Consumer Credit Act 1974: Credit Charges and APR

The Office of Fair Trading have published the final draft of a booklet “Consumer Credit Act 1974: Credit Charges and APR”.  This is obtainable as a download from the offices web www.oft.gov.uk. The booklet prints to a large document of some 50 pages and is probably one of the best government publications that I have seen. It sets out to describe the thinking behind the relevant regulations, the tolerances allowed in statement of APR, and the formulas used to calculate it. However, at this point I have to emphasize that there is no straightforward formula to calculate APR. It can only be arrived at by use of what is known as an algorithm of successive comparisons. More on this later. It is worth mentioning that the calculators supplied by various finance companies must return a result by referencing tables of preset values. A simple calculator is incapable of running an algorithm.

Firstly, how do new regulations affect us all? Well very little in seems. Basically, the only difference is the method used to round APR to first decimal. The old way was to round out the second decimal, hence 31.37 would become 31.3. The new method requires proper rounding, hence 31.37 would become 31.4, 31.33 would become 31.3.  However the existing tolerances remain unchanged; an overstatement of up to 1%, an understatement of 0.1% is acceptable. In view of this the figures supplied by existing calculators and rate tables will probably remain within legal parameters.

Calculation of APR

As previously stated, there is no easy way of doing this.

The logic of APR is based on the assumption that when you make a loan, the value of the loan at the time when it is made is the amount lent, and no more. This is called the “Present Value”. The loan with its interest is known as “Future Value”. If the present value of the loan is the amount lent, than it must follow that the present value of the installments must be the same as the present value of the advance. This known as the “Present Value Rule”

 The actual expression of the formula would be very difficult for the average person to understand. 

OK so try it:

Meaning of letters and symbols:

K         is the number identifying a particular advance of credit;

K'         is the number identifying a particular installment;

A_K        is the amount of advance K;

A'_K'       is the amount of installment K';

S         represents the sum of all the terms indicated;

m         is the number of advances of credit;

m'        is the total number of installments;

t_K         is the interval, expressed in years, between the relevant date and the date of advance K;

t_K'         is the interval,  expressed in years, between the relevant date and the date of installment K';

i           is the APR, expressed as a decimal fraction.

 I warned you!

I shall try to express it in a simple way. This is taken from the OFT publication and modified slightly

A typical application of the formula might be in the case of a loan for £1,000 repayable over three years by 36 monthly installments.  The lender charges interest at a flat annual rate of 9% (this means that 9% of £1,000 is payable for each year of the loan and the total interest charge is therefore £270).  The regular installments would therefore be £1,270 ¸ 36 =  £35.28 a month (rounded up to the nearest penny).  There is an additional £50 administration charge, payable when the agreement is made.

The statutory formula would therefore require the interest rate to be found in the following equation:

The term on the left-hand side of the equals sign and the first one on the right-hand side could be written more simply as £1,000 and £50, because they relate to an advance and installment made at the  start of the loan (time ‘zero’).  Any value raised to the power of zero is equal to one, so the bottom halves of these fractions will be equal to one for any value of i.  

Period of installment is the number of installment expressed in years. Hence monthly installments are represented by this figure divided by 12. I.e. 1/12, 2/12 etc

i is the APR expressed as a decimal fraction. 31.3 would become 0.313

You should note that (1 + i)1/12 means 1+ i raised to the power of 1/12 and all the dots represent parts taken out for simplicity, if you haven't gathered this.

People who have had some experience of dealing with credit installment agreements will notice that the principle of 1+2+3+4+5 etc is a reoccurring theme

As can be seen, there is no algebra that can produce a formula for calculating the Interest from the known values advance, installment and number of installments. An algorithm can only do this

An algorithm is a mathematical function, which produces a result by trial and error so to speak. The correct term is an algorithm of successive comparisons. The people cracking the German codes used this type of math at Bletchley Park in the last war. In those days there wasn’t any electronic computers and it all had to be done by hand with paper and pen. Rooms of people would spend days doing a task which a modern computer would be able to do in seconds.

Back to the subject in hand, we know that the present value of the advance must equal the present value of the installments, when the interest element (APR expressed as a decimal fraction) is correct.

Our code runs an algorithm, which on the average modern computer will return a result in about half a second. (Tested on Pentium 2, 333mhz) We have written similar code, which returns the present value for the installments as a benchmark test. The result of our algorithm gives installment present value to within 20 pence of the advance, this represents an APR accuracy of + - 0.001% on average. (Dependant on the value of the loan)

January 2002 addendum.

I feel that it is necessary now to add a few comments. This is has been motivated by an article in the computer magazine "PC Plus" (No 186, Feb 2002). This magazine is one of best ones available, and has attempted to address this rather complicated matter in a manner more easily understood by the consumer, for whose benefit the article was written. I was very pleasantly surprised to see an article on this in a computer magazine, which had been written in response to a readers letter. The article was meant to accompany a Microsoft Excel spreadsheet application which was supposed to be on one of the companion CD ROMs. Unfortunately, I have not had the opportunity to try this as the disk was not included with my subscription copy of the magazine, and I am therefore unable to comment on the application. I visited the PC Plus website, www.pcplus.co.uk in the hope of finding a copy available for download. Unfortunately, either this was not available or I missed something. From reading the article, it has become apparent that this web page has been visited by ordinary consumers as well as members of financial institutions. This is fine. Every financial institution has a duty to the public at large to honest and forthright in its dealings. The following has been added for the benefit of the ordinary consumer. I have always tried to be helpful to everyone.

APR as we all know stands for Annual Percentage Rate. This could be alternately described as "True Percentage Rate". Most loans are arranged by persons acting as brokers for actual lending institution. For the purpose of calculating the figures a "flat" rate of annual interest is usually applied to the amount of the loan, plus any extra charges such as documentation fees. If the amount borrowed was for a fixed term of 12 months, with the total amount repayable at the end of the term, then assuming that no extra charges had been added, the flat rate applied would be the same as the APR. However most agreements stipulate different terms for a loan, and repayment by installments. Also, there are usually some extra charges such as documentation fees. Now if one considers for example a two year loan plus interest, repayable by say 24 monthly installments, its obvious that half way through the length of the agreement (12 months) more than half the amount originally borrowed will have been repaid to the lender, who would then be free to lend it on to someone else. This makes nonsense of any use of the so called "flat" rate for the purpose of evaluation and comparison. We therefore require some accurate method whereby the total amount of charges added to a loan can be expressed as a true annual rate of interest. Regulations cite a "Total Charge for Credit" which must include ALL the charges including the aforementioned fees, and it is this provides the basis for the calculation of APR. Obviously, the full amount of the principal is initially in the lands of the borrower, who then repays this amount together with the charges by installments, the effect being that the borrower has the actual benefit of a decreasing balance. Basically, each installment consists of so much of the actual amount borrowed, or PRINCIPAL, and so much of the CHARGES. The actual amount of the CHARGES contained in each installment is the interest due on the outstanding PRINCIPAL for the time elapsed since the last installment. This is usually one month as most agreements are repayable by monthly installments. In effect, the initial payment would consist mainly of CHARGES, with the final payment being mainly repayment of PRINCIPAL. What APR attempts to do, is to ascertain the actual annual percentage rate of interest by relating the total charges to the amount borrowed for the time that it is actually in the hands of the borrower. The way that APR is calculated is based on a principle known as "The Rule Of 78".The name comes from the fact that if you add together 1+2+3 etc until you reach 12 (representing 12 months) the figure that you arrive at is 78. What this rule is actually attempting to do is fractionalize the charges on the basis of the time, i.e. MONTH 1 + MONTH 2 + MONTH 3 etc. I should add that it is unlikely in the extreme that any financial institution will actually misquote the APR on their loans. I feel that from the consumers point of view, its more important to understand what APR actually is, and to to ensure that this figure is always available when seeking credit.

Roy Fellows, January 2002