FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux"

Transcription

1 Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION Carole Bernard Chrstane Lemeux Department of Statstcs and Actuaral Scence Unversty of Waterloo Waterloo, ON, N2L 3G1, CANADA ABSTRACT In ths paper, we consder equty-lnked lfe nsurance contracts that gve ther holder the possblty to surrender ther polcy before maturty. Such contracts can be valued usng smulaton methods proposed for the prcng of Amercan optons, but the mortalty rsk must also be taken nto account when prcng such contracts. Here, we use the least-squares Monte Carlo approach of Longstaff and Schwartz coupled wth quas-monte Carlo samplng and a control varate n order to construct effcent estmators for the value of such contracts. We also show how to ncorporate the mortalty rsk nto these prcng algorthms wthout explctly smulatng t. 1 INTRODUCTION Lfe nsurance companes used to sell lfe nsurance contracts that provde a fxed captal at tme of death of the nsured. In the past ten years, ndvduals are more and more wllng to combne nvestment and lfe nsurance. Nowadays popular contracts often nvolve a return lnked to the market. Equty ndex annutes are products that offer nvestors downsde protecton because of the presence of a mnmum maturty beneft and death beneft wth a potental upsde rate n case the ndex performs well (see for nstance Palmer (2006) or Hardy (2003) for more detals on contracts currently offered). In ths paper we nvestgate some of the optons embedded n an equty-lnked lfe nsurance contract. We nclude standard optons such as a guaranteed mnmum maturty beneft and a guaranteed mnmum death beneft. But many more complex optons exst (see for nstance Ballotta and Haberman (2003) or Boyle and Hardy (2003) to cte only a few). The possblty of early wthdrawals s nvestgated by Mlevsky and Salsbury (2006). In ths study we focus on the surrender opton, whch gves polcyholders the rght to termnate ther polces before the maturty ndcated n ther contract. It s the possblty of wthdrawng the total value of the contract. Lterature on surrender optons n equty-lnked lfe nsurance products s recent snce these optons have been neglected for a long tme. In the 1990s, they were ndeed consdered as very low-rsk optons snce nterest rates n the market were much hgher than the mnmum guaranteed rate and the stock market performed very well. After several bankruptces n the nsurance sector (for nstance around 2001, the Brtsh company, Equtable Lfe went bankrupt due to optons embedded n penson annutes where mortalty rsk and fnancal rsk were underestmated and not hedged), embedded optons are now taken more serously nto account n the valuaton of the contracts. Each opton offers nvestors some addtonal rghts that mght cause some proft reducton for nsurer f optmally exercsed. It s thus of utmost mportance for companes to understand all optons embedded n contracts they sell to better prce and hedge them. Concernng the surrender opton, there are several consequences. Frst, the nsurer mght ncur fnancal losses from early payments caused by surrendered polces due to lqudty of ther assets (they mght have to lqudate deprecated assets to face surrender demand). Indeed companes face a trade-off between long term nvestment (and hgh returns) but wth few lqudty or nvestng n lqud assets (wth a lower return). Second, n the long run nsurer mght suffer from adverse selecton snce polcyholders that have health problems and thus nsurablty dffcultes wll not surrender. To avod adverse selecton, some contracts (for example contracts that have only a beneft n case of survval) do not allow surrender. To mnmze rsks assocated wth the surrender opton, nsurers can gve hgh penaltes. However nsured are keen on far surrender condtons snce these contracts are often long-term nvestments and ther needs mght change and force them to surrender. Regulators, lawyers and market competton between companes protect nvestors aganst unfar condtons n case of early termnaton. To value ths surrender opton, no unfed framework exsts n prevous lterature. Surrender decson can be of two types: an exogenous surrender (for example, dependng on /08/$ IEEE 444

2 Bernard and Lemeux personal reasons: famly problem, unemployment, sudden need of lqudty) or an economc or endogenous surrender, lnked to nterest rates fluctuatons, fnancal envronment changes, etc. Actuares estmate exogenous surrender from hstorcal data on the lapse rate. Exogenous surrender rsk can be dversfed. Thus nsurers should be more threatened by endogenous surrender rsk that s dffcult to estmate (n partcular because of lack of data) and leads to systematc rsks. It seems that there were only weak hstorcal evdence of stuatons when surrender was optmal. Any approach based on hstorcal lapses mght underestmate the real cost of surrender optons. Kuo, Tsa, and Chen (2003) study how the lapse rate s nfluenced by unemployment rate and nterest rate fluctuatons. To deal wth the second type of surrender decsons (fnancal, endogenous surrender rsk), the alternatve s to use a pure fnancal approach based on the opton theory. Surrender opton s seen as an Amercan opton. Indeed equty-lnked contracts wth a fxed term can be expressed as a portfolo of European optons (Brennan and Schwartz (1976), and Boyle and Schwartz (1977) are the frst ones proposng ths approach). The cost of the surrender opton s then the Amercan premum that s added f these optons can be exercsed before maturty. In the Black and Scholes framework, Grosen and Jørgensen (1997) gve the optmal exercse barrer by usng results from Mynen (1992). Frst studes of the surrender opton market value use numercal schemes to solve partal dfferental equatons (Jensen, Jørgensen, and Grosen 2001)) or bnomal trees (Bacnello 2003, Bacnello 2005). Bacnello takes nto account mortalty rsk, perodcal premums and annual bonus, but both of these approaches are extremely slow. Recently, Shen and Xu (2005) have also nvestgated surrender optons by way of partal dfferental equatons. These works use the no-arbtrage prncple and the market value of the surrender opton s the market value of an optmal surrender decson. New accountng rules force nsurers to evaluate ther labltes (consstng manly of the sold contracts) at ther market value even though the contracts are not really traded on a market. There s thus an mportant need for fnancal modelng of nsurance contracts. The no arbtrage approach to value the surrender opton has been crtczed snce ths opton s not traded. Moreover, t mples that f t s optmal for one nsured to surrender, t s thus optmal for all of them to surrender, whch s not realstc even f t s the greatest rsk faced by the ssung company. Albzzat and Geman (1994) propose an nterestng model to address ths ssue. The dea s that the value of the surrender opton at a gven date s known and then t can be splt over dfferent dates usng an exogenous lapse rate. In ths paper, we are also nterested n gvng a new way to tackle ths problem by ntroducng a parameter that wll depends on the ndvdual, and wll nuance hs optmal decson. All ndvduals wll thus not have the same optmal decson. Our approach s a fnancal approach and we want to obtan a market value of the surrender opton, n a smlar way to what has been done n Andreatta and Corradn (2003), Bacnello, Bffs, and Mllossovch (2008a), and Bacnello, Bffs, and Mllossovch (2008b). We use the least-squares Monte Carlo approach of Longstaff and Schwartz (2001) to perform the valuaton, but we nclude the mortalty rsk usng a dfferent model and a dfferent approach than Bacnello, Bffs, and Mllossovch (2008a), Bacnello, Bffs, and Mllossovch (2008b). In partcular, by makng the usual assumpton that the mortalty rsk s ndependent of the fnancal rsk, we show that survval tmes do not need to be smulated. Ths reduces the varablty of the obtaned estmators for the value of the surrender opton. In addton, we use the value of the European contract as a control varate and perform smulatons usng quas-random samplng. The rest of ths paper s organzed as follows. In Secton 2, we descrbe the lfe nsurance contract, a pont to pont Equty Indexed Annuty wth a guaranteed mnmum death beneft and a guaranteed mnmum surrender beneft. The surrender decson s frst based exclusvely on fnancal crtera, n other words a polcyholder surrenders hs polcy f the surrender amount gves hm at least the opportunty to buy the same contract at a lower prce on the market. But ths approach overestmates the surrender opton by lookng at the worst case that practcally never happens: all polcyholders surrender at the same tme (the optmal surrender tme). In Secton 2.3, we address ths ssue and ntroduce a parameter that represents the propensty of the ndvdual to act optmally. In Secton 3, we show how to nclude mortalty rsk n the least-squares Monte Carlo approach used to prce the contract. Then we dscuss n Secton 4 the two methods we used to mprove the effcency of ths Monte Carlo estmator: a control varate based on the European contract, and quas-monte Carlo samplng to smulate the fnancal paths. Numercal results are gven n Secton 5, and a bref concluson s provded n Secton 6. 2 FRAMEWORK We frst descrbe the contract, a pont-to-pont Equty Indexed Annuty (EIA) wth a mnmum guarantee at maturty or at tme of death of the nsured. Then, we gve our assumptons for the underlyng ndex dynamcs, and we ntroduce the guaranteed mnmum surrender beneft studed n ths paper. 2.1 The contract We consder a pont-to-pont EIA wth a Guaranteed Mnmum Maturty Beneft. The polcy has a fxed maturty date of T years. The nsured s ntal nvestment s P. We assume 445

3 Bernard and Lemeux that no addtonal payments are done after the ncepton of the contract. Most EIAs n the U.S. are purchased wth a sngle premum (see Palmer (2006)). The contract s lnked to an ndex, for nstance the S&P 500 ndex. We denote by S t the value of the ndex at tme t. We suppose a percentage α of the ntal premum P s guaranteed at a mnmum annual rate equal to g. These parameters α and g are often constraned by law. For nstance, the Unted States have adopted recently revsed nonforfeture regulatons, statng that the mnmum guarantee under newly ssued contracts must be at least 87.5% of all premums pad, P accumulated at a mnmum nterest rate, here g, between 1% and 3%. About one thrd of EIAs apply ther stated nterest rate guarantee to only 87.5% percent of the premum, one thrd to 90%, and only one ffth to 100%, see Palmer (2006). The fnal payoff of the contract at maturty tme T wrtes as: ( ) ) k V T = αp max (1 + g) T, ( ST S 0. (1) Ths contract s only for the purpose of llustraton and ths study can easly be extended to more complcated contracts. Insurance companes propose unt-lnked polces or equty-ndexed annutes wth addtonal death guarantees. Assumng that we neglect longevty rsk, mortalty rsk can be dversfed. Indeed deaths observed n a portfolo of nsureds mght happen at ndependent dates. The large law of numbers and the central lmt theorem are sutable to estmate rsks. Note that ths s not the case when mortalty s stochastc (as studed, for nstance, by Bacnello, Bffs, and Mllossovch (2008a) and Bacnello, Bffs, and Mllossovch (2008b). Here we assume the equty-lnked contract descrbed by (1) s combned wth a Guaranteed Mnmum Death Beneft. We assume the guaranteed mnmum death beneft s pad at the end of the year when the death of the nsured occurs and that t can be wrtten as: ( ( ) ) kd D t = αp max (1 + g d ) t St,, t = 1,...,T, (2) f death occurs between t 1 and t, where we assume there s no penalty but the guaranteed rate g d and the partcpaton k d n the ndex performance mght be dfferent than the ones nvolved n the contract s payoff descrbed by (1). Palmer (2006) note that few EIAs assess a surrender charge f the EIA s cashed n due to the contract owner s death. Unlke the mortalty rsk, the fnancal rsk cannot be hedged by poolng arguments. All polces depend on the same ndex and f t goes up, the nsurer wll have to pay a hgh partcpaton rate for all polces at the same tme. S 0 Ths cannot be hedged by acceptng more polces. Theoretcally, those rsks can be hedged by usng the followng decomposton of the contract s fnal value as ( (ST ) k V T = αp (1 + g) T + max αp (1 + g),0) T, (3) whch s a portfolo made up of a guaranteed amount αp usually yeldng a lower nterest rate g than the rsk-free rate r of the market plus a long poston n a call opton wrtten on the underlyng ST k. Usng Black&Scholes-type results, companes can adjust ther portfolo contnuously and replcate the above payoff. Ths decomposton and ts prcng n a Black and Scholes framework were frst done n Brennan and Schwartz (1976) and n Boyle and Schwartz (1977). 2.2 Model For the ease of exposton, here we neglect all types of costs, and assume the market s complete and perfectly lqud. We set ourselves n Black and Scholes framework, although our methodology could easly be appled to more complex models. The rsk-free nterest rate s constant and denoted by r. Snce the market s complete, there s a unque rsk-neutral measure, Q. Dynamcs of the ndex S under the Q-measure wrte as: ds t S t S 0 = rdt + σdw t where σ s the volatlty of the ndex and W t s a Q-standard Brownan moton. The far value of the contract s obtaned by takng the expectaton under the rsk-neutral probablty Q of the dscounted cashflows: ξ (S 0,g,k,T ) = e rt E [V T ] (4) wherev T s gven by (1). After straghtforward computatons n the Black and Scholes framework, we obtan a closedform formula for the European contract: ( ) ξ (S 0,g,k,T ) = e (g r)t S 0 Φ g k r σ 2 2 T kσ ( + S 0 e (k 1)rT +k(k 1) σ2 T 2 Φ α + kσ ) T, where Φ( ) s the CDF of a standard normal varable. Ths s the rsk-neutral prce the ntal amount of money needed to perfectly hedge the fnal cash-flows and provde the payoff (1) to the nsured for the European verson of the contract where surrender s not allowed and there s 446

4 Bernard and Lemeux no death beneft. Ths formula does not nclude the cost of hedgng, transacton costs and the dscrete hedgng error that wll obvously occurred n practce. We now nclude the mnmum guaranteed death beneft defned earler n (2). Let us assume the probablty that a polcyholder of age x+t wll de before the end of the year s q x+t, for t = 0,...,T 1, and we let p x+t = 1 q x+t. More precsely, f we nclude the mortalty rsk n the formula (4), then we have that the European contract s value at tme 0, denoted V 0,e, s gven by T 1 V 0,e = T p x ξ (S 0,g,k,T )+ t p x q x+t ξ (S 0,g d,k d,t +1), t=0 (5) where ξ s defned by the formula (4) and t p x = t 1 j=0 p x+ j s the probablty that an ndvdual of age x survves at least t years. 2.3 The surrender opton We are nterested n a contract where there s a Guaranteed Mnmum Surrender Beneft. That s, we assume that the above polcy also offers a mnmum surrender guarantee, and that the rght to surrender can only be exercsed at the end of each year untl the maturty of the contract. Hence ths s a Bermudan-type opton rather than a truly Amercan one. In practce, ths mnmum guarantee s often ndependent of the return of the ndex, and thus we make the assumpton that the polcyholder receves the followng amount L t f he surrenders the polcy at tme t: L t = (1 β t ) αp (1 + h) t, (6) where 0 h < r s the guaranteed rate, and β t s the penalty charged for surrenderng at tme t, for t = 1,...,T 1. A standard penalty would be for nstance a decreasng rate over years. Examples of penalty functons are gven n Palmer (2006). Sometmes a market value adjustment s appled when market performs poorly. By law, the polcy s cash surrender value cannot fall below the guaranteed mnmum value (h g). The ndvdual wll surrender because of fnancal reasons f L t > C t, where C t s the market value at tme t of the contract. In other words, f the nsured has an opportunty to make a proft f he surrenders and that at the same tme he can buy exactly the same polcy. However, and as mentoned n the ntroducton, t s plausble to assume that a polcyholder may not make a decson that s optmal from the purely fnancal pont of vew when choosng to surrender or not. For ths reason, we assume there s an extra decson parameter λ 1 such that the contract s surrendered only f L t > λc t. In some sense, t means that some agents wll react faster to the surrender crtera than others. Informed agents wll have λ = 1. For unnformed agents that do not worry about ther nvestments, they wll surrender f the surrender condton s really nterestng and thus f L t s sgnfcantly hgher than the market value of the contract C t. It should be clear that as λ ncreases, the value of the (fnancal) surrender opton goes to 0, a fact that we verfy n our experments of Secton 5. 3 LEAST-SQUARES MONTE CARLO WITH MORTALITY Let us frst revew the least-squares Monte Carlo approach as proposed n Longstaff and Schwartz (2001) appled to the problem of prcng an equty-lnked contract when there s no mortalty rsk. The method uses n realzaton paths {St,t = 0,1,...,T ; = 1,...,n} of the ndex, and then estmates for each path when s the optmal exercse tme t. Ths s done by proceedng backward from T as follows: set t = T, and the contract s value on path to VT, whch s the value of V T as n (1), but wth S T replaced by the fnal value ST of the ndex on path. Then at tme t = T 1,T 2,...,1, set t = t and Vt = Lt f Lt > Ĉt, where Ĉt s an estmate of the contnuaton value of the contract at tme t gven St, gven by e r E(V t+1 St), and Lt s the surrender value at tme t on path. Ths estmate Ĉt s obtaned by regresson of the dscounted contract s value at the next tme step aganst the current value of the ndex. More precsely, a fnte set of multvarate bass functons {ψ l ( ),l = 0,1,...,M} s chosen, and the regresson coeffcents are estmated as ( ˆβ 0,..., ˆβ M ) T = (Ψ T Ψ) 1 Ψ T (y 1,...,y n ) T, where y = e r Vt+1, and Ψ,l = ψ l (St) for = 1,...,n,l = 0,...,M. Then Ĉt = M ˆβ l=0 l ψ l (St). When Lt Ĉ(t,St), then we smply update the contract s value on path by dscountng t for one more step,.e., we set Vt = e r Vt+1. Once the optmal exercse tmes t are estmated for each path, the contract s value s approxmated by the low-based estmator ˆV 0, f n = 1 n n =1 where we used the conventon that L T = V T. e rt L t, (7) 447

5 Bernard and Lemeux We now want to nclude the mortalty rsk n the above prcng methodology, as well as the surrender parameter λ descrbed n the prevous secton. When decdng whether to surrender or not, the polcyholder must take nto account the fact that he/she mght de n the comng year, thus recevng D t+1 at tme t + 1 rather than holdng on to the contract. That s, the contnuaton value can be wrtten as E(V t+1 S t) = e r (q x+t E(V t+1 S t,death) +p x+t E(V t+1 S t,survval)) = q x+t e r E(D t+1 S t) + p x+t C t = q x+t ξ (S t,g d,k d,1) + p x+t C t. where ξ s defned by (4). Hence at tme t, the contract s surrendered f L t > C t := q x+t e r D t+1 + p x+t Ĉ t, where Ĉ t s the contnuaton value condtoned on survval untl t + 1, and s calculated as before. In ths case, we set V t = L t. However the contract s value V t must be updated dfferently when the surrender value L t does not exceed the weghted contnuaton value C t. More precsely, we have LsMCEqLnkMort() for 1 to n t () T generate S1,...,S T V [] VT for j T 1 downto 1 q q x+t p = 1 q compute ˆβ 0,..., ˆβ M for 1 to n Ct M 1 l=0 ˆβ l, j ψ l (St) C t e r qξ (St,g d,k d,1) + pct f Lt > λ Ct then t [] t V [] Lt else V [] e r (qξ (St,g d,k d,1) + pv []) for 1 to n A[] t []p x e rt [] Lt [] for t = 0 to t [] A[] A[] + e rt t p x q x+t Dt+1 return n =1 A[]/n V t = e r (q x+t ξ (S t,g d,k d,1) + p x+t V t+1 ). Once we have an estmate of the optmal exercse tme t on each path, then we must take nto account the mortalty rsk by replacng the estmator (7) by where A = ( t p x e rt L t ˆV 0 = 1 n t 1 + n =1 A, (8) t p x q x+t e r(t+1) Dt+1 t=0 and t p x = t 1 j=0 p x+ j s the probablty that an ndvdual of age x survves at least t years. The dea s that f the polcyholder des before the estmated optmal surrender tme, then a death beneft wll nstead be pad at the end of the year of death. Summng up, we proceed as n the pseudocode gven n Fgure 1. 4 EFFICIENCY IMPROVEMENT A frst way to mprove the qualty of the estmator (8) s to use the value of the correspondng European contract as a control varate. We know the European contract value when the mortalty s ncluded. It s gven by V 0,e defned earler by (5). Hence the control varate estmator s gven ), Fgure 1: Pseudocode descrbng regresson-based Monte Carlo for an equty-lnked contract wth mortalty by where ˆV 0,e s gven by where ˆV 0,cv = ˆV 0 + ˆγ(V 0,e ˆV 0,e ), (9) ˆV 0,e = 1 n E = e rt T p x VT T 1 + t=0 n =1 E, e r(t+1) t p x q x+t D t+1. The coeffcent ˆγ n (9) s gven by the estmated value of Cov(E,A )/Var(E ). The surrender opton can then be estmated as ˆX 0,cv = ˆV 0,cv V 0,e. (10) Secondly, we can use randomzed quas-monte Carlo methods to smulate the n paths of the ndex. Ths was done n the context of the least-squares Monte Carlo algorthm n Lemeux (2004), among others. Here, we do ths usng 448

6 Bernard and Lemeux a randomly dgtally shfted Sobol pont set, and apply the Brownan brdge technque (Caflsch and Moskowtz 1995) n order to reduce the effectve dmenson of the problem. More precsely, what we need here s a T -dmensonal pont set P n n [0,1) T, obtaned usng the frst n ponts of the Sobol sequence (Sobol 1967) wth the drecton numbers gven n Bratley and Fox (1988). Then we randomze ths pont set usng a random dgtal shft (L Ecuyer and Lemeux 2002). By repeatng the randomzaton process m tmes ndependently, we thus obtan m estmators ˆV 0,l of the form (8) for the contract (wth the surrender opton), but where the path S1,...,S T s based on the randomzed pont ũ,l, obtaned by addng the lth dgtal shft to the th pont of P n, for =,1...,n and l = 1,...,m. The applcaton of the Brownan brdge technque n ths case amounts to use the frst coordnate of ũ,l to generate ST, when constructng the lth estmator ˆV 0,l, then the second coordnate s used to generate S T /2, the thrd one for S T /4, and so on. Once we have these m estmators ˆV 0,l, we can construct a 95% confdence nterval for V 0 wth half-wdth m(m 1) m l=1 ( ˆV 0,l V 0 ) 2, where V 0 = m l=1 ˆV 0,l /m. A smlar approach can be used for the control varate estmator. 5 RESULTS In what follows, we gve results for dfferent equty-lnked contracts. In each case, we report the value of the contract, ˆV 0, the smulated value of the contract wthout a surrender opton, gven by ˆV 0,e, and the value ˆX 0,cv of the surrender opton. Note that we also report the exact value of the contract n the lne startng wth exact. For each estmate, we report results usng ether Monte Carlo (MC) or randomzed quas-monte Carlo (RQMC) samplng, and gve below each estmate the half-wdth of a 95% confdence nterval for the value that we are tryng to estmate. To prce the contract, we run smulatons ether wth a control varate as gven n lnes MC-cv and RQMC-cv or wthout as gven n lnes MC-nocv and RQMC-nocv. Note that snce the estmate ˆX 0,cv for the surrender opton dffers only by a constant from the one V 0,cv for the contract tself, the half-wdth of the confdence nterval s the same for both estmators. The default (benchmark) values for the parameters are: T = 10 years, σ = 20%, P = 100, α = 0.85, r = 4%, the mnmum guaranteed rates g, h and g d are all set to 2%, and the partcpatng coeffcent k and k d to 90%. The penaltes are as follows: β 1 = 0.05, β 2 = 0.04, β , β 4 = 0.01, β t = 0 for t 5. We also assume that the polcyholder s ratonal and perfectly nformed: λ = 1. For mortalty, we use a smple parametrc model: the Makeham s model. The survval functon s(x), whch gves the probablty that an ndvdual of age 0 wll survve at least x years, s gven by ( x ) s(x) = exp µ(s)ds = exp( Ax B(c x 1)) 0 where µ(x) = A + Bc x wth B > 0,A B,c > 1,x 0. We can then calculate the probablty to lve and to de as follows: t p x = s(x +t), tq x = 1 t p x. s(x) Melnkov and Romanuk (2006) calbrated Makeham s model for the mortalty n the USA and gve the followng estmates of the parameters A, B and c: A = , B = , c = All experments have been performed by generatng 25..d. groups of 8192 smulatons, for a total of 204,800 smulaton runs. Ths s about 10 tmes less than the number of smulatons 19,000 smulatons wth 140 seeds, for a total of 2, runs used n Bacnello, Bffs, and Mllossovch (2008a), Bacnello, Bffs, and Mllossovch (2008b), who get results wth about the same precson as our RQMC estmators wth the control varate (but use stochastc mortalty, as explaned before). For the bass functons of the regresson, we use the frst four Legendre polynomals. Table 1: Result wth the default parameters ˆV 0 ˆV 0,e ˆX 0 MC-nocv MC-cv RQMC-nocv RQMC-cv exact In Table 1, we report our results wth the benchmark parameters. The European contract has the value Note that the ntal premum of the contract s P = 100 and thus the market value of the contract s lower than the premum charged at tme 0. Ths s consstent wth real contracts that often nclude commssons and fees for dfferent servces and optons. Includng the surrender opton ncreases the value of the contract by approxmately

7 Bernard and Lemeux Thus, the surrender opton represents about 1.9% of the total value of the (European) contract. We now experment wth dfferent values of λ, partcpatng parameters, and model parameters. Table 2: Varyng λ ˆV 0 ˆV 0,e ˆX 0 λ = 1.05 MC-nocv MC-cv RQMC-nocv RQMC-cv exact λ = 1.15 MC-nocv MC-cv RQMC-nocv RQMC-cv exact We frst ncrease the value of λ to 1.05 or 1.15 and we report the results n Table 2. We observe that when λ s ncreasng, the polcyholder becomes less and less nterested n the surrender opton. Its value becomes almost zero. In fact, nsurers rely a lot on ths fact. Polcyholders wll not have a fnancally optmal behavour and wll be reluctant to surrender ther polces. A stuaton where λ > 1 as t s the case n Table 2 s thus more realstc. Note that t would be nterestng to look at the rsk of a portfolo of polces wth a pool of polcyholders wth dfferent parameter λ. We can also look at the mpact of changes n the partcpatng coeffcent k and k d, or n the mnmum guaranteed rates g, g d, and h. Our results suggest that the surrender opton s less valuable when the mnmum guarantees at maturty, for death, and for surrender are hgher (that s, g = h = g d s hgher) or f the partcpatng coeffcent k at maturty or k d at tme of death ncreases. Indeed, the surrender opton represents 0.82/96.73 = 0.85% of the value of the European contract when the guaranteed rates ncrease to g = h = g d = 3% nstead of the benchmark case of g = h = g d = 2%, and 1.68/ % when the partcpatng coeffcents k and k d are 95% nstead of 90%, nstead of a rato of 1.95% wth the default parameters. In the second case, we expect that an ncrease n the partcpaton at maturty and n case of death are ncentve Table 3: Varyng partcpaton parameters ˆV 0 ˆV 0,e ˆX 0 g = h = 0.03 MC-nocv MC-cv RQMC-nocv RQMC-cv exact k = k d = 0.95 MC-nocv MC-cv RQMC-nocv RQMC-cv exact h = 0.03 MC-nocv MC-cv RQMC-nocv RQMC-cv exact to not surrender. But n the frst case, there s a tradeoff between the ncreased guarantee at maturty and death whch are ncentve for not exercsng and the ncrease n the guarantee n case of surrender. Our results suggest that the ncrease at maturty and death offset the ncrease n the surrender payoff. However, f only the guaranteed rate h for the surrender opton ncreases to 3%, then, as expected, the opton s value ncreases from about 1.8 to about Fnally, we can analyze the mpact of the fnancal assumptons and n partcular the volatlty of the underlyng ndex. We can compare the value of the surrender opton equal to 1.8 when volatlty s 20% (benchmark parameter n Table 1), wth the values n Table 4, gven by 1.47 f σ = 10% or 1.91 f σ = 30%. The value of the surrender opton thus appears to be ncreasng wth the volatlty σ. Ths s consstent wth the well-known fact that the value of an opton ncreases wth the volatlty of the underlyng ndex. 450

8 Bernard and Lemeux Table 4: Varyng fnancal model ˆV 0 ˆV 0,e ˆX 0 σ = 0.1 MC-nocv MC-cv RQMC-nocv RQMC-cv exact exact σ = 0.3 MC-nocv MC-cv RQMC-nocv RQMC-cv exact CONCLUSIONS Ths paper extends the Least Squares Monte Carlo method proposed by Longstaff and Schwartz (2001) to the valuaton of surrender benefts embedded n lfe nsurance ndex lnked contracts. Wth a relatvely small number of smulatons, we get a qute precse approxmaton of the surrender beneft usng quas-monte Carlo smulatons and ncorporatng a powerful control varate. We show how to nclude mortalty rsk when ths rsk s modeled usng fxed probabltes that are the same for everybody. In practce, polcyholders who decde to surrender a lfe nsurance contract are healther than the average. The surrender opton wll ntroduce adverse selecton (worse rsks stay wth the nsurer and healthy people leave the portfolo) and thus wll ncrease a lot the potental value of the surrender benefts. We beleve that the proposed method could easly be extended, for nstance, to the case where the optmal decson of the polcyholder s taken condtonally to hs health state. We plan to study ths model n the near future. ACKNOWLEDGMENTS Both authors are grateful to the Natonal Scence and Engneerng Research Councl (NSERC) of Canada for ts fnancal support through ndvdual Dscovery Grants. We would lke to thank Phelm Boyle, Mary Hardy and Harry Panjer for useful comments and suggestons on ths work. REFERENCES Albzzat, M.-O., and H. Geman Interest rate rsk management and valuaton of the surrender opton n lfe nsurance polces. The Journal of Rsk and Insurance 61: Andreatta, G., and S. Corradn Far value of lfe labltes wth embedded optons: an applcaton to a portfolo of Italan nsurance polces. Workng Paper, RAS Spa, Panfcazone Reddtvtá d Gruppo. Bacnello, A. R Prcng guaranteed lfe nsurance partcpatng polces wth annual premums and surrender opton. North Amercan Actuaral Journal 7 (3): Bacnello, A. R Endogenous model of surrender condtons n equty-lnked lfe nsurance. Insurance: Mathematcs and Economcs 37 (2): Bacnello, A. R., E. Bffs, and P. Mllossovch. 2008a. Prcng lfe nsurance contracts wth early exercses features. Journal of Computatonal and Appled Mathematcs. In press. Bacnello, A. R., E. Bffs, and P. Mllossovch. 2008b. Regresson-based algorthms for lfe nsurance contracts wth surrender guarantees. Workng Paper. Ballotta, L., and S. Haberman Valuaton of guaranteed annuty converson optons. Insurance: Mathematcs and Economcs 33 (1): Boyle, P. P., and M. R. Hardy Guaranteed annuty optons. Astn Bulletn 33 (2): Boyle, P. P., and E. S. Schwartz Equlbrum prces of guarantees under equty-lnked contracts. Journal of Rsk and Insurance 44: Bratley, P., and B. L. Fox Algorthm 659: Implementng Sobol s quasrandom sequence generator. ACM Transactons on Mathematcal Software 14 (1): Brennan, M. J., and E. S. Schwartz The prcng of equty-lnked lfe nsurance polces wth an asset value guarantee. Journal of Fnancal Economcs 3: Caflsch, R. E., and B. Moskowtz Modfed Monte Carlo methods usng quas-random sequences. In Monte Carlo and Quas-Monte Carlo Methods n Scentfc Computng, ed. H. Nederreter and P. J.-S. Shue, Number 106 n Lecture Notes n Statstcs, New York: Sprnger-Verlag. Grosen, A., and P. L. Jørgensen Valuaton of early exercsable nterest rate guarantees. Journal of Rsk and Insurance 64 (3): Hardy, M. R Investment Guarantees: Modellng and Rsk Management for Equty-Lnked Lfe Insurance. Wley. 451

9 Bernard and Lemeux Jensen, B., P. L. Jørgensen, and A. Grosen A fnte dfference approach to the valuaton of path-dependent lfe nsurance labltes. The Geneva Papers on Rsk and Insurance Theory 26 (1): Kolkewcz, A., and K. Tan Unt-lnked lfe nsurance contracts wth lapse rates dependent on economc factors. Annals of Actuaral Scence 1 (1): Kuo, W., C. Tsa, and W.-K. Chen An emprcal study on the lapse rate: the contegraton approach. Journal of Rsk and Insurance 70 (3): L Ecuyer, P., and C. Lemeux Recent advances n randomzed quas-monte Carlo methods. In Modelng Uncertanty: An Examnaton of Stochastc Theory, Methods, and Applcatons, ed. M. Dror, P. L Ecuyer, and F. Szdarovszk, : Kluwer Academc Publshers, Boston. Lemeux, C Randomzed quas-monte Carlo methods: a tool for mprovng the effcency of smulatons n fnance. In Proceedngs of the 2004 Wnter Smulaton Conference, : IEEE Press. Longstaff, F., and E. Schwartz Valung Amercan optons by smulaton: a smple least-squares approach. The Revew of Fnancal Studes 14 (1): Melnkov, A., and Y. Romanuk Evaluatng the performance of Gompertz, Makeham and Lee-Carter mortalty models for rsk management wth unt-lnked contracts. Insurance: Mathematcs and Economcs 39: Mlevsky, M., and T. S. Salsbury Fnancal valuaton of guaranteed mnmum wthdrawal benefts. Insurance: Mathematcs and Economcs 38 (1): Mynen, R The prcng of the Amercan opton. Annals of Appled Probablty 2:1 23. Palmer, B. A Equty-ndexed annutes: Fundamental concepts and ssues. Workng Paper. Shen, W., and H. Xu The valuaton of unt-lnked polces wth or wthout surrender optons. Insurance: Mathematcs and Economcs 36 (1): Sobol, I. M On the dstrbuton of ponts n a cube and the approxmate evaluaton of ntegrals. U.S.S.R. Comput. Math. and Math. Phys. 7: from the Unversté de Montréal. Her research nterests are quas-monte Carlo methods, and how they can replace random samplng n a varety of problems. Her emal address s AUTHOR BIOGRAPHIES CAROLE BERNARD s an Assstant Professor n the Department of Statstcs and Actuaral Scence at the Unversty of Waterloo. She obtaned her Ph.D. n 2006 from Unversté de Lyon I, for whch she was rewarded wth the Best Fnance Thess Award from France the same year. Her research nterests are fnancal and nsurance modelng. Her emal address s CHRISTIANE LEMIEUX s an Assocate Professor n the Department of Statstcs and Actuaral Scence at the Unversty of Waterloo. She obtaned her Ph.D. n

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

Traffic-light extended with stress test for insurance and expense risks in life insurance

Traffic-light extended with stress test for insurance and expense risks in life insurance PROMEMORIA Datum 0 July 007 FI Dnr 07-1171-30 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffc-lght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum

More information

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Section 2.3 Present Value of an Annuity; Amortization

Section 2.3 Present Value of an Annuity; Amortization Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

The Short-term and Long-term Market

The Short-term and Long-term Market A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Stock Profit Patterns

Stock Profit Patterns Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

Pricing index options in a multivariate Black & Scholes model

Pricing index options in a multivariate Black & Scholes model Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Variable Payout Annuities with Downside Protection: How to Replace the Lost Longevity Insurance in DC Plans

Variable Payout Annuities with Downside Protection: How to Replace the Lost Longevity Insurance in DC Plans Varable Payout Annutes wth Downsde Protecton: How to Replace the Lost Longevty Insurance n DC Plans By: Moshe A. Mlevsky 1 and Anna Abamova 2 Summary Abstract Date: 12 October 2005 Motvated by the rapd

More information

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET *

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET * ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET * Amy Fnkelsten Harvard Unversty and NBER James Poterba MIT and NBER * We are grateful to Jeffrey Brown, Perre-Andre

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton

More information

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closed-end funds Abstract The mpact

More information

Trivial lump sum R5.0

Trivial lump sum R5.0 Optons form Once you have flled n ths form, please return t wth your orgnal brth certfcate to: Premer PO Box 2067 Croydon CR90 9ND. Fll n ths form usng BLOCK CAPITALS and black nk. Mark all answers wth

More information

An Interest-Oriented Network Evolution Mechanism for Online Communities

An Interest-Oriented Network Evolution Mechanism for Online Communities An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne

More information

On the pricing of illiquid options with Black-Scholes formula

On the pricing of illiquid options with Black-Scholes formula 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth Black-Scholes

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

Pragmatic Insurance Option Pricing

Pragmatic Insurance Option Pricing Paper to be presented at the XXXVth ASTIN Colloquum, Bergen, 6 9th June 004 Pragmatc Insurance Opton Prcng by Jon Holtan If P&C Insurance Company Ltd Oslo, Norway Emal: jon.holtan@f.no Telephone: +47960065

More information

Chapter 3 Research Method

Chapter 3 Research Method Chapter 3 Research Method 3.1 Framework 3.1.1 Research Desgn A two-phase study was desgned to explore the feasblty of RM n agng socetes from both supply and demand aspect. In the aspect of the borrowers,

More information

Small pots lump sum payment instruction

Small pots lump sum payment instruction For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested

More information

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMAN-JORDAN, 18 20

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Pricing Multi-Asset Cross Currency Options

Pricing Multi-Asset Cross Currency Options CIRJE-F-844 Prcng Mult-Asset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

Implied (risk neutral) probabilities, betting odds and prediction markets

Implied (risk neutral) probabilities, betting odds and prediction markets Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

Simon Acomb NAG Financial Mathematics Day

Simon Acomb NAG Financial Mathematics Day 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

More information

LIFETIME INCOME OPTIONS

LIFETIME INCOME OPTIONS LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 357-5200 Fax: (617) 357-5250 www.ersa-lawyers.com

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

Chapter 11 Practice Problems Answers

Chapter 11 Practice Problems Answers Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Lecture 14: Implementing CAPM

Lecture 14: Implementing CAPM Lecture 14: Implementng CAPM Queston: So, how do I apply the CAPM? Current readng: Brealey and Myers, Chapter 9 Reader, Chapter 15 M. Spegel and R. Stanton, 2000 1 Key Results So Far All nvestors should

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

A Master Time Value of Money Formula. Floyd Vest

A Master Time Value of Money Formula. Floyd Vest A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.

More information

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

More information

Prediction of Disability Frequencies in Life Insurance

Prediction of Disability Frequencies in Life Insurance Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but

More information

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

More information

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET Amy Fnkelsten Harvard Unversty and NBER James Poterba MIT and NBER Revsed May 2003 ABSTRACT In ths paper, we nvestgate

More information

Prediction of Disability Frequencies in Life Insurance

Prediction of Disability Frequencies in Life Insurance 1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng 1, Fran Weber 1, Maro V. Wüthrch 2 Abstract: For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Research Note APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES * Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

Scale Dependence of Overconfidence in Stock Market Volatility Forecasts

Scale Dependence of Overconfidence in Stock Market Volatility Forecasts Scale Dependence of Overconfdence n Stoc Maret Volatlty Forecasts Marus Glaser, Thomas Langer, Jens Reynders, Martn Weber* June 7, 007 Abstract In ths study, we analyze whether volatlty forecasts (judgmental

More information

An Empirical Study of Search Engine Advertising Effectiveness

An Empirical Study of Search Engine Advertising Effectiveness An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan Rmm-Kaufman, Rmm-Kaufman

More information

Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautiousness and Measuring An Investor s Tendency to Buy Options Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

More information

A Critical Note on MCEV Calculations Used in the Life Insurance Industry

A Critical Note on MCEV Calculations Used in the Life Insurance Industry A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares

More information

Mathematics of Finance

Mathematics of Finance 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

More information

Risk Model of Long-Term Production Scheduling in Open Pit Gold Mining

Risk Model of Long-Term Production Scheduling in Open Pit Gold Mining Rsk Model of Long-Term Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,

More information

Effective September 2015

Effective September 2015 Annuty rates for all states EXCEPT: NY Lock Polces Prevous Prevous Sheet Feld Bulletns Index Annuty s effectve Monday, September 28 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt

More information

An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services

An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsn-yng Wu b a Professor (Management Scence), Natonal Chao

More information

Statistical Methods to Develop Rating Models

Statistical Methods to Develop Rating Models Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and

More information

Uncrystallised funds pension lump sum payment instruction

Uncrystallised funds pension lump sum payment instruction For customers Uncrystallsed funds penson lump sum payment nstructon Don t complete ths form f your wrapper s derved from a penson credt receved followng a dvorce where your ex spouse or cvl partner had

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information