3 Life Insurance and Differential Equations

Transcription

1 3 Life Insurance and Differential Equations 3.1 Model specification Ourmodelwasspecifiedbytheassumptionthattheremaininglifetimeatage xwasa randomvariable T x withacontinuousdistribution.thatdistributioniscompletelydescribed ifweknoweither: thec.d.f. F x (t),or thep.d.f. f x (t),or theforceofmortality µ x+t. Onewayistoassumethatthefirstoftheseistrue,andthatthec.d.f. F x (t)isknown (meaning that it has been estimated using suitable data). It is summarised numerically bythelifetable l x,whichisusedtocomputeepvs. Here,weassumethethirdoftheseistrue:wearegiventheforceofmortalityatallages. We can visualise this with the following picture: 38

2 Alive µ x+t Dead Figure1:Amodelthatcanbeusedtorepresentalifeinsurancecontract. Theproblemthenbecomes:given µ x+t,howcanwefindalltheprobabilitiesand EPVs needed to find premiums and policy values? The answer lies in solving ordinary differential equations(odes). We have already seen the following ODE which allows us to find survival probabilities: d dt t p x = t p x µ x+t. 39

3 We will call this the Kolmogorov equationfor reasons that will become clear later. Combinedwiththeinitialcondition 0 p x = 1,wewishtosolvethisforall t > 0. We have also seen Thiele s differential equation, which for a whole-life contract was: d dt V (t) = V (t)δ + P x µ x+t (1 V (t)). Thisallowsustocalculatepolicyvalues,butinfacttheEPVsofanycashflowscontingent uponthemodelpicturedabovecanbefound.theboundaryconditioninthiscaseisa terminal conditionrather than an initial condition(we will see this in Section 3.5). Veryrarely, µ x+t hassuchasimpleform(e.g.aconstant)thatwecanfindexplicit solutions, i.e. simple formulaeto one of both of these ODEs. Nearly always, however, we must solve them numerically, using a computer. We will do this using the simplest possible method, an Euler scheme. 40

4 3.2 Acommentonthisapproach At first sight, this approach seems more complicated than just using a life table. Indeed it is, moreover it is distinctly modern because before today s computer power became available, the numerical solution of ODEs was a formidable task. The payoff will come later, when we consider more complicated contracts. Here is an advance look at a model that underlies disability insurance, in which someone who is too sick to work receives a regular income to replace their lost earnings. 41

5 State 1 Healthy ρ x σ x µ x ν x State 3 Dead State 2 Sick Figure 2: A model that can be used to represent a disability insurance contract. Wecouldtrytoformulatethismodelintermsoftherandomtimesatwhicheventstake place,theanaloguesoftherandomlifetime T x.thisturnsouttobeextremelydifficult and complicated, and computing probabilities and EPVs by this approach is a nightmare. But, if we take as given the forces governing transitions between the three states, the analoguesoftheforceofmortality µ x+t,itturnsoutthatversionsofthekolmogorov equation and Thiele s equation can be written down very easily, and solving them 42

6 numericallyisjustaseasyasinthesimplelife-deathmodel.thatiswhywetakethis approach. 3.3 AnEulerschemeforsolvinganODE Weshow,below,howasimplerecursiveschemeEulerschemecanbeusedtosolvean ODE, given an initial condition. We are given the differential equation f (t) = g(f(t),t) andaninitialcondition f(0) = c. For example, put: 43

7 f(t) = t p x g(f(t),t) = f(t) µ x+t f(0) = 1. This sets up an Euler scheme to solve the Kolmogorov equation, conditional on the life beingaliveatage x. First, we choose a suitable step size, denoted h.this should be as small as possible, consistent with the computing capacity available. In the tutorials there is an opportunity to experiment with different step sizes. For example, 0.1 year or 0.01 year might be chosen. TheEulerschemeadvancesthesolutionoftheODEinstepsoflength h,startingwiththe initial condition. It does so by assuming that the function f(t) is approximately linear.this assumption gets more reasonable as the step size decreases. Suppose the solution has been advanced by k steps, that is, we have found approximate 44

8 values of f(h),f(2h),...,f(kh). If f(t)actuallywaslinearon [kh,(k + 1)h],itwouldbeastraightlinewithslope f (t), and the following would be exactly true: f((k + 1)h) = f(kh) + h f (kh). Although f(t)isnot,ingeneral,linearanywhere(certainlynotinthecaseofthe Kolmogorov equation representing human mortality) if the step size h is small enough, the errorwemakebyassuming f(t)tobeapproximatelylinearmaybesmallenoughtobe acceptable. Therefore, we apply Equation(3.3) successively, starting with the initial condition f(0).thisistheeulerscheme. 45

9 3.4 Solving the Kolmogorov equation Applying Euler s method to the Kolmogorov equation we have: hp x 0 p x + h d dt t p x t=0 = 0 p x + h [ 0 p x µ x+0 ] = 1 h µ x and: 2hp x h p x + h d dt t p x t=h = h p x + h [ h p x µ x+h ] = 1 h µ x h (1 h µ x )µ x+h 46

10 and: 3hp x 2h p x + h d dt t p x t=2h = 2h p x + h [ 2h p x µ x+2h ] andsoon.thiscaneasilybeprogrammed: in a spreadsheet(see tutorials) using any programming language(e.g. Visual Basic, C) using any standard maths package(e.g. Maple, Matlab, Mathematica). Infact,anystandardmathspackagewillprobablysolveODEsasastandardfeature,sothe user need not program the Euler scheme themselves. Note that the Euler scheme is the simplest numerical method for solving ODEs(which is whyweuseit).itisalso,however,amongtheslowestandleastaccuratemethods.many better methods are available, described in any good text on numerical analysis(any maths 47

11 packagewillcertainlyuseoneofthem).wewouldprobablynotuseaneulerschemein practice. 3.5 Solving Thiele s differential equation We can solve Thiele s differential equation either: (a) forwards from the initialcondition V (0) = 0 (b) or backwards from terminalconditions like: V (n) = 0 or V (n) = 1. Note: the direction of time is immaterial when solving ODEs. Suppose the function f(t) satisfies the ODE: 48

12 f (t) = g(f(t),t) andsatisfiestheinitialcondition f(0) = c 0 andtheterminalcondition f(t) = c T for sometime T > 0.Thenwecaneither: chooseapositivestepsize handadvancethesolutionforwardfrom f(0),or chooseanegativestepsize handadvancethesolutionbackwardsfrom f(t). Inthefirstcase,whenwereachtime T weshouldobtainthecorrectvalue f(t) = c T (to withinnumericalerror)and,inthesecondcase,whenwereachtime0weshouldobtain thecorrectvalue f(0) = c 0. InsolvingThiele sequation,wemightknowaninitialvalue,butoftenwedonot.infactthe initialvalueisoftentheunknownquantitywhosevaluewewanttofind.butwealmost always know a terminal value, because: apolicywithnomaturitybenefit(e.g.atermassurance)alwayshaspolicyvalue0at expiry; and 49

13 apolicywithamaturitybenefit(e.g.anendowment)alwayshaspolicyvalueequalto the maturity benefit just before expiry. Therefore, we will solve Thiele s equation backwards from such terminal conditions. ThefollowingarethefirstfewstepsofanEulerschemewithstepsize hforaterm assurancecontractsoldtoalifeage x,withsumassured$1payableimmediatelyondeath within nyearsandpremiumpayablecontinuouslyatrate P perannum.thepolicyvalueat duration tisdenoted V (t)asusual. f(t) = V (t) g(f(t),t) = V (t)δ + P µ x+t (1 V (t)) f(n) = 0. Applying Euler s method with step size h we have: 50

14 V (n h) V (n) h d dt t=n V (t) = V (n) h[v (n)δ + P µ x+n (1 V (n))] and: V (n 2h) V (n h) h d dt t=n h V (t) = V (n h) h[v (n h)δ + P µ x+n h (1 V (n h))] and: 51

15 V (n 3h) V (n 2h) h d dt t=n 2h V (t) = V (n 2h) h[v (n 2h)δ + P µ x+n 2h (1 V (n 2h))] andsoon. 3.6 Thiele s equation as a general tool We have introduced Thiele s equation as a method of finding policy values.but, if we consideranyfuturebenefitsasbeingpaidforbyasinglepremiumatoutset,wehave: Premium = EPV[Benefits] = EPV[Benefits] EPV[Future premiums] 52

16 = Policy value at outset because there are no future premiums. In other words, we can use Thiele s equation to compute the EPV of any benefits at all,not just policy values of contracts with regular premiums. Example: Consider a non-profit endowment with term 10 years sold to (30). The sum assured of $50,000 is payable on maturity, or immediately on earlier death. The force of mortalityis µ x+t andtheforceofinterestis δ,andtherearenoexpenses.findtheannual rateofpremium P. Method 1: Set up Thiele s equation for the policy value: d dt V (t) = V (t) δ + P µ x+t (50,000 V (t)). Theboundaryconditionis V (10) = 50,000,but P isunknown.sincethiswillallbe donewithacomputerwecanfind P easilybytrialanderror. Method2:SetupThiele sequationseparatelyforabenefitof$1andforanannuityof$1 53

17 per annum. These are, respectively: d dt V A(t) = V A (t) δ + 0 µ x+t (1 V A (t)) d dt V a(t) = V a (t) δ + 1 µ x+t (0 V a (t)) withboundaryconditions V A (10) = 1and V a (10) = 0.Thenbytheusualequivalence principle: P = 50,000 V A (0)/V a (0). 54