Introduction

1.  The purpose of this paper is to provide a description of the algorithm and formulae used for the calculation of rebate on early settlement of a credit agreement by the Rule of 78 method.  Please see the Office of Fair Trading's booklet "Matters arising during the lifetime of an agreement" for details of the rules on calculating rebates and settlement dates.  Although the algorithm described here has been in use for some time, it has not previously been documented in the same way as the algorithm used for calculating APRs has been in the paper 'A MICROCOMPUTER/CALCULATOR BASED SOLUTION TO THE CALCULATION OF APR' ("the APR Paper").  This paper fills that gap but it does not try to explain the (somewhat suspect) thinking behind the Rule of 78 method.

2.  The Consumer Credit (Rebate on Early Settlement) Regulations 1980 provide the following general formula for the calculation of a rebate where the capital is repaid in instalments:

3.  You can see that the bottom half of the formula represents all the repayments under the agreement and the top half represents those the borrower doesn't pay because they settle early.  What the formula is doing is finding the fraction of the total charge which should be rebated (ie the Bb's values over the Aa's).  It's a common convention in maths to refer to a fraction as u/v and we'll use this here, so we can also say that the rebate is found by:

4.  Although the arithmetic the general formula requires is fairly simple (more so, for example, than the maths required to find an APR) the large number of factors which may have to be taken into account (particularly for long term agreements) and the possibility for error when making the calculation 'by hand' makes a computer based solution desirable in most cases.

5.  The Regulations also provide a simple formula for dealing with cases where the capital and charges are paid in equal instalment at regular intervals.  The application of that formula is fairly straightforward and a computer would probably only be used in those cases for convenience.  It is perhaps worth mentioning however that the complex formula is equally applicable to equal instalment cases (the simple formula is in fact just a special case of the complex formula, see paragraph 15 below) so the algorithm described here can be used for all cases where repayment is made in instalments.

6.  One of the main objectives in creating this algorithm was to provide a facility which can easily be used by those already familiar with the similar program developed for calculating APRs, so the same approach of describing the repayments under a credit agreement in terms of `levels' and `extra payments' will be adopted.  See the APR Paper for further details.

Levels of Repayments

7.  Most credit agreements have as the basis a series of repayments of the same amount made at regular intervals, or a combination of such series.  For example, an agreement might have regular repayments apart from the first or last repayment being of a different value, or it might have a series of repayments of one amount, followed by a series of a higher amount, and then by a series of a lower amount.  We can take advantage of this semi-regularity to reduce the amount of information the user has to enter into our program - and to reduce the number of calculations the program has to carry out.

8.  Take an agreement which is repaid by 24 instalments, the first 18 of £50 and the last 6 of £57.50.  We can describe the repayment pattern with the diagram in Figure 1 below.  Each cross on the diagram represents the amount and time of a repayment.  For obvious reasons I usually describe such a payment pattern as having two levels.

Amount
  (£) ^
      |
57.50 |                  ++++++
50.00 |++++++++++++++++++
      |
      |
      |
      |
      +------------------------->
    0                            time
       1    6    12    18    24

                (Figure 1)

9.  Here I should introduce a formal definition of the term `level' - this is taken straight from the earlier paper on APR calculations:

Point to Note

10.  The first level is regarded as starting at time 'zero' rather than time 1 and the second at time 18 (ie when the last payment in the first level is made).  The idea that the first repayment occurs one time period after the beginning of the level may seem confusing at first but it's used for a number of reasons:-

11.  The general formula given above requires us to add up the multiples of the amount and time of each repayment.  First we'll concentrate on calculating v, the bottom half of our fraction and consider the simplest example of a loan repaid by n instalments of amount A paid at regular intervals.  In that case v would be represented by ...

... which can also be written as ...

12.  There is a general formula for adding together a regular series of numbers such as this, but an explanation of how it is arrived at, tailored to our particular case, may help.  The first step to getting to the general formula is to simply write the summed digits in equation (1) the other way around, which gives ...

... the next step is to add together equations (1) and (2) to give ...

... you can see from this that each of the expressions in the inner brackets, eg (3+n-2) or (n-1+2), in fact comes to (n+1) and, since we know there are n of these terms we know that adding them all up will give us n(n+1), so we can rewrite (3) as follows ...

13.  This formula enables us to calculate v by adding up the multiples of the amounts and times of a level of n repayments of the same amount which starts at the outset of the loan (time zero).

14.  What we need however is a formula which will deal with the more general case of a level of repayments of the same amount which start at a non-zero time.  There's a way to get this.  Look at the diagram in Figure 2.  We need to be able to calculate the sum of the payment and time multiples represented by the crosses.  We can already calculate the multiples for all these payments from 1 to e (the end of the level) - ie both the noughts and the crosses - by applying equation (4) with n = e.  Also we can calculate the multiples for the payments from 1 to b (the beginning of the level) - ie just the noughts - by applying (4) with n = b ...

Amount
  (£) ^
      |

    A |oooooooooooo++++++++++++
      |           .           .
      |           .           .
      |           .           .
      |           .           .
      |

      +------------------------->
    0                            time
       1          b           e

               (Figure 2)

... so, the multiples for the crosses (which we'll call S_lev) can be calculated by subtracting the sum for the noughts from the sum for the noughts and crosses ...

... and this is the basic formula we can use to deal with levels.  Of course in practice an agreement may have several levels of repayments, but since the method requires us to calculate the sum of all the multiples, we can simply add together the results of applying equation (5) to each level.

15.  Although we've only considered v above, our formula can equally be used when calculating u, the top half of our fraction.  The only difference is that the times of the repayments in u are measured from the deferred settlement date rather than the relevant date of the agreement.  So for example, if we have a level which runs from time 15 to time 20 in an agreement where the deferred settlement date is time 10, we can apply (5) with b=15 and e=20 to find the value to be added to v and then apply it again with b=5 and e=10 to find the value to be added to u.

16.  We are not going to be able to describe every agreement we come across in terms of levels.  'Extra' repayments is the term I use for payments which don't fall neatly into a pattern of levels.  They will be described individually to our program by the user entering separately the amount of each extra repayment and the time at which it is made.  Again, since our general formula requires us to add together the multiples of the amounts and times of each repayment to get u (where appropriate) and v, we can simply calculate the multiples for any extra repayments and add them those for levels obtained using (5).

Time Periods

17.  You may have noticed that, unlike the corresponding formulae for APR calculations, those drawn up in this paper make no reference to how the times of repayments should be expressed (ie in days, weeks, months, years, etc).  This is because both u and v in our 'Rule of 78 fraction' are made up of multiples of time periods and, provided we always use the same basis for all the periods, the value of the fraction will always be the same regardless of how we express them.  For example:  5 days divided by 10 days comes to a half (NB:  not half a day) and so does 5 years divided by 10 years.

18.  Consequently, our program will not ask the user for information on the time periods being used.  More sophisticated implementations which, for example, allow the user to describe payments times or the length of levels using dates, will also require this information.

19.  It should be remembered that our formula for levels assumes that there is one repayment at the end of each time period.  So, if the user describes regular monthly repayments to the program using levels, the times of any extra repayments should also be described in months.

Deferment Period

20.  The time periods in use are particularly important when it comes to describing the deferment to the settlement date allowed by the Regulations.  Remember that this is one or two months not one or two payment periods.  For example, when dealing with an agreement with weekly repayments the deferment may be four to five or eight to nine weeks.

Other Factors to be Calculated

21.  So far we have only considered the user entering details of the repayments and how we are going to calculate u and v from them.  In order to find the rebate we also need to know the total charge for credit under the agreement.  This can be arrived at fairly simply by asking the user to enter the amount of the loan and then subtracting it from the total amount payable under the agreement (found by adding up all the repayments entered).

22. It will probably be useful for the program to provide a settlement figure (ie how much the borrower has to pay to settle) as well as a rebate.  In order to calculate this we will also need to know how much the borrower has paid up to (and including - see 'Not Yet Due' below) the time of settlement.  Again, this can be obtained fairly simple by examining the time of each repayment and adding together those which occur on or before the settlement date.

23.  Obviously we also need to know the settlement date.  But to calculate how much was paid we will need to keep a separate record of the settlement date and the deferment to be applied, since only repayments before or on the settlement date will be treated as paid, but only those after the deferred settlement date are included in u.

'Not Yet Due'

24.  I said earlier that we would return to these words.  Under the regulations the settlement date almost always falls on a repayment date and this invites two ways of dealing with that repayment.  We could, for example, assume that the borrower has not made it - so it should be included in the final settlement figure.

25.  However, the Regulations talk in terms of a rebate on the repayments 'not yet due' at the time of settlement.  Since the payment due on the settlement date is not 'not yet due', we have opted to assume that it has been paid for the purposes of calculating the settlement figure - ie we do not include it in the final settlement figure.

26.  Since, in practice, most borrowers will hand over a single sum comprising both the settlement figure and the final repayment, this decision might seem a little perverse.  However the reason for doing it is to keep distinct the additional sum the borrower has to repay to discharge his indebtedness under the agreement (the settlement figure) and, what is after all, a normal contractual repayment which would have been paid at that time even if early settlement had not taken place.

Split Levels

27.  In the case of a level where all the repayments occur after the deferred settlement date we know that all their multiples should be included in u - with the times of the repayments being measured from the deferred settlement date.  Similarly we know that for a level where all the repayments occur on or before the settlement date they should all be regarded as having been paid (so we can calculate the total paid for a level as A(e-b), ie the amount of the repayments in the level multiplied by the number of repayments in the level).

28.  However, one point which is not always immediately obvious is that we cannot guarantee that the settlement date (or the deferred settlement date) will necessarily fall exactly at the end of a level.  We could deal with this by asking the user to describe separately the repayments before and after settlement, but in order to maintain the same basic input format as the APR program I have opted to allow the user to describe all the repayments under the agreement both before and after settlement and then indicate where during those repayments settlement takes place.  It will be up to the program to work out which of the repayments should be included in u (they are all included in v) and which are treated as having been paid.

29.  We can still use our equation (5) for cases where the level straddles the deferred settlement date as shown in Figure 3 below.  However when calculating the sum of the multiples to be added to u, we should use the deferred settlement date (labelled t in Figure 3) in place of the beginning of the level (labelled b) so that only the multiples of the repayments after t are included in u.

30.  Similarly, when it comes to calculating the repayments which have been paid we should include only those up to and including t (ie by calculating A(t-b), ie the amounts of the repayments multiplied by the number of repayments up to and including t).

Amount
 (£) ^
      |
    A |      .ooooo+++++++++
      |      .    .        .
      |      .    .        .
      |      .    .        .
      |      .    .        .
      |

      +-...--------------------->
    0                           time
       1     b    t        e

               (Figure 3)

The Listing

31.  We now have everything we need to approach a calculation.  The following listing is a program applying the method described above.  As with the APR listing in the earlier paper this is kept fairly simple to make it as easy as possible to understand and convert to other languages.  However, rather than use BBC BASIC I have opted to produce it in the more generally available GWBASIC or QBASIC dialect.  Even so, it uses only a few of the more sophisticated facilities those languages offer (longer variable names, a WHILE WEND loop and the use of PRINT USING to provide a tidy output are about all).

32.  If you have received this document in an electronic format (by disc or e-mail) you should be able to simply copy the listing from the text and save it as an ASCII file for loading into GWBASIC or QBASIC (although it may be necessary to carry out some editing).  Of course, as with the APR listing, this will probably only form the starting point of your implementation.  A full implementation would require input validation, calculator and date entry facilities.

33.  Please remember that, although every effort has been made to ensure that the information contained in this paper is correct, because of the many different types of machines, languages and credit agreements in existence, we cannot guarantee that the results obtained using the method and program described here represent a complete and authoritative interpretation of the law.

ANNEX 1

THE ACTUARIAL RULE

1.  This annex shows how to adapt the approach described in the main paper to the Actuarial Rule for calculating the sum due on early settlement of a loan.  The principle on which this method is based is that the borrower, when settling early, should incur the same overall effective rate of charge as they would have done had the agreement run its full course.  Present value methods are used to calculate the settlement figure which, combined with the contractual payments due up to the time of settlement, gives the same rate.

2.  Consequently, the first step in calculating a settlement figure by the Actuarial Rule is to determine what the effective, compound annual rate for the agreement would have been if it had run to full term, based on the facts known at the time of settlement.  This calculation is the same mathematical process as that used to calculate APR, except that any variations in the agreement which took place after the agreement was made but before settlement will be known, and so are taken into account.  If the agreement is one which has fixed charges, or where no variation in charges has occurred before the time of settlement, the agreement's APR could be used in place of this first step.  The methods for making this calculation are covered in detail in the APR Paper.  The implementation described will simply ask the user to enter the rate to be used.

The Formulae

3.  The calculation by the Actuarial Rule is most conveniently dealt with in two stages in the following manner:

The Listing

4.  We now have all the formulae necessary to carry out an actuarial calculation adopting the same general approach as that used for the Rule of 78 method in the main body of this paper.  However there are a few additional points to note:

ANNEX 2

THE REDUCING BALANCE

1.  This annex shows briefly how to apply the Reducing Balance method for calculating the sum due on early settlement of a loan.  The principle on which this method is based is that the borrower is charged interest each period at the current rate on the balance then outstanding and, when settling early, simply pays the current balance.  So, this approach treats any loan rather like a bank or credit card account and could be looked at as an accounting method rather than a method for determining a rebate or settlement figure.

2.  Implementing this method by computer is actually very simple and the listing below correspondingly short.  All that is required is to obtain the opening balance on the account and loan term from the user and then, for each sequence of payments (ie sequences where the rate and payment remain constant) up to the time of settlement, obtain the rate of interest for the period at the current rate, the number or repayments and the current periodic repayment being made by the borrower.  Interest is then added to the balance and the borrower's periodic repayment deducted for each period in the sequence.

The Formulae

3.  The formulae used in this implementation are very straightforward:

The Listing

4.  Please note that this listing applies only the most simple method of determining the interest charge each period.  In practice credit agreements often use more complex methods (such as recording the exact date on what sums are paid or charged to the account and calculating interest on the average daily balance outstanding during the period) and, in such cases, your implementation should apply the method required by the agreement in order to obtain an accurate result.