PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

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1 Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004

2 Profi Tes Modelling in Life Assurance Using Spreadshees Profi es modelling in life assurance. Inroducion The aim of his brief is o demonsrae he developmen of profi es models in life assurance using spreadshees. We will work hrough several illusraive examples sep-by-sep and he degree of complexiy will increase from one example o he nex. We will sar wih a single policy excluding loadings and gradually work our way o an enire porfolio of policies where we also include invesmen reurn, expenses, loadings and mauriy benefis. In par wo we will build more advanced models where we will include surrenders, paid-ups and moraliy risk. I is very common in life assurance ha he life office has high iniial expenses when wriing a new policy. This cos could be commission o he sales agen bu could also be inernal coss for underwriing or for IT sysems. This leads o a negaive cash flow or negaive resul for he life office a he incepion of a life policy. The premiums charged by a life office are calculaed in such a way ha he presen value of he premiums should be equal o or exceed he presen value of he fuure benefis and expenses. If no, he policy is wrien a an expeced loss which, if i is done consisenly, would hreaen he solvency of he life office. Leing he presen value of he premiums being equal o he presen value of he benefis and expenses is no enough. In order for he life office o be able o wrie new business, i needs o pu up risk capial. For a proprieary company, his is done by shareholders who expec reurn on heir share capial. For a muual company his is done by he exising policyholders, who expec he surplus accumulaed in he company no o be dilued by he wriing of new policies ha do no conribue o his surplus. The life office usually ses inernal rules deermining he minimum profi o be emerging from a new life policy or new block of life policies wrien. One such rule could be ha he presen value of he premiums should exceed he presen value of he benefis by a cerain percenage. Anoher way of expressing profi requiremens for a life policy is o sae when he iniial expenses are repaid a a saed inernal discoun rae. We will mainly work wih his ype of profi requiremen in his brief. 2. Uni-linked assurance 2.. A policy Le us firs sar wih a uni-linked policy as an example. In a ypical uni-linked assurance policy he financial risk is born by he policyholder. This differs from radiional assurance where he producs ypically provide guaraneed benefis a mauriy, deah or surrender. In uni-linked assurance, premiums are invesed in a fund of he choice of he policyholder afer deducions for expenses and moraliy. The premium reserve will hus (in mos cases) be defined rerospecively, using he invesmen reurn earned by he fund. I should be noed ha here are uni-linked policies sold which provide some guaranees. One ypical guaranee is a guaraneed benefi of a leas he sum of premium paid. In our discussion, we will assume ha no guaranees are provided

3 Profi Tes Modelling in Life Assurance Using Spreadshees Le V = premium reserve a year P = premium paid a he beginning of he year i = invesmen reurn of he fund during he period Assuming zero iniial expenses and no moraliy risk he premium reserve a ime is expressed as: V = V + P ) *( + i ) () ( Expression () shows he developmen of he fund in he ime-discree case. The keen suden could here and in laer examples consruc he corresponding formula in he ime-coninuous case. This ype of policy could, jus as well as a radiional policy, be sudied analyically. We will however insead mainly sudy i in a more sraighforward way in a spreadshee environmen. The main reason for his is ha he assumpions used do ofen no lead o a nice analyical formula, like he Makeham formula. Anoher imporan advanage of his mehod is ha he suden could much easier see wha happens during he lifeime of he policy raher han jus seeing one figure being he resul of a long formula. Also, i is easier in a spreadshee o creae wha if siuaions, i.e. o es he effec of changes in assumpions and o see how hese changes affec he policy in differen periods. We will look a some principle problems and also give some pracical ips on how his ype of sudy is bes done in a spreadshee environmen. We will generally assume a level premium is paid, i.e. P = P, 0<=d, where d is he duraion of he policy and denoes ime. We choose a policy where annual premiums of 00 unis are paid for en years and express his as in he following: P = 00, 0<=0 We will ofen use only P o denoe a level premium. Le us look a a very simple able illusraing his in a able aken from an Excel (or Lous or MoSeS) spreadshee

4 Profi Tes Modelling in Life Assurance Using Spreadshees Policy duraion 0 years 00 per year Year One imporan hing o noe here is ha all parameers should be saed explicily in any Excel spreadshee used for his ype of calculaions. I should always be very easy o change he parameers and sudy he effecs of such change. If P = 50 for 0<=5 we should hen easily ge: Policy duraion 5 years 50 per year Year Le us assume ha he invesmen reurn is zero. A he end of he policy a mauriy benefi is paid, consising of he sum of he premiums: Policy duraion paymen 0 years 0 years 00 per year Year Mauriy benefi

5 Profi Tes Modelling in Life Assurance Using Spreadshees We show he mauriy benefi as negaive, since i represens an ougo for he life office. Le us also include he cash flow o he life office: Policy duraion paymen 0 years 0 years 00 per year Year Mauriy benefi Cash flow The premiums he life office receives years o 0 mus be reserved, since i will be needed o pay for he mauriy benefi in year 0. The developmen of he premium reserve is given by (assuming zero invesmen reurn): V + P or V = V = P The mauriy benefi C d = V d = d P C = 0 for? d. Cd a ime d is given by Policy duraion paymen 0 years 0 years 00 per year Year Mauriy benefi Cash flow Reserve We here use a minus sign for he mauriy benefi, since i eners he cash flow as negaive. The reserve increases wih he premiums paid and decreases wih he mauriy benefi paid ou and we noe ha i is zero afer he mauriy of he policy jus as we would expec

6 Profi Tes Modelling in Life Assurance Using Spreadshees For uni-linked business, he reserve (a leas in simple cases) consiss of savings belonging o he policyholder, where he policyholder bears he financial risks conneced wih he reserve. In such case, he reserve is ofen called he fund and he cash flow o and from he fund is no included when one sudies he cash flow from he life office's poin of view. The fund is like a bank accoun and is reaed as such in US GAAP, he general acceped accouning principles in he US. In our simplified example, he cash flow is given by CF = V + P C V = 0 or in spreadshee forma: Policy duraion paymen 0 years 0 years 00 per year Year Mauriy benefi Fund Cash flow In order o avoid having differen formulae for year one and subsequen years, we include he value of he fund a he beginning of he year and he value of he fund a he end of he year. We have divided he able ino wo pars, one showing he developmen of he fund and one showing he cash flow o he life office. Policy duraion paymen 0 years 0 years 00 per year Year Fund in Mauriy benefi Fund ou Cash flow For pracical reasons, we use he convenion ha he fund ou is shown wih a minus sign (i is posiive for he clien bu is a liabiliy for he life office)

7 Profi Tes Modelling in Life Assurance Using Spreadshees 2.2. Invesmen reurn We have up o now assumed ha he money in he fund will earn no reurn. In real life, his money is invesed in financial asses, in equiies or bonds or boh. The invesmen reurn includes dividends on shares and realised or unrealised gains on shares or bonds. We will assume ha he invesmen reurn is fixed a a rae of i per ime period, even hough he fund value migh change coninuously. The fund value and cash flows are given by: V V + P ) ( + i ) = ( CF CF = ( V + P ) ( + i ) C V or = V + P + ( V + P ) i C V Assuming a level premium, he mauriy benefi a mauriy dae d is expressed as C d = V d = P d = ( + i) Imporan o remember is ha he invesmen reurn varies over ime, depending on he developmen of he asses in he fund. Le us now assume ha he fund will earn 5% annual reurn (afer axes and afer inernal fund expenses). Policy duraion paymen Expeced increase in uni value 0 years 0 years 5% annually 00 per year Year Fund in Ineres Mauriy benefi Fund ou Cash flow (In uni linked business, he value of he fund is ofen expressed as a number of unis, muliplied by he value of a uni. When he underlying asses increase in value, he number of unis remains consan while he value of a uni increases. When new premium is added o he fund, he number of uni increases.) Problem: Wha ineres rae is needed in order o provide a mauriy benefi of 2000? Answer: 2.3% This problem could be solved by analyical mehods, bu a more pracical and faser way is o use he problem solving mehods of Excel: Goal Seeker or Solver. (Tools, Goal Seeker)

8 Profi Tes Modelling in Life Assurance Using Spreadshees 2.3. Iniial commission Up o now, he cash flow for he life office has been zero. We shall now sar o look a his cash flow. The mos imporan cos for he life office in wriing business is he iniial expenses, and especially he commission paid o he sales agens, being ies agens or brokers. This commission is ofen paid up-fron (i.e. direcly afer a sale is made) and is ofen calculaed as a percenage (or per mille) of he oal premium volume of he conrac. We call he percenage a. The iniial commission is hus given by I = α d P, I = 0, This iniial commission eners as negaive cash flow year one. The cash flow formula is now given by: CF V + P + V + P i C I V = ( ) The above expression consiss of wo pars. The firs par is inflow and ouflow of he premium reserve. This par does in realiy no affec he life office as such bu raher he clien fund. The second par is he iniial commission. Looking a cash flow ha affecs he life office separaely, we have: CF = I Le us assume ha he commission is 40 per mille of he oal premium. For our conrac, he oal premium is 0*00 = 000 and he commission is hus 40. Policy duraion paymen Expeced increase in uni value Iniial commission 0 years 0 years 5% annually 00 per year 4 % of oal premium Year Fund in Ineres Mauriy benefi Fund ou Commission Cash flow In our ables, we show commission and oher expenses as negaive, since hey mean ouflow for he life office charges We have an ouflow from he life office in he form of he commission. The life office will need o cover hese expenses and his is done by inroducing some charges ha he policyholder has o pay. One way of doing his is o charge a percenage of each premium paid o he life office. Le us inroduce such a charge and le ha charge? be he same as he commission, i.e. 4%

9 Profi Tes Modelling in Life Assurance Using Spreadshees Inroducing premium charges, he developmen of he premium reserve is given by: V = ( V = V + P P γ + P ( γ )) ( + i ) C + ( V + P γ ) i C The mauriy benefi is C d = V d = P ( γ ) d = ( + i) The cash flow o he life office is CF = P γ I Policy duraion paymen Expeced increase in uni value Iniial commission charge 0 years 0 years 5% annually 00 per year 4 % of oal premium 4 % of each premium Year Fund in Charge Ineres Mauriy benefi Fund ou Charg e Commi ssion Cash flow

10 Profi Tes Modelling in Life Assurance Using Spreadshees A profi esing sudy in a spreadshee environmen is normally done verically he way we have done i up o now. We will however do i horizonally for he remainder of his brief. Policy duraion paymen Expeced increase in uni value Iniial commission charge 0 years 0 years 5% annually 00 per year 4 % of oal premium 4 % of each premium Year Fund in Charge Ineres Mauriy Fund ou Charge Comm Cash flow The premium charge is shown wice, as an expense for he policyholder and as an income for he life office. Le us also look a he accumulaed cash flow a ime where <= d. This is given by: AccCF = CF x Year Fund in Charge Ineres Mauriy Fund ou Charge Comm Cash flow Accumulaed cash flow Ne presen value We noe in he previous able ha he accumulaed cash flow amouns o zero a mauriy dae d, which seems o show ha income and ougo for he life office are equal. The iming of he wo is however no equal. The life office has an iniial ougo while he income comes laer and he life office will need o borrow exernally or use inernal funds o finance his ougo. These funds are no free and he life office mus herefore include he effec of his cos in is calculaions. The mos common way o do his is o calculae presen values (he fuure cash flows are discouned o he presen ime.)

11 Profi Tes Modelling in Life Assurance Using Spreadshees The general formula for calculaion of Ne Presen Value as per he beginning of year is NPV ( X 0... X n ) = d k = X k v k = d k= 0 X k + v k where v = + r is he discoun facor and r is he discoun rae. X k = cash flow a ime k (i.e. a beginning of year k) One may use he NPV funcion of Excel o do his calculaion. One mus decide on an appropriae discoun ineres rae. This discoun rae should ake ino accoun he cos of money for he life office. If he commission is financed hrough new equiy in he company, he cos of money is he reurn he shareholders wan on his new equiy (including ax). This migh be 5%. If he life office has idle funds which would oherwise be invesed, he discoun rae should ake ino accoun he income which would have been received in such an alernaive invesmen, where one should include he risk involved wih invesing funds ino iniial commissions. If he iniial commission invesmen is funded hrough reinsurance, he cos of his reinsurance could be used for he discoun rae. We will here assume a discoun rae of 0%, giving us a discoun facor v= Policy duraion 0 years Discoun rae 0% paymen 0 years NPV -3 Expeced increase in uni value 5% annually 00 per year Iniial commission 4 % of oal premium charge 4 % of each premium Year Fund in Charge Ineres Mauriy Fund ou Charge Comm Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow We can see ha he Accumulaed discouned cash flow is equal o 3 a he mauriy age. This is he NPV of he cash flow valued a he discoun rae of 0%. We 2004

12 Profi Tes Modelling in Life Assurance Using Spreadshees define his as our profi and our profi goal is ha he profi should be posiive (or a leas no negaive). The profi could also be calculaed direcly by using he NPV funcion in Excel. Please noe ha he Excel formula assumes ha all paymens are made in arrears i.e.a he end of he period in quesion, while we here assume ha all paymens (excep he mauriy) are made a he beginning of he period in quesion. The value calculaed by Excel mus herefore be muliplied by (+r), in our case 0% in order o arrive a he righ answer. Using he NPV formula in Excel gives he answer 2, which muliplied by 0% gives 3 as can be found in he lower righ hand corner of he able above. We see ha we mus use a premium charge greaer han 4% in order o break-even, i.e. a profi of zero. We can calculae he premium ha is required for a break-even siuaion by seing he NPV of fuure premium charges equal o he iniial commission. This gives: n = 0 γ P v = I = α d P (n=d). where n=0, d=0, i=5% and a=4%, We ge 9 γ P v = 0 9 γ = 0 v 0 v γ = 0.4 v γ 6.76 = 0.4 = α d P = α d = 0.4 or The premium charge ha will give break-even is γ = The same answer could have been found by once again using he Goal Seek or Solver Porfolios, model poins We have up o now looked a a 0-year policy. Le us look a a 5-year policy, assuming an iniial commission of 5.9% of he oal premium

13 Profi Tes Modelling in Life Assurance Using Spreadshees Policy duraion 5 years Discoun rae 0% paymen 5 years NPV 5 Expeced increase in uni value 5% annually 00 per year Iniial commission 4 % of oal premium charge 5.9 % of each premium Year Fund in Charge Ineres Mauriy Fund ou Charge Comm Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow The able above shows ha he 5-year policy gives a profi of 5. If we insead calculae he profi of a 5-year policy, we would make a loss of. For a 20-year policy, we make a loss of 25. The iniial commission formula ges more expensive for long erm policies. Le us herefore assume ha he agen ges commission for only he firs 20 premiums, even if he policy duraion is longer. This is a common way o consruc sales commission scales. The iniial commission is given by: I = min(20; d) P α Le us now assume ha he mauriy age is 65 years, x is he age of he assured, (i.e. d=65-x) and ha he minimum iniial age is 20. Le us also assume ha he disribuion of iniial age will be even over he age band years. We could hen calculae he profiabiliy of each iniial age and sum he resul over all ages as: NPV = P 64 ( x = v γ ) α min(20;65 x) One could in principle solve for? from he above expression by seing he oal o zero o ge he break-even siuaion x NPV = ( v γ ) α min(20;65 x) = 0 20 = A rearrangemen of he erms gives α 64 min(20;65 x) = x = and hen v γ

14 Profi Tes Modelling in Life Assurance Using Spreadshees γ α = 64 min(20;65 x) 65 x 20 = v This could however be a bi complicaed o handle. Anoher problem is ha, by using his complex formula, one can no differeniae he profiable from he non-profiable policies. One ges a much beer view of he siuaion by sudying he differen policies one by one. We sudy herefore he expression for calculaing he profi for a cohor of policies 65 x NPV = ( v γ P) α P min(20;65 x) for x = 20, 2,,64. = This is sraighforward bu could be cumbersome. One common way o simplify he calculaions is o use model poins. The profis of a 25-year and a 26-year policy are raher equal and he 25-year policy could represen boh a 24-year and a 26-year policy. We herefore choose a number of model policies ha will represen he res. Using his principle and leing each 5-year age bands be represened by is middle poin, we hus sudy 65 x NPV = P γ ( v ) P α min(20;65 x) for x = 22, 27,,62. = This gives: Mauriy age 65 years Discoun rae 0% Expeced increase in uni value 5% annually 00 per year Max commission years 20 Iniial commission 4 % of oal premium max 80% charge 6 % of each premium Age Policy Profi duraion Toal -03 This calculaion could be done by esing he policy duraions one a a ime. A quicker way is o use he Daa Table funcion in Excel which gives all values a he same ime. Please noe ha ables are dynamically updaed if his funcion is no urned off (Tools, Calculaion, Auomaic excep ables), why having large ables migh lead o heavy updae imes. We find:

15 Profi Tes Modelling in Life Assurance Using Spreadshees x NPV = % ( v ) 00 α min(20;65 x) = 03 (a) 22,27... = The resul is no good, bu i is hard o see how bad i is. We wan o know how much we need o increase he premium charge in order o go break-even. We wan o find a k such ha he profi is equal o zero, i.e.: x NPV = 00 (5.9% + k) ( v ) 00 α min(20;65 x) = 0 (b) 22,27... = Insering expression (a) in (b) gives x 00 (5.9% + k ) ( v ) 22,27... = This hen gives Furher k % ( 65 x 65 x ( ) 00 k v k v = 22,27... = x 64 NPV ( P) ( = v ) = = 03 v ) = 03 We herefore also include he ne presen value of he premiums paid for each policy in our able. This gives Age Policy duraion Profi NPV of premium Toal k = 03 = The loss is hus -.33% of he NPV of he oal premium. Le us herefore increase he premium charge wih.4% o 7.4%:

16 Profi Tes Modelling in Life Assurance Using Spreadshees Mauriy age 65 years Discoun rae 0% Expeced increase in uni value 5% annually 00 per year Max commission years 20 Iniial commission 4 % of oal premium max 80% charge 7.4 % of each premium Age Policy duraion Profi NPV of premium Toal We find ha he porfolio has a break-even poin wih a premium charge of 7.4% Le us now assume ha we expec o sell more of some policies and less of ohers. Mos of our new cliens are expeced o be around 35 years and few are 20 or 60 years. We include his in our calculaion by weighing he differen policies by heir expeced sales figures: x NPV = P Wx ( v γ ) α min(20;65 x) 20 = Assume ha our porfolio has an average duraion of 23 years and has an age disribuion as shown in he able below: Age Policy duraion Number of policies Toal

17 Profi Tes Modelling in Life Assurance Using Spreadshees This gives he following resuls: Policy duraion Number of policies Profi per policy NPV premium per policy Toal profi Toal NPV of premium Toal The figures in he column Profi per policy are rounded o he neares ineger. When calculaing he oal profi, non-rounded figures are used. We have here more of he non-profiable policies and less of he profiable policies. The NPV of he loss is only 0.6% of he NPV of he oal premium, why an increase of he premium charge of 0.2% should be enough o make he porfolio profiable. We increase he premium charge o 7.6%. Mauriy age 65 years Discoun rae 0% Expeced increase in uni value 5% annually 00 per year Max commission years 20 Iniial commission 4 % of oal premium max 80% charge 7.6 % of each premium Policy duraion Number of policies Profi per policy NPV premium per policy Toal profi Toal NPV of premium Toal As shown in he previous able, we have arrived a a small profi of 987. In he real world, you migh no know he acual age disribuion of he porfolio. I is herefore ofen a good idea o es differen reasonably realisic age disribuions in he porfolio and choose he leas favourable. In our case, we assume ha we will sell eiher he evenly disribued porfolio or he one wih he weigh on duraion 23 and we choose he laer one and hus he premium charge of 7.6%. We discussed in secion 2.5. he choice of discoun rae. The resul ha we arrive a is dependen on he discoun rae chosen. Problem: How would he profiabiliy be wih a discoun rae of 2%

18 Profi Tes Modelling in Life Assurance Using Spreadshees Answer: There will be a loss of 0.66% of NPV of oal premiums. A high discoun makes i more expensive o have high iniial coss Fixed coss Up o now, we have only included he commissions o he sales agens as expenses. These commissions are defined o be proporional o premium volume, why i does no maer if we have sold small or large policies. Le us now assume ha we have an iniial fixed expense of 0 for each new policy. The inroducion of his new expense leads o a need of increase in charges. One possibiliy could be o inroduce a policy charge of he same amoun as he policy expense. Anoher would be o increase he premium charge. We will choose he laer alernaive. We herefore now wan o deermine how much we need o increase he premium charge o offse his expense. Wih a fixed cos, large policies will be more profiable han small policies. We will invesigae he effec on a porfolio of policies wih differen premium. The example below shows he case for one policy wih a premium of 00. Policy duraion 0 years Discoun rae 0% Expeced increase in uni value 5% annually NPV of profi per year NPV of premium 68 Iniial commission 4 % of oal premium max 80% Inernal iniial expenses 0 per policy Max commission years 20 charge 7.6 % of each premium Year Fund in Charge Ineres Mauriy Fund ou Charge Comm Inernal expenses Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow

19 Profi Tes Modelling in Life Assurance Using Spreadshees If we do his calculaion for differen policy premiums and duraions, we ge: Policy duraion x years Discoun rae 0% Expeced increase in uni value 5% annually NPV of profi per year NPV of premium 68 Iniial commission 4 % of oal premium max 80% Inernal iniial expenses y per policy Max commission years 20 charge 7.6 % of each premium Profi Annual premium Duraion If all policy duraions and premiums were evenly disribued, we could jus sum up a oal and ge 56. Le us now however assume ha he policies are expeced o be disribued as follows: %.60%.00% 0.40% 0.20% 8.60% 3.20% 2.00% 0.80% 0.40% % 4.80% 3.00%.20% 0.60% % 6.40% 4.00%.60% 0.80% % 8.00% 5.00% 2.00%.00% % 6.40% 4.00%.60% 0.80% % 4.80% 3.00%.20% 0.60% 38.60% 3.20% 2.00% 0.80% 0.40% %.60%.00% 0.40% 0.20% As before, we muliply he resul for each ype of policy wih he probabiliy weigh of ha policy in order o arrive a he porfolio probabiliy. We hus muliply he profi marix wih he disribuion marix and arrive a he following resul. Profi Toal 9.53 For he premium, we correspondingly muliply he premium per policy wih he weigh:

20 Profi Tes Modelling in Life Assurance Using Spreadshees Toal 074 The profi in relaion o he premium is -9.53/074 = 0.9%. Le us ry wih a premium charge of 8.5%. We ge a profi very close o zero as expeced. We have hus found our break-even poin

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

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