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Concept 9: Present ValueDiscount Rate To find the present value of future dollars, one way is to seewhat amount of money, if invested today until the futuredate, will yield that sum of future money The interest rate used to find the present value discountrate There are individual differences in discount rates Present orientation high rate of time preference high Is the value of a dollar received today the same as received adiscount rateyear from today? Future orientation low rate of time preference low A dollar today is worth more than a dollar tomorrow because ofdiscount rateinflation, opportunity cost, and risk Notation: r discount rate The issue of compounding also applies to Present Value Bringing the future value of money back to the present iscalled finding the Present Value (PV) of a future dollarcomputations.1Present Value (PV) of Lump SumMoneyPresent Value Factor To bring one dollar in the future back to present, oneuses the Present Value Factor (PVF):PVF 2 1(1 r ) nFor lump sum payments, Present Value (PV)is the amount of money (denoted as P) timesPVF Factor (PVF)PV P PVF P 1(1 r ) n3An Example Using AnnualCompoundingAn Example Using MonthlyCompounding Suppose you are promised a payment of 100,000 You are promised to be paid 100,000 in 10 years. If youafter 10 years from a legal settlement. If yourdiscount rate is 6%, what is the present value ofthis settlement?PV P PVF 100,000 4have a discount rate of 12%, using monthlycompounding, what is the present value of this 100,000? First compute monthly discount rateMonthly r 12%/12 1%, n 120 months1 55,839.48(1 6%)10PV P PVF 100,000 51 100,000 * 0.302995 30,299.50(1 1%)1206
An Example Comparing TwoOptions Your answer will depend on your discount rate: Discount rate r 10% annually, annual compounding Option (1): PV 10,000 (note there is no need to convert thisnumber as it is already a present value you receive right now). Option (2): PV 15,000 *(1/ (1 10%) 5) 9,313.82 Option (1) is better Suppose you have won lottery. You are faced with twooptions in terms of receiving the money you have won:(1) 10,000 paid now; (2) 15,000 paid five years later.Which one would you take? Use annual compoundingand a discount rate of 10% first and an discount rate of5% next. Discount rate r 5% annually, annual compounding Option (1): PV 10,000 Option (2): PV 15,000*(1/ (1 5%) 5) 11,752.89 Option (2) is better78 Annual discount rate r 10%, annual compounding Option (1): PV 10,000 Option (2): PV of money paid in 1 year 2500*[1/(1 10%)1] 2272.73 PV of money paid in 2 years 2500*[1/(1 10%)2] 2066.12 PV of money paid in 3 years 2500*[1/(1 10%)3] 1878.29 PV of money paid in 4 years 2500*[1/(1 10%)4] 1707.53 PV of money paid in 5 years 2500*[1/(1 10%)5] 1552.30 Total PV Sum of the above 5 PVs 9,476.97 Option (3): PV of money paid now (year 0) 2380 (no discounting needed) PV of money paid in 1 year 2380*[1/(1 10%) 1] 2163.64 PV of money paid in 2 years 2380*[1/(1 10%)2] 1966.94 PV of money paid in 3 years 2380*[1/(1 10%)3] 1788.13 PV of money paid in 4 years 2380*[1/(1 10%)4] 1625.57 Total PV Sum of the above 5 PVs 9,924.28Present Value (PV) of Periodical Payments For the lottery example, what if the options are (1) 10,000 now; (2) 2,500 every year for 5 years, startingfrom a year from now; (3) 2,380 every year for 5 years,starting from now? The answer to this question is quite a bit morecomplicated because it involves multiple payments fortwo of the three options. First, let’s again assume annual compounding with a10% discount rate. Option (1) is the best, option (3) is the second, andoption (2) is the worst.910Present Value Factor Sum (PVFS) Are there simpler ways to compute present If the first payment is paid right now (so the firstvalue for periodical payments?payment does not need to be discounted), it iscalled the Beginning of the month (BOM): Just as in Future Value computations, if the periodicpayments are equal value payments, then PresentValue Factor Sum (PVFS) can be used. Present Value (PV) is the periodical paymenttimes Present Value Factor Sum (PVFS). In theformula below Pp denotes the periodicalpayment:111 . (1 r )0 (1 r )1(1 r ) n 111 (1 r ) n 1 1 rPVFS PV Pp*PVFS1112
If the first payment is paid a period away from now,BOM or EOMthen the first payment needs to be discounted forone period. In this case, the end of the month(EOM) formula applies: In most cases End of the Month (EOM) is used inPVFS computation. So use EOM as the default unlessthe situation clearly calls for Beginning of the Month(BOM) calculation. Appendix PVFS Table uses EOM.11 . (1 r )1(1 r ) n11 (1 r ) n rPVFS 13Applications of Present Value:Computing Installment Payments Use PVFS to solve the example problem but use a5% discount rate: discount rate r 5% Option (1): PV 10,000 Option (2): You buy a computer. Price 3,000. No down payment. r 18% with monthlyPV 2500 PVFS ( r 5%, n 5, EOM )1 2500 14compounding, n 36 months. What is your monthlyinstallment payment M?1(1 5%) 5 2500 4.329477 10,823.695% The basic idea here is that the present value of all futureOption (3):payments you pay should equal to the computer price.PV 2380 PVFS ( r 5%, n 5, BOM )1 2380 (1 1(1 5%) 5 1) 2380 4.545951 10,819.365%Option (2) is the best.1516Application of Present Value:Rebate vs. Low Interest Rate Answer: Apply PVFS, n 36, monthly r 18%/12 1.5%, end ofthe month because the first payment usually doesnot start until next month (or else it would beconsidered a down payment) Suppose you are buying a new car. You negotiate a price of 12,000 with the salesman, and you want to make a 30%down payment. He then offers you two options in terms ofdealer financing: (1) You pay a 6% annual interest rate for afour-year loan, and get 600 rebate right now; or (2) Youget a 3% annual interest rate on a four-year loan withoutany rebate. Which one of the options is a better deal foryou, and why? What if you only put 5% down instead of30% down (Use monthly compounding)3000 M PVFS (r 1.5%, n 36, EOM ),3000PVFS (r 1.5%, n 36, EOM ) 300011 (1 1.5%)361.5% 3000 108.4627.660684M In this case because your down payment is the same for thesetwo options, and both loans are of four years, comparingmonthly payments is sufficient.1718
30% down situation 5% down situationOption 1. Amount borrowed is 12,000*(1-30%) – 600 7,800 Monthly r 6%/12 0.5%, n 48 months Option 1. Amount borrowed is 12,000*(1-5%) – 600 10,800 Monthly r 6%/12 0.5%, n 48 months7800M PVFS (r 0.5%, n 48, EOM ) 780011 (1 0.5%) 480.5% 7800 183.1842.580318M 1 10,800 Option 2. The amount borrowed: 12,000*(1-30%) 8,400 Monthly r 3%/12 0.25%, n 48 months8400PVFS (r 0.25%, n 48, EOM ) 840011 (1 0.25%) 480.25% 8400 185.9345.17869510,800PVFS ( r 0.5%, n 48, EOM ) 10,8001(1 0.5%) 480.5%42.580318 253.64 Option 2. The amount borrowed: 12,000*(1-5%) 11,400 Monthly r 3%/12 0.25%, n 48 months11,400PVFS ( r 0.25%, n 48, EOM )11,400 11 (1 0.25%) 480.25% 11,400 252.3345.178695M M Option 1 is better becauseit has a lower monthlypaymentOption 2 is better nowbecause it has a lowermonthly payment19Application of Present Value:Annuity20 Annuity calculation is an application PVFS Annuity is defined as equal periodic payments which abecause the present value of all future annuitypayments should equal to the nestegg one hasbuilt up.sum of money will produce for a specific number ofyears, when invested at a given interest rate. Example: You have built up a nest egg of 100,000which you plan to spend over 10 years. How much canyou spend each year assuming you buy an annuity at7% annual interest rate, compounded annually ?100,000 M PVFS ( r 7%, n 10, EOM ),100,000PVFS (r 7%, n 10, EOM ) 100,00011 (1 7%)107%100,000 14,237.757.023582M 2122 Approximate solution: Step 1: 10,000/ 2,000 5 Step 2: Find a PVFS that is the closest possible to 5 If you know how much money you want to haveevery year, given the interest rate and the initialamount of money, you can compute how long theannuity will last. Say you have 10,000 now, youwant to get 2,000 a year. The annual interest rate is7% with annual compounding (EOM) PVFS(r 7%, n 5, EOM) 4.100197PVFS(r 7%, n 6, EOM) 4.76654 close to 5PVFS(r 7%, n 7, EOM) 5.389289 close to 5 Because 5 is in-between PVFS(n 6) and PVFS(n 7), this annuity is going tolast between 6 and 7 years Exact solution: 10,000/ 2,000 55 PVFS (r 7%, n ?, EOM) 5 [1- 1/(1 7%) n]/7%0.35 1-1/(1.07) n0.65 1/(1.07) n1/0.65 (1.07) nLog(1/0.65) n log(1.07)n log(1/0.65)/log(1.07) 6.37 years Note: Homework, Quiz and Exam questions will ask for approximatesolution, not the exact solution, although for those who understand theexact solution the computation can be easier.2324
Appendix: A Step-by-Step Example for PVFSComputation1111 PVFS (n 5, r 7%, EOM ) 1 (1 7%) 5 7%1 (1 0.07) 5 0.071 1.07 50.0711.402552 1 0.712986 0.287014 4.1001970.070.070.0725
Bringing the future value of money back to the present is called finding the Present Value (PV) of a future dollar 1 Discount Rate To find the present value of future dollars, one way is to see what amount of money, if invested today until the future date, will yield that sum of future money The interest
