PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

Transcription

1 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon

2 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo 1. Inroducion In par one of his brief we sudied he opic of Ne Presen Value. We also saw how one could sudy a porfolio consising of policies of differen policy duraions and premium amouns. In par wo we will sudy a single policy and a porfolio of idenical policies. We will also look a how surrenders and paid-ups affec he porfolio. In par one, we worked wih a premium proporional charge. Here we will here look a oher ypes of charges, including a fund proporional charge. We will give some formulae, bu he focus will be on he furher developmen of profi es models using spreadshees. 2. Our policy We sar again wih a uni linked policy ha pays accumulaed sum of premiums plus ineres as a mauriy benefi afer en years. We have an iniial commission of 4% imes oal premium calculaed a maximum duraion of weny years. The only charge he policyholder pays is a premium proporional charge of 6% of premium. The benefi paid afer mauriy afer d years is: C d d = P ( 1 γ ) (1 + i), C = 0 for?d = 1 The expenses paid by he office are I1 = α min(20; d) P, I = 0 for?1 where α = 4% and γ = 6% d= 10 The expression for he profi is d 1 NPV = γ P v I1 = 1

3 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo In spreadshee environmen we ge: Policy duraion 10 years Discoun rae 10% paymen 10 years NPV 1 Expeced increase in uni value 5% annually Iniial commission 4 % of oal premium max 20 years charge 6 % of each premium Fund in , Charge Ineres Mauriy ,245 Fund ou ,088 0 Charge Comm Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow and surrender charges Up o now, he idea has been ha he life office pays an iniial commission year one and receives income (wih ineres) hrough he premium charge. Our quesion now is: Wha will happen if some of he policies are disconinued before he signed period of en years? I is very common ha a policyholder signs up for a cerain period bu changes his mind afer a while and wans his money back before he mauriy dae. This is called a surrender. Le us assume ha he policyholder a a surrender receives his full savings amoun as surrender value a surrender. Le us also assume ha surrender occurs a he end of he year. The surrender value a year is given by SV or = V SV k = P ( 1 γ ) *(1 + i) k = 1 = V = V 1 + P ( 1 γ ) + ( V 1 + P P γ ) If he policy were surrendered afer 5 years, i would look like his: i

4 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV -15 Expeced increase in uni value 5% annually Surrender year 5 Iniial commission 4 % of oal premium max 20 years charge 6 % of each premium Fund in Charge Ineres Mauriy Surrender Fund ou Charge Comm Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow The NPV of he cash flow over he period is -15, which means ha we make a loss. The reason for his is ha he commission is paid for en years while he premium charge is earned during five years only. No all policies surrender a he same ime. We mus once again look a he whole porfolio and make saisical assumpions, in his case regarding he proporion of he policies ha will be surrendered. We could view his as having he policy change from he acive sae o he surrender sae and need he ransiional probabiliy beween hese saes a ime. We define A P S = P( surrender a year policy is acive a year end -1) The uncondiional probabiliy of surrender a year is expressed hen as P S = 1 k = 1 (1 P A S k A ) P S The probabiliy of he policy no being surrendered before mauriy is S A S P d = (1 Pk ). k= 1 Le us assume he following surrender assumpions: - A S PS = P 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% d 1

5 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo from which we find ha he probabiliy of a policy remaining acive for en years is 49.7%. We could calculae oal profiabiliy by firs calculaing he profi assuming ha surrender akes place a year-end : k NPV() = P ( γ v ) α min(20; d) k = 1 The overall expeced profi by summing over years 1 o d: d NPV = P S P = 1 k = 1 k ( γ v ) α min(20; d) When doing he corresponding calculaions in a spreadshee, he bes way would be o look a a block of policies wrien a he same ime and o follow he proporion of policies remaining a any ime. As a sandard assumpion, we assume ha we sar wih 1000 policies which are iniially idenical bu which are surrendered a differen imes. We will also from now on sudy he accumulaed discouned cash flow (NPV of profi or profi). Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 6 % of each premium 100% of sandard Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Number of policies Mauriies Fund in Charge Ineres Mauriy Surrender Fund ou Charge Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow We find as above ha he number of policies remaining in he acive sae a mauriy is 497, which means ha jus less han half of he porfolio remains afer en years. We

6 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo also find ha we are in a loss posiion wih he NPV of annual profis of By increasing he premium charge o 8.6%, we could come back o a profi posiion. Anoher way o couner he problem of surrenders is o inroduce a surrender charge. The policyholder will no ge he full fund when surrendering he policy, bu has o pay par of i as a surrender charge, SC. The surrender value a ime is given by SV SV = V k ( 1 SC ) = (1 SC ) P (1 γ ) *(1 + i) or k= 1 = V ( 1 SC ) = (1 SC ) ( V 1 + P P γ + ( V 1 + P (1 γ )) i) This charge could be fla or i could be larger he firs years. One possibiliy could be o have i 75% year 1, 50% year 2, 25% year 3, 10% year 4 and 5% hereafer. We would hen ge he following resul: Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 6 % of each premium 100% of sandard Surrender charge 100% of sandard Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Number of policies Mauriies Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow As discussed previously, we have assumed ha surrenders are aking place a he end of he year while commission and premium charges are earned in he beginning of he

7 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo year. Therefore, when we calculae he oal charges, he surrender charge has been muliplied by v in order o arrive a a more correcly discouned cash flow. We find ha we make a nice profi, wih a NPV of profi of A more deailed sudy will however show ha for policies wih longer duraions, we will make losses. Policies wih duraion 15 years or longer show losses. To ge he picure more clear, we once again look a a porfolio wih boh shor and long policies, hey way we did in par one of his brief: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal The loss is -0.9% of premium. If we increase he premium charge o 7%, we will be back in black (i.e. make a profi).

8 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 100% of sandard Surrender charge 100% of sandard Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Number of policies Mauriies Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal This resul is buil on our surrender assumpions being correc. This assumpion may however be made from raher weak facs. Le us herefore es wha would happen if surrenders were 150% of he original assumpion.

9 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 150% of sandard Surrender charge 100% of sandard Surrender 30% 15% 12% 9% 8% 8% 6% 6% 6% 5% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Number of policies Mauriies Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow We find ha he profi for he 10-year policy has increased from o Obviously, he surrender charges are such ha we earn more from hen han we lose in premium charges a surrender. The picure for he whole porfolio looks like his: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal Since we make profis on surrenders, we mus also check o see wha happens if surrenders are fewer han assumed. Le us assume surrenders being 50% of sandard:

10 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 50% of sandard Surrender charge 100% of sandard Surrender 10% 5% 4% 3% 3% 3% 2% 2% 2% 2% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Number of policies Mauriies Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal The figure of shows ha we make a minor loss. We are obviously sensiive o surrenders being oo few. When he surrender charge is small, he opposie will happen: We will be sensiive o surrenders being oo many.

11 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo 4. Paid up policies We have up o now used wo saes for he policies: Acive and Surrendered. We will now inroduce a hird sae: Paid-up. We have sudied surrenders and found ha inroducing surrender charges will help us cope wih he negaive effec of surrenders. A surrender occurs when a policyholder wishes o disconinue premium paymen ino his policy and insead wans o wihdraw his fund. This will look unfavourable if here is a high surrender charge. Anoher possible opporuniy for hose who wish o disconinue premium paymen is he paid up policy. The accumulaed fund remains wih he life office, i.e. P = 0, V = V 1 i, pu where pu is he insan he policy becomes paid-up. A paid-up policy could laer be surrendered. Le us assume ha each year 10% of he policies will be convered o paid-ups, i.e. A P P = 10%,1=<d We also here assume ha paid-up policies are surrendered in he same percenage as acive policies, i.e. P P = S P A S. The number of paid-up policies is P P P S A A P N = N 1 ( 1 P ) + N 1 P 1 Since paid-up policies are now allowed, we assume ha he overall surrender probabiliy is 70% of he sandard assumpions used previously.

12 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo In he spreadshee environmen, we do no keep rack of he number of acives and paid-ups separaely. The reason for his is he previously menioned assumpion ha paid-ups have he same surrender probabiliy as acive policies. Therefore, we could calculae surrendered amouns as a percenage of he fund wihou keeping rack of how much of he fund ha belongs o paid-up policies. Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Mauriies from acive New paid ups Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow The 10-year policy is sill profiable bu less han wihou he paid-ups. The NPV of profi is insead of Le us also look a he porfolio:

13 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We have an overall loss of more han 5% of premium and we need o increase he premium charge o 12.6% in order o make he whole porfolio profiable: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We also here see, as we have done previously, ha policies wih a long duraion show losses while shor policies show a profi. This effec is more pronounced here, since afer a few years, he number of acive policies is low and he life offices does no receive he expeced premium charges. Le us now, as an illusraion, assume ha surrender probabiliies are differen for paidup policies han for acive policies. This means ha we need o keep rack of hem and he corresponding funds separaely.

14 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surr from p-u 10% 5% 4% 3% 3% 3% 2% 2% 2% 2% Surrender charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Acive policies Paid-ups Mauriies from acive Mauriies from paidups from acive from p-u New paid ups Acive fund in Charge Ineres Mauriy Surrender S charge To paid-ups Fund ou Paid-up fund in Ineres Mauriy Surrender S charge From acive Paid-up fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow

15 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo 5. Fund proporional charge We have found ha he premium charges needed o make he longer policies profiable (or less unprofiable) migh become raher high, especially when we expec a high percenage of paid-up policies. We herefore inroduce a new ype of charge, a fund charge β. This charge is aken ou as a fixed percenage of he oal fund, also for paidup policies. In our case, we will charge i annually, in he beginning of he year jus before he annual premium is paid. In realiy, i is normally charged once a monh. The developmen of he fund for a single acive policy is given by: V = V (1 β ) + P (1 γ )) (1 + i ) or ( 1 V = ( V 1 + P V 1 β P γ ) + ( V 1 + P V 1 β P γ ) i ) C The income earned by he life office during year, calculaed as a beginning of year, is he charges, i.e. V 1 β + P γ + SC v, where he surrender charge is muliplied by v since i is earned a he end of he year If we inroduce a fund charge of 0.75%, we can keep he premium charge o 7% and sill ge porfolio profiabiliy. The resuls for he 10-year policy looks as follows.

16 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV cash flow Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 7 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Fund charge 0.75% of fund a beginning of he year Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Mauriies from acive New paid ups Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow (The row Charge includes boh premium and fund charges. ) For he porfolio we ge he following resuls: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal

17 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo We see here ha mos long policies sill creae losses bu look quie a bi beer han wihou he fund charge. If we choose a 5% premium charge and a 1% fund charge, we will arrive a a raher good balance. No individual policy duraion shows a very high posiive or negaive resul. Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal Inernal expenses We have up o now mainly sudied exernal expenses in he form of commission. We also have inernal expenses. I is a separae science o disribue hese in a correc way. Le us assume ha his is already done and ha we know he expenses in relaion o premium, funds ec. We also assume ha all expenses are proporional o he size of he policy. The problems encounered when we have expenses ha are fixed irrespecive of policy size were discussed in par one of he profi sudy brief. Le us herefore assume ha we have he following inernal expenses: proporional expenses c = % of premium. Fund proporional expenses c = % of funds as a beginning of he year. Paymen expenses c = % of amouns paid ou as benefis. The expenses for year are S E = c P + c V + I + v c P SV + C ), ( where P is zero for paid up policies and all erms are zero for surrendered policies. Expenses incurred a end of year are muliplied by v in order o discoun hem back o he beginning of he year.

18 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surrender charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Acive policies Mauriies from acive New paid ups Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow We have assumed ha premium and fund proporional expenses are incurred a he beginning of he year, and surrender and mauriy benefi proporional expenses a he end of he year. The surrender and mauriy expenses are herefore muliplied by v in order o ge a correc discouned cash flow.

19 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Over he porfolio, he resuls are: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We have a loss of on a oal NPV of premium of , i.e. 3% of premium. We need however o increase he premium charge wih 3.4%, i.e. o 8.4%, in order o resore profiabiliy. The reason for his is ha an increased premium charge leads o less income from he fund and benefi charges. Wih a premium charge of 8.5%, we ge he following resul:

20 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surrender charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Acive policies Mauriies from acive New paid ups Fund in Charge Ineres Mauriy Surrender S charge Fund ou Charges Comm Expenses Cash flow Accumulaed cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow

21 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We can also see ha we have a reasonable balance beween he resul of shor and long policies. 7. Moraliy Up o now, we have seen he life policy as a pure savings vehicle. We will now sar o include some effecs of moraliy by including a deah benefi. Le us firs assume ha he fund value is paid as a benefi a deah. S = V The sum a risk or risk sum is defined as he difference beween he deah benefi and he reserve a he momen of deah, i.e. in his case: R = S V = 0 which means ha his produc has zero moraliy risk. Inroducing moraliy also means ha we need o define a new sae; Dead. We need o define ransiion probabiliies from he acive sae o he dead sae A P or he paid-up sae o he dead sae A P D P D = P = q = 0.2% P D P. Le us assume a fixed moraliy of 0.2% per year, i.e. For pracical reasons, we assume in our calculaions ha he deahs occur jus before mauriies and surrenders. This means ha he surrender and paid up probabiliies are defined as relaing o he number of acive, less hose who die during he year. The number of ransiions from acives o surrenders A S N is given by D A N S = ( 1 q) N A P A S

22 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo The number of ransiions from acive o paid-ups is given by A P A A P N = ( 1 q) N P The number of acives is given by A A A P A S N N (1 q) (1 P P ) = 1 The number a ime is equal o he number a ime -1 less deahs, paid-ups and surrenders. The number of paid-ups is given by P P P S A A N N 1 q) (1 P ) + N (1 q P = 1 ( 1 ) where he firs erm is he number of persons remaining in he paid-up sae afer deah and surrender. The second erm is he number of ransiions from acives o paid-ups. The number of surrenders is given by S P P S A A N N 1 q) (1 P ) + N (1 q P = 1 ( 1 ) where he firs erm is he number of ransiions from paid-ups o surrender and he second erm is he number of ransiions from acives o surrender. P S

23 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Moraliy 0.2% Deah benefi 100% of fund Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Moraliy 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Dead from acive Mauriies from acive New paid ups Fund in Charge Ineres Deahs Mauriy Surrender S charge Fund ou Charges Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow We noe ha he resuls are very close o hose where we did no have any deah benefis. The porfolio looks like his:

24 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal Le us look a wha would happen if moraliy was double he expeced: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We find ha he resul is raher insensiive o changes in moraliy. This is because we ake no moraliy risk and ha deahs are few compared o surrenders and paid-ups. Le us now assume ha he moraliy is age dependen. We will use he following funcion: q x = ( x+ 1/ 2) We have here as a simplificaion chosen o have q(x) as a Makeham formula. Alernaively, we could have chosen o have he moraliy inensiy µ(x) as a Makeham and o approximae q(x) wih his formula: q x = µ x+ ½ ( µ x+ ½ ) For he 10-year policy (incepion age 55) we ge

25 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Moraliy 100% of sandard Age 55 Deah benefi 100% of fund Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Moraliy 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Dead from acive Mauriies from acive New paid ups Fund in Charge Ineres Deahs Mauriy Surrender S charge Fund ou Charges Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow

26 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo And he porfolio Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal We ge a resul a lile bi worse han he one we had previously ( compared o and -2952) The difference is ha he shorer policies are a lile less profiable while he longer policies look a lile bi beer. This is a resul of moraliy in he higher ages being more han he 0.2% assumed earlier while moraliy in younger ages is less. One should remember ha he 3-year policy has incepion age 62. Le us now as an example look a a block of policies sold o anoher group of cliens. The major policy principle is he same, bu i is used for inheriance planning, so he end age is 100 years insead of 65 years. The 10-year policy looks like his:

27 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Moraliy 100% of sandard Age 90 Deah benefi 100% of fund Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Moraliy 21.56% 23.75% 26.15% 28.80% 31.72% 34.93% 38.47% 42.37% 46.67% 51.40% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Dead from acive Mauriies from acive New paid ups Fund in Charge Ineres Deahs Mauriy Surrender S charge Fund ou Charges Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow The policy ges unprofiable. The effec of he high moraliy is he receip of insufficien premium and fund charges.

28 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo And he porfolio looks equally bad: Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal If we decrease he commission o 2.6% insead of 4% imes oal premium (max 20 years) we ge a beer picure. Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal Le us now assume ha we ake some moraliy risk. Le us assume a deah benefi of hree imes he fund value, valid for boh acive and paid-up policies. S = 3 V We also ge he sum a risk as R = S V = 3 V V = 2 V We charge he policyholder a moraliy charge being he expeced moraliy imes he sum a risk. MC = qx+ R = qx+ 2 V

29 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Moraliy 100% of sandard Age 55 Deah benefi 300% of fund Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Moraliy 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surrender 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% charge Acive policies Dead from acive Mauriies from acive New paid ups Fund in Charge Ineres Moraliy charge Deahs Risk sums paid Mauriy Surrender S charge Fund ou Expense charges Moraliy resul Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow

30 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo We have included hree new rows in he spreadshee. The firs new row is he moraliy charge which is he sum a risk muliplied by he moraliy q x. This is he risk premium ha he insured has o pay o he life office for he deah proecion included in he policy. The second new row is he risk sums paid, i.e. he par of he moraliy benefi ha is no paid from he fund bu is raher paid from he moraliy resul of he life office. In our case wih a deah benefi of 300%, i is calculaed as 2/3 of he deah benefi, since 1/3 is paid from he fund. The hird new row is he moraliy resul. This is calculaed as he moraliy charge, minus he risk sums paid. In our example, his resul is zero, since we assume ha moraliy follows he assumpion used for calculaion of he moraliy charge. For he porfolio we ge Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) Toal The resuls are worse han wihou he moraliy benefi. We previously made a profi of 556, while we here have a loss of This is because he moraliy charge makes he funds and herefore also he fund and surrender charges smaller. Le us now assume ha we have some margins in our moraliy assumpions, so real moraliy is 75% of he moraliy used for he moraliy charge calculaion. Moraliy assumpions should be made wih margins for expenses and profi.

31 Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi Expeced increase in uni value 5% annually NPV premium Iniial commission 4 % of oal premium max 20 years charge 8.5 % of each premium 70% of sandard Surrender charge 100% of sandard Paid ups 100% of sandard Technical moraliy 100% of sandard Age 55 Acual moraliy 75% of echnical Deah benefi 300% of fund Fund charge 1.00% of fund a beginning of he year proporional expenses 1.5% Fund proporional expenses 0.2% of fund a beginning of he year Benefi proporional expenses 0.5% Surrender 14.0% 7.0% 5.6% 4.2% 3.5% 3.5% 2.8% 2.8% 2.8% 2.1% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Technical 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% moraliy Acual moraliy 0.59% 0.65% 0.71% 0.78% 0.85% 0.93% 1.02% 1.12% 1.23% 1.35% Surrender charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Acive policies Dead from acive Mauriies from acive New paid ups Fund in Charge Ineres Moraliy charge Deahs Risk sums paid Mauriy Surrender S charge Fund ou Expense charges Moraliy resul Comm Expenses Cash flow Discoun facor Discouned cash flow Accumulaed discouned cash flow