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=0.015015 1 0.06 P1 = 3000 1+ 4 P 2 = 3000 ( 1.015 ) 3 P3 = 3000(1.015) 4 P = 3000(1.015) 4 2 3 4
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, 457.78 0.015 7 8
Iterest Eared Amortzato Problem The preset value of all paymets s $22,457.78. 78 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 $22,457.78 =$1,542.22. Problem: A bak loas a customer $50,000 000 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 $50,000 000 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 ) + 11 12
Care must be take to perform the correct order of operatos. 1. eter 0.045 dvded by 12 2. 1 + step 1 result 3. Rase aswer to -180 power. 4. 1 step 3 result 5. Take recprocal (1/x) of step 4 result. Multply l by 0.045 045 ad dvde by 12. 5. Fally, multply that result by 50,000 to obta 382.50 PMT = PV 1 (1 + ) 0.045 PMT = 50,000 12 = 382.50 180 0.045 1 1+ 12 If the customer makes a mothly paymet of $382.50 to the bak for 180 paymets, the the total amout pad to the bak s the product of $382.50 ad 180 = $68,850. Thus, the terest eared by the bak s the dfferece betwee $68,850 ad $50,000 (orgal loa) = $18,850. 13 14 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 = 500 1 (1.01) 6 = $86.27 15 16
At the ed of the frst moth, the terest due s $500(0.01) = $5.00. The amortzato paymet s dvded to two parts, paymet of the terest due ad reducto of the upad balace. Mothly Paymet Iterest Due Upad Balace Reducto $86.27 = $5.00 + $81.27 The upad balace for the ext moth s Prevous Upad Bal Upad Bal Reducto New Upad Bal $500.00 $81.27 = $418.73 Ths process cotues utl all paymets have bee made ad the upad balace s reduced to zero. The calculatos for each moth are lsted the followg table, whch was doe o a spreadsheet. Paymet Paymet Iterest Ipad Bal Upad Number Reducto Balace 0 $500.00 1 86.27 $5.00 $81.27 $418.73 2 86.27 $4.19 $82.08 $336.65 3 86.27 $3.37 $82.90 $253.74 4 86.27 $2.54 $83.73 $170.01 5 86.27 $1.70 $84.57 $85.44 6 86.27 $0.85 $85.42 $0.03 I realty, the last paymet would be creased by $0.03, so that the balace s zero. 17 18 Strategy for Solvg Mathematcs of Face Problems Strategy Step 1. Determe whether the problem volves a sgle paymet or a sequece of equal perodc paymets. Smple ad compoud terest problems volve a sgle preset value ad a sgle future value. Ordary autes may be cocered wth a preset value or a future value but always volve a sequece of equal perodc paymets. Step 2. If a sgle paymet s volved, ved, determe e whether e smple or compoud terest s used. Smple terest s usually used for duratos of a year or less ad compoud terest for loger perods. Step 3. If a sequece of perodc paymets s volved, determe whether the paymets are beg made to a accout that s creasg value -a future value problem - or the paymets are beg made out of a accout that s decreasg value - a preset value problem. Remember that amortzato problems always volve the preset value of a ordary auty. 19 20