How To Value An Annuity

Transcription

1 Future Value of a Auty After payg all your blls, you have $200 left each payday (at the ed of each moth) that you wll put to savgs order to save up a dow paymet for a house. If you vest ths moey at 5% terest per year, compouded mothly, how much moey wll you have saved toward the dow paymet at the ed of 5 years? Frst, a lttle backgroud: Deftos: 1. A auty s a successo of equal paymets made at equal perods of tme. A ordary auty has paymets due at the ed of each tme perod. 2. A sequece s a lst of umbers. 3. A seres s the sum of the umbers a sequece. 4. A geometrc sequece s a lst of umbers, startg wth a umber, a, where each umber s some costat r tmes the precedg umber. 5. A geometrc seres s the sum of a geometrc sequece. Example: 1, 2, 4, 8 s a geometrc sequece where a = 1ad r = s a geometrc seres. I geeral, a geometrc seres has the form: a + ar + ar 2 + ar ar +... The sum of the frst terms of a geometrc seres ca be calculated by the formula: a( r 1) Sum = ( r 1) 1

2 So, f we vest $200 a moth, usg m t r A = P 1+ the future m maturty value.05 of the frst depost s: of the secod: of the thrd: The sum of the future value of all the deposts s: 58 + whch s a geometrc seres, where + L a = 200, r =.05 1, ad =. So the value of our auty 5 years s: ( 1) a r = = ( r 1) 1 1 $13, (rouded to the earest pey). Future Value of a Ordary Auty: Let r be the aual terest rate, m be the umber of tmes per year that terest s r compouded, ad let =. Also, R s the m paymet, ad the umber of paymets, made at the ed of each terest perod, = m. t. The the future value of a auty, S, s: I the formula s sometmes deoted s, read s agle at. Example: After 40 years of work, t s tme to retre. Your compay vested $500 at the ed of each moth a ordary auty wth aual terest rate of 8% compouded mothly. If you take a lump-sum retremet paymet, how much should t be? 2

3 .08 where R = 500, =, the umber of paymets,, s ( per year)(40 years) = 480, so your lump sum = = $1,745, Let s say you work for the same compay, ad you wat to have $2,000,000 whe you retre after 40 years. The compay wll allow you to make addtoal mothly paymets. How much more do you have to pay each moth to retre after 40 years wth 2 mllo?.08 where =, = 480, ad ow S = 2,000,000, ad we eed to solve for R where = R Smplfyg, you have: = R( ) Dvde to fd R = (to the earest pey). Keep as may decmal places as possble for greatest accuracy. Classwork: Fd the future value of a ordary auty wth paymets of $150 made semaually for 20 years a accout that pays 6% terest compouded semaually. Defto: A skg fud s a auty set up to reach a specfed value at a specfed tme. Typcally, skg fuds are set up by a orgazato to retre a debt. 3

4 Example: A departmet of trasportato wshes to buld a brdge at a cost of $1.2 mllo. They pla to face the brdge wth the sale of bods that wll come due 5 years ad have a face value at that tme of $2 mllo. At the ed of each fscal year, the departmet ca vest part of ther aual budgeted fuds a mutual fud that returs 10% per year compouded quarterly. How much do they eed to budget each quarter order to be able to repay the $2 mllo debt 5 years? We ca use the formula for the future value of a auty. R = $78, = R = R( ) = R( ) Sce we re roudg the fal aswer two decmal places (the earest pey), why hold o to 8 decmal places the termedate aswer ( )? Could we roud t to two decmal places ad get the same aswer? = $ $78, Hold o to as may decmal places as possble termedate calculatos. Whe uts are dollars, t s reasoable to roud you fal aswer to 2 decmal places. Auty due: A auty due dffers from a ordary that paymets to a auty due are made at the begg of each perod, ot at the ed of the perod. 4

5 The future value, S, of a auty due s: + 1 R where R s the paymet, s terest per perod, ad s umber of paymets (same as for a ordary auty. 5