Simple Annuities Present Value.


 Gwendoline Lyons
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1 Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities. ote that we suggest you complete the activity called Uderstadig ad workig with compoud iterest before startig this set of activities. EXPLORATORY ACTIVITIES Activity 1: Payig back a loa. Suppose I am iterestig i buyig a ew large (45 ich) Sharp LCD TV RRP $ Also, suppose I have to take out a loa for such a large purchase. It is a shame, but may people do ot uderstad the way the repaymets are calculated or how to carry out some what if scearios with loas. This ivestigatio will equip you to be able to do this, with ease. Suppose you borrow $12000 from a bak. The bak tells you that you must make regular repaymets every moth for 3 years util you have paid back the loa. Ufortuately, for you, it is ot as simple has repayig $ because the bak has allowed you to use their moey ad wat some moey i retur for the privilege (this is called iterest). The way the process works ca be described as follows: You borrow $12000 (preset value of the loa) ad the bak charges you (adds to the loa) a percetage of this for its use durig the first moth. At the ed of the moth you make a repaymet (cotributio). So at the start of the ext moth you owe the startig preset value + the iterest  your repaymet (cotributio). The the bak charges you a percetage of this moths preset value for usig it durig this moth ad you make aother repaymet (cotributio) at the ed of this moth. So at the start of the ext moth you owe the previous moths preset value + the iterest figure  your repaymet. This (repetitive) process cotiues each moth util you owe othig. Page 1 of 5
2 Let s experimet. Suppose that the baks states the iterest will be 7% per aum compouded mothly ad that you have to make mothly repaymets of $40. To see the amout you would owe at the ed of the first moth (the preset value of the loa at the start of the secod moth) we could do the followig usig the 9850GB PLUS: Eter RU mode Eter Commit it to the calculators aswer mode by pressig EXE Multiply the As(wer) (SHIFT the ()) by ( ) Subtract 40 ad calculate the aswer by pressig EXE You should fid that preset value of the auity at the start of the ext moth is $ Pressig EXE agai will repeat the process for the secod moth. You should fid you would owe (the preset value at the start of the third moth) $ (to the earest cet). You should realize that somethig is wrog here. Keep pressig the EXE butto to see the amouts you will owe moth by moth. What do you otice? Clearly the amout of your repaymet must be greater tha the iterest charged or you will ever pay the loa back. Lets suppose you pay back $50 per moth. You ca ivestigate this without reeterig the whole calculatio. First commit to the AS memory (12000 ad EXE). You must do this as the machie takes the last umber computed ad places it i the AS memory. Page 2 of 5
3 ow press AC/O to clear the RU scree ad the press the UP ARROW ( ) of the mouse twice to recall the previous calculatio we etered (uless you have tured the machie off or chaged modes). ow press to have the cursor eter the calculatio lie. You ca edit the calculatio by pressig repeatedly util you have the cursor o the 4 i the umeral 40. The press the DEL(ete) key ad the 5. You should fid that you owe $12020 at the start of the secod moth. So you are still ot makig large eough repaymets. ow you ca experimet to fid the amout you must repay to breakeve. Explore this further ad share your fidigs with your colleagues. Be sure to experimet here with cotributios greater tha the iterest amout. Ca you thik of ay other situatios related to ivestig where this process is applied? Oe such case is a retiremet fud where a retiree has, for example, $ ad ivests it i a auity ad takes out regular istallmets (egative cotributios if you like). Such cases as we have see here are traditioally called Preset Value auities because they are associated with situatios where you eed the moey presetly the value of the auity is i the preset. That is, you start with some moey i the preset ad it is eroded to zero evetually. Activity 2: Geeralisig the Preset Value auity computatio. A formula exists for computig values associated with Preset Value auities. Its use saves us from repeatedly pressig the equal sig o the calculator. The derivatio of the formula relies o kowledge of geometric series. You might like to research this. Firstly, defie each quatity as follows: Let the preset value of the auity, after compoudig periods, be Let the regular cotributios (or take outs) be M Let the percetage iterest rate per compoudig period be r (expressed as a decimal) Let the umber of compoudig periods be Page 3 of 5
4 The formula is: = M ( ) 1+ r 1 r( 1+ r) The 9850Gb PLUS ca be use to compute the value of, M, r or if all but oe of the variables is kow. I EQUA mode, after choosig SOLV(er) (F3) the formula ca be etered as P = M ( ) 1+ R 1 R( 1 + R) usig bracket carefully. ote that is replaced with P as sesible chage. So, if M = 100, r explai why? = R= , ad = =1, you should fid that P < 100, ca you Oce the result is give you ca cotiue to use the formula you have etered. Simply press REPT (F1) ad you will be prompted for the values of the variables agai. This is a useful feature if you have may of the same computatios to do where the values of the variable chage ofte. Activity 3: Usig the formula more broadly. Of course we ca use the formula for more tha just computig the preset value of the auity after iterest periods. Suppose we wated to kow how log we could remove $1500 per moth from a auity with preset value $ if the iterest paid was 4% compouded mothly This would require us to solve the equatio = for. Page 4 of 5
5 Proceed as see below: ad i the blik of a eye EXERCISES The purpose of the exercises is to give you a opportuity to do some idepedet work with simple auities ad to develop fluecy with both the mathematical cocepts ad Casio 9850GB PLUS. Exercise 1. Ralph borrows $15000 to buy a car. The coditios of the loa are that he must make repaymets $350 each moth. The iterest rate associated with the loa is 12% per aum compouded mothly. a) How log will Ralph take to pay the loa back? b) How much iterest does Ralph pay? c) After how may periods will the amout owed first fall below $ SOLUTIOS Exercise 1 a) After 57 periods (4 years ad 9 moths) he will owe less tha the paymet value ($83.80). b) $ c) 23 periods. Page 5 of 5
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