Single Premium of Equity-Linked with CRR and CIR Binomial Tree

Transcription

1 The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences, Universias Gadjah Mada, Yogyakara, Indonesia. a) Corresponding auhor: b) Absrac. Uni-linked life insurance is a life insurance produc ha is a hybrid because i provides wo benefis a once, life insurance benefi and invesmen benefi. Equiy-linked is a ype of unilinked life insurance ha inves premiums paid parly in shares. Some of hese insurance policies include opions ha give he righ o he policyholder o erminae heir policy conrac and receive some cash. In his research, we sudy how o deermine he amoun of single premium endowmen equiy-linked using Cox, Rubensein, and Ross (CRR) and Cox, Ingersoll, and Ross (CIR) binomial ree. CRR binomial ree is used o predic he sock price of several period ahead, while CIR binomial ree is used o predic he ineres rae. Keywords : equiy-linked, single premium, binomial ree, sock price, ineres rae. INTRODUCTION In he financial plans, invesmen and proecion are wo hings ha mus be possessed. Unilinked is a life insurance company produc ha combines hese wo funcions. Uni-linked offers many invesmen choices, including socks, bonds, money markes, and a mixure of socks and bonds. Equiy-linked is a uni-linked ha inves a leas 80% premiums paid in shares. Some of equiy-linked policies include an american pu opion ha gives righs o policyholder o end he policy conrac and receive some cash [1]. This opion is usually included he equiy-linked endowmen life insurance. Policyholder will ge he benefi a mauriy if he is sill alive or when he dies before mauriy during he opion is no exercised [2]. In his paper, we will deermine a single premium equiy-linked wih he binomial ree. Insurance premium which we pay once a he beginning of he policy conrac should reflec how he fuure sock price and ineres rae movemens [3]. Binomial ree is one of he mehods used o predic hem. Cox, Robensein, and Ross (CRR) binomial ree mehod is used o predic sock price and Cox, Ingersoll, and Ross (CIR) is used o predic he fuure ineres rae. In he firs secion, some backgrounds of equiy-linked are provided. In he nex secion, we explain he CIR binomial ree. The hird secion and he fourh secion discuss he CRR binomial ree and equiy-linked binomial ree respecively. And in he las secion, we give he concluion.

2 2 YUNITA WULAN SARI & GUNARDI CIR BINOMIAL TREE Cox, Ingersoll, and Ross (CIR) ineres rae model was inroduced in 1985 o address he Vasicek model ha generaes negaive value in calculaion of ineres rae. The CIR model is shown by dr r d r dw where,, brownian moion.. is unknown parameers ha are posiive and independen of ime. dr W is a sandard shows he ineres rae changes. shows he average speed of ineres rae changes. and show he ineres rae and volailiy respecively [4]. If he life insurance policy enered ino force upon signaure (=0) unil he mauriy (T), he ime inerval [0,T] can be devided ino N ime discree, ie [(i-1)h, ih], i=1,2,..., N. The lengh of each discree ime period is h T N he nex period. The CIR model, which in is discreizaion, so we need he ransformaion of r firs [5]. Choose xr x r r dw dw r obained by xr s invers.. r is a ineres rae value a ime and r +1 is a ineres rae value a, he ransformaion of r ha mees r and he volailiy of Based on he relaionship beween r and resul ha he as follows is no worh he consan and a funcion of r, is no easy dxr x r 1 dz. As a resul Z r is consan. r binomial process can be xr where he r s value canno be negaive wih he xr s value should no be negaive [5], he r s value on binomial ree is deermined 1 r xr xr 0, Figure 1 and Figure 2 show he forward binomial ree of, 0 ohers xr and r respecively. Figure 1. xr Binomial Tree

3 3 YUNITA WULAN SARI & GUNARDI Figure 2. r Binomial Tree CRR BINOMIAL TREE S and The sock price model a ime is ds S d S dw. S S S are unknown parameers and independen of ime. As well as he CIR binomial ree, he ime unil o mauriy inerval [0,T] is divided ino N discree ime seps [(i 1)h,ih], i = 1,..., N, each of lengh h = T/N. The price of he sock one period ahead wih probabiliies of 0.5 will rise by S exp u S 1 s h and decrease by d S S 1 exp h, where S is he sock price a period/ime ( 0, T 3 shows he forward CRR binomial ree. s ) and s is he volailiy of sock price [6]. Figure Figure 3. CRR Sock Price binomial ree

4 4 YUNITA WULAN SARI & GUNARDI In his paper, i is assumed ha sock price and ineres rae are dependen or inerconneced. In Kewal s Theory, wihou any oher facors ha affec, hey have a negaive relaionship [7]. The high ineres rae will affec he presen value of cash flows of he company, so ha invesmen opporuniies will no be as arracive anymore. The rising of ineres rae also couse invesors o move heir invesmen in savings or deposis. Therefore, i can be formed a new binomial ree model which is a combinaion of ineres rae and sock price binomial ree as follow: Figure 4. The Combinaion of Binomial Tree EQUITY-LINKED BINOMIAL TREE This paper use endowmen life insurance, which he policyholder will receive policy benefis if he died a he ime (0<<T) or alive a ime T. In addiion, his policy benefis jus can be received if he opion did no excercised. The backward binomial ree of equiy-linkend can be formed based on he forward binomial ree of sock price and ineres rae wih he following condiions : M In he case of deah occured beween and +1, policy benefi C = M and i is paid a ime +1. Opion can be exercised a he beginning of each subinervals > 0. If he opion is execued, he insurance company shall pay R = R and R M. If he policyholder is sill alive unil he beginning of period and he opion have no been exercised before, he policyholder has wo alernaives: 1. Exercise he opion and i mean he policy conrac expires. 2. Coninuing he conrac and if he dies in he ime inerval [, + 1], he policy conrac expires a + 1. V V C f F ns If he policyholder is sill alive unil mauriy, he will ge benefi T T T T The single premium value is he value of equiy-linked binomial ree a ime =0. From he provisions above, he coninuous value of policy conrac is and he conrac value a ime is where rh rh u d W p e M q e V V h xih h xih V max R, W

5 5 YUNITA WULAN SARI & GUNARDI h px ih hqx ih r u V 1 d V 1 : he chance of he insured aged x + ih will survive unil h years laer. : an opporuniy of insured aged x + ih will die wihin h years laer. : he risk-free inere rae. : he value of in he up posiion. : a value of V 1 in he down posiion. V 1 CONCLUTIONS The equiy-linked life insurance can be an alernaive invesmen ha i is aracive and promising because for he same ime, an invesor can proec he risks wih he benefis of life insurance and also may inves in shares desired. Exra opion righs on endowmen life insurance (surrender opion) will make his produc more ineresing because i will reduce invesor losses due o falling share prices in he marke. Besides ha, he binomial ree can be an alernaive in he fair premium life insurance calculaion for easy inerpreaion and compuaion. REFERENCES [1] Shen, W., and Xu, H., The Valuaion of Uni-Linked Policies wih or Wihou Surrender Opions, Insurance:Mahemaics and economics (IME), 36, 79 92, [2] Bacinello, A. R., A Full Mone Carlo Approach o The Valuaion of The Surrender Opion Embedded in Life Insurance Conacs, Mahemaical and Saisical Mehods for Insurance and Finance, 19 26, [3] Bacinello, A.R., Endogeneus Model of Surrender Condiions in Equiy-Linked Life Insurance, Insurance:Mahemaics and Economics (IMS), Vol.37, , [4] Cox, J.C., Ingersoll, J.E., and Ross, S.A., The Theory of Term Srucure of Ineres Rae, Journal of The Economeric Sociey (Economerica), Vol.53, No.2, , [5] Nelson, D.B., and Ramashwany, K., 1990, Simple Binomial Processes as Diffusion Approximaions in Financial Models, The Review of Financial Sudies, Vol.3, No.3, Hal [6] Wilmo, P., Paul Wilmo Inroduces Quaniaive Finance, John Wiley and Sons : England, [7] Kewal, S.S., 2012, Pengaruh Inflasi, Suku Bunga, Kurs, dan Perumbuhan PDB Terhadap Indeks Harga Saham Gabungan, Jurnal Economia, Vol.8, No.1, Hal