Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

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1 Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig the future value of moey back to the preset is called fidig the Preset Value (PV) of a future dollar Discout Rate To fid the preset value of future dollars, oe way is to see what amout of moey, if ivested today util the future date, will yield that sum of future moey The iterest rate used to fid the preset value discout rate There are idividual differeces i discout rates Preset orietatiohigh rate of time preferece high discout rate Future orietatio low rate of time preferece low discout rate Notatio: rdiscout rate The issue of compoudig also applies to Preset Value computatios. 2 Preset Value Factor To brig oe dollar i the future back to preset, oe uses the Preset Value Factor (PVF): PVF ( + r) Preset Value (PV) of Lump Sum Moey For lump sum paymets, Preset Value (PV) is the amout of moey (deoted as P) times PVF Factor (PVF) PV P PVF P ( + r) 3 4 A Example Usig Aual Compoudig Suppose you are promised a paymet of $00,000 after 0 years from a legal settlemet. If your discout rate is 6%, what is the preset value of this settlemet? PV P PVF 00,000, ( + 6%) A Example Usig Mothly Compoudig You are promised to be paid $00,000 i 0 years. If you have a discout rate of 2%, usig mothly compoudig, what is the preset value of this $00,000? First compute mothly discout rate Mothly r 2%/2%, 20 moths PV P PVF 00,000 00,000* $30, ( + %) 6

2 A Example Comparig Two Optios Suppose you have wo lottery. You are faced with two optios i terms of receivig the moey you have wo: () $0,000 paid ow; (2) $,000 paid five years later. Which oe would you take? Use aual compoudig ad a discout rate of 0% first ad a discout rate of % ext. Your aswer will deped o your discout rate: Discout rate r0% aually, aual compoudig Optio (): PV0,000 (ote there is o eed to covert this umber as it is already a preset value you receive right ow). Optio (2): PV,000 *(/ (+0%)^) $9,33.82 Optio () is better Discout rate r % aually, aual compoudig Optio (): PV0,000 Optio (2): PV,000*(/ (+%)^) $,72.89 Optio (2) is better 7 8 Preset Value (PV) of Periodical Paymets For the lottery example, what if the optios are () $0,000 ow; (2) $2,00 every year for years, startig from a year from ow; (3) $2,380 every year for years, startig from ow? The aswer to this questio is quite a bit more complicated because it ivolves multiple paymets for two of the three optios. First, let s agai assume aual compoudig with a 0% discout rate. Aual discout rate r 0%, aual compoudig Optio (): PV0,000 Optio (2): PV of moey paid i year 200*[/(+0%) ] PV of moey paid i 2 years 200*[/(+0%) 2 ] PV of moey paid i 3 years 200*[/(+0%) 3 ] PV of moey paid i 4 years 200*[/(+0%) 4 ] PV of moey paid i years 200*[/(+0%) ] 2.30 Total PV Sum of the above PVs 9, Optio (3): PV of moey paid ow (year 0) 2380 (o discoutig eeded) PV of moey paid i year 2380*[/(+0%) ] PV of moey paid i 2 years 2380*[/(+0%) 2 ] PV of moey paid i 3 years 2380*[/(+0%) 3 ] PV of moey paid i 4 years 2380*[/(+0%) 4 ] 62.7 Total PV Sum of the above PVs 9, Optio () is the best, optio (3) is the secod, ad optio (2) is the worst. 9 0 Are there simpler ways to compute preset value for periodical paymets? Just as i Future Value computatios, if the periodic paymets are equal value paymets, the Preset Value Factor Sum (PVFS) ca be used. Preset Value (PV) is the periodical paymet times Preset Value Factor Sum (PVFS). I the formula below P p deotes the periodical paymet: PVP p *PVFS Preset Value Factor Sum (PVFS) If the first paymet is paid right ow (so the first paymet does ot eed to be discouted), it is called the Begiig of the moth (BOM): PVFS ( + r) + ( + r) ( + r) + r ( + r) 2

3 If the first paymet is paid a period away from ow, the the first paymet eeds to be discouted for oe period. I this case, the ed of the moth (EOM) formula applies: PVFS ( + r) ( + r) r ( + r) BOM or EOM I most cases Ed of the Moth (EOM) is used i PVFS computatio. So use EOM as the default uless the situatio clearly calls for Begiig of the Moth (BOM) calculatio. Appedix PVFS Table uses EOM. 3 4 Use PVFS to solve the example problem but use a % discout rate: discout rate r% Optio (): PV 0,000 Optio (2): PV 200 PVFS ( r %,, EOM ) ( + %) , % Optio (3): PV 2380 PVFS ( r %,, BOM ) (+ %) 2380 ( + % ) ,89.36 Applicatios of Preset Value: Computig Istallmet Paymets You buy a computer. Price$3,000. No dow paymet. r8% with mothly compoudig, 36 moths. What is your mothly istallmet paymet M? The basic idea here is that the preset value of all future paymets you pay should equal to the computer price. Optio (2) is the best. 6 Aswer: Apply PVFS, 36, mothly r8%/2.%, ed of the moth because the first paymet usually does ot start util ext moth (or else it would be cosidered a dow paymet) 3000 M PVFS ( r.%, 36, EOM ), 3000 M PVFS ( r.%, 36, EOM ) ( +.%).% Applicatio of Preset Value: Rebate vs. Low Iterest Rate Suppose you are buyig a ew car. You egotiate a price of $2,000 with the salesma, ad you wat to make a 30% dow paymet. He the offers you two optios i terms of dealer fiacig: () You pay a 6% aual iterest rate for a four-year loa, ad get $600 rebate right ow; or (2) You get a 3% aual iterest rate o a four-year loa without ay rebate. Which oe of the optios is a better deal for you, ad why? What if you oly put % dow istead of 30% dow (Use mothly compoudig) I this case because your dow paymet is the same for these two optios, ad both loas are of four years, comparig mothly paymets is sufficiet. 7 8

4 30% dow situatio Optio. Amout borrowed is 2,000*(-30%) 600 7,800 Mothly r6%/20.%, moths 7800 M PVFS ( r 0.%,, EOM ) 7800 ( + 0.%) 0.% Optio 2. The amout borrowed: 2,000*(-30%)8,400 Mothly r3%/20.2%, moths % dow situatio Optio. Amout borrowed is 2,000*(-%) 600 0,800 Mothly r6%/20.%, moths 0,800 M PVFS ( r 0.%,, EOM) 0,800 ( + 0.%) 0.% 0, Optio 2. The amout borrowed: 2,000*(-%),400 Mothly r3%/20.2%, moths 8400 M PVFS ( r 0.2%,, EOM ) 8400 ( + 0.2%) 0.2% Optio is better because it has a lower mothly paymet 9,400 M PVFS ( r 0.2%,, EOM ),400 ( + 0.2%) 0.2%, Optio 2 is better ow because it has a lower mothly paymet 20 Applicatio of Preset Value: Auity Auity is defied as equal periodic paymets which a sum of moey will produce for a specific umber of years, whe ivested at a give iterest rate. Example: You have built up a est egg of $00,000 which you pla to sped over 0 years. How much ca you sped each year assumig you buy a auity at 7% aual iterest rate, compouded aually? Auity calculatio is a applicatio PVFS because the preset value of all future auity paymets should equal to the estegg oe has built up. 00,000 M PVFS( r 7%, 0, EOM ), 00,000 M PVFS( r 7%, 0, EOM ) 00,000 0 ( + 7%) 7% 00,000 $4, If you kow how much moey you wat to have every year, give the iterest rate ad the iitial amout of moey, you ca compute how log the auity will last. Say you have $0,000 ow, you wat to get $2,000 a year. The aual iterest rate is 7% with aual compoudig (EOM) Approximate solutio: Step : $0,000/$2,000 Step 2: Fid a PVFS that is the closest possible to PVFS(r7%,, EOM) PVFS(r7%, 6, EOM) close to PVFS(r7%, 7, EOM) close to Because is i-betwee PVFS(6) ad PVFS(7), this auity is goig to last betwee 6 ad 7 years Exact solutio: $0,000/$2,000 PVFS (r7%,?, EOM) > [- /(+7%)^]/7% 0.3-/(.07)^ 0.6/(.07)^ /0.6(.07)^ Log(/0.6) log(.07) log(/0.6)/log(.07)6.37 years Note: Homework, Quiz ad Exam questios will ask for approximate solutio, ot the exact solutio, although for those who uderstad the exact solutio the computatio ca be easier

5 Appedix: A Step-by-Step Example for PVFS Computatio ( + 7%) PVFS (, r 7%, EOM ) 7%.4022 ( + )

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