APPENDIX 21
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
Background
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 25year 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 interestfree 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)
reposted 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 nonlawyer (albeit an educated one). Once again, standard
practice within the industry probably makes the intended meaning
clear.
APRs
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 onedimensional
projection from a highdimensional 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 www.creditcardsorter.co.uk.
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.
Transparency
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 onedimensional 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
—Cash advance interest rate
—Cash advance handling fee
—Date from which interest is charged
—Order in which repayments are applied
—Tariff of charges for breaking various
aspects of the credit agreement
—Purchase protection, travel insurance
and other insurances
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 interestfree 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 interestfree 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.
Conclusion
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.
EXAMPLE CALCULATION
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
121.069 = 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 121.179 = 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 interestfree
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.
EXTRACTS FROM TERMS AND CONDITIONS OF CARD:
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  1.385%
 21.4% APR  1.462%

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.
