Stochastc ALM models - General Methodology Stochastc ALM models are generally mplemented wthn separate modules: A stochastc scenaros generator (ESG) A cash-flow projecton tool (or ALM projecton) For projectng future (probable) cash-flows of the nsurance contracts but also of assets Cash-flows and values are projected along each scenaro Modellng of polcyholders behavour : surrender / reductons / other optons (Lfe) Integraton of management actons and strateges (proft sharng, asset allocaton, dvdend polcy, hedgng polcy ncludng rensurance) Potental ncorporaton of underwrtng rsks Addtonal tools Model ponts, replcatng portfolos, curve fttng, scenaros selecton Stochastc Methods 1
ALM projecton After havng generated the dfferent stochastc processes of the dfferent market and economc varables, we have a set of scenaros at our dsposal Two sets, under the rsk neutral and real world measure These scenaros are then taken as nput of an ALM projecton tool, n whch cash-flows and values of assets and labltes wll be projected Ths wll be the bass of dfferent types of analyses Generally, the scenaro generator and the ALM projecton tool are separate tools (separate programs, possbly dfferent languages) Stochastc Methods 2
ALM projecton Purpose: smulate smultaneously the evoluton of a portfolo of nsurance contracts wth ts coverng assets Projecton of cash-flows Projecton of (market consstent) values of assets and labltes Objectves: perform dfferent analyses Valuaton of nsurance labltes (best estmate, rsk margn) Valuaton of the company / nsurance segment Valuaton of a budget Measure of the nsolvency rsk and calculaton of economc or regulatory captal Testng dfferent polces or strateges proft sharng, dvdends polces, asset allocaton strateges Searchng for optmalty Stochastc Methods 3
ALM projecton: techncal provsons Projecton of probable mathematcal reserves These can be projected by applyng e.g. Fouret formula These are then typcally weghted by: Mortalty and survval probabltes Surrender rates Reducton rates Fouret formula allows to project condtonal reserves,.e. condtonally to the fact that the contract s stll present n the portfolo Ths corresponds to the level of reserve that s contaned n the nformaton receved each year by the nsured from the nsurer The mortalty table to be used wthn Fouret formula s the techncal table, used for contracts prcng purpose These condtonal reserves are then weghted by the adequate probabltes (survval, of no surrender, ) Stochastc Methods 4
ALM projecton: techncal provsons Treatment of Reductons: Techncally In order to take reductons nto account, a parallel set (portfolo) of reduced contracts can be buld These parallel contracts wll have the same characterstcs as the ntal contracts (same age, same maturty, same guaranteed rate, ) except that there wll be no new premums pad Ths set s then growng from year to year n the projecton, wth reductons the probable reserved of the reduced contracts ncrease, whle the probable reserves of the non-reduced contracts decrease Probable reserves are weghted by survval probabltes from the experence tables, and not by techncal mortalty tables used for the prcng of the contract Stochastc Methods 5
ALM projecton: labltes cash-flows Labltes cash-flows nclude generally: Commercal premums, commssons Clams n case of death, n case of survval at maturty, or surrender value payment (n case of surrender) Admnstraton costs, clams handlng expenses Proft sharng (ncorporated progressvely n the provsons, and generally not pad as cash-flows they ncrease the clam amounts n practce) These cash-flows should be projected, based on hypotheses n terms of surrenders, proft sharng, reducton laws, modelled by the company from ts experence Stochastc Methods 6
ALM projecton: labltes cash-flows All labltes cash-flows are weghted by adequate probabltes Cash-flows n case of death: captal n case of death multpled by the probablty to be alve n the begnnng of the projecton year but to de durng the year Cash-flows n case of survval at maturty of the contract: weghted by the probablty that the nsured s stll alve at that tme and has not surrendered ts contract Premums: by the probablty that the contract has not been surrendered nor reduced, and by the survval probablty untl the begnnng of the year (premums are generally supposed to be pad at the begnnng of each perod) weghted by survval, death, (non) surrender and (non) reducton probabltes Stochastc Methods 7
ALM projecton: assets cash-flows Assets cash-flows nclude typcally: Dvdends on stocks Coupons on bonds, bank accounts and deposts Interests earnngs and repayments on mortgages Earnngs on buldngs The net cash-flow (Asset and Lablty sdes) for year t s then obtaned along each scenaro as: CF( t, CFA( t, CFP( t, wth the sgn conventon that a lablty cash outflow s postve The net cash-flow s then re-nvested n each asset followng a gven asset allocaton strategy (e.g. fxed target proportons, or other type of strategy) Stochastc Methods 8
Assets rebalancng before re-nvestment Each year wthn the projecton (loop on years wthn the ALM projecton tool), assets must be re-balanced n order to keep (possbly exstng) target proportons, before re-nvestment or des-nvestment of the total net cash-flows (total cash-flows from assets and labltes) For any asset category = 1,, n: Market values before rebalancng: VM t, sc = x t 1 S t, sc Market values after rebalancng: VMbal t, sc = VM t, sc Δ VM t, sc Where λ = target proporton nvested n asset class and VM 1 VM ( t, ( t, VM ( t,... VM 2 n ( t, Stochastc Methods 9
Assets rebalancng before re-nvestment Treatment of book values: Realsaton n book value f VCbal ( t,:) VC ( t,:) ( VM 1 0 : ( t,:) VM 2 ( t,:)... VM n VC ( t,:)) VM ( t,:) ( t,:) Re -nvestement f VCbal ( t,:) VC 0 : ( t,:) ( VM1( t,:) VM 2( t,:)... VM n( t,:)) Stochastc Methods 10
Investment of the net cash-flow of the year The net cash-flow s renvested n the dfferent assets followng target proportons λ : For any asset class : Market value after re-nvestment: VMrenv t, VMbal ( t, CF Adaptaton of the number of securtes (number of stocks for nstance) nvested asset class : ( ( t, If CF(t)>0, nvestment n asset and adaptaton of the book values: VCrenv ( t, VCbal ( t, CF( t, If CF(t)<0, desnvestment n asset and adaptaton of the book values: VCrenv ( t, VCbal x( t) VMrenv ( t, / S ( t, VCbal ( t, ( t, CF( t, VMbal ( t, Stochastc Methods 11
Proft sharng The company should frst have a clear dea of ts proft sharng polcy for each analyzed ALM segment A clear formulaton of the proft sharng polcy must be establshed On the modellng pont of vew, ths allows modellng of the proft sharng rate by an equaton Typcally, t appears as a functon of the level of nterest rates n the market, n comparson wth the guaranteed rate of the contract, also functon on the return obtaned on the assets portfolo On the rsk management pont of vew, ths allows assessng the proftablty and rsk of a gven proft sharng polcy The provsons must be adapted n order to ncorporate progressvely the proft sharng of the year In practce, proft sharng are not an outflow for the company but but are ntegrated n the reserve of the nsured, and consttute an addtonal debt w.r.t. the nsured (n practce by applyng to the Stochastc Methods reserves a PS rate) 12
Surrender laws Dfferent model types: Mcro-models: the surrender probablty s calculated at the level of each contract Macro-models: the same model (same equaton wth the same parameters) s appled to a portfolo of labltes Stochastc Methods 13
Surrender laws: Mcro-models Exogeneous determnstc models: surrender date s stochastc but the probablty to surrender s predefned (statc model, no lnk wth the economy) Endogeneous models: the behavour of polcyholders s supposed to be ratonal the surrender opton s modelled lke an Amercan opton, wth an optmal surrender tme: Optmal date of surrender = date at whch V surrender > contract far value (dynamcal approach, purely fnancal, supposng a completely ratonal behavour of polcyholders) Hazard processes based models: ntermedary soluton: lke an Amercan opton n presence of assymmetrc nformaton. Ths s more realstc, but more dffcult to understand and to mplement (dffcult calbraton) Stochastc Methods 14
Surrender laws: Macro-models Econometrc models: surrender rate explaned n a regresson model by economc varables and characterstcs of the polcyholder and of the contract (polcyholder age, age of the contract, nflaton rate, nterest rate of the market, ) Ex: s Logt model:ln V 1 q 0 1 1 s Complementary log- logmodel: ln1 q s 0 1V... nvn q ln 1... V n n Stochastc Methods 15
Rsk measurements Based on cash-flows and value projectons (market values, possbly also book values) under the real world measure Assets evolve wth an average return reflectng the return observed on hstorcal tme seres (superor to the rsk-free rates n prncple) Dfferent solvency ndcators can be calculated from the projectons: Run probabltes on dfferent tme horzons (dfferent defntons of run can be appled) NAV (or surplus) dstrbuton at the end of the projecton horzon or at ntermedary dates (1 year) Rsk measures of the NAV varaton on the tme horzon (1 year) Stochastc Methods 16
Rsk measurements Run probablty on a tme horzon Probablty to get a surplus nferor to the opposte value of the captalsed margn at any moment from ntal nstant to the tme horzon H P(Surplus(t) < Captalsed Margn (t) at some nstant t n [0,H] ) Need for the surplus at all ntermedary nstants, not only at tme horzon Based on another defnton of the surplus: Surplus *= BE (assets) mathematcal reserves Stochastc Methods
Valuatons Market consstent approach and use of arbtrage free prcng theory Best Estmate of TP: from rsk neutral projectons untl run-off of the portfolo N E Q 1 1 NbSmul CF ( ) exp( r( s) ds) 1 NbSmul labltes NbSmul scen1 1 0 scen scen1 1 k 1 NbSmul N N CF CF scen ( t ( t ) ) DFsto exp( r scen scen ( t t 1 )) Economc Value of the company (or of a segment of the company) E N Q 1 Stochastc Methods CF ( ) CF ( ) VM ( N) TP(N) exp( r( s) ds) assets labltes assets 0 18
Other analyses Asset allocaton optmsaton, hedgng / rsk mtgaton optmzaton, proft sharng polces testng, Performance analyses and comparson of ALM segments Proftablty testng for new products Stochastc Methods 19