2. The Time Value of Money
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1 2. The Time Value of Moey Problem 4 Suppose you deposit $100 i the bak today ad it ears iterest at a rate of 10% compouded aually. How much will be i the accout 50 years from today? I this case, $100 ivested at 10-percet for 50 years accumulates to FV = PV = 11739,1 Problem 5 How much moey must you deposit i a savigs accout today to have $10, years from today if the iterest rate is 5% compouded aually? 1 PV = FV (1 + = 6,139 Problem 6 A stock has paid divideds regularly for the last 10 years, startig with a $1 divided i 1997 ad icreasig to $8 i If these divideds have bee growig at a costat rate, what has that rate bee for the last 10 years? The exact aswer is obtaied by solvig for i i the expressio 10 =8; the equatio holds whe i = = 23,11% FV i = 1 PV = 23,11% Problem 7 Your sister borrows $1 ad promises to repay $1,5. If you wat at least a 6% retur o your loa, withi how may years must she pay you back? The exact aswer is obtaied by solvig for "" i the expressio (1,06) =1,5; Takig the atural log of both sides ad rearragig, FV l = PV l(1 + = 6,96 years = (l 1.5 / l 1.06) =6,96 years. 8
2 Problem 8 Compare the future value i oe year of $100,000 to be ivested today at 10% uder the followig scearios. (a) Aual compoudig (b) Quarterly compoudig (c) Cotiuous compoudig (a) 100,000*10%)^1= 110,000 (b) 100,000*2.5%)^4= 110,381 (c) 100,000*e^10%= 110,517 Problem 9 A auity due (or ret auity) is a auity whose paymets are made at the begiig of each period. What is the preset value of a 20-year auity due with $1000 paymets if the discout rate is 5%? PVA = ( PVFA.05,19 ) = Twety paymets of $1000 made at the begiig of each year is the same as $1000 received ow ad a ordiary auity of $1,000 received at the ed of years 1,2,...,19. Problem 10 Which of the followig perpetuities represets the largest preset value? Aual Discout Perpetuity amout ($) rate (%) A 100,000 5,0% B 120,000 6,0% C 160,000 8,0% D 200,000 10,0% A B C D 2 000, , , ,000 Problem 11 Calculate the preset value of a growig auity that starts off with a paymet of $100 ad the subsequet aual paymets grow at the aual rate of 8%. The discout rate is 10%, ad the security makes a total of 5 paymets. Value of auity 1 PV = i g 1 + g i = 438,31 9
3 Problem 12 A bod pays $50 iterest every year plus $1000 whe it matures i 12 years. You ca buy the bod today for $950. What is the iteral rate of retur of this ivestmet? The YTM (iteral rate of retur) discouts the cash flows to the curret market price IRR = 5.58% Rate Price 2.0% % % % % % % % % % % % % % % % % % % % % % % % % % % % 5% 10% 15% 20% 10
4 Problem 13 Loa amout $ 100,000 Iterest rate 12% per year Period 1 year, mothly equal paymets Provisio 2% (a) Show the loa amortizatio schedule. (b) Calculate the effective iterest rate with ad without a provisio $2,000. (a) PV = Period Paymet Iterest Pricipal paid off Remaiig pricipal 0 100, ,885 1,000 7,885 92, ,885 0,921 7,964 84, ,885 0,842 8,043 76, ,885 0,761 8,124 67, ,885 0,680 8,205 59, ,885 0,598 8,287 51, ,885 0,515 8,370 43, ,885 0,431 8,454 34, ,885 0,347 8,538 26, ,885 0,261 8,624 17, ,885 0,175 8,710 8, ,885 0,088 8,797 0 (b) 0-100,000-98, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8, ,885 8,885 IRR = 12,0% 15,9% 1 1 (1 + = i 1 = i(1 + 11
5 Problem 14 Brew Corporatio is cosiderig two alterative bod issues. I optio 1, Brew will make iterest paymets idefiitely at the rate of 5%. Uder optio 2, Brew will make the first iterest paymet i oe year's time at 4%, but subsequet paymets will icrease at 3%. The discout rate is 6%. Which optio should Brew choose? Assume that Brew borrows the same amout uder each alterative. Optio 1 833,333 = 5% * 1000 / 6% Optio ,33 = 4% * 1000/ ( 6% - 3%) Optio 1 is better, sice they are receivig $1,000 i exchage for promised paymets worth oly $
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