The Mathematics Of Money
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Mathematics of MoneyBeth Kirby and Carl LeeUniversity of KentuckyMA 111Fall 2009MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansMoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment Loans10.2 Simple InterestMoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Time Value of MoneyWhen you deposit 1000 into a savings account at the bank,you expect that amount to gain interest over time.A year from now, you would have more than 1000.In return for having access to the present value of your money,the bank increases the future value of the money by addinginterest.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Time Value of MoneyIf you take out a car loan for 10,000, you expect to pay itback with interest.Suppose the total amount you repay over time is 12,000.The present value is P 10, 000.The future value is F 12, 000.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansWhat determines the future value?The interest is the return the lender expects as a reward forthe use of their money.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansWhat determines the future value?The interest is the return the lender expects as a reward forthe use of their money.Since the amount of interest should depend on the amount ofthe loan, we consider an interest rate.The standard way to describe an interest rate is the annualpercentage rate or APR.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple InterestWith simple interest, only the principal (the original moneyinvested or borrowed) generates interest over time.The amount of interest generated each year will be the samethroughout the life of the loan/investment.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you buy a 1000 savings bond that pays 5% annual simpleinterest, how much is the bond worth 10 years from now?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you buy a 1000 savings bond that pays 5% annual simpleinterest, how much is the bond worth 10 years from now?The present value or principal is P 1000.Each year, the principal earns 5% interest.How much interest will be earned in one year?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you buy a 1000 savings bond that pays 5% annual simpleinterest, how much is the bond worth 10 years from now?The present value or principal is P 1000.Each year, the principal earns 5% interest.How much interest will be earned in one year? 1000 ·Money5 1000(0.05) 50.100UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleAfter one year, the bond will be worth 1000 50 1050.After two years, the bond will be worth 1000 50 50 1100.How much will the bond be worth after 10 years?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleAfter one year, the bond will be worth 1000 50 1050.After two years, the bond will be worth 1000 50 50 1100.How much will the bond be worth after 10 years? 1000 10( 50) 1500.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleAfter one year, the bond will be worth 1000 50 1050.After two years, the bond will be worth 1000 50 50 1100.How much will the bond be worth after 10 years? 1000 10( 50) 1500.How much will the bond be worth after t years? 1000 50t.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple Interest FormulaRemember that the annual interest was found by multiplying5 1000 100.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple Interest FormulaRemember that the annual interest was found by multiplying5 1000 100.In general, if the principal is P dollars and the interest rate isR%, the amount of annual interest is RP100or P · r where r MoneyR.100UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple Interest FormulaRemember that the annual interest was found by multiplying5 1000 100.In general, if the principal is P dollars and the interest rate isR%, the amount of annual interest is RP100or P · r where r R.100Over t years, the amount of interest accrued isP · r · t.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple Interest FormulaThus, the total future value will beP P ·r ·tMoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansSimple Interest FormulaThus, the total future value will beP P ·r ·tIf P dollars is invested under simple interest for t years at anAPR of R%, then the future value is:F P (1 r · t)where r is the decimal form of R%.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSuppose you want to buy a government bond that will beworth 2500 in 8 years. If there is 5.75% annual simpleinterest on the bond, how much do you need to pay now?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSuppose you want to buy a government bond that will beworth 2500 in 8 years. If there is 5.75% annual simpleinterest on the bond, how much do you need to pay now?We know the future value F 2500 and we want to find thepresent value or principal P.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSuppose you want to buy a government bond that will beworth 2500 in 8 years. If there is 5.75% annual simpleinterest on the bond, how much do you need to pay now?We know the future value F 2500 and we want to find thepresent value or principal P.Solve for P:2500 P (1 (0.0575)(8))2500 P(1.46)2500P 1.46P 1712.33.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaPage 393, #27: A loan of 5400 collects simple interest eachyear for eight years. At the end of that time, a total of 8316is paid back. Find the APR for the loan.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSolution: 5400 is the present value P, and 8316 is thefuture value F . Solve for r :8316 5400(1 8r )MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSolution: 5400 is the present value P, and 8316 is thefuture value F . Solve for r :831683162916rMoney 5400(1 8r )5400 5400 · 8r43200r0.0675.UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Simple Interest FormulaSolution: 5400 is the present value P, and 8316 is thefuture value F . Solve for r :831683162916r 5400(1 8r )5400 5400 · 8r43200r0.0675.The APR is 6.75%.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment Loans10.3 Compound InterestMoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansCompound InterestWith compound interest, the interest rate applies to theprincipal and the previously accumulated interest.Money collecting compound interest will grow faster than thatcollecting simple interest. Over time, the difference betweencompound and simple interest becomes greater and greater.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after one year?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after one year?2000 2000(.06) 2000(1 .06) 2000(1.06).MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after one year?2000 2000(.06) 2000(1 .06) 2000(1.06).After two years? The interest rate will be applied to theprevious amount, 2000(1.06).2000(1.06)(1.06) 2000(1.06)2 .MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after one year?2000 2000(.06) 2000(1 .06) 2000(1.06).After two years? The interest rate will be applied to theprevious amount, 2000(1.06).2000(1.06)(1.06) 2000(1.06)2 .After three years?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after one year?2000 2000(.06) 2000(1 .06) 2000(1.06).After two years? The interest rate will be applied to theprevious amount, 2000(1.06).2000(1.06)(1.06) 2000(1.06)2 .After three years?2000(1.06)2 (1.06) 2000(1.06)3 .MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after fifteen years?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR, how much is theinvestment worth after fifteen years?2000(1.06)15 2000(2.3966) 4793.20.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansCompound vs. Simple Interest10000800060004000200051015Blue line: 6% annual simple interestRed line: 6% annual compound interestMoney2025UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansCompound Interest FormulaIf P dollars is compounded annually for t years at an APR ofR%, then the future value isF P (1 r )twhere r is the decimal form of R%.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSuppose you invest 2000 in a fund with a 6% APR that iscompounded monthly. That is, interest is applied at the endof each month (instead of just the end of each year).Since the interest rate is 6% annually (APR), it must be6% 0.5% per month.12After one month, you’ll have 2000(1.005) 2010.After one year (twelve months), you’ll have 2000(1.005)12 2123.36.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR compoundedmonthly, how much is the investment worth after fifteen years?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleIf you invest 2000 in a fund with a 6% APR compoundedmonthly, how much is the investment worth after fifteen years?2000 (1.005)15·12 2000(1.005)180 4908.19.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansCompound Interest FormulaIf P is invested at an APR of R% compounded n times peryear, for t years, then the future value F is:F P 1 where r is the decimal form of R%.Money r ntnUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Compound Interest FormulaYou put 800 in a bank account that offers a 4.5% APRcompounded weekly. How much is in the account in 5 years?Since there are 52 weeks in a year, the interest rate isr 4.5% 0.086538% or 0.00086538.n52In 5 years, interest will be compounded nt 52 · 5 260times.800(1 0.00086538)260 800(1.00086538)260 800(1.252076) 1001.66.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Compound Interest FormulaYou want to save up 1500. If you can buy a 3 year CD(certificate of deposit) from the bank that pays an APR of 5%compounded biannually, how much should you invest now?Note: Biannually means two times per year.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansUsing the Compound Interest FormulaYou want to save up 1500. If you can buy a 3 year CD(certificate of deposit) from the bank that pays an APR of 5%compounded biannually, how much should you invest now?Note: Biannually means two times per year.Solve for P: 2·3 .051500 P 1 261500 P(1.025)1500 P(1.159693)P 1293.45.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansAnnual Percentage YieldThe annual percentage yield or APY of an investment is thepercentage of profit that is generated in a one-year period.The APY is essentially the same as the percent increase in theinvestment over one year.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExample: APYIf an investment of 575 is worth 630 after one year, what isthe APY?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExample: APYIf an investment of 575 is worth 630 after one year, what isthe APY?The profit made is 630 575 55. Thus the annualpercentage yield is: 630 575 55 0.096 9.6%. 575 575MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansComparing InvestmentsThe APY allows us to compare different investments.Use the APY to compare an investment at 3.5% compoundedmonthly with an investment at 3.8% compounded annually.The amount of principal is unimportant. Pick P 1 to makeour lives easier.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansComparing InvestmentsFor the first loan, after one year we have .0351 1 12So the APY is1.035571 11 12 1.035571. 0.035571 3.56%.For the second loan, after one year we have1(1 .038) 1.038. 0.038 3.8%.So the APY is 1.038 11The second loan is better because it has a higher APY.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansGeometric SequencesSuppose 5000 is invested with an annual interest rate of 6%compounded annually. Let GN represent the amount of moneyyou have at the end of N years. Then:G0G1G2G3 .5000(1.06)5000 5300(1.06)2 5000 5618(1.06)3 5000 5955.08GN (1.06)N 5000Note that each term is obtained from the previous one bymultiplying by 1.06. This is called the common ratio. Thenumber 5000 is the initial term.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansGeometric SequencesA geometric sequence starts with an initial term P, and fromthen on every term in the sequence is obtained by multiplyingthe preceding term by the same constant c, called thecommon ratio.Examples: 5, 10, 20, 40, 80,. . . 27, 9, 3, 1, 1 , 1 ,. . .3 9 27, 9, 3, 1, 1 , 1 ,. . .39In each case, what is the initial term P and what is thecommon ratio c?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansGeometric SequencesHere is the general form of a geometric sequence:P, cP, c 2 P, c 3 P, c 4 P, . . . , c N P, . . .We write GN to label the terms of a geometric sequence:G0 P, G1 cP, G2 c 2 P, G3 c 3 P, . . . , GN c N P, . . . MoneyG0 P and GN cGN 1 . This expresses the statementthat the initial term is a number P and each term isobtained by multiplying the preceding term by c. This iscalled a recursive formula.GN c N P. This expresses the term GN directly in termsof P and c. this is called an explicit formula.UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesA principal amount of P is invested with annual compoundinterest rate r (expressed as a decimal). Express the yearlyamounts of money as a geometric sequence.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesA principal amount of P is invested with annual compoundinterest rate r (expressed as a decimal). Express the yearlyamounts of money as a geometric sequence.P, P(1 r ), P(1 r )2 , P(1 r )3 , . . . , .MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesA principal amount of P is invested with annual compoundinterest rate r (expressed as a decimal). Express the yearlyamounts of money as a geometric sequence.P, P(1 r ), P(1 r )2 , P(1 r )3 , . . . , .What is the initial term and what is the common ratio?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesA principal amount of P is invested with annual compoundinterest rate r (expressed as a decimal). Express the yearlyamounts of money as a geometric sequence.P, P(1 r ), P(1 r )2 , P(1 r )3 , . . . , .What is the initial term and what is the common ratio?The initial term is P and the common ratio is (1 r ).MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesIn 2008 there were 1 million reported cases of the gammavirus. The number of cases has been dropping each year by70% since then. If the present rate continues, how manyreported cases can we predict by the year 2014?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplesIn 2008 there were 1 million reported cases of the gammavirus. The number of cases has been dropping each year by70% since then. If the present rate continues, how manyreported cases can we predict by the year 2014?To decrease a number by 70%, remember that we multiply by70) or 0.30. So we have a geometric sequence with(1 100initial term 1,000,000 and common ratio 0.30. We areinterested in the term after the elapse of 6 years. So we needto calculate G6 c 6 P. In this case,G6 (0.30)6 (1000000) 729.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is a formula for the sum of the first N terms in ageometric sequence (in the case that c 6 1): N c 12N 1P cP cP · · · cP P.c 1Note that we are adding the first N terms, up to c N 1 P.Note also that the power of c on the right-hand side (N) isone more than the power of c on the left-hand side (N 1).We are going to need this formula soon!MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is one way to derive this formula.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is one way to derive this formula. Let S be the sum:S P cP c 2 P c 3 P · · · c N 1 P.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is one way to derive this formula. Let S be the sum:S P cP c 2 P c 3 P · · · c N 1 P.cS cP c 2 P c 3 P c 4 P · · · c N P.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is one way to derive this formula. Let S be the sum:S P cP c 2 P c 3 P · · · c N 1 P.cS cP c 2 P c 3 P c 4 P · · · c N P.Now subtract the upper expression from the lower expression.There is lots of cancellation!cS S c N P P.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaHere is one way to derive this formula. Let S be the sum:S P cP c 2 P c 3 P · · · c N 1 P.cS cP c 2 P c 3 P c 4 P · · · c N P.Now subtract the upper expression from the lower expression.There is lots of cancellation!cS S c N P P.Finally solve for S:S(c 1) P(c N 1) N c 1S P.c 1MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaCalculate 1 2 4 8 16 · · · 263 .(This is the “rice on the chessboard” problem.)MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaCalculate 1 2 4 8 16 · · · 263 .(This is the “rice on the chessboard” problem.)Here P 1, c 2, and N 1 63.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaCalculate 1 2 4 8 16 · · · 263 .(This is the “rice on the chessboard” problem.)Here P 1, c 2, and N 1 63.23631 2(1) 2 (1) 2 (1) · · · 2 (1)64 1 (1) 22 1 18, 446, 744, 073, 709, 551, 615.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaCalculateMoney12 61 118 · · · 21 ( 31 )10 .UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaCalculate12 61 118 · · · 21 ( 31 )10 .Here P 12 , c 13 , and N 1 10. The sum is 1 11 ( ) 1( 12 ) 31 13 0.75.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaExample: Suppose we want to count the total number of casesof a particular disease. Suppose in 2008 there were 5000cases, and that for each after that the number of new caseswas 40% more than the year before. How many total casesoccurred in the 10-year period from 2008 to 2017?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaExample: Suppose we want to count the total number of casesof a particular disease. Suppose in 2008 there were 5000cases, and that for each after that the number of new caseswas 40% more than the year before. How many total casesoccurred in the 10-year period from 2008 to 2017?NumberNumberNumberetc.NumberMoneyof cases in 2008: 5000,of new cases in 2009: 5000(1.40),of new cases in 2010: 5000(1.40)2 ,of new cases in 2017: 5000(1.40)9 .UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Geometric Sum FormulaTotal number of cases:5000 5000(1.40) 5000(1.40)2 · · · 5000(1.40)910 1 5000 (1.40)1.40 1 1, 932, 101.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansFixed AnnuitiesA fixed annuity is a sequence of equal payments made orreceived over regular time intervals.Examples: making regular payments on a car or home loan making regular deposits into a college fund receiving regular payments from an retirement fund orinheritanceMoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansTwo Types of Fixed AnnuitiesA deferred annuity is one in which regular payments are madefirst, so as to produce a lump-sum payment at a later date. Example: Making regular payments to save up for college.An installment loan is an annuity in which a lump sum is paidfirst, and then regular payments are made against it later. MoneyExample: Receiving a car loan, and paying it back withmonthly payments.UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansDeferred AnnuitiesExample: A newborn’s parents set up a college fund. Theyplan to invest 100 each month. If the fund pays 6% annualinterest, compounded monthly, what is the future value of thefund in 18 years?Notice that each monthly installment has a different“lifespan”:Money The first installment will generate interest for all18 · 12 216 months. The second installment will generate interest for 215 months. The last installment will generate interest for only one month.UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansDeferred Annuities: ExampleThe first installment will generate interest for all 18 · 12 216months. Using the compound interest formula, after 18 yearsthe installment is worth: 18·12.06100 1 100(1.005)216 .12The future value of the second installment is: 215 .06 100(1.005)215 .100 1 12.The future value of the final installment is:100(1.005)1 .MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansDeferred Annuities: ExampleThe total future value is the sum of all of these future values:FMoney 100(1.005) 100(1.005)2 · · · 100(1.005)215 100(1.005)216 100(1.005) 1 1.005 · · · 1.005214 1.005215 .UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansDeferred Annuities: ExampleThe total future value is the sum of all of these future values:F 100(1.005) 100(1.005)2 · · · 100(1.005)215 100(1.005)216 100(1.005) 1 1.005 · · · 1.005214 1.005215 .Inside the brackets, we have a geometric sum with initial termP 1 and common ratio c 1.005. Use the geometric sumformula: 1.005216 1 38, 929.00F 100(1.005) 11.005 1Notice that the exponent is 216 because there are 216 terms in thesum, and because the last exponent in the sum is 215.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansThe Fixed Deferred Annuity FormulaThe future value F of a fixed deferred annuity consisting of Tpayments of P each, having a periodic interest rate p (indecimal form) is: TF L (1 p)p 1where L denotes the future value of the last payment.Note that the periodic interest rate p nr where r is the APRin decimal form and the interest is compounded n times peryear. Pay attention to what quantities the various variablesstand for!MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplePage 395, #63:Starting at age 25, Markus invests 2000 at the beginning ofeach year in an IRA (individual retirement account) with anAPR of 7.5% compounded annually. How much money willthere be in Markus’s retirement account when he retires at age65?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExamplePage 395, #63:Starting at age 25, Markus invests 2000 at the beginning ofeach year in an IRA (individual retirement account) with anAPR of 7.5% compounded annually. How much money willthere be in Markus’s retirement account when he retires at age65?Notice that the periodic interest rate p is p 0.075 andT 65 25 40.The future value of the last payment is L 2000(1.075)because the final payment will accumulate interest for oneyear.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSo the future value is:F 1.07540 12000(1.075)0.075 18.044239 12000(1.075)0.075 17.0442392000(1.075)0.0752000(1.075)(227.25652) 488601.52.Markus will have 488,601.52 in his account when he retires.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSame example as before, except that Markus invests themoney at the end of each year, after the interest for that yearhas been added to the account.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleThis time, L 2000. So the future value is:F 1.07540 120000.075 18.044239 120000.075 17.04423920000.0752000(227.25652) 454513.04.Markus will have 454,513.04 in his account when he retires.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSuppose you want to set up an 18-year annuity at an APR of6% compounded monthly, if your goal is to have 50,000 atthe end of 18 years. How much should the monthly paymentsbe?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSuppose you want to set up an 18-year annuity at an APR of6% compounded monthly, if your goal is to have 50,000 atthe end of 18 years. How much should the monthly paymentsbe?RememberF L (1 p)T 1p .We know F 50, 000, p .06 0.005, and12T 18 12 216. Let P be the unknown monthly payment.Then we know L P(1.005).MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSubstitute:50, 000 P(1.005)SoP Money (1.005)216 10.005 P(389.29).50, 000 128.44.389.29UK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSuppose you want to have 2000 at the end of 7.5 years. Youalready have 875 in the bank, invested at a 6.75% APRcompounded monthly. You want to put more money eachmonth into the bank to end up with the 2000 goal. Whatshould your monthly deposit be?MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment LoansExampleSuppose you want to have 2000 at the end of 7.5 years. Youalready have 875 in the bank, invested at a 6.75% APRcompounded monthly. You want to put more money eachmonth into the bank to end up with the 2000 goal. Whatshould your monthly deposit be?First, the 875 in the bank will grow to875(1 0.0675)7.5(12) 1449.62. So you only need12 2000 1449.62 550.38 more.MoneyUK
Simple InterestCompound InterestGeometric SequencesDeferred AnnuitiesInstallment Loan
The Time Value of Money When you deposit 1000 into a savings account at the bank, you expect that amount to gain interest over time. A year from now, you would have more than 1000. In return for having access to the present value of your money, the bank increases the future value of the money by adding interest. Money UK
May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .
On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. Crawford M., Marsh D. The driving force : food in human evolution and the future.
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
