Functions - Compound Interest

1 - Compound InterestObjective: Calculate final account balances using the formulas for com-pound and continuous application of exponential Functions is Compound Interest . When money isinvested in an account (or given out on loan) a certain amountis added to thebalance. This money added to the balance is called that Interest isadded to the balance, it will earn more Interest during the next compoundingperiod. This idea of earning Interest on Interest is called Compound Interest . Forexample, if you investS100 at 10% Interest compounded annually, after one yearyou will earnS10 in Interest , giving you a new balance ofS110. The next yearyou will earn another 10% orS11, giving you a new balance ofS121.

2 The thirdyear you will earn another 10% , giving you a new balance pattern will continue each year until you close the are several ways Interest can be paid. The first way, as described above, iscompounded annually. In this model the Interest is paid onceper year. Butinterest can be compounded more often. Some common compounds include com-pounded semi-annually (twice per year), quarterly (four times per year, such asquarterly taxes), monthly (12 times per year, such as a savings account), weekly(52 times per year), or even daily (365 times per year, such assome studentloans). When Interest is compounded in any of these ways we can calculate thebalance after any amount of time using the following formula: Compound Interest Formula:A=P(1 +rn)ntA=Final AmountP=Principle(starting balance)r= Interest rate(asadecimal)n=number of compounds per yeart=time(in years)Example you take a car loan forS25000 with an Interest rate of compounded quar-terly, no payments required for the first five years, what willyour balance be atthe end of those five years?

3 P=25000, r= , n= 4, t= 5 Identify each variableA=25000(1 + )4 5 Plug each value into formula,evaluate parenthesisA=25000( )4 5 Multiply exponentsA=25000( )20 Evaluate exponentA=25000( )MultiplyA= , Our Solution1We can also find a missing part of the equation by using our techniques forsolving principle will amount toS3000 if invested at compounded weekly for4 years?A=3000, r= , n=52, t= 4 Identify each variable3000=P(1 + )52 4 Evaluate parentheses3000=P( )52 4 Multiply exponent3000=P( )208 Evaluate exponent3000=P( )Divide each side by Our SolutionIt is interesting to compare equal investments that are madeat several differenttypes of compounds. The next few examples do just is invested in an account paying 3% Interest compoundedmonthly, whatis the balance after 7 years?

4 P=4000, r= , n=12, t= 7 Identify each variableA=4000(1 + )12 7 Plug each value into formula,evaluate parenthesesA=4000( )12 7 Multiply exponentsA=4000( )84 Evaluate exponentA=4000( )MultiplyA= Our SolutionTo investigate what happens to the balance if the compounds happen more often,we will consider the same problem, this time with Interest compounded is invested in an account paying 3% Interest compoundeddaily, what isthe balance after 7 years?P=4000, r= , n=365, t= 7 Identify each variableA=4000(1 + )365 7 Plug each value into formula,evaluate parenthesisA=4000( )365 7 Multiply exponentA=4000( )2555 Evaluate exponentA=4000( .)MultiplyA= Our SolutionWhile this difference is not very large, it is a bit higher.

5 Thetable below showsthe result for the same problem with different the table illustrates, the more often Interest is compounded, the higher thefinal balance will be. The reason is, because we are calculating Compound interestor Interest on Interest . So once Interest is paid into the account it will startearning Interest for the next Compound and thus giving a higher final next question one might consider is what is the maximum number of com-pounds possible? We actually have a way to calculate Interest compounded aninfinite number of times a year. This is when the Interest is compounded continu-ously. When we see the word continuously we will know that we cannot use thefirst formula. Instead we will use the following formula: Interest Compounded Continuously:A=P ertA=Final AmountP=Principle(starting balance)e=aconstant approximately.

6 R= Interest rate(written asadecimal)t=time(years)The variableeis a constant similar in idea to pi ( ) in that it goes on foreverwithout repeat or pattern, but just as pi ( ) naturally occurs in several geometryapplications, so doeseappear in many exponential applications, continuousinterest being one of them. If you have a scientific calculator you probably haveanebutton (often using the 2nd or shift key, then hit ln) that will be useful incalculating Interest compounded View Note:efirst appeared in 1618 in Scottish mathematician sNapier s work on logarithms. However it was Euler in Switzerland who used theletterefirst to represent this value. Some say he usedebecause his name beginswith E. Others, say it is because exponent starts withe.

7 Others say it is becauseEuler s work already had the letterain use, so e would be the next value. What-ever the reason, ever since he used it in 1731,ebecame the natural is invested in an account paying 3% Interest compoundedcontinuously,what is the balance after 7 years?P=4000, r= , t= 7 Identify each of the variablesA= 7 Multiply exponentA= ( )MultiplyA= Our SolutionAlbert Einstein once said that the most powerful force in theuniverse is com-pound Interest . Consider the following example, illustrating how powerful com-pound Interest can you in an account paying 12% Interest compounded continuouslyfor 100 years, and that is all you have to leave your children as an inheritance,what will the final balance be that they will receive?

8 P= , r= , t=100 Identify each of the variablesA= 100 Multiply exponentA= (162, )MultiplyA= 1,002, ,002, Our SolutionIn 100 years that one time investment investment grew to over one mil-lion dollars! That s the power of Compound Interest !Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( ) Practice - Compound InterestSolve1) Find each of the invested at 4% compounded annually for 10 invested at 6% compounded annually for 6 invested at 3% compounded annually for 8 invested at 4% compounded semiannually for 7 invested at 6% compounded semiannually for 5 invested at 4% compounded semiannually for 12 invested at 5% compounded quarterly for 6 invested at 4% compounded quarterly for 9 invested at 6% compounded quarterly for 12 All of the above compounded ) What principal will amount toS2000 if invested at 4% Interest compoundedsemiannually for 5 years?

9 53) What principal will amount toS3500 if invested at 4% Interest compoundedquarterly for 5 years?4) What principal will amount toS3000 if invested at 3% Interest compoundedsemiannually for 10 years?5) What principal will amount toS2500 if invested at 5% Interest compoundedsemiannually for years?6) What principal will amount toS1750 if invested at 3% Interest compoundedquarterly for 5 years?7) A thousand dollars is left in a bank savings account drawing 7% Interest ,compounded quarterly for 10 years. What is the balance at theend of thattime?8) A thousand dollars is left in a credit union drawing 7% compounded is the balance at the end of 10 years?9)S1750 is invested in an account earning Interest compounded monthlyfor a 2 year period.

10 What is the balance at the end of 9 years?10) You lend outS5500 at 10% compounded monthly. If the debt is repaid in 18months, what is the total owed at the time of repayment?11) AS10,000 Treasury Bill earned 16% compounded monthly. If the billmatured in 2 years, what was it worth at maturity?12) You borrowS25000 at Interest compounded monthly. If you areunable to make any payments the first year, how much do you owe,excludingpenalties?13) A savings institution advertises 7% annual Interest , compounded daily, Howmuch more Interest would you earn over the bank savings account or creditunion in problems 7 and 8?14) An account earns continuous Interest . IfS2500 is deposited for 5 years,what is the total accumulated?