# Stochastic Interest Model Based on Compound Poisson Process and Applications in Actuarial Science

Authors: Shilong Li; Chuancun Yin; Xia Zhao; Hongshuai Dai
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Considering stochastic behavior of interest rates in financial market, we construct a new class of interest models based on compound Poisson process. Different from the references, this paper describes the randomness of interest rates by modeling the force of interest with Poisson random jumps directly. To solve the problem in calculation of accumulated interest force function, one important integral technique is employed. And a conception called the critical value is introduced to investigate the validity condition of this new model. We also discuss actuarial present values of several life annuities under this new interest model. Simulations are done to illustrate the theoretical results and the effect of parameters in interest model on actuarial present values is also analyzed.


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## Body

## 1. Introduction

In traditional study of life insurance, interest rate is assumed to be deterministic. However, durations of policies are typically very long (often 20 or even more years) in life insurance and life annuity. So the uncertainty of future interest rate influences the accuracy of its actuarial values deeply. In addition, stochastic models have been widely used in finance and insurance (such as [1–4]). Hence, it is necessary and natural to consider stochastic interest models in life contingencies.So far, many models have been investigated in order to describe the randomness of interest in actuarial literature. Reference [5] first treated the force of interest as a random variable in his actuarial research. In [6], autoregressive models of order one are introduced to model interest rate. References [7, 8] computed moments of insurance and annuity functions using similar models. Reference [9] modeled the force of interest as an ARIMAp,d,q process and utilized this model to analyze the moments of present value functions. Reference [10] used a Markov process to model the series of interest rates in the research of ruin probability. The above literatures enriched the application of stochastic interest models in actuarial science, but they assume that the interest rate in one year is fixed, which does not always agree with the practice of financial market.To capture the randomness of interest rates in actuarial science better, the method of stochastic perturbation was proposed. In this method, the interest force at timet is expressed as (1)δt=δ+Xt,where δ is an interest force unrelated to the time t and X(t) denotes a stochastic process resulting in perturbation of fixed interest force δ. Hence, the accumulated interest force function is(2)J0t=∫0tδsds=δ·t+∫0tXsds=δ·t+Zt.There are two modeling ways about perturbation methods in actuarial literatures. The first one considersZ(t) as a stochastic process, like Wiener process, Ornstein-Uhlenbeck process, Poisson process, and so on. References [11, 12] constructed stochastic interest rate models regarding Z(t) as Ornstein-Uhlenbeck process and Wiener process, respectively, and investigated the mean and the standard deviation of continuous-time life annuities. Reference [13] studied the mean and the variance of the present value of discrete-time streams in life insurance under these models. Reference [14] also discussed the distribution of life annuities with stochastic interest models driven by Wiener process and Ornstein-Uhlenbeck process, respectively. Considering the jumps in interest process, [15, 16] expressed the accumulated force of interest by Wiener process and Poisson process and further studied the optimal dividend strategy in ruin theory and pricing perpetual options, respectively.In the second way, the researchers first describe the perturbation of interest forceX(t) as a special stochastic process and then find the accumulated interest force function by stochastic integration. References [17, 18] discussed the first three moments of homogeneous portfolios of life insurance and endowment policies by modeling the force of interest directly based on the Wiener process or the Ornstein-Uhlenbeck process and [19] also generalized these results to heterogeneous portfolios. The stochastic upper and lower bounds on the present value of a sequence of cash flows are discussed in [20]. Reference [21] also introduced a class of stochastic interest model in which the force of interest is driven by second order stochastic differential equation. Reference [22] compared these two approaches to the randomness of interest rates by modeling the accumulated force of interest rate and by modeling the force of interest.Both of these two modeling ways have attracted much attention from researchers. The first way brings convenience to calculation, but the behavior of the force of interest can not be expressed distinctly. In the second way, so far only a few special and simple processes have been considered because of great difficulty to the related stochastic calculations, especially under the case with random jumps. However, the market interest rate often jumps discontinuously and randomly (such as the Federal reserve rate and the China’s central bank benchmark interest rate). Here we introduce a new class of stochastic interest model in which the force of interest is expressed by compound Poisson process directly. This model might characterize the stochastic jumping of market interest rate more honestly and directly.This paper is organized as follows. In Section2, we give stochastic interest model based on compound Poisson process and further obtain mathematical expectation of present value of the payment paid at a future time point. In Section 3, its properties and applicability are investigated. In Section 4, we discuss actuarial present values of life annuities in discrete and continuous cases. Simulation results show the influences of the parameters on actuarial present values of annuities. In Section 5, we conclude this paper and put forward some interesting problems in the following sequel researches.


## 2. Stochastic Interest Model under Compound Poisson Process

In this section, we construct a new class of stochastic interest models. As a premise, the following assumptions are given:(
1
) the adjustment interarrival times of interest rate are random;(
2
) the adjustment direction (rise or fall) of interest rate in every stage is independent of each other;(
3
) the adjustment range of interest rate in every stage is identical.As we know, these assumptions coincide with plenty of practical finance markets. All random variables and stochastic processes under consideration are defined on an appropriate probability spaceΩ,P,F and are integrable.


### 2.1. Compound Poisson Process

Compound Poisson process has been widely used in the field of finance and actuarial science, especially in classical ruin probability model. It is described as follows:(3)St=∑n=1NtYn,t>0,S0=0,where Nt,t≥0 is a Poisson process with rate λ>0 which will indicate the adjustment number of the market interest rate on time interval 0,t in this paper. Ynn=1∞ is a sequence of i.i.d. random variables with common distribution Fx=PYn≤x, and Nt,t≥0 and Ynn=1∞ are independent. Suppose that X1,X2,…,Xn,… are the interoccurrence times between adjacent adjustments of interest rate; then they are independent and obey exponential distribution with the same parameter λ>0.For Poisson processNt,t≥0, we first introduce the following important property (see [23]).Lemma 1.
Given thatNt=n, the n arrival times T1,T2,…,Tn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval 0,t. That is, the joint density function of T1,T2,…,Tn is (4)ft1,t2,…,tn=n!tn,0<t1<t2<⋯<tn≤t.From Lemma1, we usually say that, under condition N(t)=n, the times T1,T2,…,Tn at which events occur, considered as unordered random variables, are distributed independently and uniformly on the interval 0,t.


### 2.2. Stochastic Interest Model under Compound Poisson Process

Suppose that the force of interestδt,t≥0 is expressed by(5)δt=δ0+α∑i=1NtYi,t∈0,+∞.Here, Yii=1,2,3,… is the direction of the ith adjustment of the force of the interest rate with Yi=1 for the raise and Yi=-1 for the reduction. Meanwhile PYi=1=1-PYi=-1=p0≤p≤1 and Yii=1,2,… are i.i.d. and also independent of Nt,t≥0. In this model, the adjustment range of interest force in every change is the same as that denoted by α.The processδt,t≥0 is a special continuous-time Markov process, the birth and death process. The initial status of this process is δ0, and the corresponding birth rate and death rate are qδt,δt+α=λp and qδt,δt-α=λ(1-p), respectively.Substituting formula (5) into (2), we can find corresponding expression of accumulated interest force function. Since the direct integration is extremely difficult, we use another integration technique, changing the integral direction similar to the idea of Stietjes integral. The integration procedure can be understood from Figure 1. Suppose that there are three adjustments of interest rate on the interval 0,t and the adjustment times are T1, T2, and T3, respectively. Then the integral ∫0tδsds can be expressed as δ0t-αt-T1+αt-T2+αt-T3. That is, ∫0tδsds=δ0t+α∑i=13Xit-Ti, where Xi, i=1,2,3, show adjustment directions of interest rate.Figure 1
Representing graph of the integration ofJ0t.So we can obtain(6)J0t=δ0t+α∑i=1NtXit-Ti,where Ti denotes the time of the ith adjustment of the force of the interest rate.From formula (6), the random present value of one unit payment at time t can be expressed as (7)exp⁡-J0t=exp⁡-δ0t∏i=1Ntexp⁡-αXit-Ti,and its mathematical expectation is(8)Eexp⁡-J0t=exp⁡-δ0tE∏i=1Ntexp⁡-αXit-Ti.It can be found from the law of total expectation that(9)Eexp⁡-J0t=exp⁡-δ0tEE∏i=1Ntexp⁡-αXit-Ti∣Nt.From Lemma 1, we can obtain that(10)Eexp-α∑i=1NtXit-Ti∣Nt=n=∏i=1nEexp⁡-αXit-Ui=∏i=1np∫0te-αt-x1tdx+1-p∫0teαt-x1tdx=1αntnp1-e-αt+1-peαt-1n=βtn,where random variables, U1,U2,…,Un, are distributed independently and uniformly on the interval [0,t], and (11)βt=1αtp1-e-αt+1-peαt-1=1αteαt-1+p2-e-αt-eαt.Based on formula (10), we can find that βt is nonincreasing with respect to p if αt is fixed and satisfies the following properties.(
1
) If p=0, the market interest will always decrease at the adjusting times of interest rate. Because eαt-1>αt, we can find that βt=1/αteαt-1>1. In this condition, it may happen that the interest rate will be negative if the number of adjusting interest rate times on the interval 0,t is large enough.(
2
) If p=1, the market interest will always increase at the adjusting times of interest rate. Since 1-e-αt<αt, we have βt=1/αt1-e-αt<1. In this condition, the larger the number of adjusting interest rate times on the interval 0,t is, the smaller the present value of the currency is.(
3
) If (12)p=p∗=exp⁡αt-αt-1exp⁡αt+exp⁡-αt-2=exp⁡αtexp⁡αt-αt-1exp⁡αt-12,we find that βt=1 which means that mathematical expectation of the present value of one unit currency at time t will be exp⁡-δ0t. That is, from the point of view of the mathematical expectation, the randomness of the market interest rate will not have an effect on this present value. Hence p∗ is called the equilibrium probability at time t here. At the same time, the following theorem can be obtained.Theorem 2.
The equilibrium probabilityp∗ in formula (12) satisfies the following properties:(1)
p∗ is an increasing function with respect to αt;(2)
0.5<p∗<1;(3)
limαt→+∞⁡p∗=1 and limαt→0⁡p∗=0.5.Proof.
Lety=αt and ϕ(y)=ey(ey-y-1)/(ey-1)2; then p∗=ϕ(y).
Sinceϕ′(y)=eyey-y-1+ey-1/2(ey-1)>0, we can show that ϕy is a strictly increasing function on R+ which illustrates that p∗ is an increasing function with respect to αt.
Further, it is easy to prove thatlimy→+∞⁡ϕy=1 and limy→0+⁡ϕy=0.5. The proof is completed.The following theorem shows the expression of the mathematical expectation of the random present value of one unit currency at timet given in formula (8).Theorem 3.
Under stochastic interest model (5), the mathematical expectation of the random present value of one unit currency at time t can be expressed as(13)Eexp⁡-J0t=exp⁡-δ0t+λtβt-1.Proof.
Based on the properties of the condition expectation, it can be obtained by substituting (10) into (9) that(14)Eexp⁡-J0t=exp⁡-δ0tEβtNt=exp⁡-δ0t∑n=0+∞βtnexp⁡-λtλtnn!=exp⁡-δ0t-λt∑n=0+∞βtλtnn!=exp⁡-δ0t+λtβt-1.


## 2.1. Compound Poisson Process

Compound Poisson process has been widely used in the field of finance and actuarial science, especially in classical ruin probability model. It is described as follows:(3)St=∑n=1NtYn,t>0,S0=0,where Nt,t≥0 is a Poisson process with rate λ>0 which will indicate the adjustment number of the market interest rate on time interval 0,t in this paper. Ynn=1∞ is a sequence of i.i.d. random variables with common distribution Fx=PYn≤x, and Nt,t≥0 and Ynn=1∞ are independent. Suppose that X1,X2,…,Xn,… are the interoccurrence times between adjacent adjustments of interest rate; then they are independent and obey exponential distribution with the same parameter λ>0.For Poisson processNt,t≥0, we first introduce the following important property (see [23]).Lemma 1.
Given thatNt=n, the n arrival times T1,T2,…,Tn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval 0,t. That is, the joint density function of T1,T2,…,Tn is (4)ft1,t2,…,tn=n!tn,0<t1<t2<⋯<tn≤t.From Lemma1, we usually say that, under condition N(t)=n, the times T1,T2,…,Tn at which events occur, considered as unordered random variables, are distributed independently and uniformly on the interval 0,t.


## 2.2. Stochastic Interest Model under Compound Poisson Process

Suppose that the force of interestδt,t≥0 is expressed by(5)δt=δ0+α∑i=1NtYi,t∈0,+∞.Here, Yii=1,2,3,… is the direction of the ith adjustment of the force of the interest rate with Yi=1 for the raise and Yi=-1 for the reduction. Meanwhile PYi=1=1-PYi=-1=p0≤p≤1 and Yii=1,2,… are i.i.d. and also independent of Nt,t≥0. In this model, the adjustment range of interest force in every change is the same as that denoted by α.The processδt,t≥0 is a special continuous-time Markov process, the birth and death process. The initial status of this process is δ0, and the corresponding birth rate and death rate are qδt,δt+α=λp and qδt,δt-α=λ(1-p), respectively.Substituting formula (5) into (2), we can find corresponding expression of accumulated interest force function. Since the direct integration is extremely difficult, we use another integration technique, changing the integral direction similar to the idea of Stietjes integral. The integration procedure can be understood from Figure 1. Suppose that there are three adjustments of interest rate on the interval 0,t and the adjustment times are T1, T2, and T3, respectively. Then the integral ∫0tδsds can be expressed as δ0t-αt-T1+αt-T2+αt-T3. That is, ∫0tδsds=δ0t+α∑i=13Xit-Ti, where Xi, i=1,2,3, show adjustment directions of interest rate.Figure 1
Representing graph of the integration ofJ0t.So we can obtain(6)J0t=δ0t+α∑i=1NtXit-Ti,where Ti denotes the time of the ith adjustment of the force of the interest rate.From formula (6), the random present value of one unit payment at time t can be expressed as (7)exp⁡-J0t=exp⁡-δ0t∏i=1Ntexp⁡-αXit-Ti,and its mathematical expectation is(8)Eexp⁡-J0t=exp⁡-δ0tE∏i=1Ntexp⁡-αXit-Ti.It can be found from the law of total expectation that(9)Eexp⁡-J0t=exp⁡-δ0tEE∏i=1Ntexp⁡-αXit-Ti∣Nt.From Lemma 1, we can obtain that(10)Eexp-α∑i=1NtXit-Ti∣Nt=n=∏i=1nEexp⁡-αXit-Ui=∏i=1np∫0te-αt-x1tdx+1-p∫0teαt-x1tdx=1αntnp1-e-αt+1-peαt-1n=βtn,where random variables, U1,U2,…,Un, are distributed independently and uniformly on the interval [0,t], and (11)βt=1αtp1-e-αt+1-peαt-1=1αteαt-1+p2-e-αt-eαt.Based on formula (10), we can find that βt is nonincreasing with respect to p if αt is fixed and satisfies the following properties.(
1
) If p=0, the market interest will always decrease at the adjusting times of interest rate. Because eαt-1>αt, we can find that βt=1/αteαt-1>1. In this condition, it may happen that the interest rate will be negative if the number of adjusting interest rate times on the interval 0,t is large enough.(
2
) If p=1, the market interest will always increase at the adjusting times of interest rate. Since 1-e-αt<αt, we have βt=1/αt1-e-αt<1. In this condition, the larger the number of adjusting interest rate times on the interval 0,t is, the smaller the present value of the currency is.(
3
) If (12)p=p∗=exp⁡αt-αt-1exp⁡αt+exp⁡-αt-2=exp⁡αtexp⁡αt-αt-1exp⁡αt-12,we find that βt=1 which means that mathematical expectation of the present value of one unit currency at time t will be exp⁡-δ0t. That is, from the point of view of the mathematical expectation, the randomness of the market interest rate will not have an effect on this present value. Hence p∗ is called the equilibrium probability at time t here. At the same time, the following theorem can be obtained.Theorem 2.
The equilibrium probabilityp∗ in formula (12) satisfies the following properties:(1)
p∗ is an increasing function with respect to αt;(2)
0.5<p∗<1;(3)
limαt→+∞⁡p∗=1 and limαt→0⁡p∗=0.5.Proof.
Lety=αt and ϕ(y)=ey(ey-y-1)/(ey-1)2; then p∗=ϕ(y).
Sinceϕ′(y)=eyey-y-1+ey-1/2(ey-1)>0, we can show that ϕy is a strictly increasing function on R+ which illustrates that p∗ is an increasing function with respect to αt.
Further, it is easy to prove thatlimy→+∞⁡ϕy=1 and limy→0+⁡ϕy=0.5. The proof is completed.The following theorem shows the expression of the mathematical expectation of the random present value of one unit currency at timet given in formula (8).Theorem 3.
Under stochastic interest model (5), the mathematical expectation of the random present value of one unit currency at time t can be expressed as(13)Eexp⁡-J0t=exp⁡-δ0t+λtβt-1.Proof.
Based on the properties of the condition expectation, it can be obtained by substituting (10) into (9) that(14)Eexp⁡-J0t=exp⁡-δ0tEβtNt=exp⁡-δ0t∑n=0+∞βtnexp⁡-λtλtnn!=exp⁡-δ0t-λt∑n=0+∞βtλtnn!=exp⁡-δ0t+λtβt-1.


## 3. Validity of Stochastic Interest Model

Under stochastic interest model (5), we can find that the expected present value of one unit currency at time t will be larger than that under fixed force of interest δ0 when p=0.5, and the larger the adjustment frequency intensity of the interest rate, λ, is, the larger the difference between the two above-mentioned values is. That is, when the market interest rate is adjusted frequently, the future interest rate will tend to be underestimated if stochastic interest rate model in formula (5) is considered. This phenomenon often appears in modeling the stochastic interest rate based on a Wiener processes too, such as [11, 12, 17].If we useδ∗=δ0-λβt-1 as the equivalent force of interest of the stochastic interest model, formula (13) can be expressed equivalently as Eexp⁡-J0t=exp-δ∗t.


### 3.1. Validity Condition of Interest Model

In this stochastic model, ifp<p∗, then βt>1; Eexp⁡-J0t will be larger than 1 when t is large enough, and this is not consistent with the actual situation in most cases. In this section, we will discuss how to restrict the value of the future time in this stochastic interest model.Let(15)ft=δ0t-λtβt-1;then (16)f′t=δ0+λ-λ1-peαt+pe-αt.Since f′0=δ0>0 and limt→+∞⁡f′t<0, there is at least one critical value t∗ which satisfies f′(t∗)=0. Now, we will try to find the value of t∗.Let us solve the equation(17)δ0+λ-λ1-peαt+pe-αt=0,which can be rearranged to (18)λ1-peαt2-δ0+λeαt+λp=0.Now we investigate the quadratic function(19)gy=λ1-py2-δ0+λy+λp.Obviously, g(0)=λp>0, g(1)=-δ0<0, and g+∞→+∞; then two real roots of (19) lie in the interval 0,1 and the interval 1,+∞, respectively. Because eαt>0 when the condition t>0, we can only consider the root in the interval 1,+∞ which is (20)y∗=δ0+λ+δ0+λ2-4λ2p1-p2λ1-p=eαt∗,and then the critical value of the interest rate model is (21)t∗=1αln⁡δ0+λ+δ0+λ2-4λ2p1-p2λ1-p.


### 3.2. Numerical Simulation Analysis

Based on the above analysis, whent<t∗, the expected present value Ee-J0t is decreasing with respect to investment term t. That is, only when t<t∗ can the stochastic interest model be used in practice. Figures 2 and 3 show the variation tendency of the critical value with the change of p from 0.4 to 0.7 under δ0=0.04 and 0.05, respectively, and the curves from top to bottom are based on λ=1, 1.5, 2, 2.5, and 3. In Table 1, the critical values t∗ under different δ0, λ, and p are given. From Figures 2 and 3 or Table 1, the critical value becomes larger and larger with p or δ0 increasing. On the contrary, the critical value will decrease if λ increases. Hence, while using this stochastic interest model, we should consider the values of every parameter and verify whether the term of investment is less than the corresponding critical value.Table 1
Critical valuest∗ under different δ0, λ, and p.
δ
0
λ
p
=
0.04
p
=
0.05
p
=
0.06
p
=
0.07
0.04
1.0
58.4220
112.7633
220.6081
374.8367
1.5
42.1442
92.1720
204.3302
363.6570
2.0
33.1076
79.8673
195.2936
357.7981
2.5
27.3141
71.4591
189.5001
354.1869
3.0
23.2686
65.2474
185.4547
351.7368
0.05
1.0
69.4389
125.9699
231.6250
382.8061
1.5
50.5440
102.9948
212.7300
369.3314
2.0
39.9502
89.2574
202.1362
362.2103
2.5
33.1076
79.8673
195.2936
357.7981
3.0
28.3015
72.9286
190.4876
354.7940
0.06
1
79.6701
137.8804
241.8561
390.4626
1.5
58.4220
112.7633
220.6080
374.8367
2
46.4156
97.7363
208.6016
366.5163
2.5
38.6138
87.4613
200.7999
361.3366
3
33.1076
79.8673
195.2936
357.7981Figure 2
Values oft∗ under δ0=0.04, α=0.0025, and λ=1, 1.5, 2, 2.5, 3, respectively.Figure 3
Values oft∗ under δ0=0.05, α=0.0025, and λ=1, 1.5, 2, 2.5, 3, respectively.Fortunately, from the actual conditions of adjustments of interest rate in financial markets and ordinary life insurance periods, we find that the critical valuet∗ can satisfy the application condition of this interest rate model in general case.


## 3.1. Validity Condition of Interest Model

In this stochastic model, ifp<p∗, then βt>1; Eexp⁡-J0t will be larger than 1 when t is large enough, and this is not consistent with the actual situation in most cases. In this section, we will discuss how to restrict the value of the future time in this stochastic interest model.Let(15)ft=δ0t-λtβt-1;then (16)f′t=δ0+λ-λ1-peαt+pe-αt.Since f′0=δ0>0 and limt→+∞⁡f′t<0, there is at least one critical value t∗ which satisfies f′(t∗)=0. Now, we will try to find the value of t∗.Let us solve the equation(17)δ0+λ-λ1-peαt+pe-αt=0,which can be rearranged to (18)λ1-peαt2-δ0+λeαt+λp=0.Now we investigate the quadratic function(19)gy=λ1-py2-δ0+λy+λp.Obviously, g(0)=λp>0, g(1)=-δ0<0, and g+∞→+∞; then two real roots of (19) lie in the interval 0,1 and the interval 1,+∞, respectively. Because eαt>0 when the condition t>0, we can only consider the root in the interval 1,+∞ which is (20)y∗=δ0+λ+δ0+λ2-4λ2p1-p2λ1-p=eαt∗,and then the critical value of the interest rate model is (21)t∗=1αln⁡δ0+λ+δ0+λ2-4λ2p1-p2λ1-p.


## 3.2. Numerical Simulation Analysis

Based on the above analysis, whent<t∗, the expected present value Ee-J0t is decreasing with respect to investment term t. That is, only when t<t∗ can the stochastic interest model be used in practice. Figures 2 and 3 show the variation tendency of the critical value with the change of p from 0.4 to 0.7 under δ0=0.04 and 0.05, respectively, and the curves from top to bottom are based on λ=1, 1.5, 2, 2.5, and 3. In Table 1, the critical values t∗ under different δ0, λ, and p are given. From Figures 2 and 3 or Table 1, the critical value becomes larger and larger with p or δ0 increasing. On the contrary, the critical value will decrease if λ increases. Hence, while using this stochastic interest model, we should consider the values of every parameter and verify whether the term of investment is less than the corresponding critical value.Table 1
Critical valuest∗ under different δ0, λ, and p.
δ
0
λ
p
=
0.04
p
=
0.05
p
=
0.06
p
=
0.07
0.04
1.0
58.4220
112.7633
220.6081
374.8367
1.5
42.1442
92.1720
204.3302
363.6570
2.0
33.1076
79.8673
195.2936
357.7981
2.5
27.3141
71.4591
189.5001
354.1869
3.0
23.2686
65.2474
185.4547
351.7368
0.05
1.0
69.4389
125.9699
231.6250
382.8061
1.5
50.5440
102.9948
212.7300
369.3314
2.0
39.9502
89.2574
202.1362
362.2103
2.5
33.1076
79.8673
195.2936
357.7981
3.0
28.3015
72.9286
190.4876
354.7940
0.06
1
79.6701
137.8804
241.8561
390.4626
1.5
58.4220
112.7633
220.6080
374.8367
2
46.4156
97.7363
208.6016
366.5163
2.5
38.6138
87.4613
200.7999
361.3366
3
33.1076
79.8673
195.2936
357.7981Figure 2
Values oft∗ under δ0=0.04, α=0.0025, and λ=1, 1.5, 2, 2.5, 3, respectively.Figure 3
Values oft∗ under δ0=0.05, α=0.0025, and λ=1, 1.5, 2, 2.5, 3, respectively.Fortunately, from the actual conditions of adjustments of interest rate in financial markets and ordinary life insurance periods, we find that the critical valuet∗ can satisfy the application condition of this interest rate model in general case.


## 4. Life Annuities under Stochastic Interest Model

Following the notations in [24], the symbol (x) is used to denote a life-age-x. The future lifetime and the curtate-future-lifetime of x are denoted by T(x) and K(x), respectively.


### 4.1. Actuarial Present Values for Discrete Life Annuities

There are two classes of the discrete life annuities: the discrete life annuities-due and the discrete life annuities-immediate. First of all, we consider the former. In the nomenclature, an annuity is called a whole life annuity-due if the annuity pays a unit amount at the beginning of each year that the annuitantx survives, and the actuarial present value of the annuity can be expressed as(22)a¨x=Ea¨Kx+1¯=∑k=0+∞a¨k+1¯PKx=k,where PKx=k=qxk=pxk·qx+k in actuarial theory and according to interest theory, we have(23)a¨k+1¯=E∑n=0ke-∫0nδtdt=∑n=0kEe-J0n=∑n=0ke-δ0n+λnβn-1.Combining formula (22) with formula (23), we obtain the following formula under stochastic interest model introduced here: (24)a¨x=1+∑n=1+∞e-δ0n+λnβn-1pxn.Forn-year temporary life annuity-due of 1 per year, the actuarial present value under this stochastic interest rate model can be expressed as(25)a¨x:n¯=1+∑k=1n-1e-δ0k+λkβk-1pxk.The procedures used above for annuities-due can be adapted for annuities-immediate where payments are made at the ends of the payment periods. Such that, for a whole life annuity-immediate, the actuarial present value can be given as(26)ax=∑n=1+∞e-δ0n+λnβn-1pxn,and the actuarial present value of n-year temporary life annuity for the annuitant (x) is (27)ax:n¯=∑k=1ne-δ0k+λkβk-1pxk.


### 4.2. Actuarial Present Value for Continuous Life Annuities

In order to analyze the actuarial present value of this class of annuities, we first consider the whole life annuity payable continuously at the rate of 1 per year. For an annuitant(x), the actuarial present value of this life annuity is denoted by a¯x. From [24], we can obtain the following formula:(28)a¯x=Ea¯Tx¯=∫0+∞a¯t¯dFTxt,a¯t¯=E∫0texp⁡-∫0uδsdsdu=∫0tEexp⁡-J0udt.Using Fubini’s theorem, from formulas (28), we have(29)a¯x=∫0+∞Eexp⁡-J0tpxtdt.Substituting formula (13) into formula (29), we have (30)a¯x=∫0+∞exp⁡-δ0t+λtβt-1pxtdt,and then the actuarial present value of an n-year temporary continuous life annuity for the annuitant (x) is (31)a¯x:n¯=∫0nexp⁡-δ0t+λtβt-1pxtdt.


### 4.3. Numerical Analysis

Based on the results in Sections4.1 and 4.2, we calculate the actuarial present value of both a 30-year temporary discrete life annuity-due and a 30-year temporary continuous life annuity for the annuitant (30) under different δ0, α, λ, and p. The simulation results are shown in Tables 2 and 3, respectively. In these calculations, the experience life table of the Chinese life insurance (2000–2003) (male) is used. It should be noted that the Uniform Distribution assumption—the classical fractional age assumption—is used in the calculation of the actuarial present values of the continuous life annuity.Table 2
Values ofa¨30:30¯ for different δ0, α, λ, and p under the stochastic interest rate model.
δ
0
α
λ
p
=
0.45
p
=
0.5
p
=
0.505
p
=
0.51
p
=
0.6
p
=
0.7
0.04
0.0025
1.0
17.9990
17.5461
17.5023
17.4587
16.7152
15.9727
1.5
18.2774
17.5840
17.5180
17.4525
16.3648
15.3324
2.0
18.5662
17.6222
17.5337
17.4463
16.0312
14.7529
2.5
18.8659
17.6606
17.5495
17.4401
15.7134
14.2268
3.0
19.1768
17.6992
17.5654
17.4339
15.4105
13.7477
0.0030
1.0
18.1289
17.5795
17.5266
17.4741
16.5884
15.7219
1.5
18.4807
17.6344
17.5546
17.4756
16.1843
14.9951
2.0
18.8489
17.6899
17.5828
17.4772
15.8027
14.3480
2.5
19.2345
17.74560
17.6112
17.4788
15.4420
13.7695
3.0
19.6385
17.8026
17.6398
17.4804
15.1007
13.2501
Nonrandom
17.4712
0.05
0.0025
1.0
16.0694
15.7005
15.6647
15.6291
15.0217
14.4128
1.5
16.2953
15.7308
15.6769
15.6235
14.7341
13.8854
2.0
16.5293
15.7613
15.6891
15.6178
14.4599
13.4064
2.5
16.7718
15.7919
15.7014
15.6122
14.1982
12.9699
3.0
17.0232
15.8228
15.7138
15.6066
13.9484
12.5711
0.0030
1.0
16.1746
15.7271
15.6840
15.6412
14.9174
14.2062
1.5
16.4596
15.7711
15.7060
15.6416
14.5854
13.6064
2.0
16.7575
15.8154
15.7281
15.6420
14.2713
13.0702
2.5
17.0690
15.8602
15.7504
15.6425
13.9738
12.5888
3.0
17.3950
15.9054
15.7729
15.6429
13.6917
12.1547
Nonrandom
15.6405Table 3
Values ofa-30:30¯ for different δ0, α, λ, and p under the stochastic interest rate model.
δ
0
α
λ
p
=
0.45
p
=
0.5
p
=
0.505
p
=
0.51
p
=
0.6
p
=
0.7
0.04
0.0025
1.0
17.6555
17.1858
17.1404
17.0952
16.3264
15.5610
1.5
17.9455
17.2257
17.1572
17.0894
15.9653
14.9034
2.0
18.2466
17.2658
17.1742
17.0836
15.6220
14.3099
2.5
18.5594
17.3063
17.1911
17.0777
15.2955
13.7724
3.0
18.8846
17.3470
17.2082
17.0719
14.9847
13.2841
0.0030
1.0
17.7910
17.2209
17.1661
17.1117
16.1959
15.3034
1.5
18.1578
17.2788
17.1960
17.1142
15.7799
14.5580
2.0
18.5423
17.3373
17.2262
17.1167
15.3876
13.8964
2.5
18.9457
17.3963
17.2565
17.1192
15.0174
13.3065
3.0
19.3692
17.4561
17.2871
17.1218
14.6677
12.7783
Nonrandom
17.1069
0.05
0.0025
1.0
15.6838
15.3024
15.2655
15.2288
14.6025
13.9766
1.5
15.9182
15.3342
15.2786
15.2234
14.3070
13.4364
2.0
16.1614
15.3662
15.2917
15.2180
14.0256
12.9470
2.5
16.4137
15.3984
15.3049
15.2127
13.7575
12.5021
3.0
16.6756
15.4308
15.3181
15.2073
13.5017
12.0964
0.0030
1.0
15.7932
15.3304
15.2859
15.2416
14.4955
13.7650
1.5
16.0893
15.3765
15.3092
15.2427
14.1546
13.1514
2.0
16.3993
15.4230
15.3328
15.2438
13.8326
12.6044
2.5
16.7241
15.4701
15.3565
15.2450
13.5281
12.1146
3.0
17.0646
15.5176
15.3804
15.2461
13.2398
11.6740
Nonrandom
15.2396From Tables2 and 3, we can find that the actuarial present values become smaller and smaller as the probability that the interest rate rises at the change time point, p, increases continuously when other parameters are fixed and this result is obvious because the probability of the future force of interest rising will become larger and larger with p increasing. Furthermore, all the values when p=0.5 are larger than those under the nonrandom condition (i.e., δt≡δ0 for t≥0), which verifies Theorem 2 from the quantitative aspect. At the same time, under the condition that δ0, α, and p are fixed, the actuarial present value becomes larger and larger with increasing λ, and this result illustrates that the present value will increase if the interest rate changes frequently. If other conditions are fixed, the actuarial present value also becomes larger and larger with increasing α which means that the present value will increase if the range of every interest rate change is larger. At last, we can find that the actuarial present value will become larger with decreasing δ0 which verifies that the present value of the money at future time will decrease in general if the initial interest rate increases in practice.


## 4.1. Actuarial Present Values for Discrete Life Annuities

There are two classes of the discrete life annuities: the discrete life annuities-due and the discrete life annuities-immediate. First of all, we consider the former. In the nomenclature, an annuity is called a whole life annuity-due if the annuity pays a unit amount at the beginning of each year that the annuitantx survives, and the actuarial present value of the annuity can be expressed as(22)a¨x=Ea¨Kx+1¯=∑k=0+∞a¨k+1¯PKx=k,where PKx=k=qxk=pxk·qx+k in actuarial theory and according to interest theory, we have(23)a¨k+1¯=E∑n=0ke-∫0nδtdt=∑n=0kEe-J0n=∑n=0ke-δ0n+λnβn-1.Combining formula (22) with formula (23), we obtain the following formula under stochastic interest model introduced here: (24)a¨x=1+∑n=1+∞e-δ0n+λnβn-1pxn.Forn-year temporary life annuity-due of 1 per year, the actuarial present value under this stochastic interest rate model can be expressed as(25)a¨x:n¯=1+∑k=1n-1e-δ0k+λkβk-1pxk.The procedures used above for annuities-due can be adapted for annuities-immediate where payments are made at the ends of the payment periods. Such that, for a whole life annuity-immediate, the actuarial present value can be given as(26)ax=∑n=1+∞e-δ0n+λnβn-1pxn,and the actuarial present value of n-year temporary life annuity for the annuitant (x) is (27)ax:n¯=∑k=1ne-δ0k+λkβk-1pxk.


## 4.2. Actuarial Present Value for Continuous Life Annuities

In order to analyze the actuarial present value of this class of annuities, we first consider the whole life annuity payable continuously at the rate of 1 per year. For an annuitant(x), the actuarial present value of this life annuity is denoted by a¯x. From [24], we can obtain the following formula:(28)a¯x=Ea¯Tx¯=∫0+∞a¯t¯dFTxt,a¯t¯=E∫0texp⁡-∫0uδsdsdu=∫0tEexp⁡-J0udt.Using Fubini’s theorem, from formulas (28), we have(29)a¯x=∫0+∞Eexp⁡-J0tpxtdt.Substituting formula (13) into formula (29), we have (30)a¯x=∫0+∞exp⁡-δ0t+λtβt-1pxtdt,and then the actuarial present value of an n-year temporary continuous life annuity for the annuitant (x) is (31)a¯x:n¯=∫0nexp⁡-δ0t+λtβt-1pxtdt.


## 4.3. Numerical Analysis

Based on the results in Sections4.1 and 4.2, we calculate the actuarial present value of both a 30-year temporary discrete life annuity-due and a 30-year temporary continuous life annuity for the annuitant (30) under different δ0, α, λ, and p. The simulation results are shown in Tables 2 and 3, respectively. In these calculations, the experience life table of the Chinese life insurance (2000–2003) (male) is used. It should be noted that the Uniform Distribution assumption—the classical fractional age assumption—is used in the calculation of the actuarial present values of the continuous life annuity.Table 2
Values ofa¨30:30¯ for different δ0, α, λ, and p under the stochastic interest rate model.
δ
0
α
λ
p
=
0.45
p
=
0.5
p
=
0.505
p
=
0.51
p
=
0.6
p
=
0.7
0.04
0.0025
1.0
17.9990
17.5461
17.5023
17.4587
16.7152
15.9727
1.5
18.2774
17.5840
17.5180
17.4525
16.3648
15.3324
2.0
18.5662
17.6222
17.5337
17.4463
16.0312
14.7529
2.5
18.8659
17.6606
17.5495
17.4401
15.7134
14.2268
3.0
19.1768
17.6992
17.5654
17.4339
15.4105
13.7477
0.0030
1.0
18.1289
17.5795
17.5266
17.4741
16.5884
15.7219
1.5
18.4807
17.6344
17.5546
17.4756
16.1843
14.9951
2.0
18.8489
17.6899
17.5828
17.4772
15.8027
14.3480
2.5
19.2345
17.74560
17.6112
17.4788
15.4420
13.7695
3.0
19.6385
17.8026
17.6398
17.4804
15.1007
13.2501
Nonrandom
17.4712
0.05
0.0025
1.0
16.0694
15.7005
15.6647
15.6291
15.0217
14.4128
1.5
16.2953
15.7308
15.6769
15.6235
14.7341
13.8854
2.0
16.5293
15.7613
15.6891
15.6178
14.4599
13.4064
2.5
16.7718
15.7919
15.7014
15.6122
14.1982
12.9699
3.0
17.0232
15.8228
15.7138
15.6066
13.9484
12.5711
0.0030
1.0
16.1746
15.7271
15.6840
15.6412
14.9174
14.2062
1.5
16.4596
15.7711
15.7060
15.6416
14.5854
13.6064
2.0
16.7575
15.8154
15.7281
15.6420
14.2713
13.0702
2.5
17.0690
15.8602
15.7504
15.6425
13.9738
12.5888
3.0
17.3950
15.9054
15.7729
15.6429
13.6917
12.1547
Nonrandom
15.6405Table 3
Values ofa-30:30¯ for different δ0, α, λ, and p under the stochastic interest rate model.
δ
0
α
λ
p
=
0.45
p
=
0.5
p
=
0.505
p
=
0.51
p
=
0.6
p
=
0.7
0.04
0.0025
1.0
17.6555
17.1858
17.1404
17.0952
16.3264
15.5610
1.5
17.9455
17.2257
17.1572
17.0894
15.9653
14.9034
2.0
18.2466
17.2658
17.1742
17.0836
15.6220
14.3099
2.5
18.5594
17.3063
17.1911
17.0777
15.2955
13.7724
3.0
18.8846
17.3470
17.2082
17.0719
14.9847
13.2841
0.0030
1.0
17.7910
17.2209
17.1661
17.1117
16.1959
15.3034
1.5
18.1578
17.2788
17.1960
17.1142
15.7799
14.5580
2.0
18.5423
17.3373
17.2262
17.1167
15.3876
13.8964
2.5
18.9457
17.3963
17.2565
17.1192
15.0174
13.3065
3.0
19.3692
17.4561
17.2871
17.1218
14.6677
12.7783
Nonrandom
17.1069
0.05
0.0025
1.0
15.6838
15.3024
15.2655
15.2288
14.6025
13.9766
1.5
15.9182
15.3342
15.2786
15.2234
14.3070
13.4364
2.0
16.1614
15.3662
15.2917
15.2180
14.0256
12.9470
2.5
16.4137
15.3984
15.3049
15.2127
13.7575
12.5021
3.0
16.6756
15.4308
15.3181
15.2073
13.5017
12.0964
0.0030
1.0
15.7932
15.3304
15.2859
15.2416
14.4955
13.7650
1.5
16.0893
15.3765
15.3092
15.2427
14.1546
13.1514
2.0
16.3993
15.4230
15.3328
15.2438
13.8326
12.6044
2.5
16.7241
15.4701
15.3565
15.2450
13.5281
12.1146
3.0
17.0646
15.5176
15.3804
15.2461
13.2398
11.6740
Nonrandom
15.2396From Tables2 and 3, we can find that the actuarial present values become smaller and smaller as the probability that the interest rate rises at the change time point, p, increases continuously when other parameters are fixed and this result is obvious because the probability of the future force of interest rising will become larger and larger with p increasing. Furthermore, all the values when p=0.5 are larger than those under the nonrandom condition (i.e., δt≡δ0 for t≥0), which verifies Theorem 2 from the quantitative aspect. At the same time, under the condition that δ0, α, and p are fixed, the actuarial present value becomes larger and larger with increasing λ, and this result illustrates that the present value will increase if the interest rate changes frequently. If other conditions are fixed, the actuarial present value also becomes larger and larger with increasing α which means that the present value will increase if the range of every interest rate change is larger. At last, we can find that the actuarial present value will become larger with decreasing δ0 which verifies that the present value of the money at future time will decrease in general if the initial interest rate increases in practice.


## 5. Conclusion

In this paper, we introduce a new stochastic interest model in which the force of interest is driven by compound Poisson process directly. Different from the references, the modeling method makes the interest model more reasonable and the random jumping behavior of interest rate is described directly. We investigate the validity conditions of this model and introduce a conception called the critical value of the interest rate model. Based on this model, several common life annuities are studied and the numerical results under different parameters are compared adequately.This paper proposes a new research perspective of modeling stochastic interest. Following this idea, there are several meaningful issues which deserve to study further.(1) Some continuous stochastic processes can be blended into modeling stochastic interest rate on the basis of the model in this paper. (2) This model can be generalized by using some random variables for the change ranges of the force of interest and for the frequency parameter of the Poisson process in this model. (3) Both the empirical study and the statistical analysis about this stochastic interest rate should be made. We will explore these issues in our future researches.



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*Source: 3472319-2017-05-24.xml*