Banking StandardsWritten evidence from David John Pope

Simple interest causes un-necessary inaccuracy in banking.

Following the Commission’s initial response questions of 26 July 2012:

Question 1. The accuracy in modern UK savings interest calculation is in line with Babylonian standard (1700BC) when un-advertised simple interest (Actual/365) is used instead of advertised compound interest (AER). Simple and compound growths at 6% are indistinguishable at five significant figure accuracy but modern home computers can offer over 30 significant figures.

Question 2. The use of simple interest can lead to under-payment of interest to taxed savers and under-collection of tax revenue through TDSI for the same advertised AER. My personal worst example lost me £169.72 net (and HMRC one quarter of that) on a five year bond.

Question 3. My personal trust in the banking system has been shattered by the lack of systemic improvement from my 130+ letters about simple interest since 2005 (50+ letters have yet to receive a reply). I was very disappointed when The FOS sided with one of my providers that deducted over 21% of an equivalent tax free saver’s interest from my three year account. This was a consequence of using un-advertised simple interest in an account advertised with AER (a measure of proper compound interest). HMRC also lost out.

Question 4. The existence of simple interest in banking since 1803 is due to it not being banned when interest taxation at source was introduced. The Act of 1803 failed to define which interest was being taxed (simple or compound) despite previously published warnings about simple interest and the 1624 work of Henry Briggs that made the simple interest approximation to daily basis compound interest un-necessary. The persistence of simple interest may be due to lack of proficiency in maths in banking and the wider community, lack of a desire for achievable accuracy in senior banking management, the way that providers deliver simple interest without advertising it and HMRC’s failure to define which interest is being taxed.

Question 5. Simple interest (Actual/365) must be banned from all banking. Division by 366 with a rate based on 365 days must be banned from all banking. It is unfortunate that APR suffers from this day count defect in both its statutory and EC definitions. It is unfortunate that trade body code for savings suffers from both defects. It is important that AER does not go the same way on day count as existing APR when (if) it becomes defined by statute (or EC).

Question 6. The Financial Services Bill needs an amendment that bans simple interest and insists on proper compound interest throughout banking.

My summary points:

1.The typical modern savings provider’s standard of accuracy in interest calculation is below what has been achievable since 1624. Note that I group banks and building societies together as “providers”. The BBA and BSA represent about 350 members in at least 50 countries but my experience is with UK savings interest. My providers (helpful and otherwise) will remain anonymous.

2.Compound interest is a smooth growth curve. Simple interest is a straight line.

3.Neither type of interest is defined or chosen by the government authority (HMRC) that tries to tax interest at source.

4.Providers have been free to choose which interest suits them for the purposes of taxing savings at source since 1803.

5.Providers prefer the traditional inaccurate and unfair approximation of simple interest to the achievable accuracy and fairness of proper compound interest.

6.APR is defined by statute and EC directive. AER is defined by trade bodies. Both APR and AER are measures of compound interest.

7.The joint BBA and BSA Code of Conduct for the Advertising of Interest Bearing Accounts (CCAIBA) defines Actual/365 without mentioning that:

(a)Actual/365 is simple interest.

(b)Actual/365 is an approximation to compound interest.

(c)Actual/365 is inappropriate for compounded (growth) accounts.

(d)Taxation at source generally fails to deliver The Chancellor’s rate when averaged over the years of a growth account calculated by repeated Actual/365.

8.Advertising AER (compound interest) but delivering interest based on un-advertised Actual/365 (simple interest) is mis-selling.

9.Statutory APR, EC directive APR and trade body Actual/365 all encourage division by 366 in a leap year. Division by 366 in a leap year with a rate based on 365 days disadvantages the saver (and HMRC, if the saver is taxed). A majority of the providers of my “fixed rate” monthly income accounts pay me less per day during a leap year.

10.The FSA’s BCOBS promotes CCAIBA without warning about Actual/365.

11.Providers and their trade bodies have allowed:

(a)Me to be unfairly treated by un-advertised simple interest.

(b)HMRC’s income to be reduced by repeated simple interest.

(c)The Chancellor’s 20% to be approximated by repeated simple interest.

12.My failure to get systemic improvement in banking accuracy after over 130 letters (over 50 with no reply) since 2005 hints at a need for more mathematical ability in high places and indicates widespread lack of professional courtesy.

13.Four providers have paid me more than their default calculations. This gives hope for the future but a more communicative industry would have responded to my criticisms by a group confession and a new-look promise to savers.

14.The Financial Services Bill needs an amendment that will ban simple interest, insist on proper compound interest and provide a suite of equations for the calculation of proper compound interest.

15.The overall result of the use of simple interest instead of proper compound interest is un-fairness and avoidable inaccuracy in the banking and taxation systems.

16.My general analysis provides my detailed findings on the problem of simple interest. It has been developed over several years, includes mathematical reminders, graphs and historical research and currently runs to 70 pages. I am sending a copy by post.1

17.The graph on the front page of my latest version (attached below) is one of several that are based on simulations of fixed rate accounts that are calculated by repeated simple interest and then analysed by proper compound interest. In this case, I compare outcome AER with advertised 5.00%AER. 7,400 different account timings, representing about 1% of all possible statement calculations up to around five years, are plotted. Proper compound interest is immune to intermediate timing variation. AER given to two decimal places in an advert (a BBA requirement) implies a tolerance of +/− 0.005%AER. Clearly this range is exceeded when repeated simple interest is used—even in tax free accounts. The Envelope of Inaccuracy is my name for the scatter of results due to simple interest, where proper compound interest would not have any scatter.

18.Various versions of my analysis have been sent to 24 organisations (including government, providers, trade bodies and watchdogs) over the years of my quest for accuracy. Nobody has given me a detailed comment on my analysis document. Nobody has reported a fault. Perhaps nobody actually read it all through!

19.The FOS found against me and for a provider that used un-advertised simple interest on my account (causing over £6 loss of revenue for HMRC). What hope is there for banking accuracy if even The FOS allows a provider to get away with un-advertised simple interest?

20.Henry Briggs demonstrated daily basis compound interest for a rate based on 365 days to 14 figure accuracy in London in 1624.

21.Simple interest is an un-necessary, un-fair embarrassment that has no rightful place in modern banking.

22.The following example is an aid to understanding the problem of simple interest. Nobody has faulted this example. A graphical version is attached.

(a)Consider a fixed rate growth account advertised at 5%AER for 3 years (that happen to be non-leap years, for ease of calculation). Two savers apply with £10,000 each. One saver is a tax payer the other is tax free.

(b)Nobody would dispute that the tax free saver would expect a maturity of £10,000x(1+5/100)^3=£11,576.25.

(c)The Chancellor expects providers to deduct 20% of interest at source through the TDSI scheme. A taxed saver would expect to receive 20% less interest than a tax free saver in the same product. £1,576.25 x 0.8=£1,261.00. Therefore a fair taxed maturity would be £11,261.00 with £315.25 being sent to HMRC.

(d)However, in my experience, typical providers would use repeated simple interest as follows for the taxed saver:

£10,000x(1+0.8x5/100)^3=£11,248.64 which is £12.36 less than the fair net maturity.

(e)In 2005 I realised that such a calculation disadvantages the taxed saver**.

(f)The typical provider’s method fails to achieve the Chancellor’s requirement when averaged over the life of the account. Compare the two savers’ interests:

100%x(£1,576.25-£1,248.64)/£1,576.25=20.784% by typical providers.

100%x(£1,576.25-£1,261.00)/£1,576.25=20.00% = Chancellors rate.

(g)Closer inspection of the typical provider’s repeated simple interest process also reveals a shortfall of revenue to HMRC, despite the apparently higher than 20% of interest deduction rate.

Year1: £10,000x5/100x0.8=£400.00 net interest,	£10,000x5/100x0.2=£100.00 tax.
Year2: £10,400x5/100x0.8=£416.00 net interest,	£10,400x5/100x0.2=£104.00 tax.
Year3: £10,816x5/100x0.8=£432.64 net interest,	£10,816x5/100x0.2=£108.16 tax.
Total net interest to saver=£1,248.64	Total tax to HMRC = £312.16
Proper tax would be 20% of the tax free saver’s interest = £1,576.25x0.2=£315.25

(h)In this case, the taxed saver loses £12.36 and HMRC loses £3.09.

(i)The combined saver and HMRC loss of £15.45 is absorbed by the provider.

** My first reply from The BBA (in 2006) did not dispute this effect. After 19 letters to The BBA (the last 7 had no reply), 5 to The BSA and 10 to The FSA, simple interest still pollutes UK banking.

An example showing some of the weakness of the repeated Actual/365 method—the graphical version.

ANALYSIS OF INTEREST CALCULATIONS ON SAVINGS
AUGUST 2012

17TH CENTURY TAX FREE INSIGHT EXTENDED
TO INTEREST TAXED AT SOURCE AND LEAP YEARS

For an advertised fixed rate AER/Gross over consecutive arbitrary time steps “days_n” the outcome by:

(1+Gross/100xdays₁/365)x(1+Gross/100xdays₂/365)x…

[Repeated Actual/365]

[Repeated simple interest]

in general is not equal to:

(1+Gross/100)^((days₁+days₂+…)/365)

[Time value of money]

[Compound interest, AER, APR, IRR, NPV...]

(From 1803) Taxation at source makes simple interest errors worse.

There is no mathematical need to prevent leap days from earning compound interest when a rate is based on 365 days.

In general:

Repeated simple interest fails to be proper compound interest.

Simple interest fails to deliver advertised AER.

Repeated simple interest fails to deliver 20.00% tax at source.

If this analysis is flawed then please tell me where.

If it is not flawed, then traditional inaccuracy can be avoided by eradicating simple interest and by including leap days as interest earning days.

15 August 2012