# Chapter = 3000 ( ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization

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1 Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve terest rate order to be able to wthdraw equal amouts from the accout the future utl o moey remas the accout. Here s a example: How much moey must you depost ow at 6% terest compouded quarterly order to be able to wthdraw \$3,000 at the ed of each quarter year for two years? 2 Dervato of Formula Preset Value of the Frst Four Paymets We beg by solvg for P the compoud terest formula: ( 1 ) A= P + P = A(1 + ) Iterest rate each perod s 0.06/4 06/4= P1 = P 2 = 3000 ( ) 3 P3 = 3000(1.015) 4 P = 3000(1.015)

2 Dervato of Short Cut Formula Preset Value of a Ordary Auty We could proceed to calculate the ext four paymets ad the smply fd the total of the 8 paymets. There are 8 paymets sce there wll be 8 total wthdrawals: (2 years) (four wthdrawals per year) = 8 wthdrawals. Ths method s tedous ad tme cosumg so we seek a short cut method. 1 (1+ ) PV = PMT PV = preset value of all paymets PMT = perodc paymet = rate per perod = umber of perods Note: Paymets are made at the ed of each perod. 5 6 Back to Our Orgal Problem Back to Our Orgal Problem How much moey must you depost ow at 6% terest compouded quarterly order to be able to wthdraw \$3,000 at the ed of each quarter year for two years? How much moey must you depost ow at 6% terest compouded quarterly order to be able to wthdraw \$3,000 at the ed of each quarter year for two years? : R = 3000, = 0.06/4 = 0.015, = 8 1 (1 + ) P= R 8 1 (1.015) P = 3000 = 22,

3 Iterest Eared Amortzato Problem The preset value of all paymets s \$22, The total amout of moey wthdraw over two years s 3000(4)(2)=24,000. Thus, the accrued terest s the dfferece betwee the two amouts: \$24,000 \$22, =\$1, Problem: A bak loas a customer \$50, at 4.5% terest per year to purchase a house. The customer agrees to make mothly paymets for the ext 15 years for a total of 180 paymets. How much should the mothly paymet be f the debt s to be retred 15 years? 9 10 Amortzato Problem Problem: A bak loas a customer \$50, at 4.5% terest per year to purchase a house. The customer agrees to make mothly paymets for the ext 15 years for a total of 180 paymets. How much should the mothly paymet be f the debt s to be retred 15 years? : The bak has bought a auty from the customer. Ths auty pays the bak a \$PMT per moth at 4.5% terest compouded mothly for 180 moths. We use the prevous formula for preset value of a auty ad solve for PMT: 1 (1 + ) PV = PMT PMT = PV 1 (1 )

4 Care must be take to perform the correct order of operatos. 1. eter dvded by step 1 result 3. Rase aswer to -180 power step 3 result 5. Take recprocal (1/x) of step 4 result. Multply l by ad dvde by Fally, multply that result by 50,000 to obta PMT = PV 1 (1 + ) PMT = 50, = If the customer makes a mothly paymet of \$ to the bak for 180 paymets, the the total amout pad to the bak s the product of \$ ad 180 = \$68,850. Thus, the terest eared by the bak s the dfferece betwee \$68,850 ad \$50,000 (orgal loa) = \$18, Costructg a Amortzato Schedule If you borrow \$500 that you agree to repay sx equal mothly paymets at 1% terest per moth o the upad balace, how much of each mothly paymet s used for terest ad how much s used to reduce the upad balace? Amortzato Schedule If you borrow \$500 that you agree to repay sx equal mothly paymets at 1% terest per moth o the upad balace, how much of each mothly paymet s used for terest ad how much s used to reduce the upad balace? : Frst, we compute the requred mothly paymet usg the formula PMT = PV 1 (1 + ) 0.01 = (1.01) 6 = \$

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