PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

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

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

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 0 99 202 311 425 545 671 804 942 1,088 100 100 100 100 100 100 100 100 100 100 Charge -6-6 -6-6 -6-6 -6-6 -6-6 Ineres 5 10 15 21 26 32 38 45 52 59 Mauriy 0 0 0 0 0 0 0 0 0-1,245 Fund ou -99-202 -311-425 -545-671 -804-942 -1,088 0 Charge 6 6 6 6 6 6 6 6 6 6 Comm -40 0 0 0 0 0 0 0 0 0 Cash flow -34 6 6 6 6 6 6 6 6 6 Accumulaed -34-28 -22-16 -10-4 2 8 14 20 cash flow Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -34 5 5 5 4 4 3 3 3 3 cash flow Accumulaed discouned cash flow -34-29 -24-19 -15-11 -8-5 -2 1 3. 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

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 0 99 203 312 427 0 0 0 0 0 100 100 100 100 100 0 0 0 0 0 Charge -6-6 -6-6 -6 0 0 0 0 0 Ineres 5 10 15 21 26 0 0 0 0 0 Mauriy 0 0 0 0 0 0 0 0 0 0 Surrender 0 0 0 0-547 0 0 0 0 0 Fund ou -99-203 -312-427 0 0 0 0 0 0 Charge 6 6 6 6 6 0 0 0 0 0 Comm -40 0 0 0 0 0 0 0 0 0 Cash flow -34 6 6 6 6 0 0 0 0 0 Accumulaed -34-28 -22-16 -10-10 -10-10 -10-10 cash flow Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -34 5 5 5 4 0 0 0 0 0 cash flow Accumulaed discouned cash flow -34-29 -24-19 -15-15 -15-15 -15-15 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

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 -12 046 Expeced increase in uni value 5% annually NPV premium 465 904 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 1000 800 720 662 623 592 562 539 518 497 policies Mauriies 0 0 0 0 0 0 0 0 0 497 200 80 58 40 31 30 22 22 21 0 Fund in 0 78 960 145 681 206 107 264 884 322 605 377 262 433 526 488 110 541 086 100 000 80 000 72 000 66 240 62 266 59 152 56 195 53 947 51 789 49 717 Charge -6 000-4 800-4 320-3 974-3 736-3 549-3 372-3 237-3 107-2 983 Ineres 4 700 7 708 10 668 13 419 16 171 18 910 21 504 24 212 26 840 29 391 Mauriy 0 0 0 0 0 0 0 0 0-617 212 Surrender -19 740-16 187-17 922-16 907-16 979-19 856-18 064-20 338-22 545 0 Fund ou -78 960-145 681-206 107-264 884-322 605-377 262-433 526-488 110-541 086 0 Charge 6 000 4 800 4 320 3 974 3 736 3 549 3 372 3 237 3 107 2 983 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -34 000 4 800 4 320 3 974 3 736 3 549 3 372 3 237 3 107 2 983 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -34 000 4 364 3 570 2 986 2 552 2 204 1 903 1 661 1 450 1 265 cash flow Accumulaed discouned cash flow -34 000-29 636-26 066-23 080-20 528-18 325-16 421-14 760-13 311-12 046 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

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 -12 046. 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 15 127 Expeced increase in uni value 5% annually NPV premium 465 904 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 1000 800 720 662 623 592 562 539 518 497 policies Mauriies 0 0 0 0 0 0 0 0 0 497 200 80 58 40 31 30 22 22 21 0 Fund in 0 78 960 145 681 206 107 264 884 322 605 377 262 433 526 488 110 541 086 100 000 80 000 72 000 66 240 62 266 59 152 56 195 53 947 51 789 49 717 Charge -6 000-4 800-4 320-3 974-3 736-3 549-3 372-3 237-3 107-2 983 Ineres 4 700 7 708 10 668 13 419 16 171 18 910 21 504 24 212 26 840 29 391 Mauriy 0 0 0 0 0 0 0 0 0-617 212 Surrender -4 935-8 093-13 442-15 217-16 130-18 863-17 160-19 321-21 418 0 S charge -14 805-8 093-4 481-1 691-849 -993-903 -1 017-1 127 0 Fund ou -78 960-145 681-206 107-264 884-322 605-377 262-433 526-488 110-541 086 0 Charges 19 459 12 158 8 393 5 511 4 508 4 452 4 193 4 161 4 132 2 983 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -20 541 12 158 8 393 5 511 4 508 4 452 4 193 4 161 4 132 2 983 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -20 541 11 052 6 937 4 141 3 079 2 764 2 367 2 135 1 928 1 265 cash flow Accumulaed discouned cash flow -20 541-9 489-2 552 1 589 4 668 7 432 9 799 11 934 13 862 15 127 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

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 15 127. 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) 3 100 22 082 2 208 23 223 8 200 19 460 3 892 84 132 13 300 7 157 2 147 154 605 18 400-8 194-3 278 226 284 23 500-13 249-6 625 296 280 28 400-11 362-4 545 242 751 33 300-10 139-3 042 184 353 38 200-9 341-1 868 123 716 43 100-8 815-881 62 075 Toal -12 401-11 991 1 397 420 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).

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 19 497 Expeced increase in uni value 5% annually NPV premium 465 904 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 1000 800 720 662 623 592 562 539 518 497 policies Mauriies 0 0 0 0 0 0 0 0 0 497 200 80 58 40 31 30 22 22 21 0 Fund in 0 78 120 144 131 203 914 262 066 319 173 373 249 428 914 482 918 535 330 100 000 80 000 72 000 66 240 62 266 59 152 56 195 53 947 51 789 49 717 Charge -7 000-5 600-5 040-4 637-4 359-4 141-3 934-3 776-3 625-3 480 Ineres 4 650 7 626 10 555 13 276 15 999 18 709 21 276 23 954 26 554 29 078 Mauriy 0 0 0 0 0 0 0 0 0-610 646 Surrender -4 883-8 007-13 299-15 055-15 959-18 662-16 978-19 115-21 190 0 S charge -14 648-8 007-4 433-1 673-840 -982-894 -1 006-1 115 0 Fund ou -78 120-144 131-203 914-262 066-319 173-373 249-428 914-482 918-535 330 0 Charges 20 316 12 879 9 070 6 157 5 122 5 034 4 746 4 691 4 639 3 480 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -19 684 12 879 9 070 6 157 5 122 5 034 4 746 4 691 4 639 3 480 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -19 684 11 709 7 496 4 626 3 499 3 125 2 679 2 407 2 164 1 476 cash flow Accumulaed discouned cash flow -19 684-7 976-480 4 146 7 645 10 770 13 449 15 857 18 021 19 497 Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) 3 100 24 190 2 419 23 223 8 200 23 387 4 677 84 132 13 300 12 010 3 603 154 605 18 400-2 855-1 142 226 284 23 500-7 655-3 828 296 280 28 400-5 636-2 254 242 751 33 300-4 345-1 304 184 353 38 200-3 512-702 123 716 43 100-2 968-297 62 075 Toal 1 173 1 397 420 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.

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 23 867 Expeced increase in uni value 5% annually NPV premium 385 903 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 1000 700 595 524 476 441 408 383 360 339 policies Mauriies 0 0 0 0 0 0 0 0 0 339 300 105 71 47 36 33 24 23 22 0 Fund in 0 68 355 119 109 161 186 200 541 237 814 270 787 304 689 335 904 364 603 100 000 70 000 59 500 52 360 47 648 44 074 40 768 38 322 36 023 33 862 Charge -7 000-4 900-4 165-3 665-3 335-3 085-2 854-2 683-2 522-2 370 Ineres 4 650 6 673 8 722 10 494 12 243 13 940 15 435 17 016 18 470 19 805 Mauriy 0 0 0 0 0 0 0 0 0-415 899 Surrender -7 324-10 510-16 485-17 850-18 318-20 858-18 476-20 369-22 109 0 S charge -21 971-10 510-5 495-1 983-964 -1 098-972 -1 072-1 164 0 Fund ou -68 355-119 109-161 186-200 541-237 814-270 787-304 689-335 904-364 603 0 Charges 26 974 14 454 9 160 5 468 4 212 4 083 3 738 3 657 3 579 2 370 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -13 026 14 454 9 160 5 468 4 212 4 083 3 738 3 657 3 579 2 370 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -13 026 13 140 7 571 4 108 2 877 2 535 2 110 1 877 1 670 1 005 cash flow Accumulaed discouned cash flow -13 026 114 7 685 11 793 14 670 17 205 19 315 21 192 22 861 23 867 We find ha he profi for he 10-year policy has increased from 19 497 o 23 867. 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) 3 100 31 556 3 156 21 281 8 200 28 692 5 738 70 947 13 300 15 214 4 564 125 578 18 400-1 067-427 179 983 23 500-6 829-3 414 232 714 28 400-5 470-2 188 189 223 33 300-4 640-1 392 143 047 38 200-4 130-826 95 736 43 100-3 815-381 47 959 Toal 4 829 1 106 468 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:

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 14 041 Expeced increase in uni value 5% annually NPV premium 561 634 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 1000 900 855 821 796 776 757 742 727 712 policies Mauriies 0 0 0 0 0 0 0 0 0 712 100 45 34 25 20 19 15 15 15 0 Fund in 0 87 885 171 156 252 676 335 098 418 859 502 715 589 723 677 806 767 024 100 000 90 000 85 500 82 080 79 618 77 627 75 686 74 173 72 689 71 236 Charge -7 000-6 300-5 985-5 746-5 573-5 434-5 298-5 192-5 088-4 986 Ineres 4 650 8 579 12 534 16 451 20 457 24 553 28 655 32 935 37 270 41 664 Mauriy 0 0 0 0 0 0 0 0 0-874 937 Surrender -2 441-4 504-7 896-9 327-10 203-12 246-11 433-13 141-14 871 0 S charge -7 324-4 504-2 632-1 036-537 -645-602 -692-783 0 Fund ou -87 885-171 156-252 676-335 098-418 859-502 715-589 723-677 806-767 024 0 Charges 13 658 10 395 8 378 6 688 6 061 6 020 5 845 5 821 5 800 4 986 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -26 342 10 395 8 378 6 688 6 061 6 020 5 845 5 821 5 800 4 986 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -26 342 9 450 6 924 5 025 4 140 3 738 3 299 2 987 2 706 2 115 cash flow Accumulaed discouned cash flow -26 342-16 892-9 969-4 944-804 2 934 6 233 9 220 11 926 14 041 Policy duraion Number of policies Profi per policy Toal profi (000) Toal NPV of premium (000) 3 100 16 054 1 605 25 248 8 200 16 898 3 380 99 503 13 300 7 914 2 374 190 381 18 400-5 324-2 129 285 372 23 500-9 046-4 523 379 406 28 400-6 301-2 520 313 976 33 300-4 513-1 354 239 995 38 200-3 331-666 161 729 43 100-2 538-254 81 363 Toal 9 814-4 087 1 776 972 The figure of -4 087 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.

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.

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 4 785 Expeced increase in uni value 5% annually NPV premium 373 279 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 1000 760 631 532 457 395 342 298 260 227 policies Mauriies 0 0 0 0 0 0 0 0 0 227 140 53 35 22 16 14 10 8 7 0 from acive New paid ups 100 76 63 53 46 40 34 30 26 23 Fund in 0 83 979 151 025 207 844 258 875 305 350 346 629 386 211 422 455 455 825 100 000 76 000 63 080 53 240 45 680 39 513 34 179 29 804 25 989 22 662 Charge -7 000-5 320-4 416-3 727-3 198-2 766-2 392-2 086-1 819-1 586 Ineres 4 650 7 733 10 484 12 868 15 068 17 105 18 921 20 696 22 331 23 845 Mauriy 0 0 0 0 0 0 0 0 0-500 746 Surrender -3 418-5 684-9 247-10 214-10 521-11 943-10 569-11 561-12 474 0 S charge -10 253-5 684-3 082-1 135-554 -629-556 -608-657 0 Fund ou -83 979-151 025-207 844-258 875-305 350-346 629-386 211-422 455-455 825 0 Charges 16 321 10 487 7 218 4 759 3 701 3 337 2 898 2 639 2 416 1 586 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -23 679 10 487 7 218 4 759 3 701 3 337 2 898 2 639 2 416 1 586 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -23 679 9 534 5 965 3 575 2 528 2 072 1 636 1 354 1 127 673 cash flow Accumulaed discouned cash flow -23 679-14 145-8 180-4 605-2 077-5 1 631 2 986 4 113 4 785 The 10-year policy is sill profiable bu less han wihou he paid-ups. The NPV of profi is 4 785 insead of 14 041. Le us also look a he porfolio:

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) 3 100 17 504 1 750 22 122 8 200 10 702 2 140 70 309 13 300-5 315-1 595 117 600 18 400-23 570-9 428 162 060 23 500-30 648-15 324 204 717 28 400-30 095-12 038 164 332 33 300-29 735-8 921 123 386 38 200-29 490-5 898 82 287 43 100-29 319-2 932 41 148 Toal -52 245 987 961 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) 3 100 29 048 2 905 22 122 8 200 29 299 5 860 70 309 13 300 15 478 4 643 117 600 18 400-2 091-836 162 060 23 500-8 966-4 483 204 717 28 400-8 362-3 345 164 332 33 300-7 996-2 399 123 386 38 200-7 757-1 551 82 287 43 100-7 593-759 41 148 Toal 34 987 961 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.

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 4 785 Expeced increase in uni value 5% annually NPV premium 373 279 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 1000 760 631 532 457 395 342 298 260 227 Paid-ups 0 100 171 227 274 313 344 371 394 412 Mauriies from acive 0 0 0 0 0 0 0 0 0 227 Mauriies from paidups 0 0 0 0 0 0 0 0 0 412 from acive 140 53 35 22 16 14 10 8 7 0 from p-u 0 5 7 7 7 8 7 7 8 0 New paid ups 100 76 63 53 46 40 34 30 26 0 Acive fund in 0 74 214 126 275 163 893 192 258 213 202 227 016 236 959 242 338 244 014 100 000 76 000 63 080 53 240 45 680 39 513 34 179 29 804 25 989 22 662 Charge -7 000-5 320-4 416-3 727-3 198-2 766-2 392-2 086-1 819-1 586 Ineres 4 650 7 245 9 247 10 670 11 737 12 497 12 940 13 234 13 325 13 255 Mauriy 0 0 0 0 0 0 0 0 0-278 345 Surrender -3 418-5 325-8 156-8 470-8 195-8 726-7 228-7 392-7 444 0 S charge -10 253-5 325-2 719-941 -431-459 -380-389 -392 0 To paid-ups -9 765-15 214-19 419-22 408-24 648-26 245-27 174-27 791-27 983 0 Fund ou -74 214-126 275-163 893-192 258-213 202-227 016-236 959-242 338-244 014 0 Paid-up fund in 0 9 765 24 954 44 573 67 805 94 063 122 542 153 270 185 505 218 868 Ineres 0 488 1 248 2 229 3 390 4 703 6 127 7 663 9 275 10 943 Mauriy 0 0 0 0 0 0 0 0 0-229 812 Surrender 0-256 -786-1 264-1 691-2 346-2 445-3 058-3 701 0 S charge 0-256 -262-140 -89-123 -129-161 -195 0 From acive 9 765 15 214 19 419 22 408 24 648 26 245 27 174 27 791 27 983 0 Paid-up fund ou -9 765-24 954-44 573-67 805-94 063-122 542-153 270-185 505-218 868 0 Charges 16 321 10 394 7 125 4 710 3 671 3 296 2 855 2 586 2 352 1 586 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -23 679 10 394 7 125 4 710 3 671 3 296 2 855 2 586 2 352 1 586 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -23 679 9 449 5 889 3 539 2 507 2 046 1 612 1 327 1 097 673 Accumulaed discouned cash flow -23 679-14 230-8 341-4 803-2 296-249 1 363 2 690 3 787 4 460

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.

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV cash flow 15 778 Expeced increase in uni value 5% annually NPV premium 373 279 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 1000 760 631 532 457 395 342 298 260 227 policies Mauriies 0 0 0 0 0 0 0 0 0 227 140 53 35 22 16 14 10 8 7 0 from acive New paid ups 100 76 63 53 46 40 34 30 26 0 Fund in 0 83 979 150 409 206 116 255 582 300 071 339 000 375 830 408 983 438 945 100 000 76 000 63 080 53 240 45 680 39 513 34 179 29 804 25 989 22 662 Charge -7 000-5 950-5 544-5 273-5 114-5 016-4 935-4 905-4 887-4 878 Ineres 4 650 7 701 10 397 12 704 14 807 16 728 18 412 20 036 21 504 22 836 Mauriy 0 0 0 0 0 0 0 0 0-479 566 Surrender -3 418-5 661-9 170-10 085-10 339-11 681-10 285-11 192-12 012 0 S charge -10 253-5 661-3 057-1 121-544 -615-541 -589-632 0 Fund ou -83 979-150 409-206 116-255 582-300 071-339 000-375 830-408 983-438 945 0 Charges 16 321 11 096 8 323 6 291 5 609 5 575 5 427 5 440 5 461 4 878 Comm -40 000 0 0 0 0 0 0 0 0 0 Cash flow -23 679 11 096 8 323 6 291 5 609 5 575 5 427 5 440 5 461 4 878 Discoun 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 facor Discouned -23 679 10 087 6 878 4 727 3 831 3 462 3 063 2 792 2 548 2 069 cash flow Accumulaed discouned cash flow -23 679-13 592-6 714-1 987 1 844 5 306 8 370 11 161 13 709 15 778 (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) 3 100 18 990 1 899 22 122 8 200 18 887 3 777 70 309 13 300 9 570 2 871 117 600 18 400-3 379-1 352 162 060 23 500-6 596-3 298 204 717 28 400-3 332-1 333 164 332 33 300-1 098-329 123 386 38 200 434 87 82 287 43 100 1 484 148 41 148 Toal 2 471 987 961

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) 3 100 15 403 1 540 22 122 8 200 15 151 3 030 70 309 13 300 7 346 2 204 117 600 18 400-4 100-1 640 162 060 23 500-6 173-3 086 204 717 28 400-2 113-845 164 332 33 300 651 195 123 386 38 200 2 530 506 82 287 43 100 3 804 380 41 148 Toal 2 284 987 961 6. 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 = 1.5 1 % of premium. Fund proporional expenses c = 0.2 2 % of funds as a beginning of he year. Paymen expenses c = 0.5 3 % of amouns paid ou as benefis. The expenses for year are S E = c P + c V + I + v c P SV + C ), ( 1 2 1 3 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.

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 2 809 Expeced increase in uni value 5% annually NPV premium 373 279 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 1000 760 631 532 457 395 342 298 260 227 Mauriies 0 0 0 0 0 0 0 0 0 227 from acive 140 53 35 22 16 14 10 8 7 0 New paid ups 100 76 63 53 46 40 34 30 26 0 Fund in 0 85 785 153 435 209 962 259 965 304 746 343 730 380 442 413 293 442 787 100 000 76 000 63 080 53 240 45 680 39 513 34 179 29 804 25 989 22 662 Charge -5 000-4 658-4 688-4 762-4 884-5 023-5 146-5 295-5 432-5 561 Ineres 4 750 7 856 10 591 12 922 15 038 16 962 18 638 20 248 21 692 22 994 Mauriy 0 0 0 0 0 0 0 0 0-482 882 Surrender -3 491-5 774-9 342-10 257-10 500-11 844-10 411-11 310-12 117 0 S charge -10 474-5 774-3 114-1 140-553 -623-548 -595-638 0 Fund ou -85 785-153 435-209 962-259 965-304 746-343 730-380 442-413 293-442 787 0 Charges 14 522 9 907 7 519 5 798 5 386 5 590 5 644 5 836 6 012 5 561 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 516-1 338-1 296-1 265-1 253-1 256-1 247-1 259-1 271-3 420 Cash flow -26 994 8 570 6 224 4 533 4 133 4 334 4 397 4 576 4 741 2 141 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -26 994 7 790 5 143 3 405 2 823 2 691 2 482 2 348 2 212 908 Accumulaed discouned cash flow -26 994-19 204-14 060-10 655-7 832-5 141-2 659-311 1 901 2 809 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.

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) 3 100 10 800 1 080 22 122 8 200 6 446 1 289 70 309 13 300-3 708-1 112 117 600 18 400-16 641-6 656 162 060 23 500-19 687-9 844 204 717 28 400-16 277-6 511 164 332 33 300-13 948-4 184 123 386 38 200-12 363-2 473 82 287 43 100-11 288-1 129 41 148 Toal -29 539 987 961 We have a loss of 29 539 on a oal NPV of premium of 987 961, 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:

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 14 778 Expeced increase in uni value 5% annually NPV premium 373 279 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 1000 760 631 532 457 395 342 298 260 227 Mauriies 0 0 0 0 0 0 0 0 0 227 from acive 140 53 35 22 16 14 10 8 7 0 New paid ups 100 76 63 53 46 40 34 30 26 0 Fund in 0 82 625 147 782 202 227 250 387 293 518 331 067 366 425 398 066 426 473 100 000 76 000 63 080 53 240 45 680 39 513 34 179 29 804 25 989 22 662 Charge -8 500-7 286-6 840-6 548-6 387-6 294-6 216-6 198-6 190-6 191 Ineres 4 575 7 567 10 201 12 446 14 484 16 337 17 951 19 502 20 893 22 147 Mauriy 0 0 0 0 0 0 0 0 0-465 092 Surrender -3 363-5 562-8 997-9 880-10 113-11 407-10 028-10 894-11 671 0 S charge -10 088-5 562-2 999-1 098-532 -600-528 -573-614 0 Fund ou -82 625-147 782-202 227-250 387-293 518-331 067-366 425-398 066-426 473 0 Charges 17 671 12 342 9 566 7 546 6 871 6 840 6 696 6 719 6 748 6 191 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 515-1 331-1 283-1 248-1 232-1 232-1 220-1 229-1 239-3 307 Cash flow -23 844 11 012 8 283 6 298 5 639 5 608 5 475 5 489 5 509 2 884 Accumulaed cash flow -23 844-12 833-4 549 1 748 7 387 12 995 18 470 23 960 29 469 32 353 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -23 844 10 011 6 846 4 731 3 851 3 482 3 091 2 817 2 570 1 223 Accumulaed discouned cash flow -23 844-13 834-6 988-2 256 1 595 5 077 8 168 10 984 13 554 14 778

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) 3 100 17 988 1 799 22 122 8 200 17 787 3 557 70 309 13 300 8 740 2 622 117 600 18 400-3 973-1 589 162 060 23 500-7 044-3 522 204 717 28 400-3 706-1 482 164 332 33 300-1 445-434 123 386 38 200 87 17 82 287 43 100 1 125 112 41 148 Toal 1 081 987 961 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

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

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 14 447 Expeced increase in uni value 5% annually NPV premium 371 392 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 1000 758 628 529 453 391 338 294 256 223 Dead from acive 2 2 1 1 1 1 1 1 1 0 Mauriies 0 0 0 0 0 0 0 0 0 222 from acive 140 53 35 22 16 14 9 8 7 0 New paid ups 100 76 63 53 45 39 34 29 26 0 Fund in 0 82 459 147 191 201 016 248 390 290 595 327 114 361 326 391 742 418 858 100 000 75 848 62 828 52 921 45 315 39 119 33 770 29 389 25 576 22 258 Charge -8 500-7 272-6 812-6 508-6 336-6 231-6 142-6 111-6 091-6 080 Ineres 4 575 7 552 10 160 12 371 14 368 16 174 17 737 19 230 20 561 21 752 Deahs -192-317 -427-520 -603-679 -745-808 -864-914 Mauriy 0 0 0 0 0 0 0 0 0-455 873 Surrender -3 356-5 539-8 944-9 801-10 013-11 271-9 888-10 720-11 463 0 S charge -10 068-5 539-2 981-1 089-527 -593-520 -564-603 0 Fund ou -82 459-147 191-201 016-248 390-290 595-327 114-361 326-391 742-418 858 0 Charges 17 652 12 308 9 522 7 498 6 815 6 770 6 615 6 624 6 640 6 080 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 516-1 329-1 279-1 243-1 225-1 222-1 209-1 216-1 223-3 248 Cash flow -23 864 10 978 8 243 6 256 5 590 5 548 5 406 5 408 5 417 2 833 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -23 864 9 980 6 812 4 700 3 818 3 445 3 051 2 775 2 527 1 201 Accumulaed discouned cash flow -23 864-13 883-7 071-2 371 1 447 4 892 7 943 10 719 13 246 14 447 We noe ha he resuls are very close o hose where we did no have any deah benefis. The porfolio looks like his:

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) 3 100 17 923 1 792 22 088 8 200 17 540 3 508 70 004 13 300 8 280 2 484 116 913 18 400-4 646-1 858 160 993 23 500-7 909-3 954 203 302 28 400-4 733-1 893 163 173 33 300-2 604-781 122 508 38 200-1 175-235 81 700 43 100-216 -22 40 854 Toal -960 981 536 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) 3 100 17 858 1 786 22 053 8 200 17 295 3 459 69 702 13 300 7 825 2 348 116 233 18 400-5 308-2 123 159 940 23 500-8 756-4 378 201 906 28 400-5 735-2 294 162 031 33 300-3 729-1 119 121 644 38 200-2 396-479 81 122 43 100-1 511-151 40 565 Toal -2 952 975 195 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 = 0.0006 + 0.000034 10 0.042 ( 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 ( 1+ 0.5 µ x+ ½ ) For he 10-year policy (incepion age 55) we ge

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 13 197 Expeced increase in uni value 5% annually NPV premium 373 184 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 1000 754 620 519 440 377 322 277 238 204 Dead from acive 8 7 6 5 5 5 4 4 4 4 Mauriies 0 0 0 0 0 0 0 0 0 200 from acive 139 52 34 22 15 13 9 8 7 0 New paid ups 99 75 61 51 44 37 32 27 23 0 Fund in 0 81 973 145 352 197 024 241 423 279 805 311 680 340 271 364 132 383 722 100 000 75 401 62 043 51 870 44 044 37 667 32 177 27 676 23 773 20 390 Charge -8 500-7 229-6 727-6 379-6 158-6 000-5 852-5 755-5 662-5 570 Ineres 4 575 7 507 10 033 12 126 13 965 15 574 16 900 18 110 19 112 19 927 Deahs -758-1 360-1 989-2 633-3 322-4 061-4 832-5 681-6 580-7 531 Mauriy 0 0 0 0 0 0 0 0 0-410 938 Surrender -3 336-5 470-8 766-9 526-9 641-10 739-9 312-9 965-10 501 0 S charge -10 008-5 470-2 922-1 058-507 -565-490 -524-553 0 Fund ou -81 973-145 352-197 024-241 423-279 805-311 680-340 271-364 132-383 722 0 Charges 17 598 12 202 9 383 7 341 6 619 6 514 6 297 6 232 6 164 5 570 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 519-1 326-1 270-1 227-1 202-1 192-1 170-1 167-1 163-2 975 Cash flow -23 920 10 876 8 113 6 114 5 417 5 322 5 127 5 065 5 002 2 595 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -23 920 9 887 6 705 4 594 3 700 3 304 2 894 2 599 2 333 1 101 Accumulaed discouned cash flow -23 920-14 033-7 328-2 734 965 4 270 7 164 9 763 12 097 13 197

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) 3 100 17 492 1 749 21 857 8 200 16 449 3 290 68 655 13 300 6 912 2 074 114 976 18 400-5 965-2 386 159 221 23 500-8 974-4 487 202 101 28 400-5 454-2 182 162 831 33 300-2 970-891 122 570 38 200-1 219-244 81 879 43 100 12 1 40 988 Toal -3 075 975 077 We ge a resul a lile bi worse han he one we had previously (-3 075 compared o - 960 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:

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi -11 090 Expeced increase in uni value 5% annually NPV premium 222 990 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 1000 596 377 235 144 85 48 26 13 6 Dead from acive 216 142 99 68 46 30 18 11 6 3 Mauriies 0 0 0 0 0 0 0 0 0 3 from acive 110 32 16 7 3 2 1 0 0 0 New paid ups 78 45 28 17 10 6 3 1 1 0 Fund in 0 64 809 88 392 89 327 78 748 63 033 46 261 31 503 19 722 11 268 100 000 59 613 37 730 23 517 14 366 8 485 4 776 2 562 1 288 599 Charge -8 500-5 715-4 091-2 892-2 009-1 352-869 -533-307 -164 Ineres 4 575 5 935 6 102 5 498 4 555 3 508 2 508 1 677 1 035 585 Deahs -20 716-29 597-33 507-33 249-30 341-25 736-20 266-14 919-10 145-6 317 Mauriy 0 0 0 0 0 0 0 0 0-5 972 Surrender -2 638-3 327-3 974-3 107-2 172-1 594-862 -540-308 0 S charge -7 913-3 327-1 325-345 -114-84 -45-28 -16 0 Fund ou -64 809-88 392-89 327-78 748-63 033-46 261-31 503-19 722-11 268 0 Charges 15 693 8 739 5 295 3 206 2 113 1 428 910 559 321 164 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 606-1 173-913 -697-521 -378-260 -172-106 -87 Cash flow -25 913 7 566 4 382 2 509 1 592 1 050 650 387 215 76 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -25 913 6 878 3 622 1 885 1 087 652 367 199 100 32 Accumulaed discouned cash flow -25 913-19 035-15 413-13 528-12 441-11 788-11 422-11 223-11 123-11 090 The policy ges unprofiable. The effec of he high moraliy is he receip of insufficien premium and fund charges.

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) 3 100 5 628 563 15 584 8 200-5 989-1 198 41 177 13 300-18 813-5 644 74 384 18 400-31 832-12 733 114 644 23 500-33 112-16 556 159 918 28 400-26 807-10 723 138 622 33 300-21 082-6 325 110 045 38 200-16 062-3 212 76 289 43 100-11 806-1 181 39 159 Toal -57 008 769 821 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) 3 100 9 828 983 15 584 8 200 5 211 1 042 41 177 13 300-613 -184 74 384 18 400-6 632-2 653 114 644 23 500-5 112-2 556 159 918 28 400 1 193 477 138 622 33 300 6 918 2 075 110 045 38 200 11 938 2 388 76 289 43 100 16 194 1 619 39 159 Toal 3 192 769 821 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

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 11 854 Expeced increase in uni value 5% annually NPV premium 364 366 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 1000 768 637 536 458 393 337 291 251 216 policies Dead from 8 7 6 6 5 5 5 4 4 4 acive Mauriies 0 0 0 0 0 0 0 0 0 212 139 53 35 22 16 14 9 8 7 0 from acive New paid ups 85 71 60 51 44 37 32 28 24 0 Fund in 0 80 700 141 688 189 877 229 632 262 171 287 082 307 385 321 787 330 793 100 000 75 401 62 043 51 870 44 044 37 667 32 177 27 676 23 773 20 390 Charge -8 500-7 216-6 691-6 308-6 040-5 823-5 606-5 426-5 239-5 041 Ineres 4 575 7 444 9 852 11 772 13 382 14 701 15 683 16 482 17 016 17 307 Moraliy charge -1 492-2 651-3 834-5 008-6 225-7 480-8 731-10 040-11 344-12 628 Deahs -2 238-3 977-5 751-7 512-9 338-11 221-13 096-15 061-17 016-18 941 Risk sums paid 1 492 2 651 3 834 5 008 6 225 7 480 8 731 10 040 11 344 12 628 Mauriy 0 0 0 0 0 0 0 0 0-344 508 Surrender -3 284-5 332-8 448-9 061-9 033-9 892-8 412-8 806-9 053 0 S charge -9 853-5 332-2 816-1 007-475 -521-443 -463-476 0 Fund ou -80 700-141 688-189 877-229 632-262 171-287 082-307 385-321 787-330 793 0 Expense charges 17 457 12 064 9 250 7 223 6 472 6 297 6 008 5 848 5 672 5 041 Moraliy resul 0 0 0 0 0 0 0 0 0 0 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 525-1 335-1 279-1 233-1 203-1 185-1 155-1 138-1 119-2 619 Cash flow -24 068 10 729 7 972 5 990 5 269 5 111 4 854 4 709 4 553 2 422 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -24 068 9 754 6 588 4 500 3 599 3 174 2 740 2 417 2 124 1 027 Accumulaed discouned cash flow -24 068-14 314-7 726-3 226 373 3 547 6 287 8 703 10 827 11 854

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) 3 100 16 955 1 695 21 857 8 200 15 348 3 070 68 655 13 300 5 241 1 572 114 986 18 400-8 025-3 210 159 286 23 500-11 171-5 585 202 230 28 400-7 588-3 035 162 954 33 300-4 922-1 477 122 670 38 200-2 942-588 81 948 43 100-1 481-148 41 023 Toal -7 706 975 609 The resuls are worse han wihou he moraliy benefi. We previously made a profi of 556, while we here have a loss of 7 706. 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.

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo Policy duraion 10 years Discoun rae 10% paymen 10 years NPV profi 21 595 Expeced increase in uni value 5% annually NPV premium 366 557 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 1000 756 623 522 444 381 327 282 243 209 Dead from acive 6 5 4 4 4 4 3 3 3 3 Mauriies 0 0 0 0 0 0 0 0 0 207 from acive 139 53 35 22 15 13 9 8 7 0 New paid ups 99 75 62 52 44 38 32 28 24 0 Fund in 0 80 861 142 278 191 122 231 742 265 338 291 463 313 153 329 067 339 686 100 000 75 551 62 301 52 210 44 449 38 122 32 668 28 196 24 311 20 939 Charge -8 500-7 230-6 718-6 349-6 096-5 894-5 691-5 528-5 357-5 177 Ineres 4 575 7 459 9 893 11 849 13 505 14 878 15 922 16 791 17 401 17 772 Moraliy charge -1 492-2 656-3 850-5 041-6 282-7 571-8 864-10 229-11 601-12 967 Deahs -1 678-2 988-4 331-5 671-7 068-8 517-9 972-11 507-13 051-14 588 Risk sums paid 1 119 1 992 2 887 3 781 4 712 5 678 6 648 7 672 8 701 9 725 Mauriy 0 0 0 0 0 0 0 0 0-355 391 Surrender -3 291-5 355-8 503-9 144-9 142-10 043-8 570-9 005-9 296 0 S charge -9 873-5 355-2 834-1 016-481 -529-451 -474-489 0 Fund ou -80 861-142 278-191 122-231 742-265 338-291 463-313 153-329 067-339 686 0 Expense charges 17 475 12 098 9 295 7 273 6 533 6 374 6 101 5 959 5 802 5 177 Moraliy resul 339 604 875 1 146 1 428 1 721 2 015 2 325 2 637 2 947 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 523-1 333-1 277-1 233-1 204-1 187-1 157-1 142-1 124-2 675 Cash flow -23 709 11 369 8 893 7 186 6 757 6 908 6 959 7 141 7 314 5 449 Discoun facor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discouned cash flow -23 709 10 335 7 349 5 399 4 615 4 289 3 928 3 665 3 412 2 311 Accumulaed discouned cash flow -23 709-13 373-6 024-625 3 990 8 279 12 207 15 872 19 284 21 595