profit test modelling in life assurance using spreadsheets part one

Profit Test Modelling in Life Assurance Using SpreadsheetsPROFIT TESTMODELLING IN LIFEASSURANCE USINGSPREADSHEETSPART ONEErik AlmPeter Millington2004 1

Profit Test Modelling in Life Assurance Using SpreadsheetsProfit test modelling in life assurance1. IntroductionThe aim of this brief is to demonstrate the development of profit test models in lifeassurance using spreadsheets. We will work through several illustrative examplesstep-by-step and the degree of complexity will increase from one example to thenext. We will start with a single policy excluding loadings and gradually work our wayto an entire portfolio of policies where we also include investment return, expenses,loadings and maturity benefits. In part two we will build more advanced modelswhere we will include surrenders, paid-ups and mortality risk.It is very common in life assurance that the life office has high initial expenses whenwriting a new policy. This cost could be commission to the sales agent but could alsobe internal costs for underwriting or for IT systems. This leads to a negative cash flowor negative result for the life office at the inception of a life policy.The premiums charged by a life office are calculated in such a way that the presentvalue of the premiums should be equal to or exceed the present value of the futurebenefits and expenses. If not, the policy is written at an expected loss which, if it isdone consistently, would threaten the solvency of the life office.Letting the present value of the premiums being equal to the present value of thebenefits and expenses is not enough. In order for the life office to be able to writenew business, it needs to put up risk capital. For a proprietary company, this is doneby shareholders who expect return on their share capital. For a mutual company thisis done by the existing policyholders, who expect the surplus accumulated in thecompany not to be diluted by the writing of new policies that do not contribute to thissurplus.The life office usually sets internal rules determining the minimum profit to beemerging from a new life policy or new block of life policies written. One such rulecould be that the present value of the premiums should exceed the present value ofthe benefits by a certain percentage. Another way of expressing profit requirementsfor a life policy is to state when the initial expenses are repaid at a stated internaldiscount rate. We will mainly work with this type of profit requirement in this brief.2. Unit-linked assurance2.1. A policyLet us first start with a unit-linked policy as an example.In a typical unit-linked assurance policy the financial risk is born by the policyholder.This differs from traditional assurance where the products typically provideguaranteed benefits at maturity, death or surrender. In unit-linked assurance,premiums are invested in a fund of the choice of the policyholder after deductions forexpenses and mortality. The premium reserve will thus (in most cases) be definedretrospectively, using the investment return earned by the fund.It should be noted that there are unit-linked policies sold which provide someguarantees. One typical guarantee is a guaranteed benefit of at least the sum ofpremium paid. In our discussion, we will assume that no guarantees are provided.2004 2

Profit Test Modelling in Life Assurance Using SpreadsheetsLetV = premium reserve at year ttP = premium paid at the beginning of the year ttti = investment return of the fund during the periodAssuming zero initial expenses and no mortality risk the premium reserve at time t isexpressed as:V = V−+ P ) *(1 + i )(1)t(t 1 ttExpression (1) shows the development of the fund in the time-discrete case. Thekeen student could here and in later examples construct the corresponding formula inthe time-continuous case.This type of policy could, just as well as a traditional policy, be studied analytically.We will however instead mainly study it in a more straightforward way in aspreadsheet environment. The main reason for this is that the assumptions used dooften not lead to a nice analytical formula, like the Makeham formula. Anotherimportant advantage of this method is that the student could much easier see whathappens during the lifetime of the policy rather than just seeing one figure being theresult of a long formula. Also, it is easier in a spreadsheet to create ‘what if’situations, i.e. to test the effect of changes in assumptions and to see how thesechanges affect the policy in different periods.We will look at some principle problems and also give some practical tips on how thistype of study is best done in a spreadsheet environment.We will generally assume a level premium is paid, i.e.P t= P , 0

Profit Test Modelling in Life Assurance Using SpreadsheetsPolicy durationPremium10 years100 per yearYearPremium1 1002 1003 1004 1005 1006 1007 1008 1009 10010 100One important thing to note here is that all parameters should be stated explicitly in any Excelspreadsheet used for this type of calculations. It should always be very easy to change the parametersand study the effects of such change. IfP = 50 for 0

Profit Test Modelling in Life Assurance Using SpreadsheetsWe show the maturity benefit as negative, since it represents an outgo for the lifeoffice.Let us also include the cash flow to the life office:Policy durationPremium paymentPremium10 years10 years100 per yearYear Premium Maturity benefit Cash flow1 100 1002 100 1003 100 1004 100 1005 100 1006 100 1007 100 1008 100 1009 100 10010 100 -1000 -900The premiums the life office receives years 1 to 10 must be reserved, since it will beneeded to pay for the maturity benefit in year 10. The development of the premiumreserve is given by (assuming zero investment return):V + PorV tt= Vt−1= t ∗ PtThe maturity benefitCd= Vd= d ∗ PC = 0 for t ? d.tCdat time d is given byPolicy durationPremium paymentPremium10 years10 years100 per yearYear Premium Maturity benefit Cash flow Reserve1 100 100 1002 100 100 2003 100 100 3004 100 100 4005 100 100 5006 100 100 6007 100 100 7008 100 100 8009 100 100 90010 100 -1000 -900 0We here use a minus sign for the maturity benefit, since it enters the cash flow as negative.The reserve increases with the premiums paid and decreases with the maturitybenefit paid out and we note that it is zero after the maturity of the policy just as wewould expect.2004 5

Profit Test Modelling in Life Assurance Using SpreadsheetsFor unit-linked business, the reserve (at least in simple cases) consists of savingsbelonging to the policyholder, where the policyholder bears the financial risksconnected with the reserve. In such case, the reserve is often called the fund and thecash flow to and from the fund is not included when one studies the cash flow fromthe life office's point of view. The fund is like a bank account and is treated as such inUS GAAP, the general accepted accounting principles in the US.In our simplified example, the cash flow is given byCFt= V t−t1+ P t− C t−V= 0or in spreadsheet format:Policy durationPremium paymentPremium10 years10 years100 per yearYear Premium Maturity benefit Fund Cash flow1 100 100 02 100 200 03 100 300 04 100 400 05 100 500 06 100 600 07 100 700 08 100 800 09 100 900 010 100 -1000 0 0In order to avoid having different formulae for year one and subsequent years, we include the value ofthe fund at the beginning of the year and the value of the fund at the end of the year. We have dividedthe table into two parts, one showing the development of the fund and one showing the cash flow to thelife office.Policy durationPremium paymentPremium10 years10 years100 per yearYear Fund in Premium Maturity benefit Fund out Cash flow1 0 100 -100 02 100 100 -200 03 200 100 -300 04 300 100 -400 05 400 100 -500 06 500 100 -600 07 600 100 -700 08 700 100 -800 09 800 100 -900 010 900 100 -1000 0 0For practical reasons, we use the convention that the fund out is shown with a minus sign (it is positivefor the client but is a liability for the life office).2004 6

Profit Test Modelling in Life Assurance Using Spreadsheets2.2. Investment returnWe have up to now assumed that the money in the fund will earn no return. In reallife, this money is invested in financial assets, in equities or bonds or both. Theinvestment return includes dividends on shares and realised or unrealised gains onshares or bonds. We will assume that the investment return is fixed at a rate of i tpertime period t, even though the fund value might change continuously.The fund value and cash flows are given by:V V + P ) ∗ (1 + i )t= (t−1 ttCFCF= ( V− 1+ P ) ∗ (1 + i ) − C −Vort t ttt= Vt− 1+ Pt+ ( Vt−1+ Pt)t∗itt− Ct−VtAssuming a level premium, the maturity benefit at maturity date d is expressed asCd= Vd= P ∗d∑t=1( 1+i)tImportant to remember is that the investment return varies over time, depending onthe development of the assets in the fund. Let us now assume that the fund will earn5% annual return (after taxes and after internal fund expenses).Policy durationPremium paymentExpected increase in unit valuePremium10 years10 years5% annually100 per yearYear Fund in Premium Interest Maturity benefit Fund out Cash flow1 0 100 5 0 -105 02 105 100 10 0 -215 03 215 100 16 0 -331 04 331 100 22 0 -453 05 453 100 28 0 -580 06 580 100 34 0 -714 07 714 100 41 0 -855 08 855 100 48 0 -1 003 09 1 003 100 55 0 -1 158 010 1 158 100 63 -1 321 0 0(In unit–linked business, the value of the fund is often expressed as a number ofunits, multiplied by the value of a unit. When the underlying assets increase in value,the number of units remains constant while the value of a unit increases. When newpremium is added to the fund, the number of unit increases.)Problem: What interest rate is needed in order to provide a maturity benefit of 2000?Answer: 12.3%This problem could be solved by analytical methods, but a more practical and faster way is to use theproblem solving methods of Excel: Goal Seeker or Solver. (Tools, Goal Seeker).2004 7

Profit Test Modelling in Life Assurance Using Spreadsheets2.3. Initial commissionUp to now, the cash flow for the life office has been zero. We shall now start to lookat this cash flow.The most important cost for the life office in writing business is the initial expenses,and especially the commission paid to the sales agents, being ties agents or brokers.This commission is often paid up-front (i.e. directly after a sale is made) and is oftencalculated as a percentage (or per mille) of the total premium volume of the contract.We call the percentage a. The initial commission is thus given byI1 = α ∗ d ∗ P , I t= 0,t ≠ 1This initial commission enters as negative cash flow year one. The cash flow formulais now given by:CF V + P + V + P ∗i− C − I −Vt=t− 1 t(t−1t)tttThe above expression consists of two parts. The first part is inflow and outflow of thepremium reserve. This part does in reality not affect the life office as such but ratherthe client fund. The second part is the initial commission. Looking at cash flow thataffects the life office separately, we have:CF = −tI tLet us assume that the commission is 40 per mille of the total premium. For ourcontract, the total premium is 10*100 = 1000 and the commission is thus 40.tPolicy durationPremium paymentExpected increase in unit valuePremiumInitial commission10 years10 years5% annually100 per year4 % of total premiumYear Fund in Premium Interest MaturitybenefitFund out CommissionCashflow1 0 100 5 0 -105 -40 -402 105 100 10 0 -215 0 03 215 100 16 0 -331 0 04 331 100 22 0 -453 0 05 453 100 28 0 -580 0 06 580 100 34 0 -714 0 07 714 100 41 0 -855 0 08 855 100 48 0 -1 003 0 09 1 003 100 55 0 -1 158 0 010 1 158 100 63 -1 321 0 0 0In our tables, we show commission and other expenses as negative, since they mean outflow for the lifeoffice.2.4. Premium chargesWe have an outflow from the life office in the form of the commission. The life officewill need to cover these expenses and this is done by introducing some charges thatthe policyholder has to pay. One way of doing this is to charge a percentage of eachpremium paid to the life office. Let us introduce such a charge and let that charge ?be the same as the commission, i.e. 4%.2004 8

Profit Test Modelling in Life Assurance Using SpreadsheetsIntroducing premium charges, the development of the premium reserve is given by:Vt= ( Vt−1= V + P − P ∗γt+ P ∗ (1 − γ )) ∗ (1 + i ) − C+ ( V + P ∗γ) ∗i− CttttThe maturity benefit isCd= Vd= P ∗ ( 1−γ ) ∗d∑t=1(1 + i)The cash flow to the life office isCF = P ∗γ−tI ttPolicy durationPremium paymentExpected increase in unit valuePremiumInitial commissionPremium charge10 years10 years5% annually100 per year4 % of total premium4 % of each premiumYearFundinChargePremiumInterestMaturitybenefitFundoutChargeCommissionCashflow1 0 100 -4 5 0 -101 4 -40 -362 101 100 -4 10 0 -207 4 0 43 207 100 -4 15 0 -318 4 0 44 318 100 -4 21 0 -435 4 0 45 435 100 -4 27 0 -558 4 0 46 558 100 -4 33 0 -687 4 0 47 687 100 -4 39 0 -822 4 0 48 822 100 -4 46 0 -964 4 0 49 964 100 -4 53 0 -1 114 4 0 410 1 114 100 -4 61 -1 270 0 4 0 42004 9

Profit Test Modelling in Life Assurance Using SpreadsheetsA profit testing study in a spreadsheet environment is normally done vertically the way we have done itup to now. We will however do it horizontally for the remainder of this brief.Policy durationPremium paymentExpected increase in unit valuePremiumInitial commissionPremium charge10 years10 years5% annually100 per year4 % of total premium4 % of each premiumYear 1 2 3 4 5 6 7 8 9 10Fund in 0 101 207 318 435 558 687 822 964 1 114Premium 100 100 100 100 100 100 100 100 100 100Charge -4 -4 -4 -4 -4 -4 -4 -4 -4 -4Interest 5 10 15 21 27 33 39 46 53 61Maturity 0 0 0 0 0 0 0 0 0 -1 270Fund out -101 -207 -318 -435 -558 -687 -822 -964 -1 114 0Charge 4 4 4 4 4 4 4 4 4 4Comm -40 0 0 0 0 0 0 0 0 0Cash flow -36 4 4 4 4 4 4 4 4 4The premium charge is shown twice, as an expense for the policyholder and as an income for the lifeoffice.Let us also look at the accumulated cash flow at time t where t

Profit Test Modelling in Life Assurance Using SpreadsheetsThe general formula for calculation of Net Present Value as per the beginning of year1 isNPV ( X0... Xn)=d∑k = 1Xk∗ vk −1=d −1∑k=0Xk + 1∗ vkwhere1v = 1 + ris the discount factor and r is the discount rate.Xk= cash flow at time k (i.e. at beginning of year k)One may use the NPV function of Excel to do this calculation.One must decide on an appropriate discount interest rate. This discount rate shouldtake into account the cost of money for the life office. If the commission is financedthrough new equity in the company, the cost of money is the return the shareholderswant on this new equity (including tax). This might be 15%. If the life office has idlefunds which would otherwise be invested, the discount rate should take into accountthe income which would have been received in such an alternative investment, whereone should include the risk involved with investing funds into initial commissions. Ifthe initial commission investment is funded through reinsurance, the cost of thisreinsurance could be used for the discount rate.We will here assume a discount rate of 10%, giving us a discount factor v=0.90909.Policy duration 10 years Discount rate 10%Premium payment 10 years NPV -13Expected increase in unit value 5% annuallyPremium100 per yearInitial commission4 % of total premiumPremium charge4 % of each premiumYear 1 2 3 4 5 6 7 8 9 10Fund in 0 101 207 318 435 558 687 822 964 1 114Premium 100 100 100 100 100 100 100 100 100 100Charge -4 -4 -4 -4 -4 -4 -4 -4 -4 -4Interest 5 10 15 21 27 33 39 46 53 61Maturity 0 0 0 0 0 0 0 0 0 -1 270Fund out -101 -207 -318 -435 -558 -687 -822 -964 -1 114 0Charge 4 4 4 4 4 4 4 4 4 4Comm -40 0 0 0 0 0 0 0 0 0Cash flow -36 4 4 4 4 4 4 4 4 4Accumulated -36 -32 -28 -24 -20 -16 -12 -8 -4 0cash flowDiscount1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424factorDiscounted -36 4 3 3 3 2 2 2 2 2cash flowAccumulateddiscountedcash flow-36 -32 -29 -26 -23 -21 -19 -17 -15 -13We can see that the Accumulated discounted cash flow is equal to –13 at thematurity age. This is the NPV of the cash flow valued at the discount rate of 10%. We2004 11

Profit Test Modelling in Life Assurance Using Spreadsheetsdefine this as our profit and our profit goal is that the profit should be positive (or atleast not negative).The profit could also be calculated directly by using the NPV function in Excel. Please note that theExcel formula assumes that all payments are made in arrears i.e.at the end of the period in question,while we here assume that all payments (except the maturity) are made at the beginning of the period inquestion. The value calculated by Excel must therefore be multiplied by (1+r), in our case 110% in orderto arrive at the right answer. Using the NPV formula in Excel gives the answer –12, which multiplied by110% gives –13 as can be found in the lower right hand corner of the table above.We see that we must use a premium charge greater than 4% in order to break-even,i.e. a profit of zero. We can calculate the premium that is required for a break-evensituation by setting the NPV of future premium charges equal to the initialcommission.This gives:n∑ −t=01γ∗ P ∗ vt= It= α ∗ d ∗ P (n=d).where n=10, d=10, i=5% and a=4%,We get9∑γ ∗ P ∗ vt=09γ ∗∑t=0v101−vγ ∗ = 0.41−vγ ∗ 6 .76 = 0.4tt= α ∗ d ∗ P= α ∗ d = 0.4orThe premium charge that will give break-even isγ = 0.059The same answer could have been found by once again using the Goal Seek or Solver.2.6. Portfolios, model pointsWe have up to now looked at a 10-year policy. Let us look at a 5-year policy,assuming an initial commission of 5.9% of the total premium.2004 12

Profit Test Modelling in Life Assurance Using SpreadsheetsPolicy duration 5 years Discount rate 10%Premium payment 5 years NPV 5Expected increase in unit value 5% annuallyPremium100 per yearInitial commission4 % of total premiumPremium charge5.9 % of each premiumYear 1 2 3 4 5 6 7 8 9 10Fund in 0 99 203 312 427 0 0 0 0 0Premium 100 100 100 100 100 0 0 0 0 0Charge -6 -6 -6 -6 -6 0 0 0 0 0Interest 5 10 15 21 26 0 0 0 0 0Maturity 0 0 0 0 -548 0 0 0 0 0Fund out -99 -203 -312 -427 0 0 0 0 0 0Charge 6 6 6 6 6 0 0 0 0 0Comm -20 0 0 0 0 0 0 0 0 0Cash flow -14 6 6 6 6 0 0 0 0 0Accumulated -14 -8 -2 4 10 10 10 10 10 10cash flowDiscount 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424factorDiscounted -14 5 5 4 4 0 0 0 0 0cash flowAccumulateddiscountedcash flow-14 -9 -4 1 5 5 5 5 5 5The table above shows that the 5-year policy gives a profit of 5. If we insteadcalculate the profit of a 15-year policy, we would make a loss of 11. For a 20-yearpolicy, we make a loss of 25. The initial commission formula gets more expensive forlong term policies. Let us therefore assume that the agent gets commission for onlythe first 20 premiums, even if the policy duration is longer. This is a common way toconstruct sales commission scales. The initial commission is given by:I= ∗ min(20; d)∗ P1αLet us now assume that the maturity age is 65 years, x is the age of the assured, (i.e.d=65-x) and that the minimum initial age is 20. Let us also assume that thedistribution of initial age will be even over the age band 20-64 years. We could thencalculate the profitability of each initial age and sum the result over all ages as:NPV =P ∗64∑⎜(∑x=20⎛⎝65−xt=1vt−1⎞∗γ) −α∗ min(20;65 − x)⎟⎠One could in principle solve for ? from the above expression by setting the total tozero to get the break-even situation.64 65−x⎛ t−1⎞NPV = ∑⎜(∑v∗γ) −α∗ min(20;65 − x)⎟ = 0x=20⎝t=1⎠A rearrangement of the terms givesα ∗64∑min(20;65 − x)=64 65−x∑ ∑x= 20x=20 t=1and thenvt − 1∗γ2004 13

Profit Test Modelling in Life Assurance Using Spreadsheetsγα ∗64∑x=20=64min(20;65 − x)65−x∑ ∑x=20 t=1vt−1This could however be a bit complicated to handle. Another problem is that, by usingthis complex formula, one can not differentiate the profitable from the non-profitablepolicies. One gets a much better view of the situation by studying the differentpolicies one by one. We study therefore the expression for calculating the profit for acohort of policies65xNPV = ∑ − t−1( v ∗γ ∗ P)−α∗ P ∗ min(20;65 − x)for x = 20, 21,…,64.t=1This is straightforward but could be cumbersome. One common way to simplify thecalculations is to use model points. The profits of a 25-year and a 26-year policy arerather equal and the 25-year policy could represent both a 24-year and a 26-yearpolicy. We therefore choose a number of model policies that will represent the rest.Using this principle and letting each 5-year age bands be represented by its middlepoint, we thus study65xNPV = ∑ − t−1P ∗γ ∗ ( v ) − P ∗α∗ min(20;65 − x)for x = 22, 27,…,62.t=1This gives:Maturity age 65 years Discount rate 10%Expected increase in unit value 5% annuallyPremium 100 per year Max commission years 20Initial commission 4 % of total premium max 80%Premium charge6 % of each premiumAge Policy Profitduration62 3 457 8 352 13 -547 18 -1842 23 -2137 28 -1932 33 -1727 38 -1622 43 -15Total -103This calculation could be done by testing the policy durations one at a time. A quicker way is to use theData Table function in Excel which gives all values at the same time. Please note that tables aredynamically updated if this function is not turned off (Tools, Calculation, Automatic except tables), whyhaving large tables might lead to heavy update times.We find:2004 14

Profit Test Modelling in Life Assurance Using Spreadsheets6265−x⎛t−1⎞NPV = ∑ ⎜100∗ 5.9% ∗ ( ∑v ) −100∗α ∗ min(20;65 − x)⎟ = −103(a)x=22,27... ⎝t=1⎠The result is not good, but it is hard to see how bad it is. We want to know how muchwe need to increase the premium charge in order to go break-even. We want to finda k such that the profit is equal to zero, i.e.:6265−x⎛t−1⎞NPV = ∑ ⎜100∗ (5.9% + k)∗ ( ∑v) −100∗α ∗ min(20;65 − x)⎟ = 0 (b)x=22,27... ⎝t=1⎠Inserting expression (a) in (b) gives6265−x⎛t−1⎞∑ ⎜100(5.9% + k ) ∗ ( ∑v) ⎟ −x=22,27... ⎝t=1 ⎠This then gives64∑x=20...Furtherk64x=20⎛⎜100∗5.9%∗ (⎝65−x∗ ∑ ∑65−x6265 1⎛1 ⎞ ⎛⎜100( ) ⎟ ≈ ⎜100∗ ∗⎝∗ ∗ ∑ ∑−t−k ∑vk vt=1 ⎠ x=22,27... ⎝t=1≈ ≈6465−x⎛− ⎞64t 1NPV ( P)∑x=20103⎜100∗ (⎝∑t=1v) ⎟⎠∑x=20103t−1t=1⎞⎟ = 103⎠t ⎞v ) ⎟ = 103⎠We therefore also include the net present value of the premiums paid for each policyin our table.This givesAge PolicydurationProfit NPV ofpremium62 3 4 27457 8 3 58752 13 -5 78147 18 -18 90242 23 -21 97737 28 -19 1 02432 33 -17 1 05327 38 -16 1 07122 43 -15 1 082Total -103 7 750k=103 =77500.0133The loss is thus -1.33% of the NPV of the total premium. Let us therefore increasethe premium charge with 1.4% to 7.4%:2004 15

Profit Test Modelling in Life Assurance Using SpreadsheetsMaturity age 65 years Discount rate 10%Expected increase in unit value 5% annuallyPremium 100 per year Max commission years 20Initial commission 4 % of total premium max 80%Premium charge7.4 % of each premiumAge PolicydurationProfit NPV ofpremium62 3 8 27457 8 11 58752 13 6 78147 18 -5 90242 23 -8 97737 28 -4 1 02432 33 -2 1 05327 38 -1 1 07122 43 0 1 082Total 5 7 750We find that the portfolio has a break-even point with a premium charge of 7.4%Let us now assume that we expect to sell more of some policies and less of others.Most of our new clients are expected to be around 35 years and few are 20 or 60years. We include this in our calculation by weighting the different policies by theirexpected sales figures:64 65−x⎛ t−1⎞NPV = P ∗ ∑Wx⎜(∑v∗γ) −α∗ min(20;65 − x)⎟x=20 ⎝ t=1⎠Assume that our portfolio has an average duration of 23 years and has an agedistribution as shown in the table below:Age PolicydurationNumber ofpolicies62 3 10057 8 20052 13 30047 18 40042 23 50037 28 40032 33 30027 38 20022 43 100Total 3 0002004 16

Profit Test Modelling in Life Assurance Using SpreadsheetsThis gives the following results:PolicydurationNumber ofpoliciesProfit perpolicyNPVpremium perpolicyTotal profitTotal NPV ofpremium3 100 8 274 824 27 3558 200 11 587 2 285 117 36813 300 6 781 1 746 234 41118 400 -5 902 -2 096 360 86223 500 -8 977 -3 845 488 57728 400 -4 1 024 -1 698 409 48933 300 -2 1 053 -631 315 79138 200 -1 1 071 -155 214 11843 100 0 1 082 5 108 174Total 3 000 -3 565 2 276 146The figures in the column Profit per policy are rounded to the nearest integer. When calculating the totalprofit, non-rounded figures are used.We have here more of the non-profitable policies and less of the profitable policies.The NPV of the loss is only 0.16% of the NPV of the total premium, why an increaseof the premium charge of 0.2% should be enough to make the portfolio profitable. Weincrease the premium charge to 7.6%.Maturity age 65 years Discount rate 10%Expected increase in unit value 5% annuallyPremium 100 per year Max commission years 20Initial commission 4 % of total premium max 80%Premium charge7.6 % of each premiumPolicydurationNumber ofpoliciesProfit perpolicyNPVpremium perpolicyTotal profitTotal NPV ofpremium3 100 9 274 879 27 3558 200 13 587 2 520 117 36813 300 7 781 2 215 234 41118 400 -3 902 -1 374 360 86223 500 -6 977 -2 868 488 57728 400 -2 1 024 -879 409 48933 300 0 1 053 0 315 79138 200 1 1 071 273 214 11843 100 2 1 082 221 108 174Total 3 000 987 2 276 146As shown in the previous table, we have arrived at a small profit of 987.In the real world, you might not know the actual age distribution of the portfolio. It istherefore often a good idea to test different reasonably realistic age distributions inthe portfolio and choose the least favourable. In our case, we assume that we will selleither the evenly distributed portfolio or the one with the weight on duration 23 andwe choose the latter one and thus the premium charge of 7.6%.We discussed in section 2.5. the choice of discount rate. The result that we arrive atis dependent on the discount rate chosen.Problem: How would the profitability be with a discount rate of 12%2004 17

Profit Test Modelling in Life Assurance Using SpreadsheetsAnswer: There will be a loss of 0.66% of NPV of total premiums. A high discountmakes it more expensive to have high initial costs.2.7. Fixed costsUp to now, we have only included the commissions to the sales agents as expenses.These commissions are defined to be proportional to premium volume, why it doesnot matter if we have sold small or large policies.Let us now assume that we have an initial fixed expense of 10 for each new policy.The introduction of this new expense leads to a need of increase in charges. Onepossibility could be to introduce a policy charge of the same amount as the policyexpense. Another would be to increase the premium charge. We will choose thelatter alternative. We therefore now want to determine how much we need toincrease the premium charge to offset this expense.With a fixed cost, large policies will be more profitable than small policies. We willinvestigate the effect on a portfolio of policies with different premium. The examplebelow shows the case for one policy with a premium of 100.Policy duration 10 years Discount rate 10%Expected increase in unit value 5% annually NPV of profit -9Premium 100 per year NPV of premium 68Initial commission 4 % of total premium max 80%Internal initial expenses 10 per policy Max commission years 20Premium charge7.6 % of each premiumYear 1 2 3 4 5 6 7 8 9 10Fund in 0 97 199 306 418 536 660 790 926 1 070Premium 100 100 100 100 100 100 100 100 100 100Charge -8 -8 -8 -8 -8 -8 -8 -8 -8 -8Interest 5 9 15 20 26 31 38 44 51 58MaturityFund out0 0 0 0 0 0 0 0 0-1220-1070 0-97 -199 -306 -418 -536 -660 -790 -926Charge 8 8 8 8 8 8 8 8 8 8Comm -40 0 0 0 0 0 0 0 0 0Internalexpenses -10 0 0 0 0 0 0 0 0 0Cash flow -42 8 8 8 8 8 8 8 8 8Accumulatedcash flow -42 -35 -27 -20 -12 -4 3 11 18 26Discountfactor 1 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424Discountedcash flow -42 7 6 6 5 5 4 4 4 3Accumulateddiscountedcash flow -42 -35 -29 -23 -18 -14 -9 -5 -2 12004 18

Profit Test Modelling in Life Assurance Using SpreadsheetsIf we do this calculation for different policy premiums and durations, we get:Policy duration x years Discount rate 10%Expected increase in unit value 5% annually NPV of profit -9Premium 100 per year NPV of premium 68Initial commission 4 % of total premium max 80%Internal initial expenses y per policy Max commission years 20Premium charge7.6 % of each premiumProfitAnnual premiumDuration 10 40 100 250 10003 -9 -6 -1 12 788 -9 -5 3 21 11613 -9 -7 -3 8 6418 -10 -11 -13 -19 -4423 -11 -12 -16 -24 -6728 -10 -11 -12 -15 -3233 -10 -10 -10 -10 -1038 -10 -9 -9 -7 443 -10 -9 -8 -4 12If all policy durations and premiums were evenly distributed, we could just sum up atotal and get –156. Let us now however assume that the policies are expected to bedistributed as follows:10 40 100 250 10003 0.80% 1.60% 1.00% 0.40% 0.20%8 1.60% 3.20% 2.00% 0.80% 0.40%13 2.40% 4.80% 3.00% 1.20% 0.60%18 3.20% 6.40% 4.00% 1.60% 0.80%23 4.00% 8.00% 5.00% 2.00% 1.00%28 3.20% 6.40% 4.00% 1.60% 0.80%33 2.40% 4.80% 3.00% 1.20% 0.60%38 1.60% 3.20% 2.00% 0.80% 0.40%43 0.80% 1.60% 1.00% 0.40% 0.20%As before, we multiply the result for each type of policy with the probability weight ofthat policy in order to arrive at the portfolio probability. We thus multiply the profitmatrix with the distribution matrix and arrive at the following result.Profit 10 40 100 250 10003 -0.07 -0.10 -0.01 0.05 0.168 -0.14 -0.16 0.05 0.17 0.4613 -0.22 -0.34 -0.08 0.10 0.3818 -0.33 -0.73 -0.54 -0.30 -0.3523 -0.42 -0.98 -0.79 -0.49 -0.6728 -0.33 -0.70 -0.49 -0.25 -0.2633 -0.24 -0.48 -0.30 -0.12 -0.0638 -0.16 -0.30 -0.17 -0.05 0.0143 -0.08 -0.15 -0.08 -0.02 0.02Total –9.53For the premium, we correspondingly multiply the premium per policy with the weight:2004 19

Profit Test Modelling in Life Assurance Using SpreadsheetsPremium 10 40 100 250 10003 0.22 1.75 2.74 2.74 5.478 0.94 7.51 11.74 11.74 23.4713 1.88 15.00 23.44 23.44 46.8818 2.89 23.10 36.09 36.09 72.1723 3.91 31.27 48.86 48.86 97.7228 3.28 26.21 40.95 40.95 81.9033 2.53 20.21 31.58 31.58 63.1638 1.71 13.70 21.41 21.41 42.8243 0.87 6.92 10.82 10.82 21.63Total 1074The profit in relation to the premium is -9.53/1074 = –0.9%. Let us try with a premiumcharge of 8.5%. We get a profit very close to zero as expected. We have thus foundour break-even point.2004 20