In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

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1 Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns an nterest rate of 8.25% (compounded monthly). At the end of 30 years the amount of money left s 0. In our example = r/12 =.0825/12 At the end of the frst month after your payment s receved your amount n the account, the balance, s P (1 + ) A At the end of the second month after your payment s receved the balance s (P (1 + ) A)(1 + ) A At the end of the thrd month after your payment s receved the balance s ((P (1 + ) A)(1 + ) A)(1 + ) A To smplfy notaton, denote x = 1 +. After three months, the balance 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 balance s P x n A xn 1 x 1 Substtutng x = 1 +, after n payments, the amount s P (1 + ) n A (1 + )n 1 Your monthly payment A s the quantty so that n 30 years,.e. when n = = 360, the amount left n the account s 0 so that wth n = = 360, 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 you receve each month, so that the orgnal prncpal reduces to 0 n 30 years. Durng the payout annuty, after n payments, the balance 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 to you up to ths pont would be n A. The nterest pad to you up to ths pont would be total pad the prncpal pad. Summary: after n payments: Balance 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 Balance $61, Prncpal pad $ 38, Total pad $ 180, Interest pad $ 141, If you look over the notes for the secton on amortzed loans, you wll see that a payout annuty s the same thng as an amortzed loan f you just change your pont of vew. The smple observaton s the followng: Start wth a prncpal P at the begnnng of a perod (a month). Wth an amortzed loan at the end of the perod you owe P (1 + ) and then after the loan payment of A s receved you owe P(1+)-A. Wth a payout annuty at the end of the perod your new balance s P (1 + ) because your prncpal earned nterest, and then after your payment s sent your balance s P (1 + ) A. 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 payout-annuty to expre 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 balance after 20 years, just enter these quanttes nto the formula for the balance after n payments. We know that after n payments, the balance s P (1 + ) n A (1 + )n 1 So the values for ths problem would be P = , =.0825/12, n = 240.

4 You should get $61, Example problem: Pam s gong to depost a prncpal that wll be used for a payout annuty. She wll receve twelve monthly payments of $1000, and after the twelve payments the annuty expres. The balance earns 10% compounded monthly. How much should she depost nto the annuty? You start wth an unknown prncpal P. The monthly nterest rate =.1/12, the monthly payment 1000, and the number of perods n = 12. So P (1 + ) n A (1 + )n 1 = 0 wth n = 12, Then solvng for P, rounded to the nearest penny. P = A (1 + )12 1 = $11, (1 + ) 12 To avod lots of parenthess, I frst store (1 + ) 12 n ANSWER n the calculator: (1 +.1/12) 12 ENTER Then I take 1000*(ANSWER -1)/((.1/12)*ANSWER) Notce that ths should seem reasonable. If no nterest were pad, you would need to depost $12, 000 to receve twelve payments of $1, 000. Snce the balance earns nterest, you can depost less than $12, 000. Cost of Lvng Adjustments: Cost of Lvng Adjustments: In order to keep the formulas smpler, we pretend that there s just one payment per year, and for the frst year the payment s A, for the second year the payment s (1 + c)a, for the thrd year the payment s (1 + c) 2 A,..., and for the k th year the payment s (1 + c) k 1 A. The quantty c represents an annual ncrease n the payment to compensate for nflaton. The quantty c s called the annual C.O.L.A. rate. Denote r as the annual nterest rate. Let P denote the orgnal amount. At the end of the frst year, after your payment A s sent to you, the amount n the account s P (1 + r) A At the end of the second year, after your new payment A(1 + c) s sent, the amount n the account s (P (1 + r) A)(1 + r) A(1 + c). At the end of the thrd year, after your payment A(1 + c) 2 s sent, the amount n the account s ((P (1 + r) A)(1 + r) A(1 + c))(1 + r) A(1 + c) 2.

5 At the end of the fourth year, after your payment A(1 + c) 3 s sent, the amount n the account s (((P (1 + r) A)(1 + r) A(1 + c))(1 + r) A(1 + c) 2 )(1 + r) A(1 + c) 3. Expandng, at the end of the fourth year, after the payment s sent, the amount s P (1 + r) 4 A[(1 + r) 3 + (1 + c)(1 + r) 2 + (1 + c) 2 (1 + r) + (1 + c) 3 ] = P (1 + r) 4 A[(1 + r) 3 + ( 1 + c 1 + r )(1 + r)3 + ( 1 + c 1 + r )2 (1 + r) 3 + ( 1 + c 1 + r )3 (1 + r) 3 ] = P (1 + r) 4 A(1 + r) 3 [1 + x + x 2 + x 3 ] where we factored out (1 + r) 3 and to smplfy notaton, wrte 1 + c 1 + r We use the geometrc seres formula so after the fourth year the amount s (1 + x + x 2 + x 3 ) = (1 x 4 )/(1 x) as x. P (1 + r) 4 A(1 + r) 4 [1 ( 1 + c 1 + r )4 ]/(r c) after substtutng 1+c 1+r for x and a lttle algebra. The above represents the amount of money left n the account after 4 years. After t years, the amount of money left n the account s P (1 + r) t A(1 + r) t [1 ( 1 + c 1 + r )t ]/(r c) The annual payments A, A(1 + c), A(1 + c) 2, etc. are such that after n years, wth n=30 n ths example, the amount of money left n the account s 0. So we set wth t=30, and we get and P (1 + r) t A(1 + r) t [1 ( 1 + c 1 + r )t ]/(r c) = 0 P = A[1 ( 1 + c 1 + r )30 ]/(r c) P (r c)/[1 ( 1 + c 1 + r )30 ]

6 In our example, wth P=100,000 and an annual nterest rate of 8.25% and a C.O.L.A. rate of 2%, the frst annual payment ( )/[1 ((1 +.02)/( )) 30 ] = $ rounded to the nearest penny. The second annual payment would be (1.02) the frst annual payment = $ rounded to the nearest penny, etc. Summary For a payout annuty wth a begnnng prncpal P, payment A per perod, nterest rate per perod, after k perods the balance s P (1 + ) k A (1 + )k 1 For a payout annuty wth COLA rate c, annual nterest rate r, begnnng prncpal P, frst annual payment A, second annual payment (1 + c)a, thrd annual payment (1 + c) 2 A, etc., after k years the balance s P (1 + r) k A (1 + r)k (1 ( 1+c 1+r )k ) r c You can dstrbute the term (1 + r) k n the numerator and rewrte the expresson for the balance as P (1 + r) k A (1 + r)k (1 + c) k r c Notce that f each perod s a year, and f there s no COLA adjustment (c = 0), then the formula smplfes to that of a payout annuty.