Exponential Functions and Investing. Compounded Interest. Compounded Interest

Transcription

1 Exponential Functions and Investing 1 Compounded Inteest Peiodic gowth of cash is efeed to as compounded inteest i = inteest ate pe yea (in decimal fom) t = time in yeas P = Pincipal (initial quantity) n = numbe of peiods pe yea i = ate pe peiod n nt = total numbe of peiods Compounded Inteest nt i A P1 P1 n nt This is used when inteest is eaned at peiods thoughout the yea Peiod n Annual 1 Semi-annual 2 Quately 4 Monthly 12 Banks efe to a compounded annual ate as annual pecentage ate (APR) 1

2 i = 0.07 t = 8 yeas P = $5000 n = = 12 Suppose we invest $5000 at 7% compounded monthly fo 8 yeas 0.07 A $ $ nt Move cuso to FV and Alpha Solve If Paul has $4300 that he wants to invest at 5% compounded semiannually, find the function he can use to detemine his balance at diffeent times in the futue. i = 0.05 P = $4300 n = = nt 2t $ t At $ t $ t Ralph and Rita have a five yea old daughte who aleady is planning on going to college They want to know much they should invest in a account eaning 7.25% inteest compounded monthly to have $30,000 in 12 yeas i = A = $30,000 n = = 12 nt P P P 12,

3 Ralph and Rita have a five yea old daughte who aleady is planning on going to college They want to know much they should invest in a account eaning 7.25% inteest compounded monthly to have $30,000 in 12 yeas P 12, You invested in popety nea the Rive Walk in 2006 fo $325,000 and sold it in 2010 fo $685,000. What is the annual ate of etun fo this investment? t n 1 Using P1 n 4 $325, , , , 000 ln 1 ln nt 137 4ln 1 ln ln 1 ln e ln ln 4 65 e You invested in popety nea the Rive Walk in 2006 fo $325,000 and sold it in 2010 fo $685,000. What is the annual ate of etun fo this investment? ln 1 ln 4 65 Answe: 20.49% 9 3

4 You invested in popety nea the Rive Walk in 2006 fo $325,000 and sold it in 2010 fo $685,000. What is the annual ate of etun fo this investment? Answe: 20.49% 10 Compound Inteest The powe of compounding can have an astonishing effect on the accumulation of wealth. This table shows the esults of making a one-time investment of $10,000 fo 30 yeas using 12% simple inteest, and 12% inteest compounded yealy and quately. Type of Inteest Pincipal Plus Inteest Eaned Simple 46, Compounded Yealy 299, Compounded Quately 347, Continuously Compounded Inteest Fo continuously compounded inteest we use the fomula A Pe t P = Pincipal = ate (in decimal) t = time (in yeas This is the fom we aleady leaned fo continuous gowth 4

5 $100 invested at 6% fo 5 yeas.06 Annual Semi-Annual Quately Monthly Daily Continuous e A bank advetises a 5.5% continuously compounded savings account. If Athu invests $3000 fo 3 yeas and six months, what is his expected yield? P = 3000 = t = 3.5 A $3000e $ Equivalent Annual Rate When compaing diffeent inteest ates, we used what is called the equivalent annual ate. Banks efe to this as annual equivalent ate (EAR) o effective APR. Peiodic compounded i 1 1 n i Continuously compounded e 1 n The EAR is always lage The Tuth in Lending Act of 1968 equies that Banks explain thei ates using the EAR 5

6 A bank advetises an investment pogam with 5% nominal inteest ate compounded semi-annually. What is the effective ate of this pogam? m.05 e m o % 16 PAC Bank A bank advetises an investment pogam with 12% inteest ate compounded continuously. What is the effective ate of this pogam? e 0.12 e 1 e o 12.75% 17 Time Value of Money The basic idea of time value of money is that a dolla today is woth moe than a dolla tomoow. Fo example having the money today allows you to buy some food immediately. Altenatively you may be willing to fogo cuent consumption and wait until late to puchase you food. A dolla eceived today is woth moe than a dolla eceived in the futue. The soone you money can ean inteest, the faste the inteest can ean inteest. 18 6

7 Definition Pesent Value - The amount of cash in today's dollas that is equivalent in value to futue payments, discounted at an appopiate compounded inteest ate. The time value of money pinciple says that futue dollas ae not woth as much as dollas today. Futue value the value, in the futue, of a cuent investment. If we know the futue value of an investment, we need a way to detemine the pesent value. 19 Remaks t t Compae the fomulas A P1 A P1 A Pe A P t 2 Dt A Pe t A P 2 th we also have i AP1 n nt i and A P1 n nt Ameica's Hot Dog King sells a cat fo $7, Justin wants to buy a cat in seven yeas. He plans on depositing cash into an account eaning 11% compounded quately. How much cash should he deposit? nt i This is whee we need A P1 n A $70991 $ Note: This should make sense that about $3K gows to about $8K ou answe is smalle To see Justin s hotdog cat see 7

8 Ameica's Hot Dog King sells a cat fo $7, Justin wants to buy a cat in seven yeas. He plans on depositing cash into an account eaning 11% compounded quately. How much cash should he deposit? $ $20, Find the pincipal needed to have $20,000 invested at 8%, compounded quately fo 8 yeas P= $20, We ae given t 8 n $20, $20, $10, Zeo Coupon Bonds Zeo coupon bonds ae bonds that do not pay inteest duing the life of the bonds. Instead, investos buy zeo coupon bonds at a deep discount fom thei face value, which is the amount a bond will be woth when it "matues" o comes due. When a zeo coupon bond matues, the investo will eceive one lump sum equal to the initial investment plus inteest that has accued. The matuity dates on zeo coupon bonds ae usually longtem many don t matue fo ten, fifteen, o moe yeas. 24 8

9 A zeo coupon bond can be edeemed in 15 yeas fo $5,000. How much should you be willing to pay now if you want a etun of 15% compounded quately? 0.15 P P P Remaks We sometimes find it convenient when given a ate to change it to doubling o halving time, o vice vesa ln 2 ln 2 D and D ln 2 ln 2 H and H Do not foget convet to decimal fom Alice invested $ in an account eaning 7% compounded annually. How long will it take fo Alice to double he investment? ln 2 D 9.9 yeas 0.07 Note: Shot cut quickly appoximate by using one of the following (leave in pecentage fom) D o D o D using ou appoximation to solve Alice s poblem we obtain D 70 =10 yeas 7 9

10 Remaks Fo time to double ln 2 t Fo time to tiple ln 3 t Fo time to five-times as much ln 5 t Juana wants to tiple he savings in fifteen yeas What inteest ate will she need to do this? ln 3 We solve t fo and obtain ln O 7.32% 15 ln 3 t 10