Transcription
Engineering Economy Chapter 4Single Payment Compound InterestInterest FormulasChapter 4P (P)resent sum of moneyi (i)nterest per time period (usually years)n (n)umber of time periods (usually years)F (F)uture sum of money that is equivalent toP given an interest rate i for n periodsz F P(1 i) nF P(F/P,i,n)z Single Payment Present Worth Formulay P F(1 i) -nP F(P/F,i,n)zzzzPeter O’GradyProfessorDepartment of IndustrialEngineeringUniversity of Iowa Peter O'Grady, 2001Steps to solutionKey pointsz Step 1: Identify cash flow (P and F)z Step 2: Identify interest rate (i) and numberof periodsz Step 3: Select appropriate table or formulay F P(1 i)ny F P(F/P,i,n)Chapter 4 - 2Chapter 4 - 1P F(1 i)-nP F(P/F,i,n)z Step 4: Perform calculationz All four steps are a small part of an actualengineering decisionz Time value of money, 1,000 today is not thesame as 1,000 one hundred years from nowz Equivalence provides a common language forcomparing present and future sums of moneyz Equivalence depends on the assumed interest ratez Notation for single payment compound interest:F P(F/P,i,n)P F(P/F,i,n)Chapter 4 - 3More Interest Formulae: Uniform Series Az Uniform amount A at end of time periodz Uniform series aggregation of severalpresent values (P)z F A(1 i)n-1 . A(1 i)2 A(1 i)z Superposition principle - Lego buildingChapter 4 - 4Uniform SeriesF/AA/F1. Uniform Series Compound Amount Factor(F/A,i,n)[(1 i)n -1]/i F/A2.Uniform Series Sinking Fund Factor(A/F,i,n)i/[(1 i) n -1] A/Fx See p 98 - 99 for derivationChapter 4 - 5 Peter O'Grady, 2001Chapter 4 - 61
Engineering Economy Chapter 4Uniform Series Compound Amount F/Az Determines future value (F) of periodiccontributions (A)z Example: Value of IRA given periodiccontributionsz F A(F/A,i,n)Retirement in 25 years?z Deposit 10,000 each year for 25 yearsz Interest rate is 15%, compounded annuallyz At the end of 25 years how much will youhave for retirement?Chapter 4 - 7Uniform Series Sinking Fund A/Fz Determines contribution/payment given afuture valuez Example: Periodic contribution to IRA that isrequired to achieve goalz A F(A/F,i,n)Chapter 4 - 8Uniform Series Capital Recovery A/Pz Determines contribution/payment given apresent valuez Example: Income from an IRA that ispossible given savings; loan repaymentz A P(A/P,i,n)Chapter 4 - 9A/P and P/ALoan RepaymentUniform Series Capital Recovery Factor(Simple interest)(A/P,i,n)Chapter 4 - 10z Car loan of 20,000z Interest rate is 15%, compounded annuallyz What are the annual repayments?[i(1 i)n ]/[(1 i) n -1] A/PNote: Inverse is Uniform Series PresentWorth Factor P/AChapter 4 - 11 Peter O'Grady, 2001Chapter 4 - 122
Engineering Economy Chapter 4ExamplesSteps to solving problemsExamples 4-5, 4-6Superposition principle can be used to modifycash flow descriptions to fit standard form.zzzzzzIdentify variables (F, P, A, i, n)Draw diagramConvert to workable formIdentify appropriate formulaPerform calculationsVerify against rough estimatesChapter 4 - 13Uniform series formulas covered thus farChapter 4 - 14Arithmetic Gradientz graduated payments (G)z A G(A/G,i,n) P G(P/G,i,n)z Example: Increasing maintenance costswith aging equipmentz Note: G 0 at time 1z Uniform series compoundedy F A(F/A,i,n)z Uniform series sinking fundy A F(A/F,i,n)z Uniform series capital recoveryy A P(A/P,i,n)z Uniform series present worth valueGy P A(P/A,i,n)0122G33G44G5Chapter 4 - 15Arithmetic GradientGeometric Gradient1.Arithmetic Gradient Present Worth Factor(P/G,i,n) [(1 i)n - in - 1]/[i2(1 i)n ] P/G2.Arithmetic Gradient Uniform Series(A/G,i,n)[(1 i)n - in -1]/[i(1 i)n - I] A/GChapter 4 - 17 Peter O'Grady, 2001Chapter 4 - 16z Determines uniform payments (A) givengraduated payments (G) that increase at aconstant percentagez P A(F/A,g,i,n)z g percent increase in Az Two formulas, one for i g and i gz Unlike arithmetic A starts at time 1z Example: IRA contributions increase withincomeChapter 4 - 183
Engineering Economy Chapter 4Nominal and effective interestNominal and effective interest ratesz Nominal interest rate Interest rate withoutconsideration of compoundingz Effective interest rate Nominal interest rateadjusted for compoundingz Nominal Effective IF compounding periodequals period of effective interest ratez Conversion to effective interest rateprovides a basis to make comparisonsChapter 4 - 19Nominal and effective interest ratesEffective interest rate, i P, (period ofcompounding period of interest) is used in formulas:i i P (1 is) m-1i i P (1 rP/m)m-1is interest per subperiodrP nominal interest per period Pm number of subperiods in period PNominal interest rate, rP m X isContinuous compounding: ia er -1F P(1 i a ) n P* ernChapter 4 - 21Table 4-3 NOMINAL AND EFFECTIVEINTEREST RATESNominalinterest rateEffective interest rate per year, ia ,per yearwhen nominal rate is compoundedr1%234568101525YearlySemi- Monthly 32Continuouslyi Effective interest rate per interest periodr Nominal interest rate per periodia Effective interest rate per year (annum)is Effective interest rate per sub periodm Number of compounding subperiods inthe period used to define the nominal rate“r”Chapter 4 - 20Nominal interest rate of 12% compounded monthlyz What is the effective interest rate permonth?z What is the nominal interest rate per month?z What is the effective interest rate per year?z Does (F/A, 12%, 30) (F/A, 1%, 360)?Chapter 4 - 224-63 A student bought a guitar for 75 andagreed to pay 85 after 6 months. Nominalinterest rate? Effective annual interest rate?1.0050% .12716.18316.18378.32788.328710.5156 10.517116.1798 16.183428.3916 28.4025Chapter 4 - 23 Peter O'Grady, 2001zzzzzChapter 4 - 244
Engineering Economy Chapter 4General problem-solving suggestionsz Draw the cash flow diagramz Calculate a rough guessy Use a crude model: ignore interest, ignorecompoundingy Doubling rule: an amount doubles every 70/i%yearsz Track unitsy Effective interest rate must have the same units forperiod of compounding as for period of interesty “n” must match “i” in tablesChapter 4 - 25 Peter O'Grady, 2001Overview of Chapter 4: Translation tocommon unitsz Convert between values in future and presentz Convert between single values and series of valuesz Convert between nominal interest rate and interestrate that considers effect of compounding(effective)z Effective interest rate (period ofcompounding period of interest) is used informulas: i (1 is)m-1(is interest per subperiod)(m number of subperiods)Chapter 4 - 265
Engineering Economy Chapter 4 Peter O'Grady, 2001 3 Chapter 4 - 13 Examples Examples 4-5, 4-6 Superposition principle can be used to modify cash flow descriptions .
