AN APPLICATION OF LINEAR PROGRAMMINGTO LOAN PLANS AND BONUS FUNDING

by

P. E. B. FORD

INTRODUCTION

The first four sections of this note discuss loan plans and bonusfunding generally, and some simple examples are shown. Section Vand the appendix describe a more powerful method of solutionusing linear programming techniques, and the final section drawssome practical conclusions.

No attempt has been made to describe the basic theories oflinear programming. I am indebted for the slight knowledge of thesubject I possess to the text-book mentioned in Section V; the objectof this note is to demonstrate that linear programming methods canbe applied in a somewhat unusual context to solve loan plan andbonus funding problems. Any errors in the technical content of thisnote are due to my misapplication of the text-book methods ratherthan to the methods themselves.

I. GENERAL DESCRIPTION

Many life assurance policies have been taken out in recent yearswith the primary object of providing a series of relatively large andpossibly uneven sums of money at certain specified future dates. Theprovision for future school fees for a family with several children isan obvious example.

Three possible methods of achieving this are:

(a) A series of loans on security of the policy, interest paymentseither being paid separately or by taking further loans.

(b) The cashing of declared reversionary bonus on a with-profitspolicy.

(c) The payment of maturity proceeds by instalments for anendowment assurance.

186

APPLICATION OF LINEAR PROGRAMMING 187

These three methods can usually be combined to form a paymentsplan that will meet the policyholder's requirements. Item (c) willnot vary except for changes in declared bonus levels, and will not beconsidered any further in this note. However items (a) and (jb) are toa great extent under the policyholder's control, and interact on eachother.

The problem is therefore to produce a pattern of loans and cashbonuses which will provide the desired series of future paymentswhilst retaining the maximum net proceeds for the policyholder.

We define these mixed patterns of payments as 'strategies', and thestrategy which results in maximum net proceeds to the policyholderas the 'optimum strategy'. These somewhat odd definitions are chosento link up as naturally as possible the practical problem we areinvestigating with the linear programming method of solutiondescribed in the appendix.

The loan interest and future bonus illustration rates and the cashbonus factors depend on external and to some extent independentconsiderations. However, their relative sizes may be at such levels asto leave the choice of raising money by loans or cash bonusesclosely balanced, and it is in this state, which we define as a 'criticalcombination', that the optimum strategy may well not be entirelyobvious. The conditions likely to create a critical combination andits resulting optimum strategy are considered later in this note.

In most circumstances the optimum strategy will in fact be quiteobvious,and to use linear programming methods would be to take asledgehammer to crack a nut; the simple methods of section IVwould be adequate in most situations to provide a solution. Occasion-ally, however, a recalcitrant nut needs a powerful tool to move it,and the general solution of the appendix is submitted with this inview.

II. SOME PRACTICAL RESTRICTIONS

Before examining some actual examples, there are certain con-straints to be noted.

1. The accumulated total of loans is normally kept within apercentage of the surrender value of the policy; this margin isintended to cover arrears of loan interest and office costs in the eventof default. In this note the limit is assumed to be a level 80% of thesurrender value at all durations.

P. E. B. FORD188

Since the surrender value of a with-profit policy will include anallowance for the cash value of attaching bonuses, any cashing ofthose bonuses will reduce the maximum loan level and may indeedrequire some existing loan to be repaid.

2. Reversionary bonus may not be cashed until it has been declared,and once it has been cashed, it may not normally be re-instated.Accumulated loans, on the other hand, may be wholly or partlyrepaid when the policyholder wishes.

3. As a result of the 1972 Finance Act, tax relief is allowed on loaninterest in excess of £35; this interest can therefore be treated on asemi-net basis. However, loan interest has now been changed from anet to a gross basis and partially back again in the last four years,and there must be some likelihood that this pattern will be repeatedagain in the future. The examples in this note are based on the pre-1972 position, with loan interest being given no tax relief.

4. Cash bonus factors are based on a net interest rate, increasedsomewhat as a precautionary margin against changes in futureinterest levels but also with a deduction made in the case of a com-pound bonus office for future bonus rights.

Although in theory it would be possible to cash sufficient bonus incertain years not only to pay off all existing loans but also to leave aresidue for investment elsewhere, in practice the policy holder wouldneed to earn a relatively high net interest rate on his external invest-ment in order to better the rate implied by the interest in the cashbonus factors combined with an expectation of future compoundbonus.

It seems reasonable to assume therefore for the purpose of thisnote that bonuses may be cashed to the extent of repaying existingaccumulated loans, but that the total of accumulated loans neverbecomes negative.

5. The optimum strategy defined in section I requires the policy-holder to choose the strategy which would leave him with maximumnet proceeds at the end of the payment term. At this stage we askwhat exactly we mean by 'maximum net proceeds'. For a life assuredin poor health, there will be considerable incentive to retain rever-sionary bonus and so raise money by loans. A healthy life, however,will not attach the same importance to life cover. For the purposesof this note, we assume that the life assured is in normal health, andtake as our criterion for 'maximum net proceeds' the maximum valueof the net balance, at the end of the specified payment period, of the

APPLICATION OF LINEAR PROGRAMMING 189

cash value of the then remaining reversionary bonus, less theaccumulated total of loans.

Clearly this criterion is a subjective one in regard to the mortalityused in the cash bonus factor, and different policyholders in practicemay well have different maximum net proceeds for identical strate-gies. This is, however, outside the scope of this note.

The criterion for maximum net proceeds just described will bereferred to as the 'objective function'.

III. EXAMPLES

Before setting up the full method of discovering an optimumstrategy, we first examine a very simple series of examples which willserve to quantify the ideas expressed so far, and help the under-standing of the subsequent theory.

Consider a man who at 25 took out a 40 year with profit endow-ment assurance for a sum assured of £10,000. He requires £100 ineach of the 10th, 11th, 12th, 13th and 14th policy-years. Due topersonal circumstances, he has already built up existing loans of£720 at the start of the 10th year. Reversionary bonuses are alldeclared at an annual compound rate of 4%, any loans are assumedtaken in the middle of a policy year, and any bonus cashed in thatyear is assumed to be taken before the loan is made. Reversionarybonus attaching immediately before the 10th premium has beenpaid is £4233-10.

In calculating the value of the objective function we make thearbitrary assumption that half a year's interest is credited up to theend of the 14th policy year, and the reversionary bonus then existingis cashed using the factor for 15 premiums paid.

Premiums paidCash bonus factorSurrender value%

excluding R.B.

10•33766

900

11•34957

10.25

12.36194

11.60

13.37477

1300

14•38809

14.50

15•40191

In order to produce a critical combination from the above data(remembering that we are in this note ignoring proposed tax relief onloan interest) we need to use artificially selected rates of loan interest.We select three rates.

(A) 8% (B) 7.69% (C)7%

P. E. B. FORD190

With each of these loan interest rates, we combine three types ofstrategy, namely:

(1) R.B. cashed throughout (plus loans only if supplementationneeded).

(2) Loans taken throughout (plus R.B. cashed only if loansexceed the 80% surrender value limit).

(3) A mixture of loans and cash bonus.

In situation A, no mixed strategy is apparent that can improve onthe value of the objective function produced by the straight cashbonus strategy (Al) (which also cashes additional R.B. at outset torepay the existing loan). This is therefore put forward as the optimumstrategy for situation A.

In B, however, a mixed strategy (B3) of loans and cash bonuses isfound to produce a higher objective function than (Bl) or (B2).(B3) is therefore submitted to be the optimum strategy for situationB. In C the optimum strategy (C2) comes from loans throughout.

(Al) 8%—All bonusPremiums paidTotal R.B. before declarationNew R.B.R.B. cashedTotal R.B. after cashing80% of surrender valueTotal loanNew loanNew total loanInterest on total loanValue of R.B. cashedNew loan for cashTotal proceedsTotal R.B. at end of 14th yearCash value of total R.B. at endTotal loan at end of 14th yearObjective function

of 14th year = 3319.9 x-40191 = 1334.3= 0

104233.1569.32428.52373.91361720

-72000

820-720100.

112373.9495028612582.81542

0000

1000

100= 3319.9

122582.8503.3276.32809.81742

0000

1000

100

132809.8512.4266.83055.41956

0000

1000

100

143055.4522.2257.7

3319.92190

0000

1000

100

= 1334.3

Similarly:(A2) 8%—All loansPremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

100

1007316.81628.81311.9

110

100

120

100

130

100

140

100

(A3) 8%—A possible mixturePremiums paid 10 11 12 13 14

APPLICATION OF LINEAR PROGRAMMING 191

0100

0100

0100

1000

820-7204153.9337.6

1331.9

Value of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

(Bl) 7.69%—All bonusPremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

10820

-7203319.9

01334.3

11100

0

12100

0

13100

0

14100

0

(B2) 7.69%—All loamPremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

100

1007316.8161101329.7

110

100

120

100

130

100

140

100

(B3) 7.69%—MixturePremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

10820

-72033201

01334.4

11100

0

120

100

130

100

14324

-224

{C1) 7.00%—All bonusPremium paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

10820

-72033190

01334.3

11100

0

12100

0

13100

0

14100

0

(C2) 7%—All loansPremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

100

1007316.8157201368.7

110

100

120

100

130

100

140

100

(C3) 7%—MixturePremiums paidValue of R.B. cashedNew loan for cashTotal R.B. at end =Total loan at end =Objective function =

10820

-7204153.9

332.71336.8

11100

0

120

100

130

100

140

100

D

192 P. E. B. FORD

IV. THE CRITICAL COMBINATION

We now investigate more closely the conditions that cause acritical combination to occur.

1. The maximum cash M that can be taken at any particular timeis:

Cash value of existing R.B.80% surrender value corpustotal existing loan

+(B say)(•8xVsay)(L say)

(where surrender value corpus is defined as the surrender valueexcluding the part arising from bonus); i.e. M = B + .8V—L.

If the actual cash C that is needed is such that M C > .8(B+V) - Lthen at least some of that cash must come from cash bonus. If,however, C<.8(B+V)—L, then it will be possible to produce Centirely from a loan (although it may not be the best way for thepolicyholder).

If C>M then the full cash will not be available.2. We note that if we can produce C out of a loan, and existing

R.B. is present, then we can always exchange some loan for anequal amount of cash bonus, since a release of loan of -8B makesavailable bonus of cash value B.

3. Although it may be a better strategy to cash bonus, we may beforced in early years to take some loans (which may possibly berepaid later) if bonus has not built up sufficiently quickly to providethe required amounts to be cashed.

4. Although we may wish to take loans in order to defer cashingbonus, the 80% of surrender value restriction may force us in fact tocash at least some bonus in early years.

Let us suppose for the moment that these restrictions do not apply.We require a cash sum of 1 in year x, there being (n—x) furtherconsecutive payments. If we cash bonus in year x, then the objectivefunction at the end of n years will be reduced by (1 +b)n-x An+1IAX,since we will be cashing reversionary bonus of 1/Ax which wouldotherwise have accumulated to (1 +b)n-x

/Ax at the end of the period(A, being the cash bonus factor in year x).

If, alternatively, we decide to defer cashing this bonus until year y(which may mean that it is still in reversion at the end of the paymentperiod) and take a loan at rate i instead, then we would need in year

An+l/Ax,

Ax

APPLICATION OF LINEAR PROGRAMMING 193

y to cash sufficient bonus to repay (l + i)y x. We would thereforeneed to cash (l + i)y-x

/Ax reversionary bonus in year y, whichotherwise would have accumulated to (l+b)n-y(l + i)y-x/Ax at theend of the period, and would hence reduce the objective function by

Then if e(x, y) < 1, it will lead to a lesser reduction in the objectivefunction if we take a loan in year x and cash bonus to repay it inyear y than if we cash bonus immediately in year x.

If year y is in fact beyond the final payment year, then e(x, y) willneed an adjusted definition to allow for half a year's loan interest inthe final year.

We therefore define e(x, y) as above for x^y^n and put

for y>n.Then if Y(x, ri) is the year in which bonus is to be cashed in order

to meet a payment of 1 in the xth year of an w-year sequence ofpayments, we take

e(x, Y(x, n)) = Min {e(x, y)} for y = x, x+1 . . . , n+1

noting that y = x implies immediate cashing of bonus and y = n +1implies a loan taken and held to the end of the payment period.

Considering Y(x, ri) for x = 1, 2 . . . n, we build up the optimumstrategy for the required series of payments.

So far we have assumed that the restrictions at the start of thissection are not operative; if this were always so, then we would havea simple method available for finding an optimum strategy.

When, however, surrender value and total bonus constraints arelikely to be relevant, then we will need a more powerful method ofsolution; a possible method is described in section V.

To illustrate the above theory, we examine example (B3) of sectionIII, which found its optimum strategy by cashing bonus in years1, 2 and 5 and by taking loans in years 3 and 4. Setting out thecriteria in tabular form, we show e(x, y) as

P. E. B. FORD

Bonus cashed in year y

194

Payment yearx12345

1

1

2

100021

3

1000310001

1

4

1000310001

11

5

10003100010.999970.99994

1

In force at end6

1002910028100271.002710027

and find the minimum value of each row.Thus we cash bonus immediately for payment years 1, 2 and 5;

for payment years 3 and 4 we take loans, and cash sufficient extrabonus in year 5 to repay those loans plus their accrued interest.

It will be noticed that this solution is independent of the size ofthe payments required each year. They only become relevant whenthe other constraints start to come into force and cause these inde-pendent rows to interact on each other.

V. GENERAL METHOD OF SOLVING OPTIMUM STRATEGY

In order to investigate this problem systematically, we construct aroutine method for maximizing the objective function, subject tocertain constraints. These constraints are inequalities in which thevariables are non-negative and linearly connected.

This is a typical Linear Programming situation and the Simplexmethod of solution is suitable to use. The theory is clearly describedin An Introduction to Linear Programming and the Theory of Gamesby S. Vajda, and its application to our present problem is describedin the appendix to this note. Once the first tableau has been con-structed, the method of solution is straightforward, although somerather laborious arithmetic may be involved.

Having set up the standard method of solution, our next problemis how to use its results, and we quickly find that our investigationwhich started as a ramble through the actuarial countryside has ledus to a place where it is difficult to see the wood for the trees, sincethere are a large number of variables concerned in the generalproblem, namely:

1. The size of cash payments in relation to the size of the policy.2. The number of cash payments.3. The type of policy.

APPLICATION OF LINEAR PROGRAMMING 195

4. The duration in force before payments start.5. The level of past and future bonuses.6. The size of the total loan at outset of the cash payments.7. The cash bonus factors.8. The loan interest rate and its associated rate of tax relief.9. The scale of surrender values.

10. The percentage of surrender values allowed to support loans.11. The amount of bonus cashed prior to the start of the cash

payments.

All these items can affect the shape of the optimum strategy.The reader may like to try the example given in section III B (i.e.

with loan interest at 7.69%) but with cash bonus factors taken as:

Premiums paidCash bonus factor

10•35100

11•35287

12.36908

13•37104

14•38809

15•39800

The optimum strategy for this plan cashes bonus in the first yearand thereafter takes loans and cash bonus in alternate years.

VI. CONCLUSION

While it is clearly sensible for the policyholder to search for hisoptimum strategy, it is not so obvious that an office has any obliga-tion to find that strategy for him, bearing in mind that the beststrategy for one policyholder will be the one that is worst for theremainder of the policyholders, and also that, since the problem is adynamic one, the policyholder should in theory require revision ofhis future strategy every time that for instance bonus or loan interestrates or general taxation legislation alter.

The office has a duty to all its policyholders (and possibly share-holders) to ensure that the cost of quoting and later of administeringthese strategies is kept as low as possible. Moreover, loan interestrates are often held artificially low by some offices as a service topolicyholders in need.

There is a basic conflict here between on the one hand providinga first-class service to any policyholder who may require it, and onthe other hand of ensuring that a few people do not benefit unduly atthe expense of their fellow policyholders. Loan plan and bonus-funding strategies are in this context only a part of the much widerpicture formed by all types of life assurance enquiries.

Clearly there must be considerable variation between offices in

P. E. B. FORD196

their treatment of this situation; the problem has been accentuated inrecent years owing to

(a) Sophisticated advice given to the public by the financial pressin the treatment of life assurance as an investment medium.

(b) The effect of inflation on the internal costs of an office.

One solution would be to offer without charge a standard range ofcalculations, but to impose a scale of charges for more complicatedcalculations.

This opens up a subject which would take a separate paper todiscuss adequately. However, in the case of loan plans and bonusfunding, it is worth noting that in section III the objective functionsof a straight cash bonus strategy (Bl) and a straight loan strategy(B2) differ by very little from the optimum (B3). Where thereforethere is a critical combination of loans and bonuses present, then astrategy using as far as possible either straight cash bonuses orstraight loans to provide the required payments should produce aresult differing by only a small amount from the optimum, andoffices might feel that this compromise solution would hold asatisfactory balance between the conflicting demands of the indi-vidual and his fellow policyholders. Whether or not even thesesimpler sets of calculation should be included in the free range ofstandard calculations would of course be a matter of judgment forindividual offices.

APPENDIXDefinitions

1. Loans are assumed taken in the middle of a policy year, afterany bonus is cashed in that year.

2. Surrender value factors and cash bonus factors do not varyduring a policy year.

3. New reversionary bonus is declared annually as each premiumis paid; this is assumed to be at the start of each policy year, and thebonus compounds on the Sum Assured and any reversionary bonusstill attaching at that point of time.

4. In the final year in which cash is required, the final accumu-lated proceeds are formed by adding half a year's interest to theaccumulated loan total (including any loan taken in the final year).Reversionary bonus existing at the end of the final year but beforeincluding the new declaration then due, is valued by the cash bonus

APPLICATION OF LINEAR PROGRAMMING 197

factor for the following year, a maturing endowment having afactor of 1.

5. Existing loans can be repaid by cashing additional bonus, butcannot become negative.

6. Existing reversionary bonus cannot be incremented by rein-stating bonuses previously cashed, nor can bonus be cashed to suchan extent that existing reversionary bonus becomes negative.

7. The basic part of the surrender value, ignoring any bonus, iscalled the 'corpus' of the surrender value.

8. Interest on the total accumulated loans is paid by taking furtherloans.

9. The rate of interest on all loans is assumed to be constant.10. The rate of future bonus is assumed constant.

SymbolsMT

bi

SBT

= Cash payment required in year T= Annual rate of compound bonus= Annual rate of interest on loans= Sum assured= Reversionary bonus existing at the start of year T, after

the new bonus for year T has been declared but beforeany cashing has occurred in that year.

VT = Corpus surrender value in year TAT = Cash bonus factor in year TjiT = Proportion of the cash payment in year T provided by

cash bonusLT = Total accumulated loan in middle of year T, including

the loan needed to meet the year's interest but beforeany loan needed in year T to meet the cash payment istaken.

N = Number of cash paymentsT = Count of years in which cash payments needed ( 1 ^ T < N )

OF(x) = Objective function for tableau xX j

T> = Sets of slack variables for converting inequalities toTJ equations.

forT = 1,2. . . N

P. E. B. FORD198

ConstraintsFor T = 1,2 N

Solution needed for μ1,uμ2 . . .UN

AnswerWe convert the two major sets of constraints into equations by

introducing two sets of non-negative slack variables XT and YT.In order to simplify the subsequent work, we also define:

Functions

Then we have N sets of equations, for T = 1, 2 . . . N:

(1)(2)(3)(4)

Since fi, u and Y are all non-negative, therefore all B must also benon-negative, by virtue of equation (4).

Therefore we can eliminate all B from the equations, leaving 3Nequations in 4N unknowns L, X, Y and fi.

Solving for L, X and Yin terms of n, and substituting the functionsdefined earlier, we get, for T = 1, 2 . . . N:

APPLICATION OF LINEAR PROGRAMMING 199

(5)

(6)

(7)

As a first feasible basic solution, put all /iT = 0 i.e. we raise all themoney by loans. Then, for T = 1, 2 . . . N:

Clearly all DT and YT are non-negative, but the XT may be negative.We replace any XT<0 by artificial basic variables Z^ of opposite signto XT in the relevant equation, and add co. GT to the objective func-tion OF(1), where co is taken as being so large that if it remains inthe objective function OF(x) when the final solution is reached, thenthere is no feasible solution. (In other words, we are trying to takeout too high a series of cash payments.)

The relevant equations are solved with the Zj- as basic variable;this results in the old basic variable XT becoming an additionalnon-basic variable in the first tableau and hence it is put equal tozero in the first trial solution.

The objective function for the first tableau is given by:

objective function excluding co.

We now construct the first tableau.As an example, we suppose that Xe<0 and is replaced by the

artificial basic variable Ze, with the consequent adjustments des-cribed earlier. The summation in the bottom row co is only for rowswhere artificial variables have been introduced, of course.

200

Basicvariables Constants

P. E. B. FORD

First Tableau

Mi-

Non-basic variablesμc

.μN Xe.

0

0

0

0

0

0

0

0

- 1

0

OF(X) OBF (1) 0

1

Having constructed the first tableau, we now apply successivetableaux transformations in the usual simplex method until a solutionis reached.

Summarizing this method very briefly:

1. We test if co (constant) = 0. If not, we examine at stage 2 the cobottom row. Else, we examine at stage 2 the OF(X) bottom row.

2. We choose the column of non-basic variables with the mostnegative value in the bottom row (as defined in 1). Suppose thiscolumn is column C.

3. For column C, we investigate for each row v the ratio VXC =((constant in row o)/(value ORC in row v, column C) ) for those rowswhere cRc>0.

Since (constant in row o)^0 by the definition of the basic variablesat outset, we know that vkc > 0 and we choose the row which gives thesmallest VXC, say in row v. Then CRC is called the pivotal value of thetableau.

APPLICATION OF LINEAR PROGRAMMING 201

4. We form the next tableau by exchanging the basic variable inrow v with the non-basic variable at the top of column C, and put:

for e

for g

= all columns (including constant)excluding column C.

= all rows (including OF and to)excluding row v.

for all Je except c[g except D

TABLEAU RULES

1. If no value in the bottom row (of co if co (constant) # 0, else ofOF (x) ) is negative, we have reached an optimum value.

2. If on reaching an optimum value, there are bottom row valuesequal to 0, we note that there is more than one solution (i.e. we couldchoose more than one pattern of loans and cash bonuses to providethe optimum strategy).

3. If the optimum solution still contains an artificial variable Z,,then (o will be included in the objective function and there will beno solution.

4. If in examining the ratios „!<., we can find no value ORC in columnC that is >0, then there is no finite bound to the increase in theobjective function. With sensible values in our problem, however,this is an unlikely occurrence.

5. If a basic variable = 0, we have a degenerate solution. This willnot be of great interest in this problem.

id)

(a)

(e)

NotesIf a computer program in Fortran is used to ease the calculation,

the elements of the tableaux will probably be held as Real (ratherthan Integer) variables in view of the size of some of those elements.This should be borne in mind when testing for elements equal tozero, since the Real variables may differ very slightly from zero; it issafer therefore to test that they lie within limits very close to zero.

The above method of solution could be extended to deal with anon-uniform pattern of future bonus and interest rates, when itcould be difficult to determine the optimum strategy by simplermeans.