Time Value of MoneyTypes of problemsTypes of Annuity ProblemsSlide 4Types of ProblemsSlide 6Deferred AnnuitiesCalculation VariablesAnnuity Problemsi and n must match!Single Sum FormulasSlide 13Annuity FormulasFormulas vs. TablesUsing the tablesStudy the tables . .“Formulas” for the IF TablesAdjustments to ordinary annuity tablesConversion to Annuity DueOrdinary Annuity ExampleStop and think . . .Annuity Due ExampleSlide 24Slide 25Slide 26Using PV Tables – Deferred AnnuityDeferred Annuity ExampleSlide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Spreadsheet demoTime Value of MoneyReview of Basic ConceptsTypes of problems•Single Sum. One sum ($1) will be received or paid either in the–Present (Present Value of a Single Sum or PV)–Future (Future Value of a Single Sum or FV)PV FVTypes of Annuity ProblemsOrdinary annuity (OAOA)A series of equal payments (or rents) received or paid at the end of a period, assuming a constant rate of interest.PV-OA (Present value of an ordinary annuity)PV-OAPMT PMTPMTPMT PMTTypes of Annuity ProblemsOrdinary annuity (OAOA)A series of equal payments (or rents) received or paid at the end of a period, assuming a constant rate of interest.FV-OA (Future value of an ordinary annuity)FV-OAPMT PMTPMTPMT PMTTypes of ProblemsAnnuity Due (ADAD)A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest.FV-ADFV-AD (future value of an annuity due) (future value of an annuity due)FV-ADPMTPMT PMTPMT PMT 0Note: Each rent or payment is discounted (interest removed) one less period under a FV-AD.Types of ProblemsAnnuity Due (ADAD)A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest.PV-AD PV-AD (present value of an annuity due)(present value of an annuity due)PV-ADPMTPMT PMTPMT PMT 0Note: Each rent or payment compounds (interest added) one more period in a PV-ADDeferred AnnuitiesPMTPMT PMT0 54321d = 2n = 3This is an ordinary annuity of 3 periods deferred for 2 periods.We could find either the PV or the FV of the annuity.1515Calculation Variables•There will always be at least four variables in any present or future value problem. Three of the four will be known and you will solve for the fourth.–Single sum problems:•n = number of compounding periods•i = interest rate•PV = Value today of a single sum ($1)•FV = Value in the future of a single sum ($1)•PMT = 0 (important it using PV calculator!)Annuity Problems•n = number of payments or rents•i = interest rate•PMT = Periodic payment (rent) received or paid–And either:•FV of an annuity (OA or AD) = Value in the future of a series of future payments –OR•PV of an annuity (OA or AD) = Value today of a series of payments in the futureWhen we know any three of the four amounts, we can solve for the fourth!i and n must match!•The “n” refers to periods not necessarily defined as years! The period may be annual, semi-annual, quarterly or another time frame.•The “n” and the “i” must match. That is, if the time period is semi-annual then so must the interest rate. •Interest rates are assumed to be annual unless otherwise stated so you may have to adjust the rate to match the time period.Single Sum Formulas•FV = (1+i)n•PV = FV (1+i)nSingle Sum Formulas•FV = (1+i)n•PV = FV (1+i)n•Present value calculators are generally no more expensive than those that do nth powers and nth roots!Annuity Formulas•FV-OA = •PV-OA = PMTi1(1 + i)n1 -(1 + i) - 1iPMTFormulas vs. Tables•Before fancy calculators, people had no easy way to compute nth roots and raise numbers to the nth power.•So they created tables for of sums of $1 or annuities of $1.•The values on the table, I call the “interest factor” or IF.•So we have PVIF (for n and i )and the PVIF-OA (for n and i) and so forthUsing the tableshorizontally for the ““ii””The tables are the result of the required multiplications and division at various “n” and “i” and are to be read vertically for the ““nn””and24Study the tables . . •They are very logical. –All sums in the future are worth LESS in the present. •All factors on the present value of a single sum table are less than one. All present sums are worth more than themselves in the future. •All factors on the future value of a single sum table are greater than one. –Notice how the factors change dramatically as the “i” increases and the “n” lengthens!“Formulas” for the IF Tables•PV = FV * IF {IF from PV of $ table}•FV = PV * IF {IF from FV of $ table}•PV-OA = PMT * IF {IF from PV-OA table}•FV-OA = PMT * IF {IF from FV-OA table}•PMT = (PV-OA) / IF{IF from PV-OA table}•or •PMT = (FV-OA) / IF{from FV-OA table}“IF” stands for “interest factor” from the appropriate n row and i column of the tableAdjustments to ordinary annuity tablesRules for annuity dues and deferred annuitiesConversion to Annuity Due•To find IF for FV-AD: Add one to the number of periods and look up IF on table. Then subtract one from the interest factor listed.•To find IF for PV-AD: Subtract one from the number of periods and look up IF on table. Add one to the interest factor.•Or look up the IF on the appropriate table and multiply by (1 + i).Ordinary Annuity Example•Suppose I must make three payments of $500, each at the end of each of the next three years. The interest rate is 8%. How much should I set aside today to have the required payments? •This is an ordinary annuity: PV-OA = PMT * (PVIF-OA n,i) where n = 3 payments and i = 8% PV-OA = $500 * 2.5771 = $1,289Page 480Stop and think . . .•If the first payment comes immediately instead of at the end of the first year,•Will the present value be •MORE or •LESS?36Annuity Due Example•If the first payment comes immediately, this would be an annuity due problem.•We can use one of the formulas to adjust the IF – the easiest to memorize is the “multiply by (1+i)” rule: PV-AD = PMT (PVIF-OA n,i)(1+i) where n = 3 payments and i = 8% PV-AD = $500 (2.5771)(1 + .08) = $1,39136Annuity Due Example•Alternative adjustment to the IF table is even easier – at least if you write the method at the top of your table!•Look up IF for (n-1) and add 1: PVIF-OA (n=2, i=8%) = 1.7833 + 1 = 2.7833PV-AD = $500 (2.7833) = $1,392Annuity Due Example•This second method is also the “logical” decision