Question 2: How is a loan amortized?


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1 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
2 Paymet 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 Paymet 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 Paymet The loa amout is PV 149, , , 200. For a 30 year mortgage, there are 30 or 360 periods. The iterest rate per period is i. Usig these values, the mothly paymet is 7
3 PMT 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 $ or $250, over the term of the loa. Sice the loa amout is $119,200, the additioal amout paid must be iterest, Iterest $250, $119, 200 $131, The professor pays $131, 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 The itest rate per moth is i. This leads to a paymet of 8
4 PMT 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 $ d. How much iterest is paid o the 15 year mortgage? Solutio The professor will pay a total of 180 $ or $168, i paymets. The iterest is Iterest $168, $119, 200 $48, The professor pays $48, 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
5 taxes, homeowers isurace, ad mortgage isurace. These amouts ca icrease the overall paymet by a large amout. 10
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