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Profit Test Modelling in Life Assurance Using Spreadsheets, part two

2004

PROFIT TEST MODELLING IN LIFE ASSURANCE USING

SPREADSHEETS

PART TWO

Erik Alm Peter Millington


2004

1. Introduction In part one of this brief we studied the topic of Net Present Value. We also saw how one could study a portfolio consisting of policies of different policy durations and premium amounts. In part two we will study a single policy and a portfolio of identical policies. We will also look at how surrenders and paid-ups affect the portfolio. In part one, we worked with a premium proportional charge. Here we will here look at other types of charges, including a fund proportional charge. We will give some formulae, but the focus will be on the further development of profit test models using spreadsheets. 2. Our policy We start again with a unit linked policy that pays accumulated sum of premiums plus interest as a maturity benefit after ten years. We have an initial commission of 4% times total premium calculated at maximum duration of twenty years. The only charge the policyholder pays is a premium proportional charge of 6% of premium. The benefit paid after maturity after d years is:

∑=

+∗−∗=d

t

td iPC

1

)1()1( γ , 0=tC for t?d

The expenses paid by the office are

PdI ∗∗= );20min(1 α , 0=tI for t?1 where

%4=α and %6=γ

d= 10 The expression for the profit is

NPV = 11

1 IvPd

t

t −∗∗∑=

−γ


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In spreadsheet environment we get: Policy duration 10 years Discount rate 10% Premium payment 10 years NPV 1 Expected increase in unit value 5% annually Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6 % of each premium Year 1 2 3 4 5 6 7 8 9 10 Fund in 0 99 202 311 425 545 671 804 942 1,088 Premium 100 100 100 100 100 100 100 100 100 100 Charge -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 Interest 5 10 15 21 26 32 38 45 52 59 Maturity 0 0 0 0 0 0 0 0 0 -1,245 Fund out -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 Accumulated cash flow

-34 -28 -22 -16 -10 -4 2 8 14 20

Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424

Discounted cash flow

-34 5 5 5 4 4 3 3 3 3

Accumulated discounted cash flow

-34 -29 -24 -19 -15 -11 -8 -5 -2 1

3. Surrenders and surrender charges Up to now, the idea has been that the life office pays an initial commission year one and receives income (with interest) through the premium charge. Our question now is: What will happen if some of the policies are discontinued before the signed period of ten years? It is very common that a policyholder signs up for a certain period but changes his mind after a while and wants his money back before the maturity date. This is called a surrender. Let us assume that the policyholder at a surrender receives his full savings amount as surrender value at surrender. Let us also assume that surrender occurs at the end of the year. The surrender value at year t is given by

∑=

+−∗==t

k

ktt iPVSV

1

)1(*)1( γ

or iPPVPVVSV tttt ∗∗−++−∗+== −− )()1( 11 γγ If the policy were surrendered after 5 years, it would look like this:


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV -15 Expected increase in unit value 5% annually Surrender year 5 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6 % of each premium Year 1 2 3 4 5 6 7 8 9 10 Fund in 0 99 203 312 427 0 0 0 0 0 Premium 100 100 100 100 100 0 0 0 0 0 Charge -6 -6 -6 -6 -6 0 0 0 0 0 Interest 5 10 15 21 26 0 0 0 0 0 Maturity 0 0 0 0 0 0 0 0 0 0 Surrender 0 0 0 0 -547 0 0 0 0 0 Fund out -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 Accumulated cash flow

-34 -28 -22 -16 -10 -10 -10 -10 -10 -10

Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-34 5 5 5 4 0 0 0 0 0


-34 -29 -24 -19 -15 -15 -15 -15 -15 -15

The NPV of the cash flow over the period is -15, which means that we make a loss. The reason for this is that the commission is paid for ten years while the premium charge is earned during five years only. Not all policies surrender at the same time. We must once again look at the whole portfolio and make statistical assumptions, in this case regarding the proportion of the policies that will be surrendered. We could view this as having the policy change from the active state to the surrender state and need the transitional probability between these states at time t. We define

)1- tendyear at active ispolicy t a ( yeartsurrenderPP St

A =

The unconditional probability of surrender at year t is expressed then as

∏−

=

∗−=1

1

)1(t

k

St

ASk

ASt PPP

The probability of the policy not being surrendered before maturity is ∏−

=

−=1

1

)1(d

k

Sk

ASd PP .

Let us assume the following surrender assumptions:

Year 1 2 3 4 5 6 7 8 9 10- S

tA

t PPS = 20% 10% 8% 6% 5% 5% 4% 4% 4% 3%


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from which we find that the probability of a policy remaining active for ten years is 49.7%. We could calculate total profitability by first calculating the profit assuming that surrender takes place at year-end t:

NPV(t) =

∗−∗∗ ∑

=

);20min()(1

dvPt

k

k αγ

The overall expected profit by summing over years 1 to d:

NPV = ∑ ∑= =

∗−∗∗∗

d

t

t

k

kSt dvPP

1 1

);20min()( αγ

When doing the corresponding calculations in a spreadsheet, the best way would be to look at a block of policies written at the same time and to follow the proportion of policies remaining at any time. As a standard assumption, we assume that we start with 1000 policies which are initially identical but which are surrendered at different times. We will also from now on study the accumulated discounted cash flow (NPV of profit or profit). Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit -12 046 Expected increase in unit value 5% annually NPV premium 465 904 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6 % of each premium Surrenders 100% of standard Year 1 2 3 4 5 6 7 8 9 10 Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Number of policies

1000 800 720 662 623 592 562 539 518 497

Maturities 0 0 0 0 0 0 0 0 0 497 Surrenders 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 Premium 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 Interest 4 700 7 708 10 668 13 419 16 171 18 910 21 504 24 212 26 840 29 391 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-34 000 4 364 3 570 2 986 2 552 2 204 1 903 1 661 1 450 1 265


-34 000 -29 636 -26 066 -23 080 -20 528 -18 325 -16 421 -14 760 -13 311 -12 046

We find as above that the number of policies remaining in the active state at maturity is 497, which means that just less than half of the portfolio remains after ten years. We


2004

also find that we are in a loss position with the NPV of annual profits of -12 046. By increasing the premium charge to 8.6%, we could come back to a profit position. Another way to counter the problem of surrenders is to introduce a surrender charge. The policyholder will not get the full fund when surrendering the policy, but has to pay part of it as a surrender charge, SCt. The surrender value at time t is given by

∑=

+−∗∗−=−∗=t

k

ktttt iPSCSCVSV

1

)1(*)1()1()1( γ or

)))1((()1()1( 11 iPVPPVSCSCVSV tttttt ∗−∗++∗−+∗−=−∗= −− γγ This charge could be flat or it could be larger the first years. One possibility could be to have it 75% year 1, 50% year 2, 25% year 3, 10% year 4 and 5% thereafter. We would then get the following result: Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 15 127 Expected increase in unit value 5% annually NPV premium 465 904 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6 % of each premium Surrenders 100% of standard Surrender charge 100% of standard Year 1 2 3 4 5 6 7 8 9 10 Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Number of policies

1000 800 720 662 623 592 562 539 518 497

Maturities 0 0 0 0 0 0 0 0 0 497 Surrenders 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 Premium 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 Interest 4 700 7 708 10 668 13 419 16 171 18 910 21 504 24 212 26 840 29 391 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-20 541 11 052 6 937 4 141 3 079 2 764 2 367 2 135 1 928 1 265


-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 that surrenders are taking place at the end of the year while commission and premium charges are earned in the beginning of the


2004

year. Therefore, when we calculate the total charges, the surrender charge has been multiplied by v in order to arrive at a more correctly discounted cash flow. We find that we make a nice profit, with a NPV of profit of 15 127. A more detailed study will however show that for policies with longer durations, we will make losses. Policies with duration 15 years or longer show losses. To get the picture more clear, we once again look at a portfolio with both short and long policies, they way we did in part one of this brief:

Policy duration

Number of policies Profit per policy

Total profit (000)

Total 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

Total -12 401 -11 991 1 397 420 The loss is -0.9% of premium. If we increase the premium charge to 7%, we will be back in black (i.e. make a profit).


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 19 497 Expected increase in unit value 5% annually NPV premium 465 904 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 7 % of each premium Surrenders 100% of standard Surrender charge 100% of standard Year 1 2 3 4 5 6 7 8 9 10 Surrender 20% 10% 8% 6% 5% 5% 4% 4% 4% 3% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Number of policies

1000 800 720 662 623 592 562 539 518 497

Maturities 0 0 0 0 0 0 0 0 0 497 Surrenders 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 Premium 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 Interest 4 650 7 626 10 555 13 276 15 999 18 709 21 276 23 954 26 554 29 078 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-19 684 11 709 7 496 4 626 3 499 3 125 2 679 2 407 2 164 1 476


-19 684 -7 976 -480 4 146 7 645 10 770 13 449 15 857 18 021 19 497

Policy duration


Total profit (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

Total 1 173 1 397 420 This result is built on our surrender assumptions being correct. This assumption may however be made from rather weak facts. Let us therefore test what would happen if surrenders were 150% of the original assumption.


2004


75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Number of policies

1000 700 595 524 476 441 408 383 360 339

Maturities 0 0 0 0 0 0 0 0 0 339 Surrenders 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 Premium 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 Interest 4 650 6 673 8 722 10 494 12 243 13 940 15 435 17 016 18 470 19 805 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-13 026 13 140 7 571 4 108 2 877 2 535 2 110 1 877 1 670 1 005


-13 026 114 7 685 11 793 14 670 17 205 19 315 21 192 22 861 23 867

We find that the profit for the 10-year policy has increased from 19 497 to 23 867. Obviously, the surrender charges are such that we earn more from then than we lose in premium charges at surrender. The picture for the whole portfolio looks like this:

Policy duration


Total profit (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

Total 4 829 1 106 468 Since we make profits on surrenders, we must also check to see what happens if surrenders are fewer than assumed. Let us assume surrenders being 50% of standard:


2004


75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Number of policies

1000 900 855 821 796 776 757 742 727 712

Maturities 0 0 0 0 0 0 0 0 0 712 Surrenders 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 Premium 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 Interest 4 650 8 579 12 534 16 451 20 457 24 553 28 655 32 935 37 270 41 664 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-26 342 9 450 6 924 5 025 4 140 3 738 3 299 2 987 2 706 2 115


-26 342 -16 892 -9 969 -4 944 -804 2 934 6 233 9 220 11 926 14 041

Policy

duration Number of policies Profit per

policy Total profit

(000) Total 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

Total 9 814 -4 087 1 776 972 The figure of -4 087 shows that we make a minor loss. We are obviously sensitive to surrenders being too few. When the surrender charge is small, the opposite will happen: We will be sensitive to surrenders being too many.


2004

4. Paid up policies We have up to now used two states for the policies: Active and Surrendered. We will now introduce a third state: Paid-up. We have studied surrenders and found that introducing surrender charges will help us cope with the negative effect of surrenders. A surrender occurs when a policyholder wishes to discontinue premium payment into his policy and instead wants to withdraw his fund. This will look unfavourable if there is a high surrender charge. Another possible opportunity for those who wish to discontinue premium payment is the paid up policy. The accumulated fund remains with the life office, i.e.

putiVVP ttt ≥∗== − ,,0 1 where pu is the instant the policy becomes paid-up. A paid-up policy could later be surrendered. Let us assume that each year 10% of the policies will be converted to paid-ups, i.e.

%10=Pt

AP ,1=t<d We also here assume that paid-up policies are surrendered in the same percentage as active policies, i.e.

St

ASt

P PP = . The number of paid-up policies is

Pt

AAt

St

PPt

Pt PNPNN 111 )1( −−− ∗+−∗=

Since paid-up policies are now allowed, we assume that the overall surrender probability is 70% of the standard assumptions used previously.


2004

In the spreadsheet environment, we do not keep track of the number of actives and paid-ups separately. The reason for this is the previously mentioned assumption that paid-ups have the same surrender probability as active policies. Therefore, we could calculate surrendered amounts as a percentage of the fund without keeping track of how much of the fund that belongs to paid-up policies. Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 4 785 Expected increase in unit value 5% annually NPV premium 373 279 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 7 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard

Year 1 2 3 4 5 6 7 8 9 10 Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2%

Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Surrender

charge75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies

1000 760 631 532 457 395 342 298 260 227

Maturities 0 0 0 0 0 0 0 0 0 227 Surrenders from active

140 53 35 22 16 14 10 8 7 0

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 Premium 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 Interest 4 650 7 733 10 484 12 868 15 068 17 105 18 921 20 696 22 331 23 845 Maturity 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 out -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 Discount

factor1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-23 679 9 534 5 965 3 575 2 528 2 072 1 636 1 354 1 127 673

Accumulated discounted

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 still profitable but less than without the paid-ups. The NPV of profit is 4 785 instead of 14 041. Let us also look at the portfolio:


2004

Policy duration


Total profit (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

Total -52 245 987 961 We have an overall loss of more than 5% of premium and we need to increase the premium charge to 12.6% in order to make the whole portfolio profitable:

Policy duration


Total profit (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

Total 34 987 961 We also here see, as we have done previously, that policies with a long duration show losses while short policies show a profit. This effect is more pronounced here, since after a few years, the number of active policies is low and the life offices does not receive the expected premium charges. Let us now, as an illustration, assume that surrender probabilities are different for paid-up policies than for active policies. This means that we need to keep track of them and the corresponding funds separately.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 4 785 Expected increase in unit value 5% annually NPV premium 373 279 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 7 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Year 1 2 3 4 5 6 7 8 9 10 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% Active policies 1000 760 631 532 457 395 342 298 260 227 Paid-ups 0 100 171 227 274 313 344 371 394 412 Maturities from active 0 0 0 0 0 0 0 0 0 227 Maturities from paid-ups 0 0 0 0 0 0 0 0 0 412 Surrenders from active 140 53 35 22 16 14 10 8 7 0 Surrenders 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 Active fund in 0 74 214 126 275 163 893 192 258 213 202 227 016 236 959 242 338 244 014 Premium 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 Interest 4 650 7 245 9 247 10 670 11 737 12 497 12 940 13 234 13 325 13 255 Maturity 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 out -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 Interest 0 488 1 248 2 229 3 390 4 703 6 127 7 663 9 275 10 943 Maturity 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 active 9 765 15 214 19 419 22 408 24 648 26 245 27 174 27 791 27 983 0 Paid-up fund out -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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 679 9 449 5 889 3 539 2 507 2 046 1 612 1 327 1 097 673 Accumulated discounted cash flow -23 679 -14 230 -8 341 -4 803 -2 296 -249 1 363 2 690 3 787 4 460


2004

5. Fund proportional charge We have found that the premium charges needed to make the longer policies profitable (or less unprofitable) might become rather high, especially when we expect a high percentage of paid-up policies. We therefore introduce a new type of charge, a fund charge β . This charge is taken out as a fixed percentage of the total fund, also for paid-up policies. In our case, we will charge it annually, in the beginning of the year just before the annual premium is paid. In reality, it is normally charged once a month. The development of the fund for a single active policy is given by:

)1())1()1(( 1 tttt iPVV +∗−∗+−∗= − γβ or

ttttttttttt CiPVPVPVPVV −∗∗−∗−++∗−∗−+= −−−− ))()( 1111 γβγβ The income earned by the life office during year t, calculated as at beginning of year t, is the charges, i.e.

vSCPV ttt ∗+∗+∗− γβ1 , where the surrender charge is multiplied by v since it is earned at the end of the year If we introduce a fund charge of 0.75%, we can keep the premium charge to 7% and still get portfolio profitability. The results for the 10-year policy looks as follows.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV cash flow 15 778 Expected increase in unit value 5% annually NPV premium 373 279 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 7 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Fund charge 0.75% of fund at beginning of the year Year 1 2 3 4 5 6 7 8 9 10 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%

Active policies

1000 760 631 532 457 395 342 298 260 227


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 83 979 150 409 206 116 255 582 300 071 339 000 375 830 408 983 438 945 Premium 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 Interest 4 650 7 701 10 397 12 704 14 807 16 728 18 412 20 036 21 504 22 836 Maturity 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 out -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 Discount factor

1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424


-23 679 10 087 6 878 4 727 3 831 3 462 3 063 2 792 2 548 2 069


-23 679 -13 592 -6 714 -1 987 1 844 5 306 8 370 11 161 13 709 15 778

(The row “Charge” includes both premium and fund charges. ) For the portfolio we get the following results:

Policy duration


Total profit (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

Total 2 471 987 961


2004

We see here that most long policies still create losses but look quite a bit better than without the fund charge. If we choose a 5% premium charge and a 1% fund charge, we will arrive at a rather good balance. No individual policy duration shows a very high positive or negative result.

Policy duration


Total profit (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

Total 2 284 987 961 6. Internal expenses We have up to now mainly studied external expenses in the form of commission. We also have internal expenses. It is a separate science to distribute these in a correct way. Let us assume that this is already done and that we know the expenses in relation to premium, funds etc. We also assume that all expenses are proportional to the size of the policy. The problems encountered when we have expenses that are fixed irrespective of policy size were discussed in part one of the profit study brief. Let us therefore assume that we have the following internal expenses: Premium proportional expenses %5.11 =c of premium.

Fund proportional expenses %2.02 =c of funds as at beginning of the year.

Payment expenses %5.03 =c of amounts paid out as benefits. The expenses for year t are

)(3121 ttS

ttttt CSVPcvIVcPcE +∗∗++∗+∗= − , where tP is zero for paid up policies and all terms are zero for surrendered policies. Expenses incurred at end of year are multiplied by v in order to discount them back to the beginning of the year.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 2 809 Expected increase in unit value 5% annually NPV premium 373 279 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Active policies 1000 760 631 532 457 395 342 298 260 227 Maturities 0 0 0 0 0 0 0 0 0 227 Surrenders from active 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 Premium 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 Interest 4 750 7 856 10 591 12 922 15 038 16 962 18 638 20 248 21 692 22 994 Maturity 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 out -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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -26 994 7 790 5 143 3 405 2 823 2 691 2 482 2 348 2 212 908 Accumulated discounted cash flow -26 994 -19 204 -14 060 -10 655 -7 832 -5 141 -2 659 -311 1 901 2 809 We have assumed that premium and fund proportional expenses are incurred at the beginning of the year, and surrender and maturity benefit proportional expenses at the end of the year. The surrender and maturity expenses are therefore multiplied by v in order to get a correct discounted cash flow.


2004

Over the portfolio, the results are:

Policy duration


Total profit (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

Total -29 539 987 961 We have a loss of 29 539 on a total NPV of premium of 987 961, i.e. 3% of premium. We need however to increase the premium charge with 3.4%, i.e. to 8.4%, in order to restore profitability. The reason for this is that an increased premium charge leads to less income from the fund and benefit charges. With a premium charge of 8.5%, we get the following result:


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 14 778 Expected increase in unit value 5% annually NPV premium 373 279 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Active policies 1000 760 631 532 457 395 342 298 260 227 Maturities 0 0 0 0 0 0 0 0 0 227 Surrenders from active 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 Premium 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 Interest 4 575 7 567 10 201 12 446 14 484 16 337 17 951 19 502 20 893 22 147 Maturity 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 out -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 Accumulated cash flow -23 844 -12 833 -4 549 1 748 7 387 12 995 18 470 23 960 29 469 32 353 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 844 10 011 6 846 4 731 3 851 3 482 3 091 2 817 2 570 1 223 Accumulated discounted cash flow -23 844 -13 834 -6 988 -2 256 1 595 5 077 8 168 10 984 13 554 14 778


2004

Policy duration


Total profit (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

Total 1 081 987 961 We can also see that we have a reasonable balance between the result of short and long policies. 7. Mortality Up to now, we have seen the life policy as a pure savings vehicle. We will now start to include some effects of mortality by including a death benefit. Let us first assume that the fund value is paid as a benefit at death.

tt VS = The sum at risk or risk sum is defined as the difference between the death benefit and the reserve at the moment of death, i.e. in this case:

0=−= ttt VSR which means that this product has zero mortality risk. Introducing mortality also means that we need to define a new state; Dead. We need to define transition probabilities from the active state to the dead state D

tAP or the paid-up

state to the dead state Dt

P P . Let us assume a fixed mortality of 0.2% per year, i.e.

%2.0=== qPP Dt

PDt

A For practical reasons, we assume in our calculations that the deaths occur just before maturities and surrenders. This means that the surrender and paid up probabilities are defined as relating to the number of active, less those who die during the year. The number of transitions from actives to surrenders S

tA N is given by

S

tAA

tSt

A PNqN ∗∗−= )1(


2004

The number of transitions from active to paid-ups is given by P

tAA

tPt

A PNqN ∗∗−= )1( The number of actives is given by

)1()1(1S

tAP

tAA

tAt PPqNN −−∗−∗= −

The number at time t is equal to the number at time t-1 less deaths, paid-ups and surrenders. The number of paid-ups is given by

Pt

AAt

St

PPt

Pt PqNPqNN ∗−∗+−∗−∗= −− )1()1()1( 11

where the first term is the number of persons remaining in the paid-up state after death and surrender. The second term is the number of transitions from actives to paid-ups. The number of surrenders is given by

St

AAt

St

PPt

St PqNPqNN ∗−∗+−∗−∗= −− )1()1()1( 11

where the first term is the number of transitions from paid-ups to surrender and the second term is the number of transitions from actives to surrender.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 14 447 Expected increase in unit value 5% annually NPV premium 371 392 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 0.2% Death benefit 100% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Mortality 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% 0.2% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies 1000 758 628 529 453 391 338 294 256 223 Dead from active 2 2 1 1 1 1 1 1 1 0 Maturities 0 0 0 0 0 0 0 0 0 222 Surrenders from active 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 Premium 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 Interest 4 575 7 552 10 160 12 371 14 368 16 174 17 737 19 230 20 561 21 752 Deaths -192 -317 -427 -520 -603 -679 -745 -808 -864 -914 Maturity 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 out -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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 864 9 980 6 812 4 700 3 818 3 445 3 051 2 775 2 527 1 201 Accumulated discounted cash flow -23 864 -13 883 -7 071 -2 371 1 447 4 892 7 943 10 719 13 246 14 447 We note that the results are very close to those where we did not have any death benefits. The portfolio looks like this:


2004

Policy duration


Total profit (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

Total -960 981 536 Let us look at what would happen if mortality was double the expected:

Policy duration


Total profit (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

Total -2 952 975 195 We find that the result is rather insensitive to changes in mortality. This is because we take no mortality risk and that deaths are few compared to surrenders and paid-ups. Let us now assume that the mortality is age dependent. We will use the following function:

)2/1(042.010000034.00006.0 +∗∗+= xxq

We have here as a simplification chosen to have q(x) as a Makeham formula. Alternatively, we could have chosen to have the mortality intensity µ(x) as a Makeham and to approximate q(x) with this formula:

)5.01( ½½ ++ ∗+= xxxq µµ For the 10-year policy (inception age 55) we get


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 13 197 Expected increase in unit value 5% annually NPV premium 373 184 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 100% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Mortality 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies 1000 754 620 519 440 377 322 277 238 204 Dead from active 8 7 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 200 Surrenders from active 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 Premium 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 Interest 4 575 7 507 10 033 12 126 13 965 15 574 16 900 18 110 19 112 19 927 Deaths -758 -1 360 -1 989 -2 633 -3 322 -4 061 -4 832 -5 681 -6 580 -7 531 Maturity 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 out -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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 920 9 887 6 705 4 594 3 700 3 304 2 894 2 599 2 333 1 101 Accumulated discounted cash flow -23 920 -14 033 -7 328 -2 734 965 4 270 7 164 9 763 12 097 13 197


2004

And the portfolio

Policy duration


Total profit (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

Total -3 075 975 077 We get a result a little bit worse than the one we had previously (-3 075 compared to -960 and -2952) The difference is that the shorter policies are a little less profitable while the longer policies look a little bit better. This is a result of mortality in the higher ages being more than the 0.2% assumed earlier while mortality in younger ages is less. One should remember that the 3-year policy has inception age 62. Let us now as an example look at a block of policies sold to another group of clients. The major policy principle is the same, but it is used for inheritance planning, so the end age is 100 years instead of 65 years. The 10-year policy looks like this:


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit -11 090 Expected increase in unit value 5% annually NPV premium 222 990 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 90 Death benefit 100% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Mortality 21.56% 23.75% 26.15% 28.80% 31.72% 34.93% 38.47% 42.37% 46.67% 51.40% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies 1000 596 377 235 144 85 48 26 13 6 Dead from active 216 142 99 68 46 30 18 11 6 3 Maturities 0 0 0 0 0 0 0 0 0 3 Surrenders from active 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 Premium 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 Interest 4 575 5 935 6 102 5 498 4 555 3 508 2 508 1 677 1 035 585 Deaths -20 716 -29 597 -33 507 -33 249 -30 341 -25 736 -20 266 -14 919 -10 145 -6 317 Maturity 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 out -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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -25 913 6 878 3 622 1 885 1 087 652 367 199 100 32 Accumulated discounted cash flow -25 913 -19 035 -15 413 -13 528 -12 441 -11 788 -11 422 -11 223 -11 123 -11 090

The policy gets unprofitable. The effect of the high mortality is the receipt of insufficient premium and fund charges.


2004

And the portfolio looks equally bad:

Policy duration


Total profit (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

Total -57 008 769 821 If we decrease the commission to 2.6% instead of 4% times total premium (max 20 years) we get a better picture.

Policy duration


Total profit (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

Total 3 192 769 821 Let us now assume that we take some mortality risk. Let us assume a death benefit of three times the fund value, valid for both active and paid-up policies.

tt VS ∗= 3 We also get the sum at risk as

tttttt VVVVSR ∗=−∗=−= 23 We charge the policyholder a mortality charge being the expected mortality times the sum at risk .

ttxttx VqRqMC ∗∗=∗= ++ 2


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 11 854 Expected increase in unit value 5% annually NPV premium 364 366 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Mortality 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies

1000 768 637 536 458 393 337 291 251 216

Dead from active

8 7 6 6 5 5 5 4 4 4


139 53 35 22 16 14 9 8 7 0

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 Premium 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 Interest 4 575 7 444 9 852 11 772 13 382 14 701 15 683 16 482 17 016 17 307 Mortality charge -1 492 -2 651 -3 834 -5 008 -6 225 -7 480 -8 731 -10 040 -11 344 -12 628 Deaths -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 Maturity 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 out -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 Mortality result 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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 9 754 6 588 4 500 3 599 3 174 2 740 2 417 2 124 1 027 Accumulated discounted cash flow -24 068 -14 314 -7 726 -3 226 373 3 547 6 287 8 703 10 827 11 854


2004

We have included three new rows in the spreadsheet. The first new row is the mortality charge which is the sum at risk multiplied by the mortality qx. This is the risk premium that the insured has to pay to the life office for the death protection included in the policy. The second new row is the risk sums paid, i.e. the part of the mortality benefit that is not paid from the fund but is rather paid from the mortality result of the life office. In our case with a death benefit of 300%, it is calculated as 2/3 of the death benefit, since 1/3 is paid from the fund. The third new row is the mortality result. This is calculated as the mortality charge, minus the risk sums paid. In our example, this result is zero, since we assume that mortality follows the assumption used for calculation of the mortality charge. For the portfolio we get

Policy duration


Total profit (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

Total -7 706 975 609 The results are worse than without the mortality benefit. We previously made a profit of 556, while we here have a loss of 7 706. This is because the mortality charge makes the funds and therefore also the fund and surrender charges smaller. Let us now assume that we have some margins in our mortality assumptions, so real morality is 75% of the mortality used for the mortality charge calculation. Mortality assumptions should be made with margins for expenses and profit.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 21 595 Expected increase in unit value 5% annually NPV premium 366 557 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Technical mortality 100% of standard Age 55 Actual mortality 75% of technical Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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 mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Actual mortality 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% Active policies 1000 756 623 522 444 381 327 282 243 209 Dead from active 6 5 4 4 4 4 3 3 3 3 Maturities 0 0 0 0 0 0 0 0 0 207 Surrenders from active 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 Premium 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 Interest 4 575 7 459 9 893 11 849 13 505 14 878 15 922 16 791 17 401 17 772 Mortality charge -1 492 -2 656 -3 850 -5 041 -6 282 -7 571 -8 864 -10 229 -11 601 -12 967 Deaths -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 Maturity 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 out -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 Mortality result 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 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 709 10 335 7 349 5 399 4 615 4 289 3 928 3 665 3 412 2 311 Accumulated discounted cash flow -23 709 -13 373 -6 024 -625 3 990 8 279 12 207 15 872 19 284 21 595


2004

Policy duration


Total profit (000)


3 100 20 057 2 006 21 923 8 200 23 765 4 753 69 062

13 300 16 134 4 840 115 621 18 400 3 128 1 251 159 918 23 500 -962 -481 202 746 28 400 1 199 480 163 202 33 300 2 381 714 122 772 38 200 3 018 604 81 980 43 100 3 352 335 41 028

Total 14 502 978 252 The mortality profits help us quite a lot in making the portfolio profitable. 8. Shorter periods We have up to now, for simplicity and for lack of space in this document, avoided some of the complications with payments happening at different times of the year. When doing actual calculation, looking at months instead of years is recommended. For such short periods, it will matter much less if one assumes payments at beginning or end of periods only. We will now as an illustration, look at the calculation made at the end of section 7 but assume the death benefits and surrenders are paid in the middle of the year. Premiums are assumes to be paid in the beginning of the year and maturities at the end of the year. We will have to separate the interest into two parts: One for the period before the payment of death benefits and surrenders and one for the period after these payments. We assume that the mortality charge is paid in the middle of the period, just before the death benefits. In order to keep track of the timing, we subdivide the expenses into those incurred in the beginning, the middle and the end of the period and we calculate cash flows the same way. Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Technical mortality 100% of standard Age 55 Actual mortality 75% of technical Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expensés 0.5%


2004

Year 1 2 3 4 5 6 7 8 9 10 Surrender 14% 7% 6% 4% 4% 4% 3% 3% 3% 2% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Technical mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Actual mortality 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% Active policies 1000 756 623 522 444 381 327 282 243 209 Dead from active 6 5 4 4 4 4 3 3 3 3 Maturities 0 0 0 0 0 0 0 0 0 207 Surrenders from active 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 Premium 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 Interest 1 2 260 3 684 4 886 5 852 6 670 7 348 7 864 8 293 8 594 8 778 Mortality charge -1 456 -2 592 -3 757 -4 919 -6 131 -7 388 -8 650 -9 982 -11 321 -12 655 Deaths -1 638 -2 916 -4 227 -5 534 -6 897 -8 312 -9 732 -11 230 -12 736 -14 236 Risk sums paid 1 092 1 944 2 818 3 689 4 598 5 541 6 488 7 487 8 491 9 491 Surrender -3 212 -5 226 -8 298 -8 924 -8 922 -9 801 -8 363 -8 788 -9 072 -6 919 S charge -9 635 -5 226 -2 766 -992 -470 -516 -440 -463 -477 -364 Interest 2 1 949 3 429 4 606 5 585 6 395 7 024 7 547 7 931 8 186 8 385 Maturity 0 0 0 0 0 0 0 0 0 -347 928 Fund out -80 861 -142 278 -191 122 -231 742 -265 338 -291 463 -313 153 -329 067 -339 686 0 Charge 1 8 500 7 230 6 718 6 349 6 096 5 894 5 691 5 528 5 357 5 177 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses 1 -1 500 -1 295 -1 219 -1 165 -1 130 -1 102 -1 073 -1 049 -1 023 -993 Mortality result 364 648 939 1 230 1 533 1 847 2 163 2 496 2 830 3 164 Expenses 2 -24 -41 -63 -72 -79 -91 -90 -100 -109 -106 Surrender charge 9 635 5 226 2 766 992 470 516 440 463 477 364 Expenses 3 0 0 0 0 0 0 0 0 0 -1 740 Cash flow 1 -33 000 5 935 5 499 5 184 4 965 4 791 4 618 4 479 4 334 4 183 Cash flow 2 9 974 5 833 3 643 2 149 1 923 2 272 2 512 2 858 3 199 3 422 Cash flow 3 0 0 0 0 0 0 0 0 0 -1 740 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -23 490 10 452 7 415 5 434 4 644 4 320 3 959 3 697 3 445 2 487 Accumulated discounted cash flow -23 490 -13 038 -5 623 -189 4 455 8 775 12 734 16 431 19 876 22 363

We arrive at a result that is slightly better than the one calculated previously. 9. Reinsurance We will now go back to our previous calculation method used in section 7. We will concentrate on a single policy rather than on a portfolio. Our aim is to illustrate how reinsurance could be used to improve profitability. In our examples, we have used a discount rate of 10%. For a listed company, where the shareholders expect dividends on their share, this might be a low discount rate. It may be assumed that shares should earn a risk premium of 4% above risk free interest rate. This is after tax so it should be a bit more before tax. With a risk free interest rate of 6% and a tax rate of 30%, we need to earn around 14% to give the shareholders their


2004

expect return on their investment. Let us choose one policy that is marginally profitable under a 10% discount rate and compare with how it looks with a 14% discount rate.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 920 Expected increase in unit value 5% annually NPV premium 364 366 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6.0 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Age 55 Mortality 100% of standard 100% of technical Death benefit 300% of fund Fund charge 0.90% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expensés 0.5% Year 1 2 3 4 5 6 7 8 9 10 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 mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Actual mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies 1000 754 620 519 440 377 322 277 238 204 Dead from active 8 7 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 200 Surrenders from active 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 82 905 145 638 195 281 236 305 269 952 295 787 316 909 331 976 341 497 Premium 100 000 75 401 62 043 51 870 44 044 37 667 32 177 27 676 23 773 20 390 Charge -6 000 -5 270 -5 033 -4 870 -4 769 -4 690 -4 593 -4 513 -4 414 -4 297 Interest 4 700 7 652 10 132 12 114 13 779 15 146 16 169 17 004 17 567 17 880 Mortality charge -1 532 -2 725 -3 943 -5 153 -6 410 -7 707 -9 001 -10 358 -11 711 -13 045 Deaths -2 299 -4 088 -5 915 -7 730 -9 615 -11 561 -13 502 -15 537 -17 567 -19 568 Risk sums 1 532 2 725 3 943 5 153 6 410 7 707 9 001 10 358 11 711 13 045 Maturity 0 0 0 0 0 0 0 0 0 -355 903 Surrender -3 374 -5 481 -8 688 -9 324 -9 301 -10 192 -8 673 -9 085 -9 346 0 S charge -10 122 -5 481 -2 896 -1 036 -490 -536 -456 -478 -492 0 Fund out -82 905 -145 638 -195 281 -236 305 -269 952-295 787 -316 909 -331 976 -341 497 0 Expense charges 15 202 10 253 7 666 5 812 5 214 5 177 5 008 4 947 4 861 4 297 Mortality result 0 0 0 0 0 0 0 0 0 0 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 526 -1 340 -1 288 -1 246 -1 219 -1 204 -1 175 -1 161 -1 143 -2 696 Cash flow -26 324 8 913 6 378 4 565 3 995 3 973 3 833 3 787 3 718 1 601 Discount factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -26 324 8 102 5 271 3 430 2 729 2 467 2 163 1 943 1 735 679 Accumulated discounted cash flow -26 324 -18 221 -12 951 -9 521 -6 792 -4 325 -2 161 -218 1 517 2 196


2004

Policy duration 10 years Discount rate 14% Premium payment 10 years NPV profit -2 664 Expected increase in unit value 5% annually NPV premium 334 856 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6.0 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Age 55 Mortality 100% of standard 100% of technical Death benefit 300% of fund Fund charge 0.90% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expensés 0.5% Year 1 2 3 4 5 6 7 8 9 10 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 mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Actual mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active policies 1000 754 620 519 440 377 322 277 238 204 Dead from active 8 7 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 200 Surrenders from active 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 82 905 145 638 195 281 236 305 269 952 295 787 316 909 331 976 341 497 Premium 100 000 75 401 62 043 51 870 44 044 37 667 32 177 27 676 23 773 20 390 Charge -6 000 -5 270 -5 033 -4 870 -4 769 -4 690 -4 593 -4 513 -4 414 -4 297 Interest 4 700 7 652 10 132 12 114 13 779 15 146 16 169 17 004 17 567 17 880 Mortality charge -1 532 -2 725 -3 943 -5 153 -6 410 -7 707 -9 001 -10 358 -11 711 -13 045 Deaths -2 299 -4 088 -5 915 -7 730 -9 615 -11 561 -13 502 -15 537 -17 567 -19 568 Risk sums 1 532 2 725 3 943 5 153 6 410 7 707 9 001 10 358 11 711 13 045 Maturity 0 0 0 0 0 0 0 0 0 -355 903 Surrender -3 374 -5 481 -8 688 -9 324 -9 301 -10 192 -8 673 -9 085 -9 346 0 S charge -10 122 -5 481 -2 896 -1 036 -490 -536 -456 -478 -492 0 Fund out -82 905 -145 638 -195 281 -236 305 -269 952-295 787 -316 909 -331 976 -341 497 0 Expense charges 14 879 10 078 7 574 5 778 5 199 5 160 4 993 4 932 4 846 4 297 Mortality result 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 -1 339 -1 286 -1 243 -1 216 -1 200 -1 171 -1 157 -1 139 -2 636 Cash flow -26 646 8 739 6 288 4 535 3 983 3 960 3 822 3 775 3 707 1 661 Discount factor 1 0.877 0.769 0.675 0.592 0.519 0.456 0.400 0.351 0.308 Discounted cash flow -26 646 7 666 4 838 3 061 2 358 2 057 1 741 1 509 1 300 511 Accumulated discounted cash flow -26 646 -18 980 -14 142 -11 080 -8 722 -6 666 -4 925 -3 416 -2 117 -1 606


2004

We see that the policy is unprofitable at the higher discount rate. One way to help solving this problem is to use reinsurance. Reinsurers are interested in assuming all types of insurance risk, not only mortality risk. Many reinsurers who reinsure both life and non-life risks have large reserves on the non-life side. They might for example have a large portfolio of long term liability risks. This normally matched by long term govern-ment bond earning risk free interest rate. An alternative for the reinsurer would be to invest in reinsurance of a life company with negative cash flow. The reinsurance is made as a “modified quota share”. A typical quota could be 80%. The reinsurer then participates in its 80% share of all technical terms of the policy. The reinsurer does not, however, participate in the internal expenses of the life office. He might also pay a lower initial commission than the original. Since the reinsurer thus has lower expenses, he will have a better profit situation than the life office had before the reinsurance. To compensate for this, one will keep track of the results for the reinsurer and when they have become good enough, then either the contract will be discontinued or much of the profits will be returned to the life office through a “profit commission”. There is of course always the risk that unexpected changes in surrenders or mortality will lead to the reinsurer never making any profit. Such things happen and will be taken into account when pricing the reinsurance contract. What is done in practice is that the parties keep track of the result created in the beginning of the treaty through a “deficit account” or “loss carried forward”. The reinsurer is credited with an interest on the deficit account to reimburse him for his negative cash flow. Since his alternate investment probably is government bonds, he will probably be prepared to use an interest rate that is a bit lower than the discount rate used by the life office, even with the reinsurance risk priced in. Let us assume that the reinsurer is prepared to pay an initial commission that is 90% of the original and that he want to get his negative cash flow back with 8% interest before he is prepared to allow the reinsurance contract to be terminated. The figures would then look like this. Policy duration 10 years Discount rate 14% Premium payment 10 years NPV profit 985 Expected increase in unit value 5% annually NPV premium 342 470 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 6.0 % of each premium Age 55 Surrenders 70% of standard Reinsurers’ share 80% Surrender charge 100% of standard Reinsurers’initial commission 90% of original Paid ups 100% of standard Reinsurer’s interest 8% Mortality 100% of standard Death benefit 300% of fund Fund charge 0.90% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5%


2004

Year 1 2 3 4 5 6 7 8 9 10 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 mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Actual mortality

0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80%

Surrender charge

75% 50% 25% 10% 5% 5% 5% 5% 5% 5%

Active 1000 754 620 519 440 377 322 277 238 204 Dead from

active 8 7 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 200

Surrenders from active 139 52 34 22 15 13 9 8 7 0

Paid ups 99 75 61 51 44 37 32 27 23 0 Fund in 0 82 905 145 638 195 281 236 305 269 952 295 787 316 909 331 976 341 497

Premium 100 000 75 401 62 043 51 870 44 044 37 667 32 177 27 676 23 773 20 390 Charge -6 000 -5 270 -5 033 -4 870 -4 769 -4 690 -4 593 -4 513 -4 414 -4 297 Interest 4 700 7 652 10 132 12 114 13 779 15 146 16 169 17 004 17 567 17 880

Mortality charge -1 532 -2 725 -3 943 -5 153 -6 410 -7 707 -9 001 -10 358 -11 711 -13 045 Deaths -2 299 -4 088 -5 915 -7 730 -9 615 -11 561 -13 502 -15 537 -17 567 -19 568

Risk sums 1 532 2 725 3 943 5 153 6 410 7 707 9 001 10 358 11 711 13 045 Maturity 0 0 0 0 0 0 0 0 0 -355 903

Surrender -3 374 -5 481 -8 688 -9 324 -9 301 -10 192 -8 673 -9 085 -9 346 0 S charge -10 122 -5 481 -2 896 -1 036 -490 -536 -456 -478 -492 0 Fund out -82 905 -145 638 -195 281 -236 305 -269 952-295 787 -316 909 -331 976 -341 497 0 Expense charges 14 879 10 078 7 574 5 778 5 199 5 160 4 993 4 932 4 846 4 297 Mortality

result 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 -1 339 -1 286 -1 243 -1 216 -1 200 -1 171 -1 157 -1 139 -2 636 Cash flow -26 646 8 739 6 288 4 535 3 983 3 960 3 822 3 775 3 707 1 661 Discount

factor 1 0.877 0.769 0.675 0.592 0.519 0.456 0.400 0.351 0.308 Discounted

cash flow -26 646 7 666 4 838 3 061 2 358 2 057 1 741 1 509 1 300 511 Accumulated

discounted cash flow -26 646 -18 980 -14 142 -11 080 -8 722 -6 666 -4 925 -3 416 -2 117 -1 606

Reinsurance Initial comm 28 800 0 0 0 0 0 0 0 0 0 Other cash flow -10 683 -6 991 -5 030 -3 628 -3 186 -3 168 0 0 0 0 Net reinsurer 18 117 -6 991 -5 030 -3 628 -3 186 -3 098 0 0 0 0 Interest 1 449 1 006 684 449 229 0 0 0 0 0 Deficit account 19 566 13 581 9 234 6 055 3 098 0 0 0 0 0 Net after reinsurance Cash flow -8 529 1 748 1 258 907 797 862 3 822 3 775 3 707 1 661 Discounted cash flow -8 529 1 533 968 612 472 447 1 741 1 509 1 300 511 Accumulated

disc c.f. -8 529 -6 996 -6 028 -5 416 -4 944 -4 497 -2 756 -1 247 52 563


2004

We see here that a policy that was unprofitable before reinsurance is profitable after reinsurance. 10. Analysis of results new version In this section, we will give some examples of how one could evaluate the result of a year, including changes in net present value of future profits. We will not give all details behind the figures used, thus it might be problematic for the student to reconstruct all details. We believe anyway that the principles discussed will be illustrated by our examples. We start with an example from section 7.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 11 868 Expected increase in unit value 5% annually NPV premium 373 184 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Mortality 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surr charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Active 1000 768 637 536 458 393 337 291 251 216 Dead from active

8 7 6 6 5 5 5 4 4 4


139 53 35 22 16 14 9 8 7 0

New paidups 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 Premium 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 Interest 4 575 7 444 9 852 11 772 13 382 14 701 15 683 16 482 17 016 17 307 Mort charge -1 492 -2 651 -3 834 -5 008 -6 225 -7 480 -8 731 -10 040 -11 344 -12 628 Deaths -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 Maturity 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 out -80 700 -141 688 -189 877 -229 632 -262 171-287 082 -307 385 -321 787 -330 793 0 Exp charges 17 457 12 064 9 250 7 223 6 472 6 297 6 008 5 848 5 672 5 041 Mort result 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 Disc factor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 9 754 6 588 4 500 3 599 3 174 2 740 2 417 2 124 1 027 Acc disc cash flow -24 068 -14 314 -7 726 -3 226 373 3 547 6 287 8 703 10 827 11 854

Let us look at this example in more detail, especially the first year. We expect to write business that is worth 11 854, measured in accumulated net present value of future profits. Let us now assume that we have reached the end of year one and that everything has been according to plans. We would then have a negative a cash flow, which is worth 24 068 discounted to the beginning of the year. To get the value as at the end of the year, we must multiply it by 1+i and we get 26 475. Let us look at the last part of our previous table. Let us introduce factors and calculations of discounted cash flow as per beginning of year two.


2004

Cash flow -24 068 10 729 7 972 5 990 5 269 5 111 4 854 4 709 4 553 2 422 Disc factor 1 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 9 754 6 588 4 500 3 599 3 174 2 740 2 417 2 124 1 027 Acc disc cash flow -24 068 -14 314 -7 726 -3 226 373 3 547 6 287 8 703 10 827 11 854 Disc factor 2 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 Discounted cash flow 10 729 7 247 4 950 3 959 3 491 3 014 2 658 2 336 1 130 Acc disc cash flow 10 729 17 976 22 926 26 885 30 376 33 390 36 048 38 385 39 514

Looking at the future cash flows as at end of year one (beginning of year two), we get a value of 39 514. So for year one we have the following results: Cash flow -26 475 Value of future cash flow 39 514 Result 13 039 Which is exactly the expected 11 854 multiplied with 1+i. We could express the results in the following way, introducing VBIF which means value of business in force VBIF beginning of year 0 Cash flow during the year -26 475 Value new business end of year 39 514 Result 13 039 Let us now look at year two and also look at discounting to the end of year two (beginning of year three) Cash flow -24 068 10 729 7 972 5 990 5 269 5 111 4 854 4 709 4 553 2 422 Disc factor 1 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 9 754 6 588 4 500 3 599 3 174 2 740 2 417 2 124 1 027 Acc disc cash flow -24 068 -14 314 -7 726 -3 226 373 3 547 6 287 8 703 10 827 11 854 Disc factor 2 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 Discounted cash flow 10 729 7 247 4 950 3 959 3 491 3 014 2 658 2 336 1 130 Acc disc cash flow 10 729 17 976 22 926 26 885 30 376 33 390 36 048 38 385 39 514 Disc fator 3 1.1 1.000 0.909 0.826 0.751 0.683 0.621 0.564 0.513

Discounted cash flow 11 802 7 972 5 445 4 354 3 840 3 315 2 924 2 570 1 243 Acc disc

cash flow 11 802 19 774 25 219 29 573 33 414 36 729 39 653 42 223 43 466

We start with VBIF beginning of the year of 39 514. This value is the present value of future profits as at end of year one, i.e. beginning of year two. As at the end of year two,


2004

it will be worth i% more. In our case, the change is 10% of 39 514 or 3 951. This figure is called unwinding of discount. The cash flow of year two is worth 11 802 as at the end of year 2. The VBIF as at the end of the year can be calculated as (43 466-11 802). We can express the result as this. VBIF beginning of year two -39 514 Unwinding of discount -3 951 Cash flow year two 11 802 VBIF end of year two 31 663 Result 0 In this example, nothing has changed compared with original assumptions. Let us now assume that we have to make changes in the assumptions. Let us assume that the interest rate environment has changed so we must increase the discount rate to 11% as per end of year two from the 10% previously used. We get: Cash flow -24 148 10 686 7 949 5 982 5 266 5 108 4 851 4 706 4 550 2 437 Disc factor 1 1 0.901 0.812 0.731 0.659 0.593 0.535 0.482 0.434 0.391 Discounted cash flow -24 148 9 627 6 452 4 374 3 469 3 031 2 594 2 267 1 974 952 Acc disc cash flow -24 148 -14 522 -8 070 -3 696 -227 2 804 5 398 7 665 9 639 10 592 Disc factor 2 1 0.901 0.812 0.731 0.659 0.593 0.535 0.482 0.434 Discounted cash flow 10 686 7 162 4 855 3 850 3 365 2 879 2 516 2 192 1 057 Acc disc cash flow 10 686 17 847 22 703 26 553 29 918 32 796 35 313 37 504 38 562 Disc fator 3 1.11 1 0.901 0.812 0.731 0.659 0.593 0.535 0.482


cash flow 11 861 19 810 25 200 29 474 33 209 36 404 39 197 41 630 42 803

We must therefore change the VIBF as at end of year two to 42 803-11 861 = 30 942 and we get VBIF beginning of the year -39 514 Unwinding of discount -3 951 Cash flow year two 11 802 VBIF end of year two 30 942 Result -722 Another way of describing it would be Value of new business written 0 Deviations from assumption 0 Changes in assumptions - 722 Result - 722 Let us now once again look at year two and assume no change in discount rate but that surrenders during the year have been 10% instead of 7%.


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 11 868 Expected increase in unit value 5% annually NPV premium 373 184 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 70% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 Surrender 14.0% 10.0% 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% Mortality 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surr charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Active 1000 754 598 500 425 363 310 267 229 197 Dead fr act 8 7 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 193 Surr fr active 139 75 33 21 15 13 9 7 6 0 New paidups 99 75 59 49 42 36 31 26 23 0 Fund in 0 80 700 137 117 183 539 221 839 253 187 277 181 296 736 310 602 319 265 Premium 100 000 75 401 59 800 49 995 42 452 36 305 31 014 26 676 22 914 19 653 Charge -8 500 -7 216 -6 454 -6 085 -5 827 -5 618 -5 408 -5 235 -5 054 -4 863 Interest 4 575 7 444 9 523 11 372 12 923 14 194 15 139 15 909 16 423 16 703 Mort charge -1 492 -2 651 -3 706 -4 838 -6 012 -7 222 -8 428 -9 691 -10 949 -12 187 Deaths -2 238 -3 977 -5 559 -7 257 -9 018 -10 834 -12 642 -14 537 -16 423 -18 280 Risk sums paid 1 492 2 651 3 706 4 838 6 012 7 222 8 428 9 691 10 949 12 187 Maturity 0 0 0 0 0 0 0 0 0 -332 478 Surrender -3 284 -7 618 -8 166 -8 753 -8 724 -9 551 -8 121 -8 500 -8 737 0 S charge -9 853 -7 618 -2 722 -973 -459 -503 -427 -447 -460 0 Fund out -80 700 -137 117 -183 539 -221 839 -253 187-277 181 -296 736 -310 602 -319 265 0 Exp charges 17 457 14 141 8 929 6 969 6 244 6 075 5 797 5 642 5 472 4 863 Mort result 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 -1 345 -1 234 -1 190 -1 161 -1 144 -1 114 -1 098 -1 079 -2 528 Cash flow -24 068 12 796 7 695 5 779 5 083 4 931 4 683 4 543 4 392 2 336 Disc factor 1 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 11 633 6 360 4 342 3 472 3 062 2 643 2 331 2 049 990 Acc disc cash flow -24 068 -12 435 -6 076 -1 733 1 738 4 800 7 443 9 775 11 824 12 814 Disc factor 2 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467


cash flow 12 796 19 792 24 568 28 387 31 755 34 663 37 227 39 481 40 571 Disc fator 3 1.1 1.000 0.909 0.826 0.751 0.683 0.621 0.564 0.513


cash flow 14 076 21 771 27 025 31 226 34 930 38 129 40 950 43 429 44 628


2004

We do not change our assumptions in regard to future surrenders. VBIF beginning of the year -39 514 Unwinding of discount -3 951 Cash flow year two 14 076 VBIF end of year two 30 552 Result 1 162 We have a good cash flow as a result of the high surrender charge, while the VBIF goes down since the size of the remaining portfolio decreases. In total, we make a minor profit on the increased surrender during the year. Or Value of new business written 0 Deviations from assumptions 1 162 Changes in assumptions 0 Result 1 162 Let us now also assume that also future surrenders from year three onwards will be higher. We assume them to be 100% of the standard assumption instead of 70%. We get


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 11 868 Expected increase in unit value 5% annually NPV premium 373 184 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 100% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 Surrender 14.0% 10.0% 8.0% 6.0% 5.0% 5.0% 4.0% 4.0% 4.0% 3.0% Paid-up 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Mortality 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surr charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Active 1000 754 598 486 404 339 285 242 205 173 Dead fr act 8 7 6 5 5 4 4 4 3 3 Maturities 0 0 0 0 0 0 0 0 0 170 Surr fr active 139 75 47 29 20 17 11 10 8 0 New paidups 99 75 59 48 40 34 28 24 20 0 Fund in 0 80 700 137 117 178 873 212 005 238 034 256 367 270 889 279 880 283 980 Premium 100 000 75 401 59 800 48 573 40 380 33 934 28 486 24 164 20 471 17 316 Charge -8 500 -7 216 -6 454 -5 917 -5 552 -5 265 -4 985 -4 763 -4 539 -4 312 Interest 4 575 7 444 9 523 11 076 12 342 13 335 13 993 14 515 14 791 14 849 Mort charge -1 492 -2 651 -3 706 -4 712 -5 741 -6 786 -7 790 -8 842 -9 860 -10 834 Deaths -2 238 -3 977 -5 559 -7 068 -8 612 -10 178 -11 685 -13 263 -14 791 -16 251 Risk sums paid 1 492 2 651 3 706 4 712 5 741 6 786 7 790 8 842 9 860 10 834 Maturity 0 0 0 0 0 0 0 0 0 -295 582 Surrender -3 284 -7 618 -11 666 -12 179 -11 902 -12 818 -10 723 -11 079 -11 241 0 S charge -9 853 -7 618 -3 889 -1 353 -626 -675 -564 -583 -592 0 Fund out -80 700 -137 117 -178 873 -212 005 -238 034-256 367 -270 889 -279 880 -283 980 0 Exp charges 17 457 14 141 9 989 7 148 6 122 5 878 5 498 5 293 5 077 4 312 Mort result 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 -1 345 -1 250 -1 174 -1 123 -1 090 -1 042 -1 015 -985 -2 245 Cash flow -24 068 12 796 8 740 5 974 4 999 4 788 4 456 4 278 4 092 2 067 Disc factor 1 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 11 633 7 223 4 488 3 414 2 973 2 515 2 195 1 909 876 Acc disc cash flow -24 068 -12 435 -5 212 -724 2 690 5 664 8 179 10 374 12 283 13 159 Disc factor 2 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467

Discounted cash flow 12 796 7 945 4 937 3 756 3 271 2 767 2 415 2 100 964 Acc disc



cash flow 14 076 22 815 28 246 32 377 35 975 39 019 41 675 43 984 45 045


2004

Our analysis becomes: VBIF beginning of the year -39 514 Unwinding of discount -3 951 Cash flow year two 14 076 VBIF end of year two 30 969 Result 1 580 Of this, we saw previously that 1 162 comes from deviations from assumptions during year two. We get: Value of new business written 0 Deviations from assumptions 1 162 Changes in assumptions 418 Result 1 580 Let us now once again look at year two and assume no change in discount rate but that actual mortality has been 150% of technical mortality instead of 100%. We do not change our assumptions in regard to future mortality. We have:


2004

Policy duration 10 years Discount rate 10% Premium payment 10 years NPV profit 11 868 Expected increase in unit value 5% annually NPV premium 373 184 Premium 100 per year Initial commission 4 % of total premium max 20 years Premium charge 8.5 % of each premium Surrenders 100% of standard Surrender charge 100% of standard Paid ups 100% of standard Mortality 100% of standard Age 55 Death benefit 300% of fund Fund charge 1.00% of fund at beginning of the year Premium proportional expenses 1.5% Fund proportional expenses 0.2% of fund at beginning of the year Benefit proportional expenses 0.5% Year 1 2 3 4 5 6 7 8 9 10 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% Tech mort 0.79% 0.86% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Actual mort 0.79% 1.29% 0.94% 1.03% 1.13% 1.24% 1.36% 1.49% 1.64% 1.80% Surr charge 75% 50% 25% 10% 5% 5% 5% 5% 5% 5% Active 1000 754 618 516 439 375 320 276 237 203 Dead fr act 8 10 6 5 5 5 4 4 4 4 Maturities 0 0 0 0 0 0 0 0 0 199 Surrenders from active 139 52 34 21 15 13 9 8 7 0 New paidups 99 74 61 51 43 37 32 27 23 0 Fund in 0 80 700 141 071 189 051 228 633 261 031 285 833 306 048 320 387 329 353 Premium 100 000 75 401 61 773 51 644 43 853 37 503 32 037 27 556 23 670 20 302 Charge -8 500 -7 216 -6 661 -6 280 -6 014 -5 798 -5 581 -5 403 -5 216 -5 019 Interest 4 575 7 444 9 809 11 721 13 324 14 637 15 614 16 410 16 942 17 232 Mort charge -1 492 -2 651 -3 817 -4 986 -6 198 -7 448 -8 693 -9 997 -11 295 -12 573 Deaths -2 238 -5 965 -5 726 -7 479 -9 297 -11 172 -13 039 -14 995 -16 942 -18 859 Risk sums paid 1 492 3 977 3 817 4 986 6 198 7 448 8 693 9 997 11 295 12 573 Maturity 0 0 0 0 0 0 0 0 0 -343 009 Surrender -3 284 -5 309 -8 411 -9 021 -8 994 -9 849 -8 375 -8 768 -9 013 0 S charge -9 853 -5 309 -2 804 -1 002 -473 -518 -441 -461 -474 0 Fund out -80 700 -141 071 -189 051 -228 633 -261 031-285 833 -306 048 -320 387 -329 353 0 Exp charges 17 457 12 043 9 210 7 191 6 444 6 269 5 982 5 822 5 647 5 019 Mort result 0 -1 264 0 0 0 0 0 0 0 0 Comm -40 000 0 0 0 0 0 0 0 0 0 Expenses -1 525 -1 344 -1 273 -1 228 -1 198 -1 180 -1 150 -1 133 -1 114 -2 608 Cash flow -24 068 9 435 7 937 5 964 5 246 5 089 4 833 4 689 4 533 2 411 Disc factor 1 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 Discounted cash flow -24 068 8 577 6 560 4 481 3 583 3 160 2 728 2 406 2 115 1 023 Acc disc cash flow -24 068 -15 491 -8 931 -4 450 -867 2 293 5 021 7 427 9 542 10 564 Disc factor 2 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467




cash flow 10 379 18 316 23 737 28 073 31 896 35 197 38 108 40 667 41 905


2004

VBIF beginning of the year -39 514 Unwinding of discount -3 951 Cash flow year two 10 379 VBIF end of year two 31 526 Result -1 560 The loss is higher than the mortality loss in the spread sheet, since the high mortality also leads to loss of future profits on those policies that were terminated because of the high mortality or Value of new business written 0 Deviations from assumption - 1 560 Changes in assumptions 0 Result - 1 560 The effect of deviations from assumptions could be subdivided into two parts: One is the result effect of the year in question. The other is the effect on the portfolio size, which gives an effect on the VBIF. We have: Value of new business written 0 Deviations from assumption this year -1 264 Deviations from assumptions VBIF -296 Changes in assumptions 0 Result - 1 560 Most of the effect of the high mortality is shown in mortality losses the actual year, but there is also some effect on the VBIF since the high mortality makes the portfolio smaller.

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