Appendix
to Code of Conduct for the Advertising of Interest Bearing Accounts. (31/1/07)
Calculation of the Annual Equivalent Rate (AER)
a)
The most general case of the calculation is the rate of interest which, if applied
each year to the deposits made by the customer, would result in the same endvalue as the contractual interest rates and interest bonuses (if any), ie the
solution to the following equation:
α ⎞
⎛
D n ⎜1 +
⎟
∑
100 ⎠
⎝
n= 1
m
where:
α
Dn
ij
m
T
1+ m - n
m
=
∑
n= 1
ij ⎞
⎛ m
D n ⎜ ∏ (1 +
)⎟
⎜ j= n
100 ⎟
⎠
⎝
(= T )
is the Annual Equivalent Rate
is the deposit to be made at the start of year n
is the interest rate (including bonuses, if any) payable at the end of
year j
is the number of years for which the product has to be held
is the amount the depositor will receive at the end of year m.
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 annual rate using the
formula given at (d) below.
The equation, in its general form, is soluble only by iterative computation. For specific
cases, general solutions are available and these can be used.
b)
If only one deposit is made at the start of the period, then the AER is
⎛ ⎛⎜
α = ⎜ ⎜⎜ m
⎜⎜
⎝⎝
c)
m
∏
n= 1
⎞
⎞
⎟
in
(1 +
) ⎟⎟ - 1 ⎟ × 100
⎟
100 ⎟⎠
⎠
If the interest rate is quoted as the total payable over the period (longer than one
year) on the initial deposit, then the AER is
⎛
⎜
⎝
α = ⎜
⎛
⎜
⎜ m
⎜
⎜
⎝
1+
r
100
⎞
⎟
⎟
⎟
⎟
⎠
⎞
- 1 ⎟ × 100
⎟
⎠
where: r is the total interest rate payable over m years.
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d)
Where interest is payable more frequently than annually, then the AER is
n
⎛⎛
⎞
i
⎞
α = ⎜ ⎜1+
- 1 ⎟ × 100
⎟
⎜⎝
⎟
n × 100 ⎠
⎝
⎠
where:
n is the number of times per year that interest is to be paid
i is the annual rate to be paid n times per year.
Notes
The general formula set out at (a) is complicated because it attempts to cater for any
sort of deposit. The following notes aim to clarify its assumptions and application:
⎜
It envisages a depositor making a series of payments (some of which may be zero)
into the account at annual intervals and the product provider paying or crediting
interest annually which may be at a different rate (possibly zero) each year.
⎜
Deposits are assumed to be made at the beginning of a year and interest paid at the
end of the year.
⎜
If deposits or interest payments are not made on anniversaries of the start date, the
formula can be operated using shorter periods and fractions of the annual rate(s),
and the answer compounded up to an annual rate using the formula at (d).
⎜
The interest payable in each year is the amount actually payable or to be added to
principal, not an accrual, so that, if paying 7% a year (simple) after 2 years, the
interest is 0% in year 1 and 14% in year 2.
⎜
The formula treats all interest as compounded because interest paid (say) annually
is worth more to the depositor than interest paid only after two years. Interest paid is
compounded at the contract rate to avoid the introduction of an arbitrary
reinvestment rate.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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Guidelines relating to AER calculations
In order to ensure consistency of calculation and fair comparison of products, the AER
should be derived on the following basis:
A1
The only changes to the amount deposited to be taken into account are
those that are required by the terms of the account. So, for example, on
an account from which withdrawals may be made, the AER calculation is
based on an initial deposit with no subsequent movements. On the other
hand, on a monthly savings account, each monthly deposit is taken into the
calculation. If certain deposits are required to qualify for a conditional bonus,
then the AER including conditional bonus must be calculated assuming that
the necessary deposits have been made. See paragraph 8 of the Code.
A2
The only changes to rates that are taken into account are those that are
stated at the outset. No allowance is made for change that may occur
because market rates generally move up or down. So, for example, if an
account has a rate which will increase the longer a deposit is held, the higher
rate will be used after the requisite time. Where an initial higher rate is
guaranteed for a number of months (an “unconditional bonus”), the rate after
that period will reduce to the “normal” rate applicable at the time of the
advertisement. If a monthly savings account has a tiered interest rate, 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 be made
for that period. If the term is indefinite, the calculation is to be done on the
first year, or until the first interest payment if that is after one year. If a
conditional bonus requires that you leave the deposit in the account for a
certain time, then the AER including conditional 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
customer makes a withdrawal (for example, 30 days loss of interest,
irrespective of how much notice is given by the customer) the AER must take
this into account. If the account has a fixed or minimum period the calculation
is made for that period. If the term is indefinite, the calculation is to be based
on the first year. (A4 does not apply to early withdrawals from a fixed-term
account where the account can earn the advertised AER [unmodified by A4] if
it is held for the full term. Nor does it apply to the non-payment of conditional
bonuses).
A5
All interest paid is treated as if it is invested and earns a rate of interest
equal to that being earned on the deposit. This may, in fact, happen if the
interest is added to principal. However, even on an account making monthly
interest payments, this assumption is made to illustrate the value of receiving
these payments during the year. If a deposit has a very short term (for
example, a six-month bond), for AER purposes it is assumed, again for
illustrative and comparative purposes, that the principal and interest can be
invested at the end of the period at the same rate for the rest of the year.
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A6
AERs are to be rounded and displayed to two decimal places.
A7
Where the AER varies according to the date of the deposit (for example,
where an unconditional bonus is offered until a fixed calendar date):
•
•
A8
all advertising should include a statement that the AER assumes that
investment was made on a specified date;
the date specified should be
- relevant to the date the advertisement or literature will appear or be
available; and
- not more than 1 month away from that date.
- Advertisements showing a recalculated AER will need to be amended on
a monthly basis.
Where regular monthly savings products are advertised which have a
limited life of one year and interest is only credited once a year the AER
should not take account of any interest earned after the account
matures. The AER will be the same as the nominal rate for the account.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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Worked Examples
1.
If an account pays or credits interest once a year, then the AER is equal to the
gross rate.
In this simple case of a single deposit of £100 at the beginning of Year 1 and interest of
10% payable at the end, the formula reduces to:
1 + 1 - 1(=1)
α ⎞
⎛
£100 ⎜1 +
⎟
⎝ 100 ⎠
10 ⎞
⎛
= £100 ⎜1 +
⎟ ( = £110
⎝ 100 ⎠
)
and clearly α is 10, and the AER is 10.00%, as you would expect.
It follows that if an account pays interest once a year on say the 31 March then the AER,
no matter when the account is advertised, will always equal the gross rate.
2.
If an account pays interest at intervals greater than 1 year, the AER is the rate
which will give the same answer if applied and compounded each year
In the simple interest example of 7% for two years quoted above, using £100,
1 + 2 - 1(= 2)
α ⎞
⎛
£100 ⎜ 1 +
⎟
⎝ 100 ⎠
0(= i1 ) ⎞ ⎛ 14(= i 2 ) ⎞
⎛
= £100 ⎜ 1 +
⎟ ⎜1 +
⎟
100 ⎠
100 ⎠ ⎝
⎝
( = £114 )
⎛ 114
⎞
so α = ⎜⎜ 2
- 1 ⎟⎟ × 100 = 6.771...
⎝ 100
⎠
(which can also be reached using formula (c) directly)
and the resultant AER is 6.77% to two decimal places.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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3.
Now suppose a deposit of £100 is to be made at the beginning of the first year and
£50 to be added at the start of Year 2 (as part of the product requirement, not an
option), and interest is to be 10% for Year 1 and 11% for Year 2 (eg an escalating
interest rate, not a prediction of interest rate 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 be, using the
right hand side of the formula:
⎛ 2 ⎛
i j ⎞⎞
⎟⎟ ⎟
T = ∑ Dn ⎜ ∏⎜⎜1 +
⎜
100
j
=
n
⎝
⎠ ⎟⎠
n=1
⎝
2
⎛ ⎛ 10(= i1 ) ⎞ ⎛ 11(= i2 ) ⎞ ⎞
⎛ 11(= i2 ) ⎞
= £100(= D1 ) ⎜⎜ ⎜1 +
⎟ ⎜1 +
⎟ ⎟⎟ + £50(= D2 ) ⎜1 +
⎟
100 ⎠ ⎝
100 ⎠ ⎠
100 ⎠
⎝
⎝⎝
= £177.60
Then find a value for α that satisfies the left hand side:
1+m - n
α ⎞
⎛
T = £177.60 = ∑ Dn ⎜1 +
⎟
⎝ 100 ⎠
n=1
m
α ⎞
α ⎞
⎛
⎛
= £100(= D1 ) ⎜1 +
⎟ + £50(= D2 ) ⎜1 +
⎟
⎝ 100 ⎠
⎝ 100 ⎠
2
Trying α = 10.5 gives £177.353; 10.6 gives £177.624; 10.59 gives £177.596.
This process of iterative computation yields the AER of 10.59% to two decimal places.
4.
If in the above example there were not an additional deposit contracted in the second
year, the calculation is simpler and formula (b) can be used:
⎛ ⎛⎜ ⎛ 10(= i ) ⎞ ⎛ 11(= i ) ⎞ ⎞⎟ ⎞
1
2
α = ⎜ ⎜⎜ 2 ⎜1 +
⎟ ⎜1 +
⎟ ⎟⎟ - 1⎟ × 100 = 10.498...
⎜⎜ ⎝
100
100
⎠
⎝
⎠ ⎟⎠ ⎟⎠
⎝⎝
giving an AER of 10.50% to two decimal places
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5.
If an account pays interest more often than once a year, then the AER is calculated
by adding each interest payment to the deposit and calculating the next interest
payment on the total – compounding the interest.
The treatment of monthly income accounts shows the basic use of formula (d).
Suppose a fixed deposit offers two options:
A)
interest paid annually of 6% 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
12
⎛⎛
⎞
⎞
5.8(= i )
⎜
⎟⎟ - 1⎟ ×100 = 5.956...
α = ⎜⎜1 +
⎜ ⎝ 12(months per year) ×100 ⎠
⎟
⎝
⎠
giving an AER of 5.96% to two decimal places
This demonstrates the value to the depositor of the monthly interest but also 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 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:
⎛
5.5 × 8 ⎞
⎜ 100 +
⎟
8
12
⎜
αm =
- 1 ⎟ × 100
⎜
⎟
100
⎜
⎟
⎝
⎠
= 0.451... ≡ 5.413...% per annum paid monthly
Then use formula (d) to convert this to an annually compounded rate
12
⎛⎛
⎞
⎞
5.413...
⎜
⎟⎟ - 1⎟ × 100 = 5.5501...
α = ⎜⎜1 +
⎜ ⎝ 12(months ) × 100 ⎠
⎟
⎝
⎠
giving an AER of 5.55% to two decimal places. This is higher than the gross rate
reflecting the value of interest paid after 8 months rather than a year.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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7.
Unconditional bonuses, for example a launch bonus of 0.5% paid at least until 30 June
2000 (this example was drafted in November 1999) on an account paying 5% annually on
30 April without a fixed term, are treated as a step down in the rate when the guarantee
expires. So, assuming a deposit on 1 November 1999, the depositor would receive 5.5%
for 8 months and then 5% for 4 months (following guideline A3, the calculation is for the
first year of the deposit).
Formula (b) can be applied in two half-years:
⎞
⎛ ⎛⎜ ⎛
5.5
5.5 × 2 + 5 × 4
⎞⎛
⎞ ⎟⎟ ⎞⎟
⎜⎜ ⎜
(
=
2
.
75
)
(
=
2
.
583
...)
⎟⎜
⎟⎟
⎜
12
⎟ ⎜1 +
⎟ ⎟⎟ - 1⎟ × 100
α h = ⎜⎜ ⎜⎜ 2 ⎜1 + 2
⎜ ⎜
100
100
⎟ ⎟⎟ ⎟
⎟⎜
⎜⎜ ⎜⎜ ⎜
⎟ ⎟ ⎟⎟
⎟⎜
⎠⎝
⎠ ⎟⎠ ⎠
⎝ ⎜⎝ ⎝
= 2.666... ≡ 5.333... per annum paid half − yearly
Again, use formula (d) to convert this to an annually compounded rate
2
⎛⎛
⎞
⎞
5.333...
⎟⎟ - 1⎟ × 100 = 5.4043...
α = ⎜ ⎜⎜1 +
⎜ ⎝ 2(halves) × 100 ⎠ ⎟
⎝
⎠
Giving an AER of 5.40% to two decimal places. The advertisement would contain a
statement “AER calculated assuming an investment on 1 November 1999.” If it were a
branch leaflet, it would be displayed only during November 1999 (see paragraph 11 of
the Code and Guideline A6).
8.
Finally, consider a hypothetical product with irregular (but committed) cash flows.
The pattern of this product is :
Deposit £3,000 on April 1 in year 1, then
£1,800 on January 1 each year for three years, and then
£600 on January 1 in year 5,
a total of £9,000 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 at
repayment, together with a 2% bonus of the total amount deposited (£9,000) for making
the required deposits and holding to maturity.
Because the subsequent deposits and interest are not added on the anniversary of the first deposit,
the AER including conditional bonus has to be calculated on a quarterly basis and then
compounded up to an annual rate. The calculation (in summary here, the spreadsheet overleaf
shows the full calculation) is as follows:
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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First calculate the total T to be repaid at the end of the contract, using the formula:
20 (quarters)
T =
∑
n=1
⎛ 20 ⎛
ij ⎞⎞
⎟⎟
D n ⎜ ∏ ⎜1+
⎜ j= n ⎝
100
⎠ ⎟⎠
⎝
⎛⎛
7
7
7
⎞⎛
⎞
⎛
⎞⎞
× 3(= i 3 ) ⎟ ⎜
× 4(= i 7 ) ⎟
(= i 20 ) ⎟ ⎟
⎜⎜
⎜
⎟ ⎜1 + 4
⎟ ... ⎜ 1 + 4
⎟⎟
= £3000( = D 1 ) ⎜ ⎜ 1 + 4
⎜⎜
100
100
100
⎟⎜
⎟
⎜
⎟⎟
⎟⎜
⎟
⎜
⎟⎟
⎜⎜
⎝
⎠
⎝
⎠
⎝
⎠⎠
⎝
⎛⎛
7
7
7
⎞⎛
⎞
⎛
⎞⎞
× 4(= i 7 ) ⎟ ⎜
× 4(= i11 ) ⎟
(= i 20 ) ⎟ ⎟
⎜⎜
⎜
⎟ ⎜1 + 4
⎟ ... ⎜ 1 + 4
⎟⎟
+ £ 1800( = D 4 ) ⎜ ⎜ 1 + 4
⎜⎜
100
100
100
⎟⎜
⎟
⎜
⎟⎟
⎟⎜
⎟
⎜
⎟⎟
⎜⎜
⎝
⎠
⎝
⎠
⎝
⎠⎠
⎝
...
⎛⎛
7
7
⎞⎛
⎞⎞
(= i 20 ) ⎟ ⎟
× 4(= i19 ) ⎟ ⎜
⎜⎜
⎟ ⎜1 + 4
⎟⎟
+ £ 600( = D 16 ) ⎜ ⎜ 1 + 4
⎜⎜
100
100
⎟⎜
⎟⎟
⎟⎜
⎟⎟
⎜⎜
⎠⎝
⎠⎠
⎝⎝
+ £180 (bonus of 2% of £9,000)
= £11,785.78
Then find the AER including conditional bonus by solving
α ⎞
⎛
T = £11,785.78 = ∑ Dn ⎜1 +
⎟
⎝ 100 ⎠
n=1
20
21- n
20
17
α ⎞
α ⎞
⎛
⎛
= £3000(= D1 ) ⎜1 +
⎟ + £1800(= D4 ) ⎜1 +
⎟
⎝ 100 ⎠
⎝ 100 ⎠
5
α ⎞
⎛
+ ... + £600(= D16 ) ⎜1 +
⎟
⎝ 100 ⎠
which, by trying various values, yields α = 1.8127… ≡ 7.2511...% per annum paid quarterly.
Use formula (d) to convert this to an annually compounded rate
4
⎛⎛
⎞ ⎞⎟
7.2511...
⎜
⎟ - 1 × 100 = 7.4506...
α = ⎜⎜1 +
⎜ ⎝ 4(quarters ) × 100 ⎟⎠ ⎟
⎝
⎠
giving an AER including conditional bonus of 7.45% to two decimal places.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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Detail of the calculation of T:
n
Dn (£)
1
3000
in (%)
1 + in / 100
0
2
0
3
5.25
4
1800
0
6
0
7
7
8
1800
0
10
0
11
7
1800
2,400.72
1.07
2,243.67
1.07
0
13
0
14
0
15
7
16
1.0525
0
9
12
4,211.27
0
5
600
T = sum of:
2,096.88
1.07
0
653.24
17
0
18
0
19
7
1.07
20
1.75
1.0175
Bonus
180.00
Total
11,785.78
Note: It is possible that the AER may very slightly depending on when the initial deposit is made
relative to the first interest payment date. Where this is the case, either the assumption about this
relationship should be clearly stated (see guideline A6) or the lowest potential result should be
used (this usually results when the period from initial deposit to first interest period is at a
maximum).
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Explanation of the AER
The following 2 pages give a simple explanation of the AER which is intended for distribution to
staff and/or customers.
Any comments on the explanation or on this appendix should be sent to the BBA, preferably by
e-mail to [email protected] .
The BBA is not in a position to undertake AER calculations but will do its best to advise on the
interpretation of this appendix.
An AER calculator for PCs is available from the BBA Publications Department.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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AER – an explanation
The Annual Equivalent Rate is a notional rate quoted in advertisements for interest-bearing
accounts which illustrates the contractual (gross) interest rate (excluding any bonus interest
payable) as if paid and compounded on an annual basis.
Advertisements may also quote an AER including conditional bonus clearly identified as
such.
¾ If an account pays or credits interest once a year, then the AER is equal to the gross rate.
¾ If an account pays interest more often than once a year, then the AER is calculated by
adding each interest payment to the deposit and calculating the next interest payment on
the total – compounding the interest.
For example, an account offering 5% gross interest paid quarterly on £100 pays
•
•
•
•
£1.25 (1.25% (¼ of 5%) of £100) after 3 months,
£1.26 (1.25% of £101.25 (£100 + £1.25)) after six months*,
£1.28 (1.25% of £102.51) after nine months, and
£1.30 at the end of the year (1.25% of £103.79),
giving a total including interest of £105.09.
The AER is thus 5.09%.
*In practice, the calculation is worked to more decimal places to avoid rounding errors.
¾ If an account pays interest at intervals greater than 1 year, we are looking for a rate which
will give us the right answer if applied and compounded each year.
For example, an account which pays 5% for five years but pays it only at the end of the
five years will pay back £125 after the five years on £100 deposited (the original £100
plus £25 interest).
The AER is 4.56% and we can see how this works as follows:
•
•
•
•
•
£4.56 (4.56% of £100) would be the interest at the end of year 1,
£4.77 (4.56% of £104.56) at the end of year 2,
£4.99 (4.56% of £109.33) at the end of year 3,
£5.21 (4.56% of £114.32) at the end of year 4, and
£5.45 (4.56% of £119.53) at the end of year 5,
giving a final total of £124.98.
Not exactly £125 but as close as we can get to 2 decimal places (try the calculation with
4.57% instead – you will get £125.05).
¾ The AER for more complicated patterns of changing deposits and/or interest rates can
really be solved only by trial and error – a process best left to computers!
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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The AER is a notional rate – it is not necessarily 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 those that are
required by the terms of the account. So, for example, on an account from which you can
make withdrawals, the assumption is that you make just an initial deposit and leave it
there. On the other hand, on a monthly savings account, it is assumed that you will make
the deposits each month. If certain deposits are required to qualify for a conditional
bonus, then the AER including conditional bonus will be calculated assuming that you
have made the necessary deposits.
•
The only changes to rates that are taken into account are those that are stated at the outset.
No allowance is made for change that may occur because market rates generally move up
or down. So, for example, if an account has a rate which will increase the longer a
deposit is held, the higher rate will be used after the requisite time. Where an initial
higher rate is guaranteed for a number of months, the rate after that will be assumed to
reduce to the “normal” rate applicable at the time of the advertisement. If a monthly
savings account has a tiered interest rate, 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 that you leave
your deposit there for that period. If the term is indefinite, the calculation is done on the
first year, or until the first interest payment if that is after one year. If a conditional bonus
requires that you leave the deposit in the account for a certain time, then the AER
including conditional bonus will be calculated over that period.
•
All interest paid is treated as if it is invested and earns a rate of interest equal to that being
earned on your deposit. This may, in fact, happen if the interest is added to principal.
However, even on an account making monthly interest payments, this assumption is made
to illustrate the value to you of receiving these payments during the year. If a deposit has
a very short term (for example, a six-month bond), for AER purposes, it is assumed that
you can invest the principal and interest at the 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 Advertising of Interest-bearing
Accounts. This explanation is intended to help you understand the AER, not as a substitute for the Code.
AER Appendix to the Code of Conduct for the Advertising of Interest-bearing Accounts
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