5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso will ever ecouter is a istallmet loa such as a car or home loa. For these types of loas, a amout of moey is borrowed. This amout plus iterest is paid back over with fixed paymets. I geeral, paymets are made o a mothly basis. The legth of time over which the paymets are made (the term) may be as short as 36 to 72 moths for a auto loa. Or the paymets may be made over a 15 to 30 year term for a home loa. Sice these loas behave like a decreasig auity, we ca use the formulas we have developed i earlier sectios to compute the paymet o the loa. I this sectio we will compute the paymet for several differet loas ad track those paymets i a special type of table called a amortizatio table. 1
Questio 1: How do you fid the preset value of a auity? I Sectio 5.3, we were able to calculate the future value or paymets of auities that were icreasig i value or decreasig i value. For a ordiary auity whose preset value is PV, the future value is 1 i 1 FV PV 1i PMT i if the paymets PMT are made ito the auity which ears iterest per period i over periods. Sice the paymets are made ito the auity, the secod term is added. The future value of the auity icreases. If the paymets are made from the auity, the secod term is subtracted to give 1 i 1 FV PV 1i PMT i I this case, the future value of the auity decreases sice moey is removed from the auity. I some applicatios, we wish to fid the preset value (what must be i the accout today) so that the accout eds up with some amout i the future. The ext two examples illustrate how to fid the preset value i cases like this. Example 1 Fid the Amout Needed to Establish a Trust Fud A wealthy idividual wishes to create a trust fud for his gradso so that he may withdraw $5000 at the ed of every quarter for te years. At the ed of te years, the gradso will receive the rest of the trust which cotais $50,000. If the trust ears 8% iterest compouded quarterly, how much should be put ito the trust iitially? Solutio I this problem, the amout i the auity is decreasig sice withdrawals are beig made. However, we wish the future value of the auity to be $50,000 i te years. This meas that a larger amout 2
must be placed i the trust ow so that paymet may be made from it. 0.08 Substitute FV 50000, PMT 5000, i 4 0.02, ad 410 40ito to give 1 i 1 FV PV 1i PMT i 40 40 1 0.02 1 50000 PV 10.02 5000 0.02 Now solve this equatio for the preset value. 40 10.02 1 50000 5000 PV 10.02 0.02 40 10.02 1 50000 5000 0.02 PV 40 1 0.02 159421.92 PV This is calculated i a TI Graphig Calculator as show below. 40 The trust must be established with a iitial deposit of $159,421.92. 3
Example 2 Reachig a Retiremet Goal A fifty-five year old ivestor wishes to retire at age 67. The ivestor has budgeted $1000 a moth that she may deposit i a ordiary auity that ears 5% iterest compouded mothly. If she wishes to accumulate $2,000,000 for retiremet, what must be i the accout today to reach that goal? Solutio His is a icreasig auity sice regular paymets are beig made a accout. Substitute PMT 1000, i, 144 ad FV 2000000 ito 1 i 1 FV PV 1i PMT i ad solve for the preset value PV. This yields 144 144 1 1 2000000 PV 1 1000 1 1 2000000 1000 PV 1 144 144 144 1 1 2000000 1000 PV 144 1 Subtract the secod term o the right from both sides 1 Divide both sides by 144 990,871.44 PV This is calculated i a TI Graphig Calculator as show below. To simplify the calculatio, the umerator is calculated first. The that aswer is divided by the deomiator. 4
There must be $990,871.44 i the auity today, for the value to grow to $2,000,000 i twelve years. 5
Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued iterest. This process is called amortizatio. I the laguage of fiace, a loa is said to be amortized if the amout of the loa ad iterest are paid usig fixed regular paymets. From the perspective of the leder, this type of loa is a decreasig auity. The amout of the loa is the preset value of the auity. The paymets from the auity (to the leder) reduce the value of the auity util the future value is zero. This iterpretatio allows us to determie the paymet PMT o a loa of PV dollars. Start with the decreasig auity formula ad set the future value FV equal to zero, This equatio is simplified to give 1 i 1 0PV1i PMT i 0i PV 1i PMT 1i 1 i 1i 1 0i PVPMT 1 0i PVPMT1 1 i Clear the fractio by multiplyig each term by i Divide each term by 1 i Simplify the fractio by dividig 1 i ito each term i the umerator Now solve this equatio for the paymet PMT: PMT 1 1i i PV Add 11 PMT i to both sides i PV PMT= 1 1 i Divide both sides by 11 i 6
o a Amortized Loa Suppose a loa of PV dollars is amortized by periodic paymets of PMT at the ed of each period. If the loa has a iterest rate of i per period over periods, the paymet is PMT= i PV 1 1 i We ca use this formula to calculate the paymet o ay loa that is amortized. Pay special attetio to the loa amout. Ofte the loa amout is ot the same as the purchase price because of a dow paymet. A dow paymet is a amout paid up frot that reduces the amout that must be borrowed. This amout must be subtracted from the purchase price to give the loa amout. Whe a loa is amortized for the purchase of a home, the loa is called a mortgage. A typical mortgage is paid back over a 15 or 30 year period with mothly paymets. Example 3 o a Amortized Loa A youg professor purchases a home for $149,000. He plas to take out a 30 year mortgage at a aual iterest rate of 5.75%. The mortgage requires a dow paymet of 20% of the purchase price. a. Fid the mothly paymet o this mortgage. Solutio To qualify for this loa, the professor must put 20% dow, Dow 0.20149000 29800 The loa amout is PV 149, 000 29,800 119, 200. For a 30 year mortgage, there are 30 or 360 periods. The iterest rate per 75 period is i. Usig these values, the mothly paymet is 7
75 PMT 1 1 119200 75 360 695.62 This calculatio may be carried out o a TI graphig calculator as show below. This paymet is usually rouded up to the earest pey to isure the loa is paid off. I practice, this meas the fial paymet will be slightly less tha all other paymets. b. How much iterest is paid o this mortgage? Solutio Accordig to part a, the professor will pay a total of 360 $695.62 or $250,423.20 over the term of the loa. Sice the loa amout is $119,200, the additioal amout paid must be iterest, Iterest $250, 423.20 $119, 200 $131, 223.20 The professor pays $131,223.20 i iterest o this 30 year mortgage. c. The professor has also discovered that he qualifies for a 15 year loa at a aual iterest rate of 4.85%. This mortgage also requires a 20% dow paymet. Fid the mothly paymet o this mortgage. Solutio For this mortgage, the umber of periods is 15 or 180. 0.0485 The itest rate per moth is i. This leads to a paymet of 8
0.0485 PMT 1 1 119200 0.0485 180 933.34 The calculatio is show below o a TI Graphig Calculator. Although the iterest rate is lower for this mortgage, the shorter term leads to a higher mothly paymet of $933.34. d. How much iterest is paid o the 15 year mortgage? Solutio The professor will pay a total of 180 $933.34 or $168,001.20 i paymets. The iterest is Iterest $168, 001.20 $119, 200 $48,801.20 The professor pays $48,801.20 i iterest o this 15 year mortgage. Eve though the 15 year mortgage has a lower iterest rate, the shorter term leads to higher paymets tha the 30 year mortgage. However, because of the lower iterest rate ad shorter term, the amout of iterest paid to the leder for the 15 year loa is almost a third of the iterest paid o the 30 year loa. I geeral, loas with shorter terms have lower iterest rates. This leads to less iterest paid for shorter term loas. The paymets calculated above are the portio of a mortgage paymet that applies to the loa. A typical mortgage paymet also icludes other amouts to cover property 9
taxes, homeowers isurace, ad mortgage isurace. These amouts ca icrease the overall paymet by a large amout. 10
Questio 3: How do you make a amortizatio table? A amortizatio table (also called a amortizatio schedule) records the portio of the paymet that applies to the pricipal ad the portio that applies to iterest. Usig this iformatio, we ca determie exactly how much is owed o the loa at the ed of ay period. Amortizatio tables are useful whe a loa is to be paid off. Recall that whe we calculated the paymet, we rouded the amout of the paymet up to the pey. Over the term of the loa, we might pay a additioal amout each moth leadig to the pricipal beig reduced more quickly tha aticipated. Whe the fial loa paymet is made, it eeds to be adjusted to isure the balace is paid off properly. Differet leders roud paymets ad iterest differetly. This may lead to slightly differet umbers i the amortizatio table. Suppose you wat to borrow $10,000 for a automobile. Navy Federal Credit Uio offers a loa at a aual rate of 1.79% amortized over moths. The paymet would be 0.0179 PMT 1 1 841.44 10000 0.0179 Sice paymets are made to the pey, a paymet of $841.44 would lead to a overpaymet of almost a half of a pey. While this may ot seem like much, over the term of the loa it ca add up. Fiacial istitutios eed to accurately accout for these small amouts to isure their books are balaced. A amortizatio table helps them to do this. Amortizatio tables geerally have five colums. These colums track the paymet umber, the amout of the paymet, the iterest paid i the paymet, the portio of the paymet applied to the balace, ad the outstadig balace o the loa after the 11
paymet is made. Let s look at how the amouts i the table are calculated. We ll do this by lookig at the differet rows of the table, oe at a time. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 The first row of the table helps us to establish the iitial balace o the loa. We call it paymet 0 sice it does ot correspod to a actual paymet. Usig this balace, we ca determie the portio of the paymet, $841.44, that is applied to the balace ad the portio that is iterest. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $841.44 $14.92 $826.52 $9173.48 The iterest i the paymet is calculate by multiplyig the iterest rate per period times the balace at the ed of the previous period, 0.0179 Iterest i 1 $10,000 $14.92 I this amortizatio table, we will roud iterest amouts to the earest pey. I practice, you should check with the leder to see how they roud iterest i the table. Sice the amout applied to balace is the differece betwee the paymet ad the iterest, Amout i 1 Applied to Balace $841.44 $14.92 $826.52
This amout reduces the balace at the ed of the period, Balace at the Ed of the First Period $10,000 $826.52 $9173.48 This strategy is also used to fill i the amouts for the secod paymet. However, i this case, the iterest is calculated usig the balace after the previous period. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $841.44 $14.92 $826.52 $9173.48 2 $841.44 $13.68 $827.76 $8345.72 0.0179 9173.48 841.44 13.68 9173.48 827.76 As the balace decreases, the iterest also decreases. This meas that a larger ad larger portio of the paymet goes to payig off the balace. s 3 through 11 are carried out i a similar fashio to give the ext few rows. Remember, i this table we are roudig iterest amouts to the earest pey. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $841.44 $14.92 $826.52 $9173.48 2 $841.44 $13.68 $827.76 $8345.72 3 $841.44 $.45 $828.99 $7516.73 4 $841.44 $11.21 $830.23 $6686.50 5 $841.44 $9.97 $831.47 $5855.03 6 $841.44 $8.73 $832.71 $5022.32 7 $841.44 $7.49 $833.95 $4188.37 13
8 $841.44 $6.25 $835.19 $3353.18 9 $841.44 $5.00 $836.44 $2516.74 10 $841.44 $3.75 $837.69 $1679.05 11 $841.44 $2.50 $838.94 $840.11 For the last paymet, we eed to pay off the outstadig balace of $840.11. This meas the amout of the last paymet applied to the balace must be $840.11. The iterest i the last paymet is 0.0179 Iterest i $840.11 $1.25 Combiig these two amouts gives the amout of the last paymet, Amout of $840.111.25 $841.36 With these amouts, we ca complete the amortizatio table. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $841.44 $14.92 $826.52 $9173.48 2 $841.44 $13.68 $827.76 $8345.72 3 $841.44 $.45 $828.99 $7516.73 4 $841.44 $11.21 $830.23 $6686.50 5 $841.44 $9.97 $831.47 $5855.03 6 $841.44 $8.73 $832.71 $5022.32 7 $841.44 $7.49 $833.95 $4188.37 8 $841.44 $6.25 $835.19 $3353.18 9 $841.44 $5.00 $836.44 $2516.74 14
10 $841.44 $3.75 $837.69 $1679.05 11 $841.44 $2.50 $838.94 $840.11 $841.36 $1.25 $840.11 $0 If we add the iterest amouts, we fid the total amout of iterest paid is $97.20. If we roud the paymet or iterest amouts differetly, the amortizatio table yields differet amouts of iterest. I the ext example, we roud all paymets ad iterest amouts up to the earest pey to see how these chage the total amout of iterest paid. Example 4 Make a Amortizatio Table Suppose Navy Federal Credit Uio rouds all iterest ad paymet amouts up. a. Fid the amortizatio table o a loa of $10,000 amortized at a aual rate of 1.79% over moths with mothly paymets. Solutio The terms of the loa are the same as was described above. If the paymet is rouded up, we still get a paymet of $841.44. Whe we carry out the process described earlier, we get the table below. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $841.44 $14.92 $826.52 $9173.48 2 $841.44 $13.69 $827.75 $8345.73 3 $841.44 $.45 $828.99 $7516.74 4 $841.44 $11.22 $830.22 $6686.52 5 $841.44 $9.98 $831.46 $5855.06 6 $841.44 $8.74 $832.70 $5022.36 15
7 $841.44 $7.50 $833.94 $4188.42 8 $841.44 $6.25 $835.19 $3353.23 9 $841.44 $5.01 $836.43 $2516.80 10 $841.44 $3.76 $837.68 $1679. 11 $841.44 $2.51 $838.93 $840.19 $841.45 $1.26 $840.19 $0 I this table, several of the paymets iclude a slightly higher amout of iterest. This meas that less of the paymet goes towards the outstadig balace. This amout is made up i the last paymet where $840.19 is paid to brig the balace to zero. This causes the fial paymet to be slightly higher. b. Add the iterest amout i the third colum to fid the total amout of iterest paid. Solutio The sum of the iterest amouts is $97.29. This is slightly higher tha whe iterest amouts are rouded to the earest pey. This is to be expected sice we rouded all iterest amouts up. The paymets ad iterest amouts may be rouded to the earest pey, rouded up to the earest pey, or rouded dow to the earest pey. I all cases, ay discrepacies are made up i the fial paymet. 16