Financial Math Lesson #2

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Donna Hudson
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1 Compoud Iteest Fiacial Math Lesso #2 Compoud iteest is a method of calculatig iteest i which the iteest is added to the picipal each peiod so that the picipal cotiues to gow thoughout the life of the loa o ivestmet (ulike simple iteest whee the picipal emais costat thoughout the peiod of the loa o ivestmet). Thus, the iteest geeated i oe peiod the eas iteest itself i the ext peiod. Calculate the iteest paid o a deposit of $6000 at 8% p.a. compouded aually fo 3 yeas. Compouded aually meas iteest is added to the picipal aually so we eed to calculate iteest each yea. We will use the simple iteest fomula ad set out ou calculatios i table fom. Yea Picipal () Iteest ( = C ) (2) Balace () + (2) $ $ = $80.00 $ $ $ = $58.0 $ $ $ = $ $ Theefoe, the $ gows to $ afte 3 yeas, i.e., $ $ = $ is iteest. Notice that the amout of iteest paid iceases fom yea to yea. ($80.00, $58.0, $559.87). Ca you see why? Note: Total Iteest eaed = fial balace picipal.

2 Diffeet Compoudig Peiods Iteest ca be compouded moe tha oce pe yea. Commoly, iteest ca be compouded: o half-yealy (two times pe yea) o quately (fou times pe yea) o mothly ( times pe yea) o daily! (365 o 366 times a yea) Calculate the fial balace of a $00 ivestmet at 6% p.a. whee iteest is compouded quately fo oe yea. We eed to calculate the iteest geeated each quate. Quate Picipal () Iteest ( = C ) (2) $00.00 $ = $ $ $ = $ $ $ = $5.53 $ $ = $56.85 Balace () + (2) $ $ $ $ Theefoe, the fial balace would be $ Compoud Iteest Fomulae A C ad I= C + C whee o A is the futue value (o fial balace) o C is the peset value o picipal (amout oigially ivested) o is the iteest ate pe compoud peiod o is the umbe of peiods (i.e., the umbe of times the iteest is compouded).

3 Example : Calculate the fial balace of a $00 ivestmet at 6% p.a. whee iteest is compouded quately fo oe yea. 6% C = 00, = =. 5% (pe quate), = = Now, A C So, A = 00 ( ) Theefoe, A = $063.6 i.e., the fial balace is $ Example 2: How much iteest is eaed if $8800 is placed i a accout % that pays p.a. compouded mothly fo 2.5 C= 8800, = = , = 3 = 2 2 Now, I= C + C So, I = ( ) 2 3 yeas? 2 Theefoe, I = $98.08 i.e., the iteest eaed is $ Example 3: Holly ivests $5000 i a accout that pays.25% p.a. compouded mothly. How much is he ivestmet woth afte 5 yeas? To aswe this usig the TVM fuctio o the calculato, fist set up the TVM scee. Note: The iitial ivestmet is cosideed as a outgoig ad is eteed as a egative value. Holly has $85.53 afte 5 yeas.

Fidig The Peset Value The peset value (picipal) is the amout of moey that we ivest ow, at a give compoud ate ove a give time iteval, so that it will amout to a give futue value.

4 Fidig The Peset Value The peset value (picipal) is the amout of moey that we ivest ow, at a give compoud ate ove a give time iteval, so that it will amout to a give futue value. How much does Halea eed to deposit ito a accout to collect $ at the ed of 3 yeas if the accout is payig 5.2% p.a. compouded quately? Fomula Solutio: 5.2 Give A = 50000, = =.3, = 3 = Usig A C.3 So, C Theefoe, C = {usig solve o the TI-83} i.e., $282 eeds to be deposited. Gaphig Calculato Solutio: To aswe this usig the TVM fuctio o the calculato, set up the TVM scee as show: Note: Thee ae 3 = quate peiods. Theefoe, $282 eeds to be deposited.

Fidig The Time Peiod Ofte we wish to kow how log we must ivest moey fo it to gow to a specified sum i the futue. Fo how log must Magus ivest $000 at 6.5% p.a. compouded half-yealy if it is to amout to $00?

5 Fidig The Time Peiod Ofte we wish to kow how log we must ivest moey fo it to gow to a specified sum i the futue. Fo how log must Magus ivest $000 at 6.5% p.a. compouded half-yealy if it is to amout to $00? Fomula Solutio: Give A = 00, C = 000, Usig A C 6.5 = = So, 00 = Theefoe, = 28.9 {usig solve o the TI-83} We fid that 29 half yeas ae equied, i.e.,.5 yeas. Gaphig Calculato Solutio: To aswe this usig the TVM fuctio o the calculato, set up the TVM scee as show. We the eed to fid the umbe of peiods equied. We fid that = Theefoe, 29 half yeas ae equied, i.e.,.5 yeas.

Fidig The Aual Rate Of Icease Whe compaig two ivestmets whee the peset ad fial values ae kow ove a paticula time peiod, we ca calculate the aual ate of icease.

6 Fidig The Aual Rate Of Icease Whe compaig two ivestmets whee the peset ad fial values ae kow ove a paticula time peiod, we ca calculate the aual ate of icease. If Ima deposits $5000 i a accout that compouds iteest mothly ad 2.5 yeas late the accout totals $6000, what aual ate of iteest was paid? Fomula Solutio: Give A = 6000, C= 5000, = 2.5 = 30 Usig A C So, = pe moth = pe yea 7.32% Theefoe, 7.32% p.a. is equied. 30 Gaphig Calculato Solutio: To aswe this usig the TVM fuctio o the calculato, set up the TVM scee as show. Note: 2.5 = 30 moths. We fid the aual iteest ate equied. Theefoe, 7.32% p.a. is equied.

7 Tem (moths) Fixed Tem Deposits As the ame suggests, deposits ca be locked away fo a fixed time peiod (fom oe moth to te yeas) at a fixed iteest ate. The iteest is calculated o the daily balace ad ca be paid mothly, quately, half-yealy o aually. The iteest ca be compouded so that the picipal iceases duig the fixed tem. Howeve, the iteest ca also be paid out; may etiees live off iteest that fixed tem deposits geeate. Geeally, the ate offeed iceases if the moey is locked away fo a loge peiod of time. We will coside sceaios whee the iteest is compouded as a applicatio of compoud iteest. Below is a typical schedule of ates offeed by a fiacial istitutio: $5000 to $ ad $ to $ 000 Iteest at Mothly Quately Half Yealy Matuity Iteest Iteest Iteest $5k - $25k - $5k - $25k - $5k - $25k - $5k - $25k - $25k $k $25k $k $25k $k $25k $k 3.80%.50% 2.5%.75% % 5.20% 5.0% 6.00% 5* 5.5%* 6.25%* 6 5.5% 5.50% 5.5% 5.50% 7 8* 6.0%* 6.25%* 6.05%* 6.30%* % 5.90% 5.65% 5.90% 7* 6.30* 6.0%* 6.5%* 6.20%* 6.5%* 6.25%* 6.20%* 6.30%* % 6.0% 5.95% 6.20% 5.95% 6.25% 6.00% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% % 6.0% 6.05% 6.20% 6.05% 6.25% 6.0% 6.30% * idicates special offes Please ote: All iteest ates ae pe aum. Fo tems of moths o moe, iteest must be paid at least aually.

Example : Fo the fiacial istitutio whose ates ae listed o the pevious page, compae the iteest offeed if $5000 is deposited fo 5 moths ad the iteest is compouded: (a.

8 Example : Fo the fiacial istitutio whose ates ae listed o the pevious page, compae the iteest offeed if $5000 is deposited fo 5 moths ad the iteest is compouded: (a.) mothly (b.) quately. (a.) $5000 deposited fo 5 moths with the iteest compouded mothly eceives 6.2% p.a. Usig a gaphig calculato, we solve fo FV. Iteest = $ $5000 = $ (b.) $5000 deposited fo 5 moths with the iteest compouded quately eceives 6.25% p.a. Usig a gaphig calculato, we solve fo FV. Iteest = $ $5000 = $ Theefoe, the quately optio is moe by $0.72. Example 2: $00 is ivested i a fixed tem deposit fo 2 moths, with iteest paid mothly. Fid the effective afte tax etu if the ivesto s tax ate is 8.5 cets o the dolla. $00 deposited fo 2 moths with the iteest compouded mothly eceives 6.05% p.a. {fom the Fixed Tem Deposit Rate table} Usig a gaphig calculato, we solve fo FV.

9 Iteest = $82.82 $00 = $82.82 Tax = 8.5% of $82.82 = $622.7 Afte tax etu = $82.82 $622.7 = $ (which is a aveage of 3.30% p.a.) The Effective Iteest Rate O A Ivestmet Because iteest ates ae applied i diffeet ways, compaig them ca be misleadig. Coside $0 000 ivested at Now coside $00 ivested 6% compouded aually. at 5.85% compouded mothly. Afte yea, Afte yea, C = 00 + C = 00 + C = 00 (.06) C = 00 (.00875) C = 0600 C = Hece, 5.85% p.a. compouded mothly is equivalet to 6% p.a. compouded aually. We say 5.85% p.a. compouded mothly is a omial ate (the amed ate) ad it is equivalet to a effective ate of 6% p.a. compouded o a aual basis. The effective ate is the equivalet aualised ate (i.e., the iteest ate compouded aually). Covesio of Nomial (Compoud) Rates to a effective ate: c = (+ i) o is the effective ate o i is the ate pe compoud iteest peiod o c is the umbe of compoud peiods pe aum

10 Which is the bette ate offeed: % p.a. compouded mothly o.2% p.a. compouded quately? 0.0 c Give i = = = (+ i) c = = i.e., the effective ate is.07% p.a. = ( ) Calculato Covesio Of Nomial Rate To Effective Rate A gaphics calculato ca be used to do the calculatios above. Thee is a effective ate covesio fuctio i the fiace pogam. The sytax is Eff(omial ate, umbe of compoud peiods p.a.) Fo example, Eff(, ) =.07 ad Eff(.2, ) =.27 Note: If effective ate is give, use aual compouds. Covesio Of A Simple Iteest Rate To A Effective Rate A ivestmet offeig a simple iteest ate ove a umbe of yeas ca also be coveted to a equivalet ate compouded aually. Calculate the effective iteest ate fo a simple iteest ate of 6.5% p.a. applied fo 5 yeas. Coside $ ivested. Iteest = C = $ = $32.50 Theefoe, the $ gows to $ + $32.50 = $32.50 If the effective ate was % p.a., the usig the fomula C + = A, we ca fid the value of. 5 So, + = Theefoe, = 5.79 {usig solve o the TI-83} i.e., the effective iteest ate is 5.79% p.a.

Understanding Financial Management: A Practical Guide Problems and Answers

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