PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO


 Derek Dixon
 1 years ago
 Views:
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 paidups 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 yearend : 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 10year 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: Paidup. 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 paidup. A paidup policy could laer be surrendered. Le us assume ha each year 10% of he policies will be convered o paidups, i.e. A P P = 10%,1=<d We also here assume ha paidup policies are surrendered in he same percenage as acive policies, i.e. P P = S P A S. The number of paidup policies is P P P S A A P N = N 1 ( 1 P ) + N 1 P 1 Since paidup 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 paidups separaely. The reason for his is he previously menioned assumpion ha paidups 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 paidup 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% Paidup 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 10year policy is sill profiable bu less han wihou he paidups. 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% Paidup 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surr from pu 10% 5% 4% 3% 3% 3% 2% 2% 2% 2% Surrender charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Acive policies Paidups Mauriies from acive Mauriies from paidups from acive from pu New paid ups Acive fund in Charge Ineres Mauriy Surrender S charge To paidups Fund ou Paidup fund in Ineres Mauriy Surrender S charge From acive Paidup 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 paidup 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 10year 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% Paidup 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% Paidup 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% Paidup 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 paidup 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 paidups 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, paidups and surrenders. The number of paidups 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 paidup sae afer deah and surrender. The second erm is he number of ransiions from acives o paidups. 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 paidups 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% Paidup 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 paidups. 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 10year 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% Paidup 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 3year 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 10year 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% Paidup 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 paidup 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% Paidup 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% Paidup 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
PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationDuration 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.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationCLASSIFICATION OF REINSURANCE IN LIFE INSURANCE
CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationWhat is a swap? A swap is a contract between two counterparties who agree to exchange a stream of payments over an agreed period of several years.
Currency swaps Wha is a swap? A swap is a conrac beween wo counerparies who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiyindexlinked
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationDiagnostic Examination
Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes  ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he kvalue for he middle erm, divide
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationChapter Four: Methodology
Chaper Four: Mehodology 1 Assessmen of isk Managemen Sraegy Comparing Is Cos of isks 1.1 Inroducion If we wan o choose a appropriae risk managemen sraegy, no only we should idenify he influence ha risks
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationCapital budgeting techniques
Capial budgeing echniques A reading prepared by Pamela Peerson Drake O U T L I N E 1. Inroducion 2. Evaluaion echniques 3. Comparing echniques 4. Capial budgeing in pracice 5. Summary 1. Inroducion The
More informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More informationEquities: Positions and Portfolio Returns
Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationACTUARIAL FUNCTIONS 1_05
ACTUARIAL FUNCTIONS _05 User Guide for MS Office 2007 or laer CONTENT Inroducion... 3 2 Insallaion procedure... 3 3 Demo Version and Acivaion... 5 4 Using formulas and synax... 7 5 Using he help... 6 Noaion...
More informationPricing Single Name Credit Derivatives
Pricing Single Name Credi Derivaives Vladimir Finkelsein 7h Annual CAP Workshop on Mahemaical Finance Columbia Universiy, New York December 1, 2 Ouline Realiies of he CDS marke Pricing Credi Defaul Swaps
More informationGoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:
For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationGMWB For Life An Analysis of Lifelong Withdrawal Guarantees
GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany daniela.holz@gmx.de Alexander Kling *) Insiu für Finanz und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationMSCI Index Calculation Methodology
Index Mehodology MSCI Index Calculaion Mehodology Index Calculaion Mehodology for he MSCI Equiy Indices Index Mehodology MSCI Index Calculaion Mehodology Conens Conens... 2 Inroducion... 5 MSCI Equiy Indices...
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationDescription of the CBOE S&P 500 BuyWrite Index (BXM SM )
Descripion of he CBOE S&P 500 BuyWrie Index (BXM SM ) Inroducion. The CBOE S&P 500 BuyWrie Index (BXM) is a benchmark index designed o rack he performance of a hypoheical buywrie sraegy on he S&P 500
More informationSignal Rectification
9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, halfwae and fullwae. Le s firs consider he ideal
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationImagine a Source (S) of sound waves that emits waves having frequency f and therefore
heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationA Further Examination of Insurance Pricing and Underwriting Cycles
A Furher Examinaion of Insurance ricing and Underwriing Cycles AFIR Conference, Sepember 2005, Zurich, Swizerland Chris K. Madsen, GE Insurance Soluions, Copenhagen, Denmark Svend Haasrup, GE Insurance
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationRotational Inertia of a Point Mass
Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha
More informationMOTION ALONG A STRAIGHT LINE
Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More information13. a. If the oneyear discount factor is.905, what is the oneyear interest rate?
CHAPTER 3: Pracice quesions 3. a. If he oneyear discoun facor is.905, wha is he oneyear ineres rae? = DF = + r 0.905 r = 0.050 = 0.50% b. If he woyear ineres rae is 0.5 percen, wha is he woyear discoun
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationThe Time Value of Money
THE TIME VALUE OF MONEY CALCULATING PRESENT AND FUTURE VALUES Fuure Value: FV = PV 0 ( + r) Presen Value: PV 0 = FV  ( + r) THE EFFECTS OF COMPOUNDING The effecs/benefis
More informationSKF Documented Solutions
SKF Documened Soluions Real world savings and we can prove i! How much can SKF save you? Le s do he numbers. The SKF Documened Soluions Program SKF is probably no he firs of your supplier parners o alk
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms)
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationAnalysis of tax effects on consolidated household/government debts of a nation in a monetary union under classical dichotomy
MPRA Munich Personal RePEc Archive Analysis of ax effecs on consolidaed household/governmen debs of a naion in a moneary union under classical dichoomy Minseong Kim 8 April 016 Online a hps://mpra.ub.unimuenchen.de/71016/
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationMortality Variance of the Present Value (PV) of Future Annuity Payments
Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationC FastDealing Property Trading Game C
If you are already an experienced MONOPOLY dealer and wan a faser game, ry he rules on he back page! AGES 8+ C FasDealing Propery Trading Game C Y Original MONOPOLY Game Rules plus Special Rules for his
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationMarket Liquidity and the Impacts of the Computerized Trading System: Evidence from the Stock Exchange of Thailand
36 Invesmen Managemen and Financial Innovaions, 4/4 Marke Liquidiy and he Impacs of he Compuerized Trading Sysem: Evidence from he Sock Exchange of Thailand Sorasar Sukcharoensin 1, Pariyada Srisopisawa,
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationSOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3.
SOLI MEHNIS TUTORIL GER SYSTEMS This work covers elemens of he syllabus for he Edexcel module 21722P HN/ Mechanical Principles OUTOME 3. On compleion of his shor uorial you should be able o do he following.
More informationTHE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS
THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationPlanning heating systems and building renovation with emissions targets. Peter Ahcin, entechma GmbH
Planning heaing sysems and building renovaion wih emissions arges Peer Ahcin, enechma GmbH Conens 1. Inroducion 2. The energy sysem porfolio framework 3. The benefi of he sochasic approach o planning heaing
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationDeveloping Equity Release Markets: Risk Analysis for Reverse Mortgage and Home Reversion
Developing Equiy Release Markes: Risk Analysis for Reverse Morgage and Home Reversion Daniel Alai, Hua Chen, Daniel Cho, Kaja Hanewald, Michael Sherris Developing he Equiy Release Markes 8 h Inernaional
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationTopic Overview. Learning Objectives. Capital Budgeting Steps: WHAT IS CAPITAL BUDGETING?
Chaper 10: THE BASICS OF CAPITAL BUDGETING Should we build his plan? Topic Overview Projec Types Capial Budgeing Decision Crieria Payback Period Discouned Payback Period Ne Presen Value () Inernal Rae
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More information