Time Value of Money. First some technical stuff. HP10B II users

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1 Time Value of Moey Basis for the course Power of compoud iterest \$3,600 each year ito a 401(k) pla yields \$2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle module The time value of moey is the cocept that bids the whole course together If you do ot have your fiacial calculator yet, tur off the PC ad buy oe ow at Staples or Office Max or fid a former Fi 225 studet ad borrow his or hers HP12C or HP10BII is recommeded HP10B II users To chage umber of decimal places, press DISP key followed by a iteger 0 to 9 Always iterally to 9 places If BEGIN idicator ever appears, press BEG/END key to toggle it off END is the default but there is o END idicator Before ay ew calculatio, clear the etire calculator with CLEAR ALL key Pressig the C key oly erases the display 1

3 Our symbols PV 0 = preset value at time 0 (today) FV = future value at time ( periods from today) i = iterest rate per period (like.06 or 6%) = umber of periods Compoud iterest Ivest \$100 (PV 0 ) today at a iterest rate of 6%/yr for 1 year FV 1 = (.06) = 100(1+.06) = \$ Leave it all i for a secod year ad ear 6% o the origial \$100 agai plus 6% o the first year s \$6.00 of iterest FV 2 = (.06) = 106(1+.06) = \$ FV 2 = 100(1+.06)(1+.06) = 100(1+.06) 2 FV = 100(1+.06) ad FV = PV 0 (1+i) Most importat equatio i fiace FV = PV 0 (1+i) FV 2 = 100(1+.06) 2 100=>PV 6=>i 2=> solve for FV = Iterest rate is etered as 6 ad ot.06 For ow disregard the egative sig Ivest \$2,000 for 40 years at i=8% FV 40 = 2,000(1.08) 40 40=> 8=>i 2000=>PV solve for FV=-43,449 Negative sig is a covetio used by HP & Excel 3

4 Same problem oly differet You ivest \$2,000 for 40 years ad emerge at the ed with \$43,449 What was the aual growth (iterest) rate? FV = PV 0 (1+i) 43,449 = 2,000( ,449=>FV 2,000=>PV 40=> solve for i Error 5 or No solutio Now the mius sig covetio matters PV & FV must have opposite sigs (use CHS or +/-) -43,449=>FV 2,000=>PV 40=> solve for i=8% Discoutig - PV What s a future sum worth today? Ivestmet promises lump-sum payoff of \$10,000 i 20 years; what s it worth today? How much would you be willig to pay today for this promise? How much would you have to ivest today to amass \$10,000 i 20 years? Need to kow the iterest rate expected rate of retur let s assume it s 6% a year Same formula FV = PV 0 (1+i) Now we kow FV ad are lookig for PV 0 PV 0 = FV / (1+i) PV 0 = 10,000 / (1+.06) =>FV 6=>i 20=> solve PV= \$3, ivested today at 6% will grow to \$10,000 i 20 years You d be willig to pay \$3, today for the promise if you wated a 6% aual retur 4

5 Oe formula three ways Give a PV today, you ca fid what it ll be worth at some poit i the future by FV = PV 0 (1+i) Give a FV at some poit i the future, you ca fid what it is worth today by PV 0 = FV / (1+i) Give both the FV ad PV, you ca fid the iterest rate with either versio Just remember to switch oe of the sigs Fidig i You ca buy a isurace policy today for \$3,000 ad the redeem it i 20 years for \$10,000. To fid your rate of retur 10,000 = 3,000(1+i) 20-3,000=>PV 20=> 10,000=>FV solve i =? Gotta be higher tha 6% At 6% it took \$3, to grow to \$10,000 Startig with oly \$3,000 so rate must be higher i = 6.20% (6.2047%) Auities Series of equal paymets at equal itervals O your child s 1 st birthday deposit \$4,000 i ivestmet earig 8% ad cotiue to ivest \$4,000 thru her 18 th birthday. What s the fial amout you have saved for college? Could just tediously add up all the FV s First deposit + secod dep + + last dep FV 18 = 4000(1.08) (1.08) First is compouded oly 17 ad last oe ot at all Auity simplifies calculatios Auity of \$4,000 => equal amts, regular fixed (aual) itervals for 18 years 5

6 FV of auity math ad otatio FV i = i FV = PMT FV = 4000 [ FVIF ] [ FVIFa ] [ FVIF 8% 18] a a FV of auity math ad otatio FV i = i FV = PMT FV = 4000 [ FVIF ] [ FVIFa ] [ FVIF 8% 18] a a FV of auity o the calculator FV = PMT [FVIF a -i%-] FV 18 = 4,000 [FVIF a -8%-18] 4,000=>PMT 8=>i 18=> FV=\$149, Notice that 18x4,000=\$72,000 More tha half of the savigs is from iterest Compoudig at work Icidetally 18.08) FV = 4000 = 4000[ ] = 149,

7 Preset value of a auity A ivestmet (life isurace policy) promises to pay you \$10,000 a year for 20 years startig oe year from today. What s the ivestmet worth ow? What s its preset value? How much would you be willig to pay for it today? How much would you have to deposit today to be able to withdraw 20 paymets of \$10,000? All questios have same aswer Eter the star of the show To aswer ay of these questios, eed to kow: The required growth rate The goig rate of retur The iterest rate you ca ear Let s assume a iterest rate of 8% Depedet o risk Depedet o expected iflatio Restate the problem Receive \$10,000 a year for 20 years startig oe year from ow Fid the preset value of the auity discouted at i=8% Could just tediously add up all the PV s First paymet + secod paymet + +last paymet PV 0 =10000/(1.08) /(1.08) /(1.08) 20 First is discouted 1 ad last oe is discouted 20 Auity simplifies calculatios Auity of \$10,000 => equal amts, regular itervals for 20 years Sice the \$10,000 is costat ad iterval is regular (oce a year) ca use the PV of auity formula 7

8 PV of auity math ad otatio i(1 = i(1 PV = PMT 0 PV = [ PVIF ] [ PVIFa ] [ PVIF 8% 20] a a PV of auity o the calculator PV 0 = 10,000 [PVIF a - 8% - 20] 10,000=>PMT 8=>i 20=> PV=98, Notice that you receive 20x10,000=\$200,000 More tha half of the beefits is from iterest Compoudig at work Icidetally 20.08) PV = = 10000[ ] = 98, (1.08) What s the 98, mea? You could deposit \$98,181 today ito a ivestmet earig 8%/yr ad be able to withdraw \$10,000 each year for 20 years If someoe or some ivestmet promises you that for \$98,181 today, you would receive \$10,000 a year for 20 years, you d be makig a 8% aual retur 8

9 Recap ad some examples Lump sum FV = PV (1 0 FV = (1 ( oe paymet) Auity ( series of paymets) FV = PMT ( FVIFa ) i = PMT ( PVIFa ) i(1 Nothig special about a year Formulas work eve if ot aual compoudig Let =umber of periods ad i=iterest rate per period Lots of very commo examples Applicatio Bods Savig accouts Mortgages & car loas Visa & MC credit cards Frequecy Semiaually Quarterly Mothly Daily Periods per year Car loa example #1 You ca afford \$300 mothly car paymet Take out a 4-year loa (chage to 48 moths) How much ca you sped o a car today, ot icludig the dow paymet? Iterest rate = 12%/yr compouded mothly = 12%/12 = 1%/moth PV 0 =PMT(PVIF a - i% - ) PV 0 =300(PVIF a -1%-48) 300=>PMT 1=>i 48=> solve PV = \$11,

10 Car loa example #2 Dream car costs \$25,000, you put \$5,000 dow Borrow remaiig \$20,000 from dealer It rate = 8%/yr comp moth=> 8/12 =.667%/mo 4 year loa (48 moths) PV 0 = PMT(PVIF a -i%-) 20,000 = PMT(PVIF a -.667%-48) 20,000=>PV 8/12=.667=>i 48=> PMT= Make 48 mothly paymets of \$ ad car is yours Car loa example #3 Bak will led you the 20,000 but requires 36 mothly paymets of \$ Fid bak s iterest rate PV 0 = PMT(PVIF a - i% - ) 20,000 = (PVIF a -i%-36) =>PMT -20,000=>PV 36=> solve i =.55%/moth or.55x12=6.6%/year Bak has lower rate tha dealer Bod example #1 Bod (corporate IOU) maturig i 15 years promises a \$70 coupo paymet every six moths plus \$1,000 at maturity PV 0 = PMT(PVIF a -i%-) + FV /(1+i) PV 0 = 70(PVIF a -i%-30) /(1+i) 30 Let s say we re give i=12% a year PV 0 = 70(PVIF a -6%-30) /(1.06) 30 70=>PMT 6=>i 30=> 1000=>FV PV = -\$1, (bod s price today) 10

11 Bod example #2 What if the same bod sells for \$794.53? Fid the iterest rate or yield to maturity PV 0 = PMT(PVIF a -i%-) + FV /(1+i) PV 0 = 70(PVIF a -i%-30) /(1+i) = 70(PVIF a -i%-30) /(1+i) =>PV -70=>PMT -1000=>FV 30=> (careful of the sigs) Solve i=9.00% per period or 18% per year Lump sum FV = PV (1 0 FV = (1 Big 4 ( oe paymet) Auity ( series of paymets) FV = PMT ( FVIFa ) i = PMT ( PVIFa ) i(1 Lump sum FV = PV (1 0 FV = (1 Big 4 ( oe paymet) Auity ( series of paymets) FV = PMT ( FVIFa ) i = PMT ( PVIFa ) i(1 11

12 Big 4 all you eed Mortgages Mothly paymets, maximum loa, 15-years vs. 30- years, bak or credit uio Car loas Mothly paymets, maximum loa, 3, 4, 5 or eve 6 years, bak or dealer Retiremet plas Mothly cotributios, how soo ca you retire, term vs. whole life isurace Get the exact aswers, ot approximatios All the math of the course You do t eed ay more math tha what we ve covered i this module All you eed is PV ad FV of sigle paymets ad of auities But PLEASE do ot go o to ay other modules util you feel comfortable with this oe 12

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