In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

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1 Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars at the end of each perod untl the loan s pad off. In ths example a perod s one month, and after = 360 payments the loan s pad off. In our example = r/12 =.0825/12 At the end of the frst month after your payment s receved your amount owed s P (1 + ) A At the end of the second month after your payment s receved your amount owed s (P (1 + ) A)(1 + ) A At the end of the thrd month after your payment s receved your amount owed s ((P (1 + ) A)(1 + ) A)(1 + ) A To smplfy notaton, denote x = 1 +. After three months, you amount owed s now P x 3 Ax 2 Ax P x 3 A(x 2 + x + 1) = P x 3 A x3 1 x 1 by the geometrc seres formula from last tme. (Recall the formula for a geometrc seres: 1 + x + x x n = xn+1 1 x 1. ) Contnung n ths fashon, after n payments, the amount owed s P x n A xn 1 x 1 Substtutng x = 1 +, after n payments, the amount owed s P (1 + ) n A (1 + )n 1 In a conventonal 30 year mortgage, your monthly payment A s the quantty so that the debt s pad off n 30 years, so that wth n = = 360, the amount owed s 0, so P (1 + ) n A (1 + )n 1 = 0

2 Then solvng for A, P (1 + )n (1 + ) n 1 Summarzng, f P s the orgnal prncpal, s the nterest per perod, and n s the number of perods, then the payment per perod s P (1 + )n (1 + ) n 1 In our example, =.0825/12, P = , and n = = 360. Plug those numbers n and get P (1 + )n (1 + ) n 1 = $ So ths s the monthly payment that wll pay off the loan n exactly thrty years. We showed earler that durng the loan, after n payments, the amount owed s P (1 + ) n A (1 + )n 1 The prncpal pad up to ths pont would be P the amount owed. The total amount of money pad up to ths pont would be n A. The nterest pad up to ths pont would be total pad the prncpal pad. Summary: after n payments: Amount owed P (1 + ) n A (1 + )n 1 Prncpal pad P amount owed Total pad Interest pad n A Total pad prncpal pad Wth our same example, after 20 years, so n = = 240, you should check that

3 Amount owed $61, Prncpal pad $ 38, Total pad $ 180, Interest pad $ 141, Notce that ths s depressng news. After payng for 20 years,.e., two thrds of the tme of the loan, your equty s only $ 38, Look at how much nterest you pad, compared to how much you have pad to reduce the prncpal. Is there a way to beat the system? Read on: Suppose you send an extra amount each month along wth your regular payment. For example, you mght decde to send n a payment of $ each month nstead of $ untl the loan s pad off. If you ever do ths n real lfe, you should nclude a note tellng them to apply the extra $100 to the prncpal just to make sure t doesn t get put nto an escrow account. So now A=$ n our example. Wth ths value of A, after k payments you owe P (1 + ) k A (1 + )k 1 Snce the monthly payment A has been ncreased, we expect the balance to be zero before 30 years. Set P (1 + ) k A (1 + )k 1 = 0 and solve for k. After some algebra, A k = ln( A P ) ln(1 + ) You mght prefer to use your graphng calculator and plot P (1 + ) x A (1 + )x 1 and see where x = 0. as a functon of x Whchever way you choose, use your calculator and check that wth P = $100, 000 and $ and r = 8.25 percent compounded monthly, that after payments,.e., after = years, 12 the 30 year mortgage wll be pad off. By ncludng the extra payment of $100 per month, not only s the mortgage pad off approxmately 10 years earler, but also your total payments are changed to hence the total nterest pad durng the loan s $ = $

4 If no extra payment s sent n, then the loan expres at the end of thrty years, and the total amount pad s = and so the total nterest pad durng the loan s $ methods to compute the monthly payment, etc. We demonstrate wth our example of P = 100, 000 and annual nterest rate of 8.25% compounded monthly, wth the loan to be pad off n thrty years. What wll the monthly payment be? Usng Excel: Open a worksheet. In a cell enter =pmt(.0825/12,360,100000) If you forget the formula, clck on a cell and then clck on the paste functon f Then choose Fnancal, then clck on PMT and clck on that. You wll fll n the values for rate (the nterest rate per perod=.0825/12), the number of perods (360), the present value (100,000). The last two quanttes, the future value and payment type are assumed to be zero unless entered otherwse. Usng a TI-83: The quckest and easest way would be to just use the formula P (1 + )n (1 + ) n 1 = $ and use the values P = , =.0825/12, n = 360. If you want to calculate the amount owed after 20 years, just enter these quanttes nto the formula for the amount owed after n payments. We showed earler that durng the loan, after n payments, the amount owed s P (1 + ) n A (1 + )n 1 So the values for ths problem would be P = , =.0825/12, n = 240. Ths s not recommended, but f you prefer, after turnng on the TI-83, clck on APPS Fnance CALC TVM Solver

5 TVM stands for TIME-VALUE-OF-MONEY There are fve basc quanttes. You fll n the four quanttes and the calculator wll compute the unknown quantty. The quanttes are N= the number of payments I% = annual nterest rate PV = present value PMT = payment per perod FV = future value In addton to these basc quanttes, you fll n P/Y = number of payments per year. In ths case you have somethng lke ths N=360 I % = 8.25 PV= PMT=0 FV=0 P/Y=12 C/Y=12 Pmt: END After Pmt, you hghlght END for a conventonal loan where the payment s made at the end of each perod. The unknown s the PMT, whch rght now s flled n as 0. Now move the cursor up to the row that contans PMT. Push the alpha key. Then push enter. That wll make the calculator solve for that varable. You should see An alternatve would be as follows: After turnng on the TI-83, choose APPS Fnance CALC tvm Pmt Then tvm Pmt appears on your screen. You have to enter the quanttes tvm Pmt(N,%, PV, FV, P/Y, C/Y) n that order. So you would enter tvm Pmt(360, 8.25,100000,0,12,12) and agan you should get The defect wth ths method, n contrast to the prevous method, s that you must enter the quanttes n the correct order, rather than fllng n the quanttes after the prompts. Suppose you want the amount owed after 20 years. If you have done nothng n the meantme, the values for the nterest, prncpal, etc. are stored n the calculator. Ths tme after

6 APPS Fnance CALC you choose bal. Ths wll gve you the balance after you enter the number of payments. For example, after 20 years, the number of payments s = 240 so you enter bal(240) and you should see

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