# An Annuitization Problem in the Tax-Deferred Annuity Model

Authors: Yanan Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper examines the optimal annuitization, investment, and consumption strategies of an individual facing a time-dependent mortality rate in the tax-deferred annuity model and considers both the case when the rate of buying annuities is unrestricted and the case when it is restricted. At the beginning, by using the dynamic programming principle, we obtain the corresponding HJB equation. Since the existence of the tax and the time-dependence of the value function make the corresponding HJB equation hard to solve, firstly, we analyze the problem in a simpler case and use some numerical methods to get the solution and some of its useful properties. Then, by using the obtained properties and Kuhn–Tucker conditions, we discuss the problem in general cases and get the value functions and the optimal annuitization strategies, respectively.


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## Body

## 1. Introduction

Referring to [1], we know that, with the aging population, the basic endowment insurance has been unable to meet people’s insurance needs. In order to guarantee the sustainable operation of the old age insurance system, many governments begin to adopt preferential tax policies and attract people to buy tax-deferred annuities. In the United States, tax-deferred annuities have benefited more than 60% of families and become the main source of continuous growth of pension assets in the past 30 years. Tax-deferred annuities also reduce the pension burden in many countries such as Germany and France. However, in some countries, the pilot of tax extension type endowment insurance is not successful. Since tax-deferred annuities can not only play a positive role in improving the three pillar endowment insurance system, but can also alleviate social problems caused by the aging population, prosper the commercial endowment insurance market, and provide investment funds for the securities market, the promotion of tax-deferred annuities is extremely essential and urgent in the whole world.In order to attract more and more people to buy tax-deferred annuities, the government should know how to adopt tax preferential policies, and insurance companies should know how to design and publicize products. In this paper, by using stochastic optimal methods and the Kuhn–Tucker conditions, we get the public’s strategies on how to buy tax-deferred annuities. Then, we can give some suggestions for the government and insurance companies to promote tax-deferred annuities.Up till now, there have been a lot of papers investigating individuals’ optimal annuitization strategies in a taxable annuity model. Reference [2] investigates the optimal annuitization strategy for an infinitely lived individual who faces a choice between voluntary annuitization and discretionary management of assets with systematic withdrawal for consumption purposes. Reference [3] examines the optimal annuitization, investment, and consumption strategies of a utility-maximizing retiree facing a stochastic time of death under an all-or-nothing arrangement and an open-market structure arrangement, respectively. Reference [4] solves a problem of finding the optimal time of annuitization for a retiree having the possibility of choosing her investment and consumption strategy. Reference [5] aims at maximizing the expected utility of consumption with commutable life annuities and finds that the optimal annuitization strategy depends on the size of proportional surrender charge. For a future reference, we can refer to [6–9].However, there are only a few articles that investigate the tax-deferred annuities. Reference [10] gives some comparisons of different tax regimes applied to private pensions. Reference [11] identifies four potential risks associated with tax-deferred. Reference [12] concerns with the risk of fluctuating tax rates and changing tax brackets; it also examines how the saver’s tax credit changes optimal tax-deferral choices of individuals. These papers are all analyzed by using the data processing method, and they are focusing on comparing the tax-deferred annuity and an ordinary one. This paper investigates the optimal annuitization, investment, and consumption strategies for an individual under EET tax payment mode (in which individuals deposit the premium into an independent account, and the insurance company invests it in the financial market and returns the income to the individual in the form of annuities on retirement. Both the taxes of premiums and investment incomes are deferred until receiving annuities) by using stochastic optimal control theory. We consider both the case when the rate of buying annuities is unrestricted and the case when it is restricted. We get the value functions and the optimal annuitization strategies, respectively.This paper is organized as follows. In Section2, the mathematical formulation of the problem is presented. In Section 3, we investigate the optimal investment, consumption, and annuitization strategies when the rate of buying annuities is unrestricted. In Section 4, the same problem is considered when the rate of buying annuities is restricted. Finally, the numerical method is provided in Appendix.


## 2. Problem Formulation

Let us consider the financial market first. Suppose that there are a risk-free asset and a risky asset in the financial market. Their prices per shareBt and Rt follow the following equations:(1)dBt=rBtdt,dRt=Rtμdt+σdWt.Here,r>0 represents the risk-free rate of return, μ>r is the drift coefficient of the risk asset, σ>0 is the volatility coefficient of the risk asset, Wt is a standard Brownian motion on a complete probability space Ω,ℱ,P, and ℱt is the natural filtration generated by Wt.Letb1,t be the wealth of the individual invested in the risky asset, and pt the rate of buying tax-deferred annuities. Setting aside taxes, we can get that the individual’s wealth process under these policies denoted by xtb1,p satisfies(2)dxtb1,p=rxtb1,p+b1,tμ−r−ptdt+σb1,tdWt.Like a general tax policy, we suppose that there is a thresholdb>0; if the income rate is less that b, nothing will be taxed; otherwise, the part above b will be taxed with a tax rate k. Here, we also assume that the income rate is greater than b since, otherwise, there is no tax preference to buy tax-deferred annuities for the individual. Then, the taxed wealth dXtb1,p satisfies(3)dXtb1,p=rXtb1,p+b1,tμ−r−ptdt+σb1,tdWt−krXtb1,p+b1,tμ−r−pt−bdt+σb1,tdWt=r^Xtb1,p+b1,tμ^−r^−p^tdt+σ^b1,tdWt+kbdt.Here,(4)r^=1−kr,μ^=1−kμ,σ^=1−kσ,p^t=1−kpt.Taking consumptions into consideration, we have(5)dXtb1,p,c=r^Xtb1,p,c+b1,tμ^−r^−p^tdt+σ^b1,tdWt+kb−ctdt,where ct is the consumption rate of the individual at time t.In the pension market, we assume that the insurance company evaluates the residual lifetime using exponentially distributed random variables. Denote the hazard rate evaluated by the individual at timet by λts and the hazard rate evaluated by the insurance company at time t by λto, respectively. The wealth in the pension fund will also be invested in the financial market. The difference is that the income will not be taxed until the retirement time T. So, the income process in the pension fund under investment policy b2,t is(6)dYtb2=rYtb2+b2,tμ−r+ptdt+σb2,tdWt.Suppose that the individual has to pay taxes at a tax ratek2 before he receives annuities. Since the insurance company assumes that the hazard rate of the individual is λto, we can get that the annuity received by the individual is(7)c0=1−k2YTA,where YT is the total pension fund at the retirement time T, A=∫T∞e−λuo+rdu. Then, the individual’s expected cumulative discounted pension is(8)gYT=∫T∞e−λus+rc0du=B1−k2YTA,where B=∫T∞e−λus+rdu.In this paper, we assume that the short selling is allowed, and then,bi,t,i=1,2 can be negative. Now, let us give the definition of admissible controls. If π=b1,tπ,b2,tπ,ptπ,ctπ satisfies(i)
bi,tπ,i=1,2 are ℱt predictable control processes, and ptπ, ctπ are nonnegative ℱt predictable control processes.(ii)
∫0∞ctπdt<∞, ∫0∞bi,tπ2dt<∞,i=1,2, ∫0∞ptπdt<∞.Then, we say thatπ is admissible. Denote all the admissible strategies by Π.Next, let us focus on this question: What are the optimal consumption/investment strategies and the best annuitization strategy for the individual to maximize the sum of his expected accumulated discounted consumption before retirement and the expected discounted wealth at retirement? Suppose that the individual’s wealth at timet is x and his money in his pension account at time t is y. Denote the objective function under policy π by Jπt,x,y, and then, we have(9)Jπt,x,y=E∫tTΓt,scsπds+Γt,TXTπ+gYTπ|Xtπ=x,Ytπ=y,where Γt,s=e−∫tsr+λusdu. Thus, the value function Vt,x,y is(10)Vt,x,y=supπ∈ΠJπt,x,y.In the following, we give some initial condition assumptions.(i)
Assumption 1(11)Vxt,0,y>1,Vxt,0,y≥Vyt,0,y1−k,∀y≥0.(ii)
Assumption 2(12)Vyt,0,01−k>1.


## 3. The Optimal Strategies whenpt Is Unrestricted

In this section, we investigate the optimal strategy of an individual whenpt is unrestricted. According to 3.4.2 in [13], we can get the following theorem.Theorem 1.
Vt,x,y satisfies(13)minr+λtsVt,x,y−supb1≥0,b2≥0r^x+μ^−r^b1+kbVxt,x,y+σ^2b122Vxxt,x,y+σσ^b1b2Vxy+ry+b2μ−rVyt,x,y+σ2b222Vyyt,x,y−Vtt,x,y,Vxt,x,y−1,1−kVxt,x,y−Vyt,x,y=0,t<T,(14)VT,x,y=x+gy,(15)Vt,0,0=0,t≥0.By taking derivatives with respect tob1,b2 in (13), we can get the maximum point:(16)b1∗t,x,y=μ^−r^Vyyt,x,yVxt,x,y−Vyt,x,yVxyt,x,yσ^2Vxy2t,x,y−Vxxt,x,yVyyt,x,y,(17)b2∗t,x,y=μ−rVxxt,x,yVyt,x,y−Vxt,x,yVxyt,x,yσ2Vxy2t,x,y−Vxxt,x,yVyyt,x,y.The following theorem gives us the solution of (13) in the case when Vxt,x,y>1, and Vyt,x,y<1−kVxt,x,y, and it is essential in solving (13).Theorem 2.
ForVxt,x,y>1, Vyt,x,y<1−kVxt,x,y, we have(18)Vt,x,y=V1t,x+V2t,y,0≤t≤T,where V1t,x satisfies (23) and (20), and V2t,y satisfies (24) and (21).Proof.
Denote the optimal rate of buying tax-deferred annuities bypt∗ and the optimal consumption rate by ct∗. For Vxt,x,y>1, Vyt,x,y<1−kVxt,x,y, we can deduce that ct∗=0,pt∗=0. The individual will never buy annuities in this situation, and there is no interaction between x and y. This implies that(19)Vt,x,y=V1t,x+V2t,y,where V1t,x are the benefits obtained in the financial market, and V2t,y are the benefits obtained from the annuity fund. Then, we can obtain that(20)V1t,0=0,t≥0,V1T,x=x,x≥0,(21)V2t,0=0,t≥0,V2T,y=gy,y≥0.
Taking (16) and (17) into (13), we can get(22)r+λtsVt,x,y−Vtt,x,y=r^xVxt,x,y+ryVyt,x,y+kbVxt,x,y−R22Vx2t,x,yVxxt,x,y+Vy2t,x,yVyyt,x,y,where R=μ−r/σ. Then, the following two differential equations are satisfied immediately:(23)−r+λtsV1t,x+V1tt,x+r^xV1xt,x+kbV1xt,x−R2V1x2t,x2V1xxt,x=0,(24)−r+λtsV2t,y+V2tt,y+ryV2yt,y−R2V2y2t,y2V2yyt,y=0.
According to [14], we know that (23) and (20) or (24) and (21) may be solved by using dual methods. However, the items r+λtsV1t,x, r+λtsV2t,y in (23) and (24) combined with the time dependence of λts make those partial differential equations hard to solve. In this paper, we use the finite difference method to get V1t,x and V2t,y in Appendix A. Considering Figures 1 and 2, we get the following theorem.Figure 1
V1t,xFigure 2
V2t,yTheorem 3.
V1t,x is a concave increasing positive function with respect to x, V2t,y is a concave increasing positive function with respect to y. The growth rates of V1t,x and V2t,y are increasing with respect to t.Next, Let us consider the solutions of (13) in other cases. Firstly, the boundlessness of pt and ct tells us that the individual’s optimal consumption and the optimal amount of buying annuities are lump-sum payments. Denote Δ1∗ to be the individual’s optimal consumption at time t, denote Δ2∗ to be the optimal amount of buying annuities at time t, and considering the right hand of (13), we can obtain that(25)Vt,x,y=Vt,x−Δ1∗−Δ2∗,y+Δ2∗1−k+Δ1∗,(26)Vxt,x−Δ1∗−Δ2∗,y+Δ2∗1−k=1,(27)Vxt,x−Δ1∗−Δ2∗,y+Δ2∗1−k=1−kVyt,x−Δ1∗−Δ2∗,y+Δ2∗1−k.Considering (26) and (27) and using Theorem 2, we know that(28)Vt,x−Δ1∗−Δ2∗,y+Δ21−k=V1t,x−Δ1∗−Δ2∗+V2t,y+Δ2∗1−k.Combining it with (25), we can get that(29)Vt,x,y=V1t,x−Δ1∗−Δ2∗+V2t,y+Δ2∗1−k+Δ1∗.This implies thatVt,x,y is the solution of the following restricted optimal problem:(30)P:supΔ1≥0,Δ2≥0,x−Δ1−Δ2≥0V1t,x−Δ1−Δ2+V2t,y+Δ21−k+Δ1.Here,Δ1 is the individual’s consumption strategy, and Δ2 is the wealth of buying annuities. Denote the optimal policy by Δ1∗t,x,y,Δ2∗t,x,y.In order to solve Problem P, we need to do some explicit case analysis. Firstly, definex0t,x1t,y as(31)V1xt,x0t=1,1−kV1xt,x1t,y=V2yt,y,y≥0,Let(32)D1=0≤t≤T,y≥0|x1t,y<x0t,D2=0≤t≤T,y≥0|x1t,y≥x0t.Then, we can get the following two theorems.Theorem 4.
Fort,y∈D1, we have(33)Δ1∗t,x,y=0,Δ2∗t,x,y=0,x≤x1t,y,x−x1t,y,x>x1t,y.Proof.
Define(34)LΔ1,Δ2=V1t,x−Δ1−Δ2+V2t,y+Δ21−k+Δ1.
On one hand, we can obtain that(35)∂LΔ1,Δ2∂Δ1=1−V1′t,x−Δ1−Δ2,∂LΔ1,Δ2∂Δ2=V2′t,y+Δ2/1−k1−k−V1′t,x−Δ1−Δ2.
On the other hand, fort,y∈D1, we have x1t,y<x0t. Then, for x≤x1t,y<x0t, we can get(36)V1′t,x−Δ1−Δ2>V1′t,x0t=1,V2′t,y+Δ21−k=1−kV1′t,x1t,y≤1−kV1′t,x−Δ1−Δ2.
That is,(37)∂LΔ1,Δ2∂Δ1<0,∂LΔ1,Δ2∂Δ2≤0.
Clearly, in this case, we have(38)Δ1∗t,x,y=0,Δ2∗t,x,y=0.
Forx>x1t,y, the signs of ∂LΔ1,Δ2/∂Δ1and∂LΔ1,Δ2/∂Δ2 are uncertain. Since(39)∂2LΔ1,Δ2∂Δ12=∂2LΔ1,Δ2∂Δ22=∂2LΔ1,Δ2∂Δ1∂Δ2=V1″t,x−Δ1−Δ2≤0.
We can get that(40)∂2LΔ1,Δ2∂Δ12=∂2LΔ1,Δ2∂Δ22≤0,∂2LΔ1,Δ2∂Δ12∂2LΔ1,Δ2∂Δ22−∂2LΔ1,Δ2∂Δ1∂Δ2=0.
Then, the Hessian matrix ofLΔ1,Δ2 is seminegative definite, which implies the concavity of LΔ1,Δ2. So, the necessary and sufficient conditions for the global maximum point are the corresponding Kuhn–Tucker conditions:(41)∂LΔ1∗t,x,y,Δ2∗t,x,y∂Δi≤0,Δi≥0,∂LΔ1∗t,x,y,Δ2∗t,x,y∂ΔiΔi=0,i=1,2.
ForΔ1=0,Δ2=x−x1t,y>0, we can get(42)∂LΔ1,Δ2∂Δ1<0,∂LΔ1,Δ2∂Δ2=0.
Clearly, the corresponding Kuhn–Tucker conditions are satisfied at0,x−x1t,y. So, it is the global maximum point.Theorem 5.
Fort,y∈D2, we have(43)Δ1∗t,x,y=0,x≤x0t,x−x0t,x>x0t,Δ2∗t,x,y=0.Proof.
The proof is similar to the one in Theorem4. Now, we omit it.Remark 1.
It is shown from the above two theorems that(i)
InD1, the individual will not consume. There is a boundary x1t,y, for x>x1t,y, putting the wealth above x1t,y to buy annuities is optimal, and for x≤x1t,y, putting all the wealth to invest is optimal.(ii)
InD2, the individual will not buy annuities. There is a boundary x0t, for x>x0t, putting the wealth above x0t to consume is optimal, and for x≤x0t, putting all the wealth to invest is optimal.(iii)
People of different ages and wealth have different optimal strategies, so, in order to promote tax-deferred annuities, the government should adopt different tax preferential policies for different people, and insurance companies should take different publicity strategies for different people.
Clearly, this is practical, and these theorems give us exact barriers to make decisions.Remark 2.
Fork≥1−V2yt,y, we have(44)V2yt,y1−k≥1.
The concavity ofV1t,x implies that x1t,y≤x0t; thus, the individual will not consume, and he will put all the wealth above x1t,y to buy annuities. This indicates that raising tax rates properly can stimulate people to buy tax-deferred annuities.Remark 3.
According to Theorem3, we know that as time goes by, x1t,y gets bigger, and the lower bound of purchasing annuities becomes more and more difficult to reach. So, young people are more likely to buy insurance than old ones. Thus, insurance companies should broaden young people’s annuity market and design more products for young people.Now, let us give the value function.Theorem 6.
Fort,y∈D1, we have(45)Vt,x,y=V1t,x+V2t,y,x≤x1t,y,V1t,x1+V2t,y+x−x11−k,x>x1t,y,and for t,y∈D2, we have(46)Vt,x,y=V1t,x+V2t,y,x≤x0t,V1t,x0+x−x0t+V2t,y,x>x0t.Up till now, we have obtained the expression of the value function, the optimal investment, consumption, and annuitization strategies in terms ofV1t,x and V2t,y. In the next section, we will analyze the problem when the rate of buying tax-deferred annuities is restricted.


## 4. The Optimal Strategies Whenpt≤p¯

The above analysis considers the optimal annuitization strategies under the assumption that buying annuities are not restricted. In fact, in order to ensure the role of Taxation and reduce the financial pressure, many countries stipulate that the number of tax-deferred annuities purchased by a person shall not exceed some point. In this section, let us consider the problem under the assumption thatpt≤p¯ for some constant p¯>0. Using the results in Theorem 1, we can get that the value function satisfies (14) and (15) and(47)minr+λtsVt,x,y−Vtt,x,y−supb1≥0,b2≥0r^x+μ^−r^b1+kbVxt,x,y+σ^2b122Vxxt,x,y+σσ^b1b2Vxy+ry+b2μ−rVyt,x,y+σ2b222Vyyt,x,y−sup0≤p≤p¯pVyt,x,y−p^Vxt,x,y,Vxt,x,y−1=0,t<T.Thus, in this case, the optimal investment strategies are still determined by (16) and (17).It follows from (47) that for 1−kVxt,x,y≥Vyt,x,y, we have(48)r+λtsVt,x,y≥supb1≥0,b2≥0r^x+μ^−r^b1+kbVxt,x,y+σ^2b122Vxxt,x,y+σσ^b1b2Vxy+ry+b2μ−rVyt,x,y+σ2b222Vyyt,x,y+supc≥0c1−Vxt,x,y+Vtt,x,y,Vxt,x,y≥1,with at least one strict equal sign. For 1−kVxt,x,y<Vyt,x,y, we have(49)r+λtsVt,x,y≥supb1≥0,b2≥0r^x+μ^−r^b1+kbVxt,x,y+σ^2b122Vxxt,x,y+σσ^b1b2Vxy+ry+b2μ−rVyt,x,y+σ2b222Vyyt,x,y+supc≥0c1−Vxt,x,y+p¯Vyt,x,y−1−kp¯Vxt,x,y+Vtt,x,y,Vxt,x,y≥1,with at least one strict equal sign.LetU1t,x satisfy(50)−r+λtsU1t,x+U1tt,x+r^xU1xt,x+kb−1−kp¯U1xt,x−R2U1x2t,x2U1xxt,x=0,(51)U1T,x=x,(52)U1t,0=0.LetU2t,y satisfy(53)−r+λtsU2t,y+U2tt,y+ry+p¯U2yt,y−R2U2y2t,y2U2yyt,y=0,(54)U2T,y=gy,y≥0,(55)U2t,0=V1t,x1t,y−U1t,x1t,y,t≥0.Definex˜0t by(56)U1xt,x˜0t=1.Appendix B gives us numerical solutions ofU1t,x and U2t,y. Figures 3 and 4 give us the following theorem.Figure 3
U1t,xFigure 4
A comparison ofV1t,x and U1t,xTheorem 7.
U1t,x is a concave increasing positive function with respect to x, and U1xt,x is increasing with respect to t and(57)U1xt,x>V1xt,x,x≥0.According to Theorem7, we can obtain that(58)V1′t,x˜0<U1′t,x˜0=1=V1′t,x0t,Then, we can get(59)x˜0t>x0t,∀t≥0.Next, we consider the form of the value function inD1 and D2, respectively.Fort∈D1, define(60)wt,x,y=V1t,x+V2t,y,x≤x1t,y,U1t,x+U2t,y,x1t,y<x<x˜0t,U1t,x˜0t+x−x˜0t+U2t,y,x≥x˜0t,Δ1∗t,x,y=0,x≤x˜0t,x−x˜0t,x>x˜0t,p∗t,x,y=0,x≤x1t,y,p¯,x>x1t,y.Fort∈D2, define(61)wt,x,y=V1t,x+V2t,y,x≤x0t,V1t,x0t+x−x0t+V2t,y,x>x0t,Δ1∗t,x,y=0,x≤x0t,x−x0t,x>x0t,p∗t,x,y=0.Then, we have the following corollary.Lemma 1.
wt,x,y defined above satisfies (14), (15), and (47).Proof.
According to (20), (21), (51), (52), (54), and (55), we can get that wt,x,y satisfies (14) and (15).
Fort∈D1, x≤x1t,y, we have that(62)1−kwxt,x,y=1−kV1′t,x≥1−kV1′t,x1t,y=V2′t,y=wyt,x,y,wxt,x,y=V1′t,x>1.
Thus, we can get that(63)ct∗=0,pt∗=0.
SinceV1t,x satisfies (23), and V2t,y satisfies (24), it follows immediately that wt,x,y satisfies (47).
Fort∈D1, x1t,y<x<x˜0t, we have(64)wxt,x,y=U1′t,x>1,1−kwxt,x,y=1−kU1′t,x≤1−kV1′t,x≤1−kV1′t,x1t,y,1−kV1′t,x1t,y=V2′t,y=U2′t,y=wyt,x,y.
It implies that(65)ct∗=0,pt∗=p¯.
SinceU1t,x satisfies (50), and U2t,y satisfies (53), it follows immediately that wt,x,y satisfies (47).
Fort∈D1, x≥x˜0t, we have(66)1−kwxt,x,y=1−kU1′t,x˜0t=1−kV1′t,x0t<1−kV1′t,x1t,y,1−kV1′t,x1t,y=V2′t,y=U2′t,y=wyt,x,y,wxt,x,y=1.
Thus, we can getpt∗=p¯. Clearly, wt,x,y is the solution of (47).
The cases fort,y∈D2 can be proved with a similar analysis, and we can obtain that wt,x,y satisfies (47) finally.
The concavity ofV1t,x,U1t,x, V2t,y, and U2t,y indicates that wt,x,y satisfies the polynomial growth condition. So, using Theorem 3.5.2 in [13], we can immediately get that wt,x,y=Vt,x,y.Remark 4.
Obviously, the boundary of buying annuities in this section is the same as the one in Section3. So, the suggestions proposed in Section 3 can also apply when the rate of buying annuities is restricted.Remark 5.
The form ofp∗ indicates that properly raising the upper bound p¯ can increase the rate of buying annuities.



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