From: Subject: BBA - British Bankers' Association - Calculation of the Annual Equivalent Rate (AER) Date: Thu, 8 Nov 2007 15:45:44 -0000 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0007_01C8221E.6C84B9A0" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3198 This is a multi-part message in MIME format. ------=_NextPart_000_0007_01C8221E.6C84B9A0 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=135&a=1575&view=print BBA - British Bankers' Association - Calculation of = the Annual Equivalent Rate (AER)
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3D"BBA

Printed from=20 = http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=3D135&a=3D1575


Calculation of the Annual Equivalent Rate = (AER)


31/01/2007

a) The most general case of the calculation is the rate of interest = which, if=20 applied each year to the deposits made by the customer, would result in = the same=20 end-value as the contractual interest rates and interest bonuses (if = any), ie=20 the solution to the following equation:

3D"formula_for

If deposits are made at more frequent intervals than whole years, the = calculation can be made using monthly interest and the result expressed = as an=20 annual rate using the formula given at (d) below.

The equation, in its general form, is soluble only by iterative = computation.=20 For specific cases, general solutions are available and these can be = used.

b)only one deposit is made at the start of the period, then the AER = is

3D"formula_for

c) If the interest rate is quoted as the total payable over the = period=20 (longer than one year) on the initial deposit, then the AER is

3D"formula_for

d) Where interest is payable more frequently than annually, then the = AER=20 is

3D"formula_for

Notes

The general formula set out at (a) is complicated because it attempts = to=20 cater for any sort of deposit.  The following notes aim to clarify = its=20 assumptions and application: 

  • It envisages a depositor making a series of payments (some of = which may be=20 zero) into the account at annual intervals and the product provider = paying or=20 crediting interest annually which may be at a different rate (possibly = zero)=20 each year.=20
  • Deposits are assumed to be made at the beginning of a year and = interest=20 paid at the end of the year.=20
  • If deposits or interest payments are not made on anniversaries of = the=20 start date, the formula can be operated using shorter periods and = fractions of=20 the annual rate(s), and the answer compounded up to an annual rate = using the=20 formula at (d). =20
  • The interest payable in each year is the amount actually payable = or to be=20 added to principal, not an accrual, so that, if paying 7% a year = (simple)=20 after 2 years, the interest is 0% in year 1 and 14% in year 2. =20
  • The formula treats all interest as compounded because interest = paid (say)=20 annually is worth more to the depositor than interest paid only after = two=20 years.  Interest paid is compounded at the contract rate to avoid = the=20 introduction of an arbitrary reinvestment rate.

Guidelines relating to AER calculations

In order to ensure consistency of calculation and fair comparison of=20 products, the AER should be derived on the following basis:

A1. The only changes to the amount deposited to be taken into = account=20 are those that are required by the terms of the account. So, = for=20 example, on an account from which withdrawals may be made, the AER = calculation=20 is based on an initial deposit with no subsequent movements.  On = the other=20 hand, on a monthly savings account, each monthly deposit is taken into = the=20 calculation.  If certain deposits are required to qualify for a = conditional=20 bonus, then the AER including conditional bonus must be = calculated=20 assuming that the necessary deposits have been made. See paragraph 8 of = the=20 Code.

A2. The only changes to rates that are taken into account are = those=20 that are stated at the outset. No allowance is made for change = that may=20 occur because market rates generally move up or down.  So, for = example, if=20 an account has a rate which will increase the longer a deposit is held, = the=20 higher rate will be used after the requisite time.  Where an = initial higher=20 rate is guaranteed for a number of months (an =93unconditional = bonus=94), the rate=20 after that period will reduce to the =93normal=94 rate applicable at the = time of the=20 advertisement.  If a monthly savings account has a tiered interest = rate,=20 then the appropriate rate will be used as the balance builds up.

A3. If an account has a fixed or minimum term, the = calculation is to=20 be made for that period.If the term is indefinite, the = calculation is=20 to be done on the first year, or until the first interest payment if = that is=20 after one year.  If a conditional bonus requires that you leave the = deposit=20 in the account for a certain time, then the AER including = conditional=20 bonus will be calculated over that period. See paragraph 8 of the = Code.

A4. If there is an unconditional charge that is payable = whenever the=20 customer makes a withdrawal (for example, 30 days loss of = interest,=20 irrespective of how much notice is given by the customer) = the=20 AER must take this into account. If the account has a fixed or minimum = period=20 the calculation is made for that period. If the term is indefinite, the=20 calculation is to be based on the first year. (A4 does not apply to = early=20 withdrawals from a fixed-term account where the account can earn the = advertised=20 AER [unmodified by A4] if it is held for the full term. Nor does it = apply to the=20 non-payment of conditional bonuses).

A5. All interest paid is treated as if it is invested and = earns a=20 rate of interest equal to that being earned on the deposit. = This may,=20 in fact, happen if the interest is added to principal.  However, = even on an=20 account making monthly interest payments, this assumption is made to = illustrate=20 the value of receiving these payments during the year.  If a = deposit has a=20 very short term (for example, a six-month bond), for AER purposes it is = assumed,=20 again for illustrative and comparative purposes, that the principal and = interest=20 can be invested at the end of the period at the same rate for the rest = of the=20 year.

A6. AERs are to be rounded and displayed to two decimal=20 places.

A7. Where the AER varies according to the date of the deposit = (for=20 example, where an unconditional bonus is offered until a fixed calendar=20 date):

  • all advertising should include a statement that the AER assumes = that=20 investment was made on a specified date;=20
  • the date specified should be=20
  • -relevant to the date the advertisement or literature will appear = or be=20 available; and=20
  • -not more than 1 month away from that date.=20
  • -Advertisements showing a recalculated AER will need to be amended = on a=20 monthly basis.

A8. Where regular monthly savings products are advertised = which have=20 a limited life of one year and interest is only credited once a year the = AER=20 should not take account of any interest earned after the account=20 matures. The AER will be the same as the nominal rate for the = account.=20

Worked Examples

1. If an account pays or credits interest once a year, then = the AER=20 is equal to the gross rate.

In this simple case of a single deposit of =A3100 at the beginning of = Year 1=20 and interest of 10% payable at the end, the formula reduces to:

3D"formula_for

It follows that if an account pays inteest once a year on say the = 31st March=20 then the AER, no matter when the acount is advertised, will always equal = the=20 gross rate.

2. If an account pays interest at intervals greater than 1 year, the = AER is=20 the rate which will give the same answer if applied and compounded each=20 year.

In the simple interest example of 7% for two years quoted above, = using=20 =A3100,

3D"formula_for

(which can also be reached using formula (c) directly)and the = resultant AER=20 is 6.77% to two decimal places.

3. Now suppose a deposit of =A3100 is to be made at the beginning of = the first=20 year and =A350 to be added at the start of Year 2 (as part of the = product=20 requirement, not an option), and interest is to be 10% for Year 1 and = 11% for=20 Year 2 (eg an escalating interest rate, not a prediction of interest = rate=20 movements). The calculation should be approached as follows:

First work out what the eventual return at the end of year 2 (T) is = going to=20 be, using the right hand side of the formula:

3D"formula_for

This process of iterative computation yields the AER of 10.59% to two = decimal=20 places.

4. If in the above example there were not an additional deposit = contracted in=20 the second year, the calculation is simpler and formula (b) can be = used:

3D"formula_for

giving an AER of 10.50% to two decimal places.

If an account pays interest more often than once a year, then = the AER=20 is calculated by adding each interest payment to the deposit and = calculating the=20 next interest payment on the total =96 compounding the = interest.

  • The treatment of monthly income accounts shows the basic use of = formula=20 (d).

Suppose a fixed deposit offers two options:
A)interest paid = annually of 6%=20 per annum
B)interest paid monthly at a rate of 5.8% per annum.

Option A will have an AER of 6.00% (see example 1 = above)

For option B, the AER is calculated using formula (d) as=20

3D"formula_for

giving an AER of 5.96% to two decimal places.

This demonstrates the value to the depositor of the monthly interest = but also=20 shows that the two options are not quite identical in terms of = return.

6. A short-term bond, for example an 8-month bond paying 5.5% per = annum, has=20 to be treated using a combination of formulae (c) and (d).

First of all, use formula (c) to find the AER on a monthly basis:

3D"formula_for

Then use formula (d) to convert this to an annually compounded = rate

3D"formula_for

giving an AER of 5.55% to two decimal places. This is higher than the = gross=20 rate reflecting the value of interest paid after 8 months rather than a=20 year.

7. Unconditional bonuses, for example a launch bonus of 0.5% paid at = least=20 until 30 June 2000 (this example was drafted in November 1999) on an = account=20 paying 5% annually on 30 April without a fixed term, are treated as a = step down=20 in the rate when the guarantee expires.  So, assuming a deposit on = 1=20 November 1999, the depositor would receive 5.5% for 8 months and then 5% = for 4=20 months (following guideline A3, the calculation is for the first year of = the=20 deposit).

  • Formula (b) can be applied in two half-years:

3D"formula_for

Again, use formula (d) to convert this to an annually compounded = rate

3D"formula_for

Giving an AER of 5.40% to = two=20 decimal places. The advertisement would contain a statement "AER = calculated=20 assuming an investment on 1 November 1999." If it were a branch leaflet, = it=20 would be displayed only during November 1999 (see paragraph 11 of the = Code and=20 Guideline A6).

Finally, consider a hypothetical product with irregular (but = committed) cash=20 flows.

The pattern of this product is :

Deposit =A33,000 on April 1 in year 1, then
=A31,800 on January 1 = each year=20 for three years, and then
=A3600 on January 1 in year 5,
a total = of =A39,000=20 to be repaid on April 1 in year 6. 

Interest at 7% per annum is added to the account on December 31 each = year and=20 at repayment, together with a 2% bonus of the total amount deposited = (=A39,000)=20 for making the required deposits and holding to maturity.

Because the subsequent deposits and interest are not added on the = anniversary=20 of the first deposit, the AER including conditional = bonus has=20 to be calculated on a quarterly basis and then compounded up to an = annual=20 rate.  The calculation (in summary here, the spreadsheet overleaf = shows the=20 full calculation) is as follows:

The pattern of this product is :=20

  • Deposit =A33,000 on April 1 in year 1, then=20
  • =A31,800 on January 1 each year for three years, and then=20
  • =A3600 on January 1 in year 5,=20
  • a total of =A39,000 to be repaid on April 1 in year 6.

Interest at 7% per annum is added to the account on December 31 each = year and=20 at repayment, together with a 2% bonus of the total amount deposited = (=A39,000)=20 for making the required deposits and holding to maturity.

Because the subsequent deposits and interest are not added on the = anniversary=20 of the first deposit, the AER including conditional bonus has to be = calculated=20 on a quarterly basis and then compounded up to an annual rate. The = calculation=20 (in summary here, the spreadsheet overleaf shows the full calculation) = is as=20 follows:

First calculate the total T to be repaid at the end of the contract, = using=20 the formula:

3D"formula_for

Then find the AER including conditional bonus by solving

3D"formula_for

which, by trying various values, yields a =3D 1.8127=85 = ≡ 7.2511...% per=20 annum paid quarterly.

Use formula (d) to convert this to an annually compounded = rate
giving an=20 AER including conditional bonus of 7.45% to = two=20 decimal places.

3D"formula_for

giving an AER including conditional bonus of 7.45% to two decimal = places.

Detail of the calculation of T:

3Dcalculating_T

Note: It is possible that the AER may very slightly depending on when = the=20 initial deposit is made relative to the first interest payment = date.  Where=20 this is the case, either the assumption about this relationship should = be=20 clearly stated (see guideline A6) or the lowest potential result should = be used=20 (this usually results when the period from initial deposit to first = interest=20 period is at a maximum).

Explanation of the AER

The following 2 pages give a simple explanation of the AER which is = intended=20 for distribution to staff and/or customers.

Any comments on the explanation or on this appendix should be sent to = the=20 BBA, preferably by e-mail to aer@bba.org.uk=20 . 

The BBA is not in a position to undertake AER calculations but will = do its=20 best to advise on the interpretation of this appendix.

An AER calculator for PCs is available from the BBA Publications=20 Department.

AER - an explanation

The Annual Equivalent Rate is a notional rate quoted in = advertisements for=20 interest-bearing accounts which illustrates the contractual (gross) = interest=20 rate (excluding any bonus interest payable) as if paid and compounded on = an=20 annual basis.

Advertisements may also quote an AER including conditional = bonus=20 clearly identified as such.

  • If an account pays or credits interest once a year, then the AER = is equal=20 to the gross rate.
  • If an account pays interest more often than once a year, then the = AER is=20 calculated by adding each interest payment to the deposit and = calculating the=20 next interest payment on the total =96 compounding the = interest.

For example, an account offering 5% gross interest paid quarterly on = =A3100=20 pays=20

  • =A31.25 (1.25% (=BC of 5%) of =A3100) after 3 months,=20
  • =A31.26 (1.25% of =A3101.25 (=A3100 + =A31.25)) after six months*, =
  • =A31.28 (1.25% of =A3102.51) after nine months, and=20
  • =A31.30 at the end of the year (1.25% of =A3103.79),=20
  • giving a total including interest of =A3105.09.

The AER is thus 5.09%.

*In practice, the calculation is worked to more decimal places to = avoid=20 rounding errors.

  • If an account pays interest at intervals greater than 1 year, we = are=20 looking for a rate which will give us the right answer if applied and=20 compounded each year.

For example, an account which pays 5% for five years but pays it only = at the=20 end of the five years will pay back =A3125 after the five years on = =A3100 deposited=20 (the original =A3100 plus =A325 interest).

The AER is 4.56% and we can see how this works as follows:=20

  • =A34.56 (4.56% of =A3100) would be the interest at the end of year = 1,=20
  • =A34.77 (4.56% of =A3104.56) at the end of year 2,=20
  • =A34.99 (4.56% of =A3109.33) at the end of year 3,=20
  • =A35.21 (4.56% of =A3114.32) at the end of year 4, and=20
  • =A35.45 (4.56% of =A3119.53) at the end of year 5,

giving a final total of =A3124.98.

Not exactly =A3125 but as close as we can get to 2 decimal places = (try the=20 calculation with 4.57% instead - you will get =A3125.05).

The AER is a notional rate =96 it is not = necessarily=20 equal to the cash return because, in calculating it, we make = assumptions:

  • The only changes to the amount deposited that are taken into = account are=20 those that are required by the terms of the account.  So, for = example, on=20 an account from which you can make withdrawals, the assumption is that = you=20 make just an initial deposit and leave it there.  On the other = hand, on a=20 monthly savings account, it is assumed that you will make the deposits = each=20 month.  If certain deposits are required to qualify for a = conditional=20 bonus, then the AER including conditional bonus will be = calculated=20 assuming that you have made the necessary deposits.

  • The only changes to rates that are taken into account are those = that are=20 stated at the outset.  No allowance is made for change that may = occur=20 because market rates generally move up or down.  So, for example, = if an=20 account has a rate which will increase the longer a deposit is held, = the=20 higher rate will be used after the requisite time.  Where an = initial=20 higher rate is guaranteed for a number of months, the rate after that = will be=20 assumed to reduce to the =93normal=94 rate applicable at the time of = the=20 advertisement.  If a monthly savings account has a tiered = interest rate,=20 then the appropriate rate will be used as the balance builds up.

  • If an account has a fixed or minimum term, the assumption will be = made=20 that you leave your deposit there for that period.  If the term = is=20 indefinite, the calculation is done on the first year, or until the = first=20 interest payment if that is after one year.  If a conditional = bonus=20 requires that you leave the deposit in the account for a certain time, = then=20 the AER including conditional bonus will be calculated  = over=20 that period.

  • All interest paid is treated as if it is invested and earns a rate = of=20 interest equal to that being earned on your deposit.  This may, = in fact,=20 happen if the interest is added to principal.  However, even on = an=20 account making monthly interest payments, this assumption is made to=20 illustrate the value to you of receiving these payments during the = year. =20 If a deposit has a very short term (for example, a six-month bond), = for AER=20 purposes, it is assumed that you can invest the principal and interest = at the=20 end of the period at the same rate for the rest of the year.

The full specification of the AER is contained in the Code of Conduct = for the=20 Advertising of Interest-bearing Accounts, please see link below. This=20 explanation is intended to help you understand the AER, not as a = substitute for=20 the Code.


Related Links

C= ode of=20 conduct for the advertising of interest bearing accounts (Internal Link)

=A9 1998-2007 British Bankers' Association & BBA = Enterprises Ltd. All=20 rights reserved.
For permission to copy please contact BBA Publication = Unit.
Any=20 reproduction, republication, transmission or reuse in whole or = part=20 requires our consent.

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