepolicyholder at the starting time of the policy, which is fixed to be 50 in this example. We still taker0 = b = 0.05,L = 0 andM = 1. We also fixa = 0.5. Table 3provides the results for the fixed continuous level premium for theVasicek and CIR models of the interest rate withk = 0.0006 (so that the average remaining life time is about 24years).

The continuous premium for the fixed interest rate casert ≡ 0.05 is 0.0325988. Although the differences aremore subtle, we note that the same trend is present as in the last example, in that the premium increases asσ

increases. We note also the differences between the two interest rate models for this example and, as above, notethe comparability of the respective results of the Vasicek model withσ = 0.01 or 0.02 with those of the CIR modelwith σ = 0.05 or 0.09.

4. Probabilistic characteristics of the discounted reserve

In this section we present a martingale associated with the discounted reserve process.We first point out a result which will be used in the proposition below and in the next section as well. Consider

the random variable∫ T

0 λ(t) dt. Regardless of the form of the hazard rateλ(t), we have

E

∫ T

0λ(t) dt = −

∫ ∞

0

[∫ t

0λ(s) ds

]de−

∫ t0λ(s) ds =

∫ ∞

0y e−y dy = 1.


Moreover, note that∫∞

0 yne−y dy < ∞ for any integern, and so we conclude that∫ T

0 λ(t) dt has moments of allorders.

We now reconsider expression (4). The present value of the reserve with respect to the natural premium is

Rt = M

[∫ t

0z0,sλ(s)(1 −Ns) ds−

∫ t

0z0,s dNs

].

Denote

It =∫ t

0z0,s dNs −

∫ t

0z0,sλ(s)(1 −Ns) ds. (12)

We have

Proposition 1. If max0≤t≤L Erz0,t < ∞, thenIt,Ft ,0 ≤ t ≤ L is a zero-mean martingale in(Ω,FL, Pr).Remark 1. The condition max0≤t≤L Erz0,t < ∞ seems unnecessary since the interest rate in reality is nevernegative and hencez0,t ≤ 1 for anyt > 0. But some short rate models do not prevent the appearance of negativevalues, for example the Vasicek model, and thus we retain this condition in the proposition.

Proof of Proposition 1. We need to prove thatEr|It| < ∞ for all t and that, fort > s, Er[It − Is|Fs] = 0.We can writeIt = z0,T1T≤t − ∫ t∧T0 z0,uλ(u) du. By the independence of the interest rate process and the random

variableT, we have

Er|It| <(

max0≤u≤L

Erz0,u

)[∫ t

0λ(u)e−

∫ u0λ(τ) dτ du+ Er

∫ t∧T

0λ(u) du

],

which is finite from the condition max0≤t≤L Er|z0,t| < ∞ and the result stated at the beginning of this section.The second part is equivalent toEr[(It − Is)1A] = 0 for any fair t > s and for any eventA ∈ Fs. Clearly

It − Is = 0 if T ≤ s. So, it is enough to proveEr[(It − Is)1A] = 0 for events of formA = A1 × T > s, whereA1is determined only byru,0 ≤ u ≤ s (A1 × T > s = A1 × Ns = 0 ∈ Fs). For an event of this form, we have

(It − Is)1A = z0,T1A1 · 1s<T≤t −∫ t∧T

s

z0,u1A1λ(u) du

= 1s<T≤t[z0,T1A1 −

∫ T

s

z0,u1A1λ(u) du

]− 1T>t

∫ t

s

z0,u1A1λ(u) du.

Denote ¯z0,u = Er[z0,u1A1]. Noting that the density ofT is λ(u)e−∫ u

0λ(τ) dτ , we have by the independence of the

interest rate process and the random variableT

Er[(It − Is)1A] =∫ t

s

λ(u)e−∫ u

0λ(τ) dτ

[z0,u −

∫ u

s

z0,wλ(w) dw

]du− e−

∫ t0λ(τ) dτ

∫ t

s

z0,uλ(u) du.

Interchanging the order of integration, we obtain∫ t

s

[λ(u)e−

∫ u0λ(τ) dτ

∫ u

s

z0,wλ(w) dw

]du =

∫ t

s

[z0,wλ(w)

∫ t

w

λ(u)e−∫ u

0λ(τ) dτ du

]dw

=∫ t

s

z0,wλ(w)

[e−∫ w

0λ(τ) dτ − e−

∫ t0λ(τ) dτ

]dw.

HenceEr[(It − Is)1A] = 0.


We can better understand the situation now from Proposition 1. The discounted reserveRt for the naturalpremium is a zero mean martingale. Thus, for anyt ∈ [0, L] and any finite stopping timeτ of the filtrationFt ,Er(Rt) = Er(Rτ) = 0, which means that the contract can be terminated early at any time or according to anystopping rule without breaking the equivalence principle.

For the case of a continuous level premium, the discounted reserveRt can be expressed in terms ofIt as

Rt =∫ t

0z0,s(v−Mλ(s))(1 −Ns) ds−MIt.

As stated inSection 3, there exists a pointξ ∈ [0, L] such thatv = Mλ(ξ). Suppose the interest rate is always positiveandλ(t) is strictly increasing. Then,

∫ t0 z0,s(v−Mλ(s))(1 −Ns) ds is an increasing process on [0, ξ] (each path of

the process is non-decreasing on [0, ξ]) and a decreasing process on [ξ, L]. Hence,Rt,0 ≤ t ≤ ξ is a submartingaleandRt, ξ ≤ t ≤ L is a supmartingale. For this premium, as well as the deposit premium,ERt > ERL = 0 andthe balance is met only at the end point of the contract.

5. Risk analysis on a collective basis

In this section we assume that the interest rate admits only non-negative values.Suppose at time 0 there aren policyholders of a similar life insurance policy. The hazard rate of thei-th poli-

cyholder is denoted byλi(t). In addition to the natural premiumMλi(t),1 ≤ i ≤ n, suppose each policyholder alsopaysd at time 0 which serves as a positive loading. LetR

(i)t (R(i)

t ) be the reserve (discounted reserve) with respectto thei-th policyholder, i.e.,

R(i)t = z−1

0,t

[d +

∫ t

0z0,sMλi(s)(1 −N(i)

s ) ds−M

∫ t

0z0,s dN(i)

s

]

whereN(i)t is the counting process with respect to thei-th policyholder’s survival status. The life times of then

policyholders are assumed to be independent of each other (the probability space is extended to be a product ofn+ 1 sub-spaces). The risk of issuing this policy is measured here by the probability that the portfolio value (sumof then reserves) becomes negative at some time point in (0, L], i.e.,Pr(min0≤t≤L

∑ni=1R

(i)t < 0). In the following

we derive a bound of exponential type for this probability.Denote

R(i)t = d +

∫ t

0Mλi(s)(1 −N(i)

s ) ds−M

∫ t

0dN(i)

s , 1 ≤ i ≤ n,

which are then reserves without interest rate growth. Note that the claim process is the same for the reserves withand without the interest rate. So, at a time pointt, if

∑ni=1 R

(i)t , the total reserve without interest rate growth, is not

negative,∑ni=1R

(i)t , the total reserve with interest rate growth, must not be negative as well. Therefore

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)≤ Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

).

A bound for this probability can be obtained from here by employing the martingale inequality.


Proposition 2. If the death times of the n policyholders are independent, then, for any fixed integer p, the probabilityof ruin in the time period[0, L] is bounded by

2p(2p)!np−1M2p

p!(nd)2p

n∑i=1

H(i)L , (13)

where

H(i)L =

∫ Λi(L)−1

−1e−(1+y)y2p dy + e−Λi(L)[Λi(L)]2p

andΛi(L) = ∫ L0 λi(t) dt.

Proof. Let

I(i)t =

∫ t

0dN(i)

s −∫ t

0λi(s)(1 −N(i)

s ) ds, 1 ≤ i ≤ n

which, by Proposition 1, are zero-mean martingales independent of each other. By the result stated in the beginningof Section 4, Er(

∑ni=1 I

(i)t )2p < ∞ for anyt ∈ [0, L] and for any integerp. Therefore

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)= Pr

(max

0≤t≤L

n∑i=1

I(i)t >

nd

M

)≤ M2pEr(

∑ni=1 I

(i)L )2p

(nd)2p. (14)

Applying the Marcinkiewicz–Zygmund inequality (see, for example, page 386 ofChow and Teicher, 1997), wehave

Er

(n∑i=1

I(i)L

)2p

≤ B2pEr

[n∑i=1

(I

(i)L

)2]p

whereB2p can be taken as

B2p = 2p(2p)!

p!.

From Holder’s inequality we also have

n∑i=1

(I

(i)L

)2 ≤ n(p−1)/p

[n∑i=1

(I

(i)L

)2p]1/p

.

Hence,

Er

(n∑i=1

I(i)L

)2p

≤ B2pnp−1Er

[n∑i=1

(I

(i)L

)2p]

= B2pnp−1

n∑i=1

[Er(I

(i)L

)2p].


Bringing this inequality back to (14), we obtain

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)≤ 2p(2p)!np−1M2p∑n

i=1[Er(I(i)L )2p]

p!(nd)2p.

A direct calculation gives

Er(I

(i)L

)2p = Er(

1Ti≤L −∫ Ti∧L

0λi(t) dt

)2p

=∫ L

0λi(t)e

−∫ t

0λi(s) ds

[1 −

∫ t

0λi(s) ds

]2p

dt + e−∫ L

0λi(t) dt

[∫ L

0λi(t) dt

]2p

=∫ Λi(L)−1

−1e−(1+y)y2p dy + e−Λi(L) [Λi(L)]2p ,

whereΛi(L) = ∫ L0 λi(t) dt. This completes the proof.

Remark 2. AsL → ∞,H (i)L converges to a finite limitH∞ = ∫∞

−1 e−(1+y)y2p dy. Thus, the bound (13) is of orderO(n−p) for any fixedp, d andM.

Remark 3. In case that the interest rate is fixed, sayrt = r, we can start directly fromPr(

min0≤t≤L∑ni=1R

(i)t < 0

)to get

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)= Pr

(max

0≤t≤L

n∑i=1

I(i)t >

nd

M

)≤ M2pEr(

∑ni=1 I

(i)L )2p

(nd)2p

whereI(i)t = ∫ t0 e−rs dN(i)

s − ∫ t0 e−rsλi(s)(1 −N(i)s ) ds, 1 ≤ i ≤ n. With the same procedure as in the proof of Propo-

sition 2, it can be shown that

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)≤

2p(2p)!np−1M2p∑ni=1

[Er(I

(i)L

)2p]

p!(nd)2p,

where

Er(I

(i)L

)2p = Er(

e−rTi1Ti≤L −∫ Ti∧L

0e−rtλi(t) dt

)2p

=∫ L

0λi(t)e

−∫ t

0λi(s) ds

[e−rt −

∫ t

0e−rsλi(s) ds

]2p

dt + e−∫ L

0λi(t) dt

[∫ L

0e−rtλi(t) dt

]2p

.

This bound is an improvement on (13).

From Proposition 2, we can derive a bound of exponential form. Let

mi(p) = Er(I(i)L )2p1/2p,


which increases as the integerp increases (due to the Holder inequality). Denotemi = limp→∞mi(p), which is

actually the essential supremum1 of the random variable| I(i)L | that can be shown to be max(1,

∫ L0 λi(t) dt). Let

m = max1≤i≤n mi. We have:

Proposition 3. Under the condition of Proposition 2,

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)≤

√2e−p∗

, (15)

wherep∗ is the largest integer which is not greater than(nd)2/8m2M2n.

Proof of Proposition 3. From Proposition 2, for any integerp, a bound of the ruin probability is

[2(max1≤i≤n m2i (p))]p(2p)!npM2p

p!(nd)2p.

For an integerj, Stirling’s formula tells that

j! = jj+0.5e−j+εj√2π

where (12j + 1)−1 < εj < (12j)−1. Thus

[2(max1≤i≤n m2i (p))]p(2p)!npM2p

p!(nd)2p<

[2(max1≤i≤n m2i (p))]p(2p)2p+0.5e−pnpM2p

pp+0.5(nd)2p

=√

2

[8(max1≤i≤n m2

i (p))pnM2

(nd)2

]pe−p.

Since inequalitymi(p) ≤ mi holds for any integerp, we have

√2

[8(max1≤i≤n m2

i (p))pnM2

(nd)2

]pe−p ≤

√2

[8(max1≤i≤n m2

i )pnM2

(nd)2

]pe−p =

√2

[8pn(mM)2

(nd)2

]pe−p.

Takingp to bep∗ given in the Proposition produces the result. Remark 4. The bound in (15) decreases to zero asn → ∞ for fixedL. The dependence onL, however is containedin the quantitymandm → ∞ asL → ∞. If n is fixed, the bound in (15) is not helpful for sufficiently large valuesof L.

Remark 5. As in the classical collective risk model, we may assume that the insurer has an initial reserveU. Thebound of the ruin probability (13) will become

2p(2p)!np−1M2p

p!(U + nd)2p

n∑i=1

H(i)L (16)

and thep∗ in bound (15) will be the largest integer which is not greater than (U + nd)2/8m2M2n.

The bounds given in Proposition 2 and 3 may help an insurer to make a quick check of the level of risk of a lifeportfolio. For example, suppose

1 The essential supremum of a random variableX is infc : P(X > c) = 0.


(1) the insurer has a life portfolio with 2000 policyholders whose remaining life time follows an exponentialdistribution withλ = 0.04 (the average remaining life time is 25 years);

(2) insurer will pay one unit of money if a policyholder dies within a period of 10 years;(3) each policyholder pays the natural premium and a positive loading at time zero ofd = 0.01.

Regardless of the interest rate in the coming 10 years, the ruin probability is less than

1.53× 10−2 if initial fund U = 180 units

3.50× 10−4 if initial fund U = 280 units

2.69× 10−6 if initial fund U = 380 units.

These bounds are obtained from (16) by takingp = 3, 6, 12, respectively.As H (i)

L is numerically available for alli, the bound selected from (16) by taking integerp which minimizesthe expression is better than the bound provided by Proposition 3. This is clear from the fact that the normsmi(p),1 ≤ i ≤ n are enlarged first tomi,1 ≤ i ≤ n and then tom in the derivation of Proposition 3. For the aboveexample, using Proposition 3 (as explained in Remark 5), the three bounds will be 1.16× 10−1, 5.10× 10−3 and6.42× 10−5, respectively.

The bounds provided by Proposition 2 and 3 may not necessarily be sharp bounds, especially for smallU andn.We illustrate this feature by taking the above example with the number of policyholders being in the range [10,400]and the initial fund (including those from both the insurer and the policyholders) being 0.2 per head.Figs. 1–3show

the distance of the bound (obtained from (16) by taking the optimalp) andPr(

min0≤t≤L∑ni=1 R

(i)t < 0

), from

which the bound is derived.The upper curve in each figure is the bound and the lower curve is the simulated probability.If λi(t),1 ≤ i ≤ nare identical (as in the above examples), by the central limit theorem, we can get an approximate

bound from (14) as

Pr

(min

0≤t≤L

n∑i=1

R(i)t < 0

)≤ M2pEr(

∑ni=1 I

(i)L )2p

(nd)2p≈ M2pσ

2pI N(2p)

npd2p (17)

for any positivep, whereσ2I = Er

(I

(1)L

)2andN(2p) is the 2p-th order moment of a standard normal variable. Note

that I(1)L has moments of any order, and therefore the sequence(∑n

i=1 I(i)L /

√n)2p, n ≥ 1 is uniformly integrable

(see for example Proposition 1 ofAras and Woodroofe (1993)). Hence, tegether with the central limit theorem, wehave the moment convergence

Er

[∑ni=1 I

(i)L√

nσI

]2p

→ N(2p), asn → ∞.

As in Remark 5, if we take into account the initial fund from the insurer (U), we will have an approximate bound

npM2pσ2pI N(2p)

(U + nd)2p.

As a simple example, we compare this approximate bound with that inFig. 3 whereλ = 0.06 for all i andn is inthe range [10,400]. The outcomes are plotted below inFig. 4.


Fig. 1.λ = 0.04. Fig. 2.λ = 0.05. Fig. 3.λ = 0.06.

Fig. 4. Comparison of boundsλ = 0.06.


The approximate bound (the curve in the middle) acts as a bound and it is much sharper than the original one.From these figures, as well as the propositions, one can see clearly the importance for solvency of increasing the

population of policyholders and the importance of the initial fund.

6. Final remarks

We have considered in this work the influence of interest rate fluctuations on a temporary life policy from twoaspects, the premium determined by the equivalence principle and the risk of ruin. The influence is related to thedesign of the policy and the form of the premium charged. Except for the case of the natural premium, it is generallytrue that the randomness of the interest rate matters in determining a premium, expecially a deposit premium. Onecan not simply ignore this randomness without a careful analysis. For the risk assessment, the martingale presentedin Proposition 1 and the martingale inequalities are helpful tools, which can be applied for any type of premiumcharging by rewriting the reserve (4) into the form

Rt = D+∫ t

0z0,s(v(s) −Mλ(s))(1 −Ns) ds−MIt

and assessing the bound of|v(t) −Mλ(t)|.The formula for the equivalence premium is also model related (see (10) and (11)). Although the numerical

examples inSection 3show that the differences caused by adopting the Vasicek model and the CIR model are notsevere, the problem of model choice and the robustness of the equivalence premium (say, the actuarial risk causedby the premium being set according to a mis-specification of the model, or based on inappropriate parameter values)deserves rigorous and further studies.

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Economics 3, 195–213.Chow, Y.S., Teicher, H., 1997. Probability Theory—Independence, Interchangeability, Martingales, third edition. Springer.Cox, J.C., Ingersoll, J.E., Ross, S.A., 1985. A theory of the term structure of interest rates. Econometrica 53, 385–407.De Schepper, A., De Vijlder, F., Goovaerts, M.J., Kaas, R., 1992. Interest randomness in annuities certain. Insurance: Math. Econ. 11, 271–281.Duffie, D., 1996. Dynamic Asset Pricing Theory. Princeton.Dufresne, D. (1990). The distribution of a perpetuity, with application to risk theory and pension funding. Scand. Actuarial J. 39–79.Frostig, E., Haberman, S., Levikson, B., 2003. Ruin probabilities in life insurance. Scand. Actuarial J., 136–152.Grosen, A., Jørgensen, P.L., 2000. Fair valuation of life insurance liabilities: the impact of interest rate guarantees, surrender options and bonus

policies. Insurance: Math. Econ. 26, 37–57.Haberman, S., Pitacco, E., 1999. Actuarial Models for Disability Insurance. Chapman & Hall/CRC.Heath, D., Jarrow, R., Morton, A., 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation.

Econometrica 60, 77–105.Karatzas, I., Shreve, S.E., 1991. Brownian Motion and Stochastic Calculus. Springer.Parker, G., 1994. Two stochastic approaches for discounting actuarial functions. ASTIN Bull 24, 167–181.Perry, D., Stadje, W., 2001. Function space integration for annuities. Insurance: Math. Econ. 29, 73–82.Perry, D., Stadje, W., Yosef, R., 2003. Annuities with controlled random interest rates. Insurance: Math. Econ. 32, 245–253.Vasicek, O., 1977. An equilibrium characterization of the term structure. J. Finan. Econ. 5, 177–188.

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The premium and the risk of a life policy in the presence of interest rate fluctuations