# Money Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version

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1 Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio

2 This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio Foudatio, Chael Oe News ad Ameica Saves. Special thaks to Rudy Gawo fo pepaig the lesso ad to Jill Sulam of Tasfomatios Editig LLC fo editoial guidace. Moey Math fo Tees. Copyight 2014 by the FINRA Ivesto Educatio Foudatio o FINRA Foudatio. Repoductio fo opofit, educatioal puposes is pemitted ad ecouaged. All ights eseved.

4 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest 4. Compoud iteest fomula: Itoduce the compoud iteest fomula: A = P (1+ ) t Whee: A = Accumulated balace P = Picipal = APR expessed as a decimal = Numbe of compoudig peiods pe yea t = Numbe of yeas the ivestmet lasts Review the sceaio outlied i Example 3, usig the compoud iteest fomula to calculate Michael s picipal, iteest ad APY at the ed of oe yea of quately compoudig. The fomula poduces the same accumulated value (\$8,161.20). The iteest is still \$ The APY is still 2.015%. Now You Ty studet woksheet: pactice aual vs. quately iteest calculatios usig the compoud iteest fomula (page 11). 5. Moey maket accout: Compouds mothly. Wok though pecalculated example of mothly compoudig (see teache woksheet, Example 4). Note: The calculatios peseted do ot take ito accout that diffeet moths have diffeet umbes of days. Mothly calculatios ae doe by computig aual iteest, the dividig by 12. Recalculate usig the compoud iteest fomula. 6. Aual vs. quately vs. mothly compoudig: If the picipal ad APR of ivestmets that compoud aually, quately ad mothly ae the same, which ivestmet will have the geatest etu i iteest (i.e., APY)? Show that mothly compoudig yields highest APY. Now You Ty studet woksheet: pactice aual, quately ad mothly iteest calculatios usig the compoud iteest fomula (page 12). 7. Discussio: Why might a bak advetise deposit accouts usig APY istead of APR? Why might they advetise APR istead of APY o cedit cads? 8. Evaluate studets compehesio (see assessmet woksheet). 2 Itoductio

5 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Teache Woksheet Vocabulay Picipal: A amout of moey owed by a ivesto ad held by a fiacial istitutio such as a bak. Deposit(s): The act of establishig, o addig to, existig picipal i a accout (veb); the moey placed i the accout (ou). Iteest: The amout of moey you ea by leavig deposits i a bak o fiacial istitutio. Iteest is a pecetage of you picipal. Tem: The peiod of time a ivestmet lasts. Aual pecetage ate (APR): The pecetage ate at which iteest is calculated aually. Cetificate of deposit (CD): A ageemet betwee a ivesto ad a bak (o fiacial istitutio) wheeby the ivesto agees to put a cetai amout of moey o deposit (this is the picipal) fo a cetai amout of time without withdawig it (this is the tem) ad the bak agees to pay the ivesto iteest at a ageed-upo pecetage ate, kow as the aual pecetage ate (APR). Compoudig peiod: The amout of time that elapses betwee iteest paymets. Aual compoudig: oce pe yea Quately compoudig: oce evey thee moths Jauay Mach: 1st quate Apil Jue: 2d quate July Septembe: 3d quate Octobe Decembe: 4th quate Mothly compoudig: oce pe moth Compoud iteest: Iteest calculated o both the picipal you have o deposit ad o iteest that has accumulated i the past. 3 Teache Woksheet

6 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Picipal ad Iteest Example 1 Michael is savig moey to buy a ca. He takes \$8,0 to the bak ad opes a aual CD upo which the bak agees to pay him 2% iteest. Picipal = 80 Tem = 1 yea APR = 2% = x 0.02 = \$ = \$8,160 Begiig Balace 2% Iteest Edig Balace \$8,0 \$160 \$8,160 Afte oe yea, Michael has eaed \$160 i iteest o his iitial deposit of \$8,0, so his balace is ow \$8,160. Aual Compoudig: Cetificate of Deposit Example 2 Now, let s say Michael leaves his moey i the bak fo fou yeas. The tem of the aual CD is fou yeas, so he will be eaig 2% iteest pe yea fo fou yeas. Sice this is a aual CD, iteest will be added to the picipal at the ed of evey yea. This is called aual compoudig. Begiig Balace 2% Iteest Edig Balace Yea 1 \$8,0. \$160. \$8,160. Yea 2 \$8,160. \$ \$8, Yea 3 \$8, \$ \$8, Yea 4 \$8, \$ \$8, Teache Woksheet

8 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Let s compae these two choices. Which type of accout would have eaed Michael moe iteest? (Statemet savigs accout) How much moe iteest would Michael ea by opeig the statemet savigs accout istead of the aual CD? (\$1.20) Why? Afte all, both accouts pay the same 2% APR. If 2% of \$8,0 is \$160, ad he eaed \$ o his picipal i the statemet savigs accout, the he must have actually eaed moe tha 2% i the statemet savigs accout. This is because he eaed iteest o his iteest duig the yea. His statemet savigs accout yielded moe tha 2% fo the yea. This exta eaig because of compoudig iteest is called aual pecetage yield, o APY. APY is the actual ate you moey eas, takig compoudig ito cosideatio. To calculate the APY, we divide the amout of iteest Michael eaed fo the yea by his oigial picipal deposit: APY = = = 2.015% 80 So a statemet savigs accout that pays a APR of 2% will ea a APY of 2.015% because of the effect of compoud iteest. Now You Ty Ask studets to do page 10 of the Now You Ty studet woksheet to pactice calculatig multiyea aual compoud iteest ad quately compoud iteest. 6 Teache Woksheet

9 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Compoud Iteest Fomula As you ca see, compoudig seveal times pe yea ad holdig a ivestmet fo multiple yeas would make fo a lot of maual calculatios. Luckily, thee is a fomula called the compoud iteest fomula that allows us to calculate the accumulated balace of a ivestmet acoss multiple yeas ad multiple compoudig peiods. Itoduce the compoud iteest fomula: A = P (1+ ) t Whee: A = Accumulated balace P = Picipal = APR expessed as a decimal = Numbe of compoudig peiods pe yea t = Numbe of yeas the ivestmet lasts Review the sceaio outlied i Example 3: Let s say Michael takes his \$8,0 to the bak ad opes a statemet savigs accout istead of a cetificate of deposit. The bak is goig to pay him the same 2% iteest o this accout that it was offeig fo the CD. Michael s begiig picipal amout is agai \$8,0. Also, his APR is still 2%. Howeve, Michael gets iteest compouded quately o this accout. Picipal (P) = 80 A = P (1+ ) t APR () = 2% = = 4 (quately) A = 80(1+ ) 4x1 4 t = 1 yea A = 80( ) 4 A = 80(1.5) 4 A = 80( ) A = \$ Compae the iteest calculated usig the fomula with the iteest calculated whe we did the calculatio the log way, by had. Notice that the amout of iteest (\$161.20) matches. Now You Ty Ask studets to do page 11 of the Now You Ty studet woksheet. 7 Teache Woksheet

10 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Mothly Compoudig: Moey Maket Savigs Accout Example 4 Michael s bak offes aothe type of ivestmet accout simila to the statemet savigs accout. This accout is called a moey maket savigs accout. This type of accout woks just like a statemet savigs accout except that the compoudig peiod is mothly istead of quately. This meas that Michael will eceive a iteest paymet deposit ito his accout at the ed of evey moth. Whe that happes, the iteest Michael eaed i the pevious moth is added to his picipal, ad the ew balace becomes Michael s ew picipal balace fo the ext moth. So ow Michael will begi eaig iteest o his iteest mothly! This time, Michael takes his \$8,0 to the bak ad opes a moey maket savigs accout istead of a statemet savigs accout. The bak is goig to pay him the same 2% o this accout that it was offeig fo the statemet savigs accout. Michael s begiig picipal amout is agai \$8,0. Also, his APR is still 2%. Howeve, Michael gets iteest compouded mothly o this accout. Afte thee moths, o oe quate, what will Michael s picipal balace be? 1. Fist, calculate his aual iteest: 80 x 0.02 = \$ Next, calculate what his 1st moth s iteest paymet will be: 160/12 = \$ At the ed of the 1st moth, Michael s ew picipal balace will be \$8, If we cotiue fo all 12 moths of the yea: Begiig Balace 2% Iteest Edig Balace 1st moth \$8,0. \$13.33 \$8, d moth \$8, \$13.36 \$8, d moth \$8, \$13.38 \$8, th moth \$8, \$13.40 \$8, th moth \$8, \$13.42 \$8, th moth \$8, \$13.44 \$8, th moth \$8, \$13.47 \$8, th moth \$8, \$13.49 \$8, th moth \$8, \$13.51 \$8, th moth \$8, \$13.53 \$8, th moth \$8, \$13.56 \$8, th moth \$8, \$13.58 \$8, Note: We ouded up the mothly iteest calculatio befoe we added a moth s iteest to the balace at the begiig of the moth. Thee may be as much as a \$0.03 diffeece at the ed of the yea, as ca be see below i the accumulated balace calculated by usig the fomula. This \$0.03 diffeece still yields the same APY. 8 Teache Woksheet

11 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Michael s \$8,0 oigial picipal deposit, put ito a moey maket savigs accout at 2% APR, compoudig mothly, would be woth \$8, at the ed of oe yea. What APY does this accout yield? Agai, divide the amout of iteest Michael eaed fo the yea by his oigial picipal deposit: APY = = = 2.018% 80 Now, let s ty usig the fomula to calculate Michael s balace at the ed of oe yea: Picipal (P) = 80 A = P (1+ ) t APR () = 2% = = 4 (quately) A = 80(1+ ) 12x1 12 t = 1 yea A = 80( ) 12 The fomula yields a accumulated balace of \$8, o \$0.03 less tha it would be if the calculatio wee doe by had. Compae Aual vs. Quately vs. Mothly Compoudig We ve see that the legth of the compoudig peiod fo a ivestmet affects its APY. If the APR fo a ivestmet is 2%, the APY will be: 2% fo aual compoudig 2.015% fo quately compoudig 2.018% fo mothly compoudig. Now You Ty Ask studets to do page 12 of the Now You Ty studet woksheet. Discussio Questios A = 80(1.1666) 12 A = 80( ) A = \$ Why might a bak advetise deposit accouts usig APY istead of APR? Deposit accout iteest eas people moey, so they would be attacted to the highe ate. Why might they advetise APR istead of APY o cedit cads? Cedit cad iteest costs people moey, so they would athe see the lowe ate. Assessmet Ask studets to complete the assessmet woksheet. 9 Teache Woksheet

12 Name Date Studet Woksheet NOW YOU TRY: APR, APY AND COMPOUND INTEREST Aual ad Quately Iteest Taylo wats to ivest he \$7,5 college fud at he bak. She picks a aual cetificate of deposit that will pay he 3% iteest each yea. She wo t eed he college moey fo thee moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) A. Fill i the table to fid out how Taylo s ivestmet gows. Begiig Balace 3% Iteest Edig Balace Yea 1 \$7,5. Yea 2 Yea 3 B. If Taylo chose a statemet savigs accout that compouds quately istead, how would he accout gow i the fist yea? Begiig Balace 3% Iteest Edig Balace 1st quate \$7,5. 2d quate 3d quate 4th quate C. Compae the APYs Taylo s accouts would ea depedig o which accout type she chooses. Aual CD APY: Quately statemet savigs accout APY: 10 Studet Woksheet

13 Name Date Aual vs. Quately Iteest: Compoud Iteest Fomula Thee is a fomula you ca use to calculate the edig balace of a ivestmet if you kow cetai facts about the ivestmet, such as its picipal, APR, compoudig peiods ad the umbe of yeas the ivestmet lasts. A = P (1+ ) t Whee: A = Accumulated balace P = Picipal = APR expessed as a decimal = Numbe of compoudig peiods pe yea t = Numbe of yeas the ivestmet lasts Let s look agai at Taylo s situatio: Taylo wats to ivest he \$7,5 college fud at he bak. She has two ivestmet optios: a CD that compouds aually, o a statemet savigs accout that compouds quately. Both ae offeig a APR of 3%. She wo t eed he college moey fo thee moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) 2. What ae the Picipal = APR = Numbe of compoudig peiods pe yea = Numbe of yeas the ivestmet lasts = 3 fo the CD ad the statemet savigs accout? Use the compoud iteest fomula to calculate he balace at the ed of the thee-yea ivestmet peiod fo both the aual CD ad the quately statemet savigs accout. Aual CD A = P (1+ ) t A = P (1+ ) t Quately Statemet Savigs Accout 11 Studet Woksheet

14 Name Date Aual vs. Quately vs. Mothly Iteest: Compoud Iteest Fomula Adea wats to ivest \$2,5 at he bak. She has thee ivestmet optios: a CD that compouds aually, a statemet savigs accout that compouds quately o a moey maket savigs accout that compouds mothly. All ae offeig a APR of 3%. She wo t eed the fuds i he accout fo five moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) A. Use the compoud iteest fomula to calculate he balace at the ed of the five-yea ivestmet peiod fo each optio: the aual CD, quately statemet savigs accout ad mothly moey maket savigs accout. A = P (1+ ) t Aual CD Quately Statemet Savigs Accout Mothly Moey Maket Savigs Accout B. Explai what ivestmet advice you d give Adea ad why. 12 Studet Woksheet

15 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Studet Woksheet: Solutios NOW YOU TRY: SOLUTIONS Aual ad Quately Iteest 1. Taylo wats to ivest he \$7,5 college fud at he bak. She picks a aual cetificate of deposit that will pay he 3% iteest each yea. She wo t eed he college moey fo thee moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) A. Fill i the table to fid out how Taylo s ivestmet gows. Begiig Balace 3% Iteest Edig Balace Yea 1 \$7,5. \$225. \$7,725. Yea 2 \$7,725. \$ \$7, Yea 3 \$7, \$ \$8, B. If Taylo chose a statemet savigs accout that compouds quately istead, how would he accout gow i the fist yea? Begiig Balace 3% Iteest Edig Balace 1st quate \$ 7,5. \$56.25 \$7, d quate \$7, \$56.67 \$7, d quate \$7, \$57.10 \$7, th quate \$7, \$57.53 \$7, C. Compae the APYs Taylo s accouts would ea depedig o which accout type she chooses. Aual CD APY: 225 APY = = 0.03 = 3% 75 Quately statemet savigs accout APY: APY = = = 3.034% Studet Woksheet: Solutios

16 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Aual vs. Quately Iteest: Compoud Iteest Fomula Thee is a fomula you ca use to calculate the edig balace of a ivestmet if you kow cetai facts about the ivestmet, such as its picipal, APR, compoudig peiods ad the umbe of yeas the ivestmet lasts. Whee: A = Accumulated balace A = P (1+ ) t P = Picipal = APR expessed as a decimal = Numbe of compoudig peiods pe yea t = Numbe of yeas the ivestmet lasts Let s look agai at Taylo s situatio: Taylo wats to ivest he \$7,5 college fud at he bak. She has two ivestmet optios: a CD that compouds aually, o a statemet savigs accout that compouds quately. Both ae offeig a APR of 3%. She wo t eed he college moey fo thee moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) 2. What ae the Picipal = 75 APR = 3% = 0.03 Numbe of compoudig peiods pe yea = 1 fo the CD, 4 fo the statemet savigs accout Numbe of yeas the ivestmet lasts = 3 fo the CD ad the statemet savigs accout? Use the compoud iteest fomula to calculate he balace at the ed of the thee-yea ivestmet peiod fo both the aual CD ad the quately statemet savigs accout. Aual CD A = P (1+ ) t 0.03 A = 75(1+ 1 A = 75(1.03) 3 A = 75( ) A = \$ ) 1x3 Quately Statemet Savigs Accout A = P (1+ ) t 0.03 A = 75(1+ ) 4x3 4 A = 75( ) 12 A = 75(1.75) 12 A = 75( ) A = \$ Studet Woksheet: Solutios

17 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Aual vs. Quately vs. Mothly Iteest: Compoud Iteest Fomula 3. Adea wats to ivest \$2,5 at he bak. She has thee ivestmet optios: a CD that compouds aually, a statemet savigs accout that compouds quately o a moey maket savigs accout that compouds mothly. All ae offeig a APR of 3%. She wo t eed the fuds i he accout fo five moe yeas. (Remembe, 3% is 0.03 whe witte as a decimal.) A. Use the compoud iteest fomula to calculate he balace at the ed of the five-yea ivestmet peiod fo each optio: the aual CD, quately statemet savigs accout ad mothly moey maket savigs accout. A = P (1+ ) t Aual CD Quately Statemet Mothly Savigs Accout Moey Maket Savigs Accout 0.03 A = 25(1+ 1 A = 25(1.03) 5 A = 25( ) A = \$2, ) 1x A = 25(1+ ) 4x5 4 A = 25( ) 20 A = 25(1.75) 20 A = 25( ) A = \$2, A = 25(1+ 12 A = 25( ) 60 A = 25(1.25) 60 A = 25( ) A = \$2, ) 12x5 B. Explai what ivestmet advice you d give Adea ad why. Adea should put he moey ito the moey maket savigs accout that compouds mothly because it will yield the geatest etu. 15 Studet Woksheet: Solutios

18 Name Date Studet Assessmet Savig moey ad plaig fo you futue ae ot oly smat thigs to do, they e essetial! Makig smat decisios about ivestmets begis by educatig youself about what types of ivestmets ae available to you ad which will wok the best fo you i you paticula situatio. The compoud iteest fomula is a tool you ca use to evaluate diffeet ivestmets. Make it wok fo you. A = P (1+ ) t Goal: You wat to have \$10,0 saved i a five-yea time fame. How much picipal would you eed to put away i a bak accout offeig a 4% APR ad compoudig mothly i ode to each that goal? A = P = = = t = Challege You have \$8,250 to ivest, ad you wo t eed the moey util five yeas fom ow. You decide you will put the moey ito a bak accout compoudig mothly fo that peiod of time. If you goal is to have \$10,0 whe the ivestmet matues, what APR do you eed to achieve you goal? A = P = = = t = 16 Studet Assessmet

19 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest Studet Assessmet: Solutios Savig moey ad plaig fo you futue ae ot oly smat thigs to do, they e essetial! Makig smat decisios about ivestmets begis by educatig youself about what types of ivestmets ae available to you ad which will wok the best fo you i you paticula situatio. The compoud iteest fomula is a tool you ca use to evaluate diffeet ivestmets. Make it wok fo you. A = P (1+ ) t Goal: You wat to have \$10,0 saved i a five-yea time fame. How much picipal would you eed to put away i a bak accout offeig a 4% APR ad compoudig mothly i ode to each that goal? A = 1 P = ukow = 4% = 0.04 = 12 t = = P (1+ ) 12x = P (1.333) 60 1 = P ( ) 1 P = = \$8, You d have to ivest \$8, to accumulate \$10,0 five yeas fom ow. Challege You have \$8,250 to ivest, ad you wo t eed the moey util five yeas fom ow. You decide you will put the moey ito a bak accout compoudig mothly fo that peiod of time. If you goal is to have \$10,0 whe the ivestmet matues, what APR do you eed to achieve you goal? A = 1 P = 8250 = ukow = 12 t = 5 17 Studet Assessmet: Solutios

20 Itoductio to Eaig Iteest: APR, APY ad Compoud Iteest 60 1 = 8250(1+ ) 12x = (1+ ) =(1+ ) = 60 (1+ ) 60 =(1+ ) =(1+ ) = x 12 = = = % You d eed the accout to have a APR of % to accumulate \$10,0 five yeas fom ow. 18 Studet Assessmet: Solutios

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