Fnte Math - Fall 2016 Lecture Notes - 9/19/2016 Secton 3.3 - Future Value of an Annuty; Snkng Funds Snkng Funds. We can turn the annutes pcture around and ask how much we would need to depost nto an account each perod n order to get the desred fnal value. It s smple to solve for P MT n the annutes formula to get Defnton 1 (Snkng Funds). P MT = F V (1 + ) n 1 where all the varables have the same meanng as for annutes. Example 1. Let s revst those new parents who are tryng to save for ther chld s college and examne the more lkely case that they wll make payments nto a savngs account. They stll want to save up $80, 000 and they have found an account that wll pay 8% compounded quarterly. How much wll they have to depost every year n order to have a value of $80, 000? Soluton. In ths case, = r m = 0.08 4 = 0.02, n = 4(17) = 68 and F V = $80, 000, so the requred payment s 0.02 P MT = $80, 000 (1.02) 68 1 = $562.54. Thus the parents would have to make a depost of $562.54 every 3 months n order to have the desred $80, 000 after 17 years. Example 2. A bond ssue s approved for buldng a marna n a cty. The cty s requred to make regular payments every 3 months nto a snkng fund payng 5.4% compounded quarterly. At the end of 10 years, the bond oblgaton wll be retred wth a cost of $5, 000, 000. How much wll the cty have to pay each quarter? Soluton. $95, 094.67 Secton 3.4 - Present Value of an Annuty; Amortzaton Present Value of an Annuty. In the next concept, we wll look at makng a large depost n order to have a fund whch we can make constant wthdraws from. We make an ntal depost, then make wthdraws at the end of each nterest perod. We should have a balance of $0 at the end of the predetermned amount of tme the fund should last. Example 3. How much should you depost nto an account payng 6% compounded semannually n order to be able to wthdraw $2000 every 6 months for 2 years? (At the end of the 2 years, there should be a balance of $0 n the account.) 1
2 Soluton. Ths problem s solved smlarly to how the future value of an annuty was, except ths tme, nstead of fndng the future value of each depost, we have to fnd the present value of each wthdraw. We do ths because we want to only depost enough money to be able to wthdraw the $2000 at the specfed tme. We can collect these agan n a table: Wthdraw Term Number of tmes Present Wthdrawn Compounded Value $2000 1 1 $2000 ( 1 + 0.06 ) 1 1 $2000 2 2 $2000 ( 1 + 0.06 ) 2 2 $2000 3 3 $2000 ( 1 + 0.06 ) 3 3 $2000 4 4 $2000 ( 1 + 0.06 ) 4 4 So addng up the present values of all these wll gve us the amount of money we should depost nto the account now D = $2000(1.03) 1 + $2000(1.03) 2 + $2000(1.03) 3 + $2000(1.03) 4 = $7434.20 Of course, just as wth fndng the future value of an annuty, snce these ft nto a pattern, we can fnd a formula for t; and we actually do t n the exact same way as before by computng 1.03D D: 1.03D = $2000(1.03) 0 + $2000(1.03) 1 + $2000(1.03) 2 + $2000(1.03) 3 D = $2000(1.03) 1 $2000(1.03) 2 $2000(1.03) 3 $2000(1.03) 4 Ths gves and solvng for D 1.03D D = 0.03D = $2, 000 $2000(1.03) 4 D = $2, 000 1 (1.03) 4 0.03 = $2, 000 1 ( 1 + 0.06 ) 4 2. 0.06/2 Ths gves rse to the followng formula Defnton 2 (Present Value of an Ordnary Annuty). where P V = P MT 1 (1 + ) n P V = present value P M T = perodc payment = rate per perod n = number of payments (perods) Note that the payments are made at the end of each perod.
In the above formula, = r, where r s the nterest rate (as a decmal) and m m s the number compoundng perods per year and n = mt where t s the length of tme of the annuty. We can rewrte the formula wth r and m nstead of P V = P MT 1 ( 1 + r m r/m Example 4. How much should you depost n an account payng 8% compounded quarterly n order to receve quarterly payments of $1, 000 for the next 4 years? Soluton. $13, 577.71 An nterestng applcaton of ths n conjuncton wth snkng funds s savng for retrement. Example 5. The full retrement age n the US s 67 for people born n 1960 or later. Suppose you start savng for retrement at 27 years old and you would lke to save enough to wthdraw $40, 000 per year for the next 20 years. If you fnd a retrement savngs account (for example, a Roth IRA) whch pays 4% nterest compounded annually, how much wll you have to depost per year from age 27 untl you retre n order to be able to make your desred wthdraws? Soluton. Frst, we should fgure out how much money we need to have n the account at the tme we retre n order to be able to make the wthdraws each year. For ths stuaton, we have ) n P MT = $40, 000, m = 1, r = 0.04, t = 20, n = mt = 20 and so the present value of the retrement account at the tme of retrement needs to be P V = $40, 000 1 ( 1 + 0.04 ) 20 1 = $543, 613.05. 0.04/1 So, now that we know how much we need to have n the account at the tme of retrement, we can fgure out how much we need to depost nto the savngs account per year n order to acheve that amount n the 40 years we have to save. To do ths, we use the snkng fund formula. The future value here s the value we want at the tme of retrement, so and the other numbers are Pluggng ths n the formula gves F V = $543, 613.05 r = 0.04, m = 1, t = 40, n = mt = 40 0.04/1 P MT = $543, 613.05( ) 1 + 0.04 40 = $5, 720.71. 1 1 So we wll have to depost $5, 720.71 per year from age 27 untl retrement nto ths account n order to be able to wthdraw $40, 000 per year for 20 years. 3
4 Example 6. Lncoln Beneft Lfe offered an ordnary annuty earnng 6.5% compounded annually. If $2, 000 s deposted annually for the frst 25 years, how much can be wthdrawn annually for the next 20 years? Soluton. $10, 688.87 Amortzaton. Amortzaton s the process of payng off a debt. The formula for present value of an annuty wll allow us to model the process of payng off a loan or other debt. The reason the formula s the same s because recevng payments from your savngs account s essentally the bank repayng you the money you loaned them by depostng t nto a savngs account. Example 7. Suppose you take out a 5-year, $25, 000 loan from your bank to purchase a new car. If your bank gves you 1.9% nterest compounded monthly on the loan and you make equal monthly payments, how much wll your monthly payment be? Soluton. Snce the loan was $25, 000 and t s beng pad off, the present value wll be P V = $25, 000. The nterest rate s r = 0.019 and s compounded monthly, m = 12. The loan lasts for 5 years, so we get $25, 000 = P MT 1 ( 1 + 0.019 12 Thus, solvng for P MT gves P MT = 0.019 12 ) 60 $25, 000 57.19500 = $437.10 = 57.19500P MT whch means our monthly payment would be $437.10 for 5 years. We get the followng formula Defnton 3 (Amortzaton). P MT = P V 1 (1 + ) n where all the varables have the same meanng as for annutes. Example 8. If you sell your car to someone for $2, 400 and agree to fnance t at 1% per month on the unpad balance, how much should you receve each month to amortze the loan n 24 months? How much nterest wll you receve? Soluton. P MT = $112.98, I = $311.52 Amortzaton Schedules. Suppose you are amortzng a debt by makng equal payments, but then decded to pay off the debt wth one lump-sum payment. How do you fnd the pay-off balance of the debt? (E.g., you take out a 5-year loan wth monthly payments for a car, but after 3-years of makng payments you decde to just make one fnal payment to retre the debt.) Ths pay-off s very useful, even f you are not retrng the debt, but refnancng t. When refnancng a debt, you are essentally takng out a new loan to pay-off the prevous debt, so you need
to know how much unpad balance remans on the account. When you are makng payments nto an amortzaton, at the begnnng, a large part of your payment goes towards nterest, whle later, a larger part goes towards the unpad balance. We can see how much of each payment goes towards nterest and how much towards unpad balance by creatng an amortzaton schedule. Example 9. Construct the amortzaton schedule for a $1, 000 debt that s to be amortzed n sx equal monthly payments at 1.25% nterest per month on the unpad balance. Soluton. The frst step n ths process s to compute the requred monthly payment usng the amortzaton formula 0.0125 P MT = $1, 000 = $174.03 1 (1 + 0.0125) 6 Now, to fgure out how much of the payment goes towards nterest and how much towards unpad balance, we compute the nterest due at the end of the frst month: $1, 000(0.0125) = $12.50 and so the amount of the payment that goes towards the unpad balance s: $174.03 $12.50 = $161.53. Thus, the unpad balance at the end of the frst month s $1, 000 $161.53 = $838.47. To compute the breakdown for the next month, we do the same thng, but wth the new unpad balance. The nterest due at the end of month 2: $838.47(0.0125) = $10.48 amount of payment towards unpad balance: $174.03 $10.48 = $163.55 and so the unpad balance at the end of 2 months s $838.47 $163.55 = $674.92. Payment Payment Interest Unpad Balance Unpad Number Reducton Balance 0 $1, 000 1 $174.03 $12.50 $161.53 $838.47 2 $174.03 $10.48 $163.55 $674.92 3 $174.03 $8.44 $165.59 $509.33 4 $174.03 $6.37 $167.66 $341.67 5 $174.03 $4.27 $169.76 $171.91 6 $174.03 $2.15 $171.91 $0.00 Total $1, 044.21 $44.21 $1, 000 5