Appedix to Code of Coduct for the Advertisig of Iterest Bearig Accouts. (31/1/0) Calculatio of the Aual Equivalet Rate (AER) a) The most geeral case of the calculatio is the rate of iterest which, if applied each year to the deposits made by the customer, would result i the same edvalue as the cotractual iterest rates ad iterest bouses (if ay), ie the solutio to the followig equatio: m 1+ m - m m α i j D 1 + = D + (1 ) = 1 = 1 j= ( = T ) where: α is the Aual Equivalet Rate D is the deposit to be made at the start of year i j is the iterest rate (icludig bouses, if ay) payable at the ed of year j m is the umber of years for which the product has to be held T is the amout the depositor will receive at the ed of year m. If deposits are made at more frequet itervals tha whole years, the calculatio ca be made usig mothly iterest ad the result expressed as a aual rate usig the formula give at (d) below. The equatio, i its geeral form, is soluble oly by iterative computatio. For specific cases, geeral solutios are available ad these ca be used. b) If oly oe deposit is made at the start of the period, the the AER is α = m m i (1 ) 1 + - = 1 c) If the iterest rate is quoted as the total payable over the period (loger tha oe year) o the iitial deposit, the the AER is α = m r 1 + 1 - where: r is the total iterest rate payable over m years.
d) Where iterest is payable more frequetly tha aually, the the AER is i α = 1 1 + - where: is the umber of times per year that iterest is to be paid i is the aual rate to be paid times per year. Notes The geeral formula set out at (a) is complicated because it attempts to cater for ay sort of deposit. The followig otes aim to clarify its assumptios ad applicatio: It evisages a depositor makig a series of paymets (some of which may be zero) ito the accout at aual itervals ad the product provider payig or creditig iterest aually which may be at a differet rate (possibly zero) each year. Deposits are assumed to be made at the begiig of a year ad iterest paid at the ed of the year. If deposits or iterest paymets are ot made o aiversaries of the start date, the formula ca be operated usig shorter periods ad fractios of the aual rate(s), ad the aswer compouded up to a aual rate usig the formula at (d). The iterest payable i each year is the amout actually payable or to be added to pricipal, ot a accrual, so that, if payig % a year (simple) after years, the iterest is 0% i year 1 ad 1% i year. The formula treats all iterest as compouded because iterest paid (say) aually is worth more to the depositor tha iterest paid oly after two years. Iterest paid is compouded at the cotract rate to avoid the itroductio of a arbitrary reivestmet rate.
Guidelies relatig to AER calculatios I order to esure cosistecy of calculatio ad fair compariso of products, the AER should be derived o the followig basis: A1 The oly chages to the amout deposited to be take ito accout are those that are required by the terms of the accout. So, for example, o a accout from which withdrawals may be made, the AER calculatio is based o a iitial deposit with o subsequet movemets. O the other had, o a mothly savigs accout, each mothly deposit is take ito the calculatio. If certai deposits are required to qualify for a coditioal bous, the the AER icludig coditioal bous must be calculated assumig that the ecessary deposits have bee made. See paragraph 8 of the Code. A A3 A A5 The oly chages to rates that are take ito accout are those that are stated at the outset. No allowace is made for chage that may occur because market rates geerally move up or dow. So, for example, if a accout has a rate which will icrease the loger a deposit is held, the higher rate will be used after the requisite time. Where a iitial higher rate is guarateed for a umber of moths (a ucoditioal bous ), the rate after that period will reduce to the ormal rate applicable at the time of the advertisemet. If a mothly savigs accout has a tiered iterest rate, the the appropriate rate will be used as the balace builds up. If a accout has a fixed or miimum term, the calculatio is to be made for that period. If the term is idefiite, the calculatio is to be doe o the first year, or util the first iterest paymet if that is after oe year. If a coditioal bous requires that you leave the deposit i the accout for a certai time, the the AER icludig coditioal bous will be calculated over that period. See paragraph 8 of the Code. If there is a ucoditioal charge that is payable wheever the customer makes a withdrawal (for example, 30 days loss of iterest, irrespective of how much otice is give by the customer) the AER must take this ito accout. If the accout has a fixed or miimum period the calculatio is made for that period. If the term is idefiite, the calculatio is to be based o the first year. (A does ot apply to early withdrawals from a fixed-term accout where the accout ca ear the advertised AER [umodified by A] if it is held for the full term. Nor does it apply to the o-paymet of coditioal bouses). All iterest paid is treated as if it is ivested ad ears a rate of iterest equal to that beig eared o the deposit. This may, i fact, happe if the iterest is added to pricipal. However, eve o a accout makig mothly iterest paymets, this assumptio is made to illustrate the value of receivig these paymets durig the year. If a deposit has a very short term (for example, a six-moth bod), for AER purposes it is assumed, agai for illustrative ad comparative purposes, that the pricipal ad iterest ca be ivested at the ed of the period at the same rate for the rest of the year.
A6 A AERs are to be rouded ad displayed to two decimal places. Where the AER varies accordig to the date of the deposit (for example, where a ucoditioal bous is offered util a fixed caledar date): all advertisig should iclude a statemet that the AER assumes that ivestmet was made o a specified date; the date specified should be - relevat to the date the advertisemet or literature will appear or be available; ad - ot more tha 1 moth away from that date. - Advertisemets showig a recalculated AER will eed to be ameded o a mothly basis. A8 Where regular mothly savigs products are advertised which have a limited life of oe year ad iterest is oly credited oce a year the AER should ot take accout of ay iterest eared after the accout matures. The AER will be the same as the omial rate for the accout.
Worked Examples 1. If a accout pays or credits iterest oce a year, the the AER is equal to the gross rate. I this simple case of a sigle deposit of at the begiig of Year 1 ad iterest of 10% payable at the ed, the formula reduces to: 1+1-1(=1) α 10 1+ = 1+ 110 ( = ) ad clearly α is 10, ad the AER is 10.00%, as you would expect. It follows that if a accout pays iterest oce a year o say the 31 March the the AER, o matter whe the accout is advertised, will always equal the gross rate.. If a accout pays iterest at itervals greater tha 1 year, the AER is the rate which will give the same aswer if applied ad compouded each year I the simple iterest example of % for two years quoted above, usig, 1+ -1(= ) α 1 + 0(= i1 ) 1(= i ) = 1 + 1 + ( = 11 ) so 11 α = - 1 = 6.1... (which ca also be reached usig formula (c) directly) ad the resultat AER is 6.% to two decimal places.
3. Now suppose a deposit of is to be made at the begiig of the first year ad 50 to be added at the start of Year (as part of the product requiremet, ot a optio), ad iterest is to be 10% for Year 1 ad 11% for Year (eg a escalatig iterest rate, ot a predictio of iterest rate movemets). The calculatio should be approached as follows: First work out what the evetual retur at the ed of year (T) is goig to be, usig the right had side of the formula: T = =1 D j= i j 1+ 10( = i1 ) 11( = i ) 11( = i ) = ( = D1 ) 1+ 1+ + 50( = D ) 1+ = 1.60 The fid a value for α that satisfies the left had side: T = 1.60 = m =1 D α 1+ 1+m- α = ( = D1 ) 1+ + 50( = D α ) 1+ Tryig α = 10.5 gives 1.353; 10.6 gives 1.6; 10.59 gives 1.596. This process of iterative computatio yields the AER of 10.59% to two decimal places.. If i the above example there were ot a additioal deposit cotracted i the secod year, the calculatio is simpler ad formula (b) ca be used: α = 10( = i1 ) 11( = i ) 1+ 1+ -1 =10.98... givig a AER of 10.50% to two decimal places
5. If a accout pays iterest more ofte tha oce a year, the the AER is calculated by addig each iterest paymet to the deposit ad calculatig the ext iterest paymet o the total compoudig the iterest. The treatmet of mothly icome accouts shows the basic use of formula (d). Suppose a fixed deposit offers two optios: A) iterest paid aually of 6% per aum B) iterest paid mothly at a rate of 5.8% per aum. Optio A will have a AER of 6.00% (see example 1 above) For optio B, the AER is calculated usig formula (d) as 5.8(= i) α = 1+ 1(moths per year) givig a AER of 5.96% to two decimal places 1-1 = 5.956... This demostrates the value to the depositor of the mothly iterest but also shows that the two optios are ot quite idetical i terms of retur. 6. A short-term bod, for example a 8-moth bod payig 5.5% per aum, has to be treated usig a combiatio of formulae (c) ad (d). First of all, use formula (c) to fid the AER o a mothly basis: 5.5 8 + α = 8 1 m = 0.51... 5.13...% -1 per aum paid mothly The use formula (d) to covert this to a aually compouded rate 5.13... α = 1+ 1( ) moths 1-1 = 5.5501... givig a AER of 5.55% to two decimal places. This is higher tha the gross rate reflectig the value of iterest paid after 8 moths rather tha a year.
. Ucoditioal bouses, for example a lauch bous of 0.5% paid at least util 30 Jue 000 (this example was drafted i November 1999) o a accout payig 5% aually o 30 April without a fixed term, are treated as a step dow i the rate whe the guaratee expires. So, assumig a deposit o 1 November 1999, the depositor would receive 5.5% for 8 moths ad the 5% for moths (followig guidelie A3, the calculatio is for the first year of the deposit). Formula (b) ca be applied i two half-years: 5.5 5.5 + 5 ( =.5) ( =.583...) α = 1 1 1 h + + =.666... 5.333... per aum paid half yearly -1 Agai, use formula (d) to covert this to a aually compouded rate α = 5.333... 1+ ( ) halves -1 = 5.03... Givig a AER of 5.0% to two decimal places. The advertisemet would cotai a statemet AER calculated assumig a ivestmet o 1 November 1999. If it were a brach leaflet, it would be displayed oly durig November 1999 (see paragraph 11 of the Code ad Guidelie A6). 8. Fially, cosider a hypothetical product with irregular (but committed) cash flows. The patter of this product is : Deposit 3,000 o April 1 i year 1, the 1,800 o Jauary 1 each year for three years, ad the 600 o Jauary 1 i year 5, a total of 9,000 to be repaid o April 1 i year 6. Iterest at % per aum is added to the accout o December 31 each year ad at repaymet, together with a % bous of the total amout deposited ( 9,000) for makig the required deposits ad holdig to maturity. Because the subsequet deposits ad iterest are ot added o the aiversary of the first deposit, the AER icludig coditioal bous has to be calculated o a quarterly basis ad the compouded up to a aual rate. The calculatio (i summary here, the spreadsheet overleaf shows the full calculatio) is as follows:
First calculate the total T to be repaid at the ed of the cotract, usig the formula: T = 0 (quarters) = 1 = 3000( = + 1800( =... D D 0 j= D1 ) 1 + i j 1+ ) 1 + 3(= (= ) 1 + (= i19 ) + 600( = D ) 1 + 16 1 + + 180 (bous of % of 9,000) i3 ) 1 + i (= i (= i (= i 0 )... 11 ) ) 1 +... 1 + (= i 0 (= ) i 0 ) = 11,85.8 The fid the AER icludig coditioal bous by solvig T = 11,85.8 = 0 = 1 D α 1+ 1- α = 3000( = D1 ) 1+ + 1800( = D 5 α +...+ 600( = D16 ) 1+ 0 α ) 1+ which, by tryig various values, yields α = 1.81.511...% per aum paid quarterly. Use formula (d) to covert this to a aually compouded rate α =.511... 1+ ( ) quarters -1 =.506... givig a AER icludig coditioal bous of.5% to two decimal places. 1
Detail of the calculatio of T: D ( ) i (%) 1 + i / T = sum of: 1 3000 0,11. 0 3 5.5 1.055 1800 0,00. 5 0 6 0 1.0 8 1800 0,3.6 9 0 10 0 11 1.0 1 1800 0,096.88 13 0 1 0 15 1.0 16 600 0 653. 1 0 18 0 19 1.0 0 1.5 1.015 Bous 180.00 Total 11,85.8 Note: It is possible that the AER may very slightly depedig o whe the iitial deposit is made relative to the first iterest paymet date. Where this is the case, either the assumptio about this relatioship should be clearly stated (see guidelie A6) or the lowest potetial result should be used (this usually results whe the period from iitial deposit to first iterest period is at a maximum).
Explaatio of the AER The followig pages give a simple explaatio of the AER which is iteded for distributio to staff ad/or customers. Ay commets o the explaatio or o this appedix should be set to the BBA, preferably by e-mail to aer@bba.org.uk. The BBA is ot i a positio to udertake AER calculatios but will do its best to advise o the iterpretatio of this appedix. A AER calculator for PCs is available from the BBA Publicatios Departmet.
AER a explaatio The Aual Equivalet Rate is a otioal rate quoted i advertisemets for iterest-bearig accouts which illustrates the cotractual (gross) iterest rate (excludig ay bous iterest payable) as if paid ad compouded o a aual basis. Advertisemets may also quote a AER icludig coditioal bous clearly idetified as such. If a accout pays or credits iterest oce a year, the the AER is equal to the gross rate. If a accout pays iterest more ofte tha oce a year, the the AER is calculated by addig each iterest paymet to the deposit ad calculatig the ext iterest paymet o the total compoudig the iterest. For example, a accout offerig 5% gross iterest paid quarterly o pays 1.5 (1.5% (¼ of 5%) of ) after 3 moths, 1.6 (1.5% of 101.5 ( + 1.5)) after six moths*, 1.8 (1.5% of 10.51) after ie moths, ad 1.30 at the ed of the year (1.5% of 103.9), givig a total icludig iterest of 105.09. The AER is thus 5.09%. *I practice, the calculatio is worked to more decimal places to avoid roudig errors. If a accout pays iterest at itervals greater tha 1 year, we are lookig for a rate which will give us the right aswer if applied ad compouded each year. For example, a accout which pays 5% for five years but pays it oly at the ed of the five years will pay back 15 after the five years o deposited (the origial plus 5 iterest). The AER is.56% ad we ca see how this works as follows:.56 (.56% of ) would be the iterest at the ed of year 1,. (.56% of 10.56) at the ed of year,.99 (.56% of 109.33) at the ed of year 3, 5.1 (.56% of 11.3) at the ed of year, ad 5.5 (.56% of 119.53) at the ed of year 5, givig a fial total of 1.98. Not exactly 15 but as close as we ca get to decimal places (try the calculatio with.5% istead you will get 15.05). The AER for more complicated patters of chagig deposits ad/or iterest rates ca really be solved oly by trial ad error a process best left to computers!
The AER is a otioal rate it is ot ecessarily equal to the cash retur because, i calculatig it, we make assumptios: The oly chages to the amout deposited that are take ito accout are those that are required by the terms of the accout. So, for example, o a accout from which you ca make withdrawals, the assumptio is that you make just a iitial deposit ad leave it there. O the other had, o a mothly savigs accout, it is assumed that you will make the deposits each moth. If certai deposits are required to qualify for a coditioal bous, the the AER icludig coditioal bous will be calculated assumig that you have made the ecessary deposits. The oly chages to rates that are take ito accout are those that are stated at the outset. No allowace is made for chage that may occur because market rates geerally move up or dow. So, for example, if a accout has a rate which will icrease the loger a deposit is held, the higher rate will be used after the requisite time. Where a iitial higher rate is guarateed for a umber of moths, the rate after that will be assumed to reduce to the ormal rate applicable at the time of the advertisemet. If a mothly savigs accout has a tiered iterest rate, the the appropriate rate will be used as the balace builds up. If a accout has a fixed or miimum term, the assumptio will be made that you leave your deposit there for that period. If the term is idefiite, the calculatio is doe o the first year, or util the first iterest paymet if that is after oe year. If a coditioal bous requires that you leave the deposit i the accout for a certai time, the the AER icludig coditioal bous will be calculated over that period. All iterest paid is treated as if it is ivested ad ears a rate of iterest equal to that beig eared o your deposit. This may, i fact, happe if the iterest is added to pricipal. However, eve o a accout makig mothly iterest paymets, this assumptio is made to illustrate the value to you of receivig these paymets durig the year. If a deposit has a very short term (for example, a six-moth bod), for AER purposes, it is assumed that you ca ivest the pricipal ad iterest at the ed of the period at the same rate for the rest of the year. The full specificatio of the AER is cotaied i the Code of Coduct for the Advertisig of Iterest-bearig Accouts. This explaatio is iteded to help you uderstad the AER, ot as a substitute for the Code.