Time Value of Money Formulas:Types of problemsSingle Sum. One sum ($1) will be received or paid either in thePVFVTypes of Annuity ProblemsOrdinary annuity (OA)PV-OAPMTFV-OAPMTAnnuity Due (AD)V-AD (future value of an annuity due)V-AD (present value of an annuity due)PV-ADPMTPMTFV-ADPMTPMTAnnuity Due vs. Ordinary AnnuityThe difference between an ordinary annuity and an annuity due is that:Types of Annuity ProblemsExample ProblemsOrdinary Annuity ExampleSuppose 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:What if the first payment comes immediately instead of at the end of the first year, how much would I need to set aside today in order to have the required payments?This is an annuity due:We can use one of the formulas to adjust the IF – the easiest to memorize is the “multiply by (1+i)” rule: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:This second method is also the “logical” decision you would make from looking at the time-line.EXAMPLE PROBLEMSUsing Time Value of Money TablesFV of Ordinary Annuity = PMT * IF {from FV of ord. annuity table}CONVERSION TO DEFERRED ANNUITYAcct 414 Prof. Teresa GordonTime Value of Money Formulas:Types of problemsSingle Sum. One sum ($1) will be received or paid either in thePresent (Present Value of a Single Sum or PV)Future (Future Value of a Single Sum or FV)PV FVSingle Sum Formulas•FV = (1+i)n•PV = FV(1+i)nThere 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 periodsi = interest ratePV = Value today of a single sum ($1)FV = Value in the future of a single sum ($1) 1/14/19 Page 1Acct 414 Prof. Teresa GordonTypes of Annuity ProblemsOrdinary annuity (OA)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 PMT PMT PMT PMT PMT PMT PMTFV-OA (Future value of an ordinary annuity)FV-OAPMT PMT PMT PMT PMT PMT PMT PMTAnnuity Due (AD)A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest.V-AD (future value of an annuity due)V-AD (present value of an annuity due)PV-AD (Present value of an ordinary annuity)PV-ADPMT PMT PMT PMT PMT PMT PMT PMTFV-AD (Future value of an ordinary annuity)FV-ADPMT PMT PMT PMT PMT PMT PMT PMT 1/14/19 Page 2Acct 414 Prof. Teresa GordonAnnuity Due vs. Ordinary AnnuityThe difference between an ordinary annuity and an annuity due is that:Given the same i, n and periodic payment, the annuity due will always yield a greater present value (less interest removed) and a greater future value(more interest added).Annuity Formulas•FV-OA = •PV-OA = PMTi1(1 + i)n1 -(1 + i) - 1iPMTThere 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.Annuity Problems:n = number of payments or rentsi = interest ratePMT = Periodic payment (rent) received or paidAnd either:FV of an annuity (OA or AD) = Value in the future of a series of future payments ORPV of an annuity (OA or AD) = Value today of a series of payments in the futureNote: To convert answer into an annuity due, multiply by (1 + n) 1/14/19 Page 3Acct 414 Prof. Teresa GordonTypes of Annuity ProblemsDeferred 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. 1/14/19 Page 4Acct 414 Prof. Teresa GordonExample ProblemsOrdinary Annuity ExampleSuppose 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:What if the first payment comes immediately instead of at the end of the first year, how much would I need to set aside today in order to have the required payments?This is an annuity due:We can use one of the formulas to adjust the IF – the easiest to memorize is the “multiply by (1+i)” rule:Alternative adjustment to the IF table is even easier – at least if you write the method at thetop of your table!Look up IF for (n-1) and add 1:This second method is also the “logical” decision you would make from looking at the time-line. 1/14/19 Page 5Acct 414 Prof. Teresa GordonEXAMPLE PROBLEMSDeferred annuity:Assume that you are to receive three payments of $100 each beginning 3 years from now. What isthis series of payments worth using a 12% discount rate?Draw time line:Analyze time-line: remember to identify the ordinary annuity involved before deciding the numberof periods deferred.Alternative 1 – Work as two part problemAlternative 2 – Adjust the ordinary annuity table IF:Look up PV-OA IF for (d+n) and then subtract the PV-OA IF for dAlternative 3 – Adjust the ordinary annuity table IF:Look up PV-OA IF for n and then multiply by the PV IF for d 1/14/19 Page 6Acct 414 Prof. Teresa GordonUsing Time Value of Money TablesLet “IF” (interest factor) stand for the number from the appropriate time value of money table for n periods and interest rate “i”PV of a lump sum = FV * IF {from PV of $ table}FV of a lump sum = PV * IF {from FV of $ table}PV of Ordinary Annuity = PMT * IF {from PV of ord. annuity table}FV of Ordinary Annuity = PMT * IF {from FV of ord. annuity table}PMT = (PV of Ordinary Annuity) / IF {from PV of ord. annuity table}or (FV of Ordinary Annuity) / IF {from FV of ord. annuity table}ADJUSTMENTS to ordinary annuity tables CONVERSION TO ANNUITY DUE:To find IF for Future Value of an Annuity Due: 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 Present Value of an Annuity Due: 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). CONVERSION TO DEFERRED ANNUITYLet d = number of periods deferred and n = number of periodic paymentsLook up (d+n) on the appropriate table. Look up d on the same table. Subtract the smaller interest factor from the