Int. Journal of Math. Analysis, Vol. 7, 03, no. 59, 93-99 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ijma.03.3033 Actuarial Present Values of Annuities under Stochastic Interest Rate Zhao Xia and Lv Huihui School of Insurance, Shandong University of Finance and Economics Jinan, Shandong, P. R. China, 5004 zhaoia-w@63.com Copyright 03 Zhao Xia and Lv Huihui. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. Astract. We studied actuarial present values of annuities under stochastic interest driven y reflected Brownian motion and Poisson jump process. We otained their eplicit epressions and discussed the effect of stochastic interest on actuarial present value y data analysis. Mathematics Suject Classification: 6P05, 60J8 Keywords: reflected Brownian motion, annuity, actuarial present value. Introduction In traditional actuarial theory, it is assumed that interest rate is fied, ut this is not the fact. Interest fluctuates with the change of financial environment and policy. So the study of actuarial theory under stochastic interest has ecome a hot topic in recent years. The Methods to model stochastic interest involve stochastic perturation, point process, semi-martingale and time series etc. Since 990 s, some researchers have constructed interest models y stochastic perturation methodology and applied it to actuarial science, see Beekman & Fueling [].
94 Zhao Xia and Lv Huihui Then many researchers used this method to study some prolems in life insurance, annuity, risk theory, option pricing etc., see [5, 8, 9] and their references. Standard Brownian motion is an important tool in modeling stochastic interest in the forgoing researches. But in reality it may e negative, which is conflict with the fact. So the reflected Brownian motion is used to model the perturation change instead of Brownian motion and the related actuarial prolems in life insurance were studied, see [, 3, 4, 6, 7]. However, few studies have een done on the analysis of annuity under stochastic interest modeled y the reflected Brownian motion and Poisson process. This paper is organized as follows. Section descries the stochastic model. Section 3 gives the eplicit epression of APV of life annuity. Section 4 shows the analysis numerically. Section 5 concludes this paper.. Stochastic Interest Model The accumulated interest rate function r ( t) is modeled as r ( t) δ t + βwt ( ) + γpt =, (.) whereδ, β andγ are positive constants, w(t) is a standard Brownian motion and independent of Poisson process P t. Let the discount δt β w ( t ) γpt functionv( t) = e. From the calculation of MGF, we can get δ λ ( ) ( e ) β σ t δt βw t γp t EV( t) = Ee = e ϕ( βσ t) γ (.) whereϕ( ) t =. e dt π Since the stochastic fluctuation of interest cannot e large in reality, without loss any generalization we can assume that > + ( e γ ) (.3) δ β σ λ
Actuarial present values of annuities 95 3. Actuarial Present Value of Life annuity Under Stochastic Interest Life annuity is a series of payments ased on the survival of the recipients within a certain period. This paper discussed temporary life annuity, life annuity and deferred life annuity. To illustrate conveniently, we let β σ δ + λ( e γ ) A e, B β σ δ + λ. ( e γ ) Theorem Suppose that the following two conditions are satisfied, (I) the accumulated interest rate is descried y the reflected Brownian motion and Poisson process, see formula (.), and (II) the payment period is from a to and every payment is. Then the actuarial present value of annuity paid at the eginning of the year is: ( ( ) ) ( ) β F = tpb ϕ t δ ϕ tδ + a δ Especially, ( t ) t ϕ ( βσ ) ϕ ( βσ ) } + + A t + A t (3.) whena = 0, = n, F = aɺɺ is the n-year temporary life annuity paid at the : n eginning of the year; year; when a = 0, =, F = aɺ is the life annuity paid at the eginning of the when a = m, = m + n, F = ɺ is the m deferred n years annuity paid at a mn the eginning of the year. Proof Let Y e the random variale of the present value of life annuity when the payment is one unit. F = E ( Y ) = EE V ( t) a tpdt = t t a t+ δv βw( v) γpv p E e dv Sustituting formula. into the aove formula, we have
96 Zhao Xia and Lv Huihui ( ) t+ β σ v γ δv λv( e ) F = tp e e ϕ βσ v e dv t. a t+ β σ v γ δv λv( e ) Note that ϕ ( βσ ) t e e v e dv βσ t+ βσ t = β 0 v e d Adv π ( ( ) ) ( ) + t+ t+ t + βσ v e d Adv π B β ϕ t δ ϕ tδ t t = + + A + ϕ βσ t A ϕ βσ t δ +, ( ) ( ) ( ) } the proof is completed. Theorem Suppose that the following two conditions are satisfied, (I) the accumulated interest rate is descried y the reflected Brownian motion and Poisson process, see formula (.), and (II) the payment period is from a to and every payment is. Then the actuarial present value of the annuity paid at the end of the year is: ( ( ) ) ( ) β F = tpb ϕ t δ ϕ tδ + a δ Especially: ( t ) t ϕ ( βσ ) ϕ ( βσ ) } + + A t + A t d (3.) when a = 0, = nd, =, F = a is the n-year term life annuity paid at the : n end of the year; year; when a = 0, =, d =, F = a when a = m +, = m + nd, = 0, F = paid at the end of the year; is the life annuity paid at the end of the a mn is the m deferred n years annuity Proof Note that the time of every payment is at the end of the year, we can complete the proof y following the similar deduction as in Theorem.
Actuarial present values of annuities 97 4. Data Analysis To find the effect of stochastic interest on APV, here we give some data analysis ased on the result in Theorem. We take n-year temporary life annuity paid at the eginning of the year as an eample. Let a = 0, = n in formula (3.), we can get ( ( ) ) ( ) n β aɺɺ = n : tpb ϕ t δ ϕ tδ + 0 δ ( t ) t ϕ ( βσ ) ϕ ( βσ ) } + + A t + A t (4.) For convenience, we might as well suppose that λ = 0, σ =, the constant interest forceδ is assumed to e 0.00 and 0.003; the coefficient β starts at 0 and ends at 0.04 evenly spaced y the value 0.0; the coefficientγ values at 0, 0.00 and 0.00, n = 0. The survival proaility is from Eperience Life Tale of China Life Insurance (000-003). The result is shown in Tale. Tale APVs of n-years term annuity δ = 0.00 δ = 0. 003 β γ = 0 γ = 0. 00 γ = 0. 00 γ = 0 γ = 0. 00 γ = 0. 00 0 0.4995003 0.49703 0.0 0.4906803 0.475989 7 0.0 0.47796 0.449970 5 0.03 0.468839 0.404 6 0.04 0.446459 0.390600 3 0.4945450 0.499507 0.496765 3 0.4640040 0.4894553 0.47500 0.494959 0.475767 0.448489 8 0.393479 0.459885 0.4947 6 0.3569758 0.446377 0.388877 0.494988 0.469674 0.47889 0.395000 0.354837
98 Zhao Xia and Lv Huihui From Tale, we can find that APV of life annuity decreases significantly as δ and γ increases respectively when other parameters keep unchanged. These results ehiit the importance to introducing stochastic interest in pricing life annuity. 5. Conclusion In this paper, we studied life annuity in classical theory under stochastic interest rate. We assumed the interest rate to e a stochastic process pertured y reflected Brownian motion and Poisson process. We otain unified eplicit epressions for APVs of life annuities. Further the data result provides insights into the effect of stochastic interest on life annuity, and show the importance of introducing stochastic interest. Acknowledgements This work was partially supported y NSFC (No. 707088). References [] Beekman, J. A., Fueling, C. P., Interest and mortality randomness in some annuities, Insurance: Mathematics and Economics, 990, 85-96. [] Chen, H., Han, S., Study on a class of models aout variale payments life insurance under the stochastic interest [J]. Mathematical Theory and Applications, 3(008), -4. [3] Liu, H., Tan, L., Zhang, L., Increasing life insurance model with interest randomness [J]. Mathematical Theory and Applications, 6 (007), 3-6. [4] Perry, D., Stadje, W., Yosef, R, Annuities with controlled random interest rates[j], Insurance: Mathematics and Economics, 003, 45-53. [5] Parker, G, A portfolio of endowment policies and its limiting distriution, Astin Bulletin, (996), 5-33.
Actuarial present values of annuities 99 [6] Wang, L., Hao, Y., Zhang, H., Increasing endowment assurance policy under stochastic rates of interest. Journal of Dalian University of Technology, 9(00), 87-830. [7] Xin, H., Actuarial study of continuous increasing life insurance under stochastic interest rate, Statistics and Decision,7(00), 8-3. [8] Zhao, X., Zhang, B., Mao, Z., Optimal Dividend Payment Strategy under Stochastic Interest Force. Quality & Quantity, 4 (007), 97-936. [9] Zhao, X., Zhang, B., Pricing perpetual options with stochastic discount interest rates. Quality & Quantity, (0), 34-349. Received: Octoer, 03