Presentation on theme: "Time Value of Money Discounted Cash Flow Analysis"— Presentation transcript:
1 Time Value of Money Discounted Cash Flow Analysis MBA 220
2 Which would you Choose?On December 31, 2003 Norman and DeAnna Shue of Columbia, South Carolina had reason to celebrate the coming new year after winning the Powerball Lottery. They had 2 options.$110 MillionPaid in 30 yearlypayments of$3,666,666$60 Million
3 Time Value of MoneyA dollar received (or paid) today is not worth the same amount as a dollar to be received (or paid) in the future WHY?You can receive interest on the current dollar
4 How much will you have in one year? A Simple ExampleYou deposit $100 today in an account that earns 5% interest annually for one year.How much will you have in one year?Value in one year = Current value + interest earned= $ (.05)= $100(1+.05) = $105The $105 next year has a present value of $100 orThe $100 today has a future value of $105
5 Using a Time Line An easy way to represent this is on a time line Time year5%$100 $105Beginning ofFirst YearEnd of First year
6 What would the $100 be worth in 2 years? You would receive interest on the interest you received in the first year (the interest compounds)Value in 2 years = Value in 1 year + interest= $ (.05)= $105(1+.05) = $110.25Or substituting $100(1+.05) for $105= [$100(1+.05)](1+.05)= $100(1+.05)2 =$110.25
7 On the time line Time 0 1 2 Cash -$100 $105 110.25 Flow Beginning of year 1End of Year 1Beginning ofYear 2End of Year 2
8 Generalizing the Formula = (100)(1+.05)2This can be written more generally:Let t = The number of periods = 2r = The interest rate per period =.05PV = The Present Value = $100FV = The Future Value = $110.25FV = PV(1+r)t($110.25) = ($100)( )2This works for any combination of t, r, and PV
9 Future Value Interest Factor FV = PV(1+r)t (1+r)t is called theFuture Value Interest Factor (FVIFr,t)FVIF’s can be found in tables or calculatedInterest RatePeriods1231.1025OR (1+.05)2 =Either way original equation can be rewritten:FV = PV(1+r)t = PV(FVIFr,t)
10 Calculation Methods FV = PV(1+r)t Tables using the Future Value Interest Factor (FVIF)Regular CalculatorFinancial CalculatorSpreadsheet
11 Plugging it into our equation Using the tablesFVIF5%,2 =Plugging it into our equationFV = PV(FVIFr,t)FV = $100(1.1025) = $110.25
12 Using a Regular Calculator Calculate the FVIF using the yx key(1+.05)2=1.1025Proceed as BeforePlugging it into our equationFV = PV(FVIFrr,t)FV = $100(1.1025) = $110.25
13 Financial Calculator Financial Calculators have 5 TVM keys N = Number of Periods = 2I = interest rate per period =5PV = Present Value = $100PMT = Payment per period = 0FV = Future Value =?After entering the portions of the problem that you know, the calculator will provide the answer
14 Financial Calculator Example On an HP-10B calculator you would enter:2 N 5 I PV 0 PMT FVand the screen shows
15 Spreadsheet Example Excel has a FV command =FV(rate,nper,pmt,pv,type) =110.25note: Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year
16 FV = PV(1+r)t = PV(FVIFr,t) Practice ProblemIf you deposit $3,000 today into a CD that pays 4% annually for a period of five years, what will it be worth at the end of the five years?FV = PV(1+r)t = PV(FVIFr,t)FVIF0.4,5 = (1+0.04)5 =FV = $3,000(1+.04)5=$3,000( )FV = $3,
17 Calculating Present Value We just showed that FV=PV(1+r)tThis can be rearranged to find PV given FV, i and n.Divide both sides by (1+r)twhich leaves PV = FV/(1+r)t
18 ExampleIf you wanted to have $ at the end of two years and could earn 5% interest on any deposits, how much would you need to deposit today?PV = FV/(1+r)tPV = $110.25/(1+0.05)2 = $100.00
19 Present Value Interest Factor PV = FV/(1+r)t /(1+r)t is called thePresent Value Interest Factor (PVIFr,t)PVIF’s can be found in tables or calculatedInterest RatePeriods 0123OR 1/(1+.05)2 =Either way original equation can be rewritten:PV = FV/(1+r)t = FV(PVIFr,t)
20 Calculating PV of a Single Sum Tables - Look up the PVIFPVIF5%,2 = PV = (0.9070) =100.00Regular calculator -Calculate PVIFPVIF =1/ (1+r)t PV = (0.9070) =Financial Calculator2 N I FV 0 PMT PV =SpreadsheetExcel command =PV(rate,nper,pmt,fv,type)Excel command =PV(.05,2,0,110.25,0)=100.00
21 How much would you need in the bank today if you were 25? ExampleAssume you want to have $1,000,000 saved for retirement when you are 65 and you believe that you can earn 10% each year.How much would you need in the bank today if you were 25?PV = 1,000,000/(1+.10)40=$22,094.93
22 What if you are currently 35? Or 45? If you are 35 you would needPV = $1,000,000/(1+.10)30 = $57,308.55If you are 45 you would needPV = $1,000,000/(1+.10)20 = $148,643.63This process is called discounting (it is the opposite of compounding)
23 AnnuitiesAnnuity: A series of equal payments made over a fixed amount of time. An ordinary annuity makes a payment at the end of each period.Example A 4 year annuity that makes $100 payments at the end of each year.TimeCF’s
24 Future Value of an Annuity The FV of the annuity is the sum of the FV of each of its payments. Assume 6% a yearTimeFV of CF100(1+.06)0=100.00100(1+.06)1=106.00100(1+.06)2=112.36100(1+.06)3=119.10FV =
25 FV of An Annuity This could also be written or for any n, r, payment, and t
26 FVIF of an Annuity (FVIFAr,t) Just like for the FV of a single sum there is a future value interest factor of an annuityThis is the FVIFAr,tFVannuity=PMT(FVIFAr,t)
27 Calculation Methods Tables - Look up the FVIFA FVIFA6%,4 = FV = 100( ) =Regular calculator -Approximate FVIFAFVIFA = [(1+r)t-1]/r FV = 100( ) =Financial Calculator4 N I 0 PV PMT FV =SpreadsheetExcel command =FV(rate,nper,pmt,pv,type)Excel command =FV(.06,4,100,0,0)=
28 Practice ProblemYour employer has agreed to make yearly contributions of $2,000 to your Roth IRA. Assuming that you have 30 years until you retire, and that your IRA will earn 8% each year, how much will you have in the account when you retire?
29 Present Value of an Annuity The PV of the annuity is the sum of the PV of each of its paymentsTime100/(1+.06)1=100/(1+.06)2=100/(1+.06)3=100/(1+.06)4=PV =
30 PV of An Annuity This could also be written or for any r, payment, and t
31 PVIF of an Annuity PVIFAr,t Just like for the PV of a single sum there is a future value interest factor of an annuityThis is the PVIFAr,tPVannuity=PMT(PVIFAr,t)
32 Calculation Methods Tables - Look up the PVIFA PVIFA6%,4 = FV = 100( ) =Regular calculator -Approximate FVIFAPVIFA = [(1/r)-1/r(1+r)t] FV = 100( ) =Financial Calculator4 N 6 I 0 FV PMT PV =SpreadsheetExcel command =PV(rate,nper,pmt,fv,type)Excel command =PV(.06,4,100,0,0)=
33 Annuity DueThe payment comes at the beginning of the period instead of the end of the period.TimeCF’s AnnuityCF’s Annuity DueHow does this change the calculation methods?
34 So what about the Shue Family? The PV of the 30 equal payments of $3,666,666 is simply the summation of the PV of each payment. This is called an annuity due since the first payment comes today.Lets assume their local banker tells them they can earn 3% interest each year on a savings account. Using that as the interest rate what is the PV of the 30 payments?
35 Present Value of an Annuity Due The PV of the annuity due is the sum of the PV of each of its paymentsTime3.6M M M M M3.6M/(1+.03)0=3.6M3.6M/(1+.03)1=3.559M3.6M/(1+.03)2=3.456M3.6M/(1+.03)3=3.355M3.6M/(1+.03)29=1.555MPV =$ 74,024,333
36 Wrong Choice?It would cost $74,024,333 to generate the same annuity payments each year, the Shue’s took the $60 Million instead of the 30 payments, did they made a mistake?Not necessarily, it depends upon the interest rate used to find the PV.The rate should be based upon the risk associated with the investment. What if we used 6% instead?
37 Present Value of an Annuity Due Time3.6M M M M M3.6M/(1+.06)0=3.6M3.6M/(1+.06)1=3.459M3.6M/(1+.06)2=3.263M3.6M/(1+.06)3=3.078M3.6M/(1+.06)29=676,708PV =$ 53,499,310
38 What is the right rate?The Lottery invests the cash payout (the amount of cash they actually have) in US Treasury securities to generate the annuity since they are assumed to be free of default.In this case a rate of 4.87% would make the present value of the securities equal to $60 Million (20 year Treasury bonds currently yield 5.02%)
39 IntuitionOver the last 50 years the S&P 500 stock index as averaged over 9% each year, the PV of the 30 payments at 9% is $41,060,370If you can guarantee a 9% return you could buy an annuity that made 30 equal payments of $3.6Million for $41,060,370 and used the rest of the $60 million for something else….
40 FV an PV of Annuity DueFVAnnuity Due There is one more period of compounding for each payment, Therefore:FVAnnuity Due = FVAnnuity(1+r)PVAnnuity Due There is one less period of discounting for each payment, ThereforePVAnnuity Due = PVAnnuity(1+r)
41 Uneven Cash Flow Streams What if you receive a stream of payments that are not constant? For example:TimeFV of CF200(1+.06)0=200.00200(1+.06)1=212.00100(1+.06)2=112.36100(1+.06)3=119.10FV =
42 FV of An Uneven CF Stream The FV is calculated the same way as we did for an annuity, however we cannot factor out the payment since it differs for each period.
43 PV of an Uneven CF Streams Similar to the FV of a series of uneven cash flows, the PV is the sum of the PV of each cash flow. Again this is the same as the first step in calculating the PV of an annuity the final formula is therefore:
44 Quick Review FV of a Single Sum FV = PV(1+r)t PV of a Single Sum PV = FV/(1+r)tFV and PV of annuities and uneven cash flows are just repeated applications of the above two equations
45 PerpetuityCash flows continue forever instead of over a finite period of time.
46 Growing PerpetuityWhat if the cash flows are not constant, but instead grow at a constant rate?The PV would first apply the PV of an uneven cash flow stream:
47 Growing PerpetuityHowever, in this case the cash flows grow at a constant rate which impliesCF1 = CF0(1+g)CF2 = CF1(1+g) = [CF0(1+g)](1+g)CF3 =CF2(1+g) = CF0(1+g)3CFt = CF0(1+g)t
49 Semiannual Compounding Often interest compounds at a different rate than the periodic rate.For example:6% yearly compounded semiannualThis implies that you receive 3% interest each six monthsThis increases the FV compared to just 6% yearly
50 Semiannual Compounding An Example You deposit $100 in an account that pays a 6% annual rate (the periodic rate) and interest compounds semiannuallyTime 0 1/ % 3%FV=100(1+.03)(1+.03)=100(1.03)2=106.09
51 Effective Annual RateThe effective Annual Rate is the annual rate that would provide the same annual return as the more often compoundingEAR = (1+inom/m)m-1m= # of times compounding per periodOur exampleEAR = (1+.06/2)2-1= =.0609