Select Committee on Treasury Appendices to the Minutes of Evidence


Memorandum submitted by Dr R. E. Hunt, Deputy Director, Isaac Newton Institute for Mathematical Sciences Lecturer in the Department of Applied Mathematics and Theoretical Physics University of Cambridge


  1.  Following the meeting of the Select Committee on Tuesday 18 June, at which the credit card business was discussed, I was asked by various national newspapers (notably the Financial Times) and other media organisations to produce examples of the calculations involved in determining interest charges on these cards. I am a professional mathematician (Deputy Director of the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge, and a Fellow of Christ's College, Cambridge) with interests in personal finance products, and I welcome this opportunity to submit evidence to the Select Committee in this capacity. I have no connection with the banking business.

  2.  At the meeting of the Select Committee on Tuesday 18 June, Mr Harley referred (question 358) to "assiduous research" which would be required in order to pin down every aspect of the calculations involved in credit card interest repayments. My purpose in this memorandum is to indicate the difficulties involved in such research and to comment on whether a customer could be expected to carry it out; to indicate those elements of the charging regime which increase the complexity; and to explain how cards with the same APR can result in different interest charges.

Example calculation

  3.  I have included as an addendum an example of an interest calculation for an extremely simple (and unrealistic) case. This example uses the interest rates and the terms and conditions of one of the UK's most popular cards, issued by one of the "Big Four". For several steps of the calculation, it was necessary to refer to the small print of the terms and conditions.

  4.  Several newspapers have reported that Mr Harley said that it was necessary to use calculus to work out interest repayments. This is a misrepresentation, and is incorrect. Mr Harley's statement (question 358) that "it may be that with assiduous research you can pin down every aspect of the calculus, but I am afraid that the product does not sell on the calculus" uses the word "calculus" not in its modern meaning of "integral and differential calculus" but rather its more traditional meaning of "a system of calculation". In fact, all that is required to perform the calculations is a solid understanding of simple and compound interest, and the ability to interpret the technicalities of the small print.

  5.  Unfortunately, most customers do not possess such an understanding, and so would be unable to complete the calculation themselves. However, neither would they be able to calculate, say, the monthly payments on a repayment mortgage over a 25-year term (unless they were prepared to take the appropriate mathematical formula on trust). In both cases, they rely on the banks to perform the calculation for them. The problem is that with a mortgage the customer can find out the level of repayments in advance of taking out the mortgage, and therefore make a choice in full possession of the facts; whereas with a credit card the repayments are only made clear once the statements start arriving.

Difficulties in the calculation

  6.  The need to read the small print significantly increases the difficulty of the calculation. Various vital facts are buried there: for example, the list of the order in which various parts of the outstanding balance will be paid off, and the monthly interest rates (as opposed to the APRs; the one cannot necessarily be deduced from the other). However, some banks (but not all) include these facts on each statement, and when they do this removes one of the reasons for consulting the small print.

  7.  It is to be applauded that all banks now include an indication of how much interest you should expect to be charged on your next statement if you make only the minimum payment. This mostly obviates the need to perform by hand calculations of the type given in the example.

  8.  Some aspects of the charging regime increase the complexity of the calculation significantly, yet it is not clear why these aspects exist. For instance, cash advances are generally charged at a higher interest rate, together with a handling fee, and without any interest-free period. I cannot see any justification for a higher interest rate, given that the handling fee effectively covers the bank's costs in providing immediate cash, unless customers who make cash withdrawals are considered higher risk. Cash advances could even be treated in the same way as normal purchases, with the handling fee adjusted to reflect any increased costs to the bank, which would greatly increase the transparency to the customer.

  9.  Despite "assiduous research", there are still some obscure aspects of the charging regime which it has been impossible to deduce even from the small print. In these cases, one would need to analyse carefully sample credit card statements to confirm one's guesses. For instance:

    The terms and conditions of the credit card used in the example refer to interest being charged on a "daily basis". No clear indication is given as to whether this daily interest is charged on a simple or compound basis, though a banker would probably state that it's obvious that daily interest is simple, compounded monthly at the statement date, as this is standard practice within the industry. What appears obvious to a banker may not be obvious even to an educated customer.

    Sometimes transactions are posted to an account late; for example, when a transaction has been posted to the wrong customer's account and is subsequently (when the error is detected) re-posted to the correct customer's account. In such cases the transaction can appear on the account several months late. No clear indication is given of the exact method of interest charging in such a case, though again standard practice probably exists for this case. Standard practice may, however, not concur with "fairness" in the eye of the consumer.

  10.  The small print (for the particular card used in the example) also contains slight ambiguities, at least for a non-lawyer (albeit an educated one). Once again, standard practice within the industry probably makes the intended meaning clear.


  11.  The purpose of an APR is to attempt to reflect all fees and interest charges in a single measure. A card with a substantial annual fee but a relatively low interest rate might have the same APR as a card with no annual fee but a higher interest rate. In more complex examples, two cards may differ in many technical aspects, yet the overall APR may be the same. In mathematical terminology, the APR is a one-dimensional projection from a high-dimensional space of credit card charging regimes; it must, therefore, be regarded as only a simple overall measure.

  12.  The calculation of the APR, which is governed by a tightly regulated mathematical definition, assumes a particular spending pattern: namely that the customer takes out the maximum credit available, and makes only minimum repayments while refraining from making any further purchases. The number of customers to whom the APR is directly relevant is therefore very small. In particular, the length of the interest free period is not reflected in the APR at all, though it is of great interest to those customers who regularly pay off their balance in full.

  13.  "Best buy" tables of the kind seen in Sunday newspapers and in financial magazines are generally useless to the majority of consumers; a credit card provider can manoeuvre their product to the top of these tables by adjusting the APR downwards while adjusting other, compensatory, features of the product in the opposite direction.

  14.  To produce "best buy" tables which would be useful, several different tables would be needed for different types of customer. For instance, those who always pay their balance in full; those who usually do so, but sometimes let their balance roll over; those who let their balances roll over for a few months at a time but then pay it off in full; and those who maintain a continuous outstanding balance. Which? magazine recently produced such a set of tables.

  15.  There are also web sites which allow consumers to make comparisons between cards; for instance These sites are usually extensive and give a lot of useful information, certainly sufficient to give the consumer a fully informed choice; but the sheer volume of information is likely to scare off most consumers.


  16.  The Consumers' Association recently published research demonstrating that credit cards with similar APRs can nevertheless charge very different amounts of interest; indeed, cards with identical APRs used by customers with identical spending and repayment patterns can make charges differing by over 40%. This should not be a surprise given the fact that the APR is a one-dimensional measure of the many different aspects of a credit card's charging regime.

  17.  These differences between credit cards are, in general, a result of both diversity and competition in the sector. A list of common differences would include

    —Introductory interest rate

    —Balance transfer interest rate

    —Purchase interest rate

    —Cash advance interest rate

    —Cash advance handling fee

    —Annual fee

    —Credit limit

    —Minimum repayment

    —Date from which interest is charged

    —Interest-free period

    —Order in which repayments are applied

    —Tariff of charges for breaking various aspects of the credit agreement

    —Purchase protection, travel insurance and other insurances


    —Reward schemes

    —Affinity schemes

  18.  It is not surprising given the length of this list of differences that consumers find that the charges made on credit cards lack transparency. Some of these differences are widely promoted on advertising materials (e.g. APRs, cashback, reward schemes) while others are not (e.g. date from which interest is charged [transaction date or posting date], tariff of charges), and are usually to be found only in the small print. Of the latter group, the interest-free period is the most important to customers in terms of potential charges, and transparency would be well served if this were more clearly advertised, a point already accepted at the meeting of the Select Committee on Tuesday 18 June by Mr Crosby (question 343) and others.

  19.  Mr Harley (question 341 of the same meeting) suggested that there are "two choices: either standardisation, so that everyone charges the same interest on the same intervals with the same interest free periods, that is you diminish product choice; or through better communication with customers, more transparency in the literature. Those are the only two polar choices."

  20.  Regardless of how good communication is with customers, they are still likely to be confused by the huge number of differences between credit cards, and will find it hard to make an informed choice. This leaves the option of standardisation. However, it is not necessary to standardise every single aspect of the product (which would indeed diminish product choice); instead, only those aspects which are usually left buried in the small print need be standardised in order to improve transparency. Product choice would not significantly be reduced.

  21.  Therefore, there is scope for improving transparency to the consumer by standardising some of the less highly promoted aspects (for instance, the date from which interest is charged, and potentially the interest-free period), as cards do not in general try to compete on these aspects. There is no reason to standardise those aspects which are highly promoted, thereby still allowing considerable diversity and competition between cards.


  22.  It is indeed possible by "assiduous research" to compute the interest which will be charged to a credit card account. The complexity of the calculation is increased by the need to read the small print. Even with assiduousness, there are a few obscure aspects of the small print which remain unclear.

  23.  The APR is only a crude measure of the potential charges made by a credit card provider. Different consumers need to look at different aspects of the charging regime.

  24.  Several aspects of the terms and conditions vary between credit card providers but are not well advertised, and standardising these in some way would improve transparency without significantly diminishing product choice.


Based on the current terms and conditions of a major credit card issued by one of the "Big Four" banks

  Situation: on 23 July I transfer a balance of £2000 from my old credit card to a new card. On 1 August I buy goods worth £1000. On 2 August I withdraw £200 cash from an ATM. I receive my first statement dated 23 August, and on the payment date of 19 September I repay £1000. What will be the balance on my next statement, dated 23 September?

  This is clearly a very simple example.

Interest rates

  The calculation will involve three monthly interest rates:

    —  Balance transfers: 0.561%. This can be found in the terms and conditions (section 3.2(b)(i)), or can be calculated approximately from the advertised APR of 6.9% (since 12-1.069 = 1.0056, taking monthly compounding into account).

    —  Purchases: 1.385%. Again, this can be found in the terms and conditions (section 3.2(b)(ii)), or can be calculated approximately from the advertised APR of 17.9% (since 12-1.179 = 1.0138).

    —  Cash advances: 1.462%. In this case the APR cannot be used to deduce the monthly rate, because there is a fee involved which has been factored into the APR. The terms and conditions (section 3.2(b)(ii)) must therefore be consulted.

First month of the calculation (23 July - 23 August)

  The balance transfer of £2000 earns interest of £2000 x 0.561% = £11.22.

  Initially, no interest is charged on the purchase (section 5.1), in case I wish to take advantage of the interest-free period. However, when it becomes clear on the payment date of 19 September that I have not paid off the balance in full, interest is charged retrospectively on the entire balance and added to the account (section 5.2). I am charged interest on the purchase on a daily basis from the date of the transaction (section 5.4); it is therefore charged for 22 days, out of 31: £1000 x 22/31 x 1.385% = £9.83. (This amount will not appear on my first statement but only on the second).

  The handling fee for the cash advance is £4 (2%, minimum £2; section 3.2(a)). This forms part of the cash advance balance (section 3.2(b)(ii)). I will be charged interest on the full amount for 21 days: £204 x 21/31 x 1.462% = £2.02.

Second month of the calculation (23 August - 23 September)

  The payment of £1000 will be applied to the balance transfer (section 5.5). I will therefore be charged interest on £2011.22 for 27 days, and on £1011.22 for 4 days: (£2011.22 x 27/31 + £1011.22 x 4/31) x 0.561% = £10.56.

  The standard balance will continue to earn interest for the whole month: £1009.83 x 1.385% = £13.99.

  Similarly for the cash balance: £206.02 x 1.462% = £3.01.

Final balance on 23 September

  The final balance will therefore be £2254.63, of which the balance transfer balance is £1021.78, the standard balance is £1023.82 and the cash balance is £209.03. I will need this breakdown for the next month's calculation.


  3.2 (a)  We will charge a handling fee each time you or any additional cardholder make a cash withdrawal, purchase travellers' cheques or use [our] cheque. The handling fee is 2 per cent of the amount with a minimum fee of £2.00.

  3.2 (b)(i)  If you apply for a card and/or a balance transfer on an application form contained in promotional material offering an APR of 6.9 per cent, we will charge 0.561 per cent (6.9 per cent APR) on the balance transfer fixed until the balance transfer is fully repaid.

  3.2 (b)(ii)  For the standard balance and cash advance balance we will charge the rates we work out as follows:

Standard Balance APR
Standard Balance Monthly Interest Rate
Cash Advance Balance APR (includes the handling fee)
Cash advance Balance Monthly Interest Rate
17.9% APR
21.4% APR

  5.1  We will always charge interest on the cash advance balance and the transfer balance up to the statement date even if you repay them in full on or before the payment date. We will not charge interest on any other items shown in that statement as part of the standard balance if you pay the standard balance in full on or before the payment date.

  5.2  If you do not pay the standard balance in full on or before the payment date, we will charge interest at the standard balance rate on the whole standard balance and add it to your account on the next statement date.

  5.4  We charge interest on a daily basis from the date of the transactions using [our] cheques when we will charge interest from the date the item is put on the account.

  5.5  If you do not pay the statement balance in full on the payment date, we will apply the amount you do pay to reduce what you owe us in the following order:

    against any transfer balances and promotional balances (including interest) which an interest rate applies to for an open ended period;

    against any transfer balances and promotional balances (including interest) which an interest rate applies to for a fixed period;

    against any other interest charges and other charges made under this agreement;

    against the cash advance balance;

    against the standard balance.

  If you have more than one transfer balance or promotional balance those with the lowest interest rate will be paid off before other balances of the same category. If we change this order for any balance transfer or special promotion we will tell you at the time.

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