Time Value of Money Discounted Cash Flow Analysis

Time Value of Money Discounted Cash Flow Analysis MBA 220 UNIVERSITE D’AUVERGNE Drake DRAKE UNIVERSITY

Which would you Choose? UNIVERSITE D’AUVERGNE Drake University On December 31, 2003 Norman and De. Anna Shue of Columbia, South Carolina had reason to celebrate the coming new year after winning the Powerball Lottery. They had 2 options. $110 Million Paid in 30 yearly payments of $3, 666 $60 Million

Time Value of Money UNIVERSITE D’AUVERGNE Drake University A 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

A Simple Example UNIVERSITE D’AUVERGNE Drake University You 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 = $100 + 100(. 05) = $100(1+. 05) = $105 The $105 next year has a present value of $100 or The $100 today has a future value of $105

Using a Time Line UNIVERSITE D’AUVERGNE Drake University An easy way to represent this is on a time line Time 0 1 year 5% $100$105 Beginning of First Year End of First year

What would the $100 be worth in 2 years? UNIVERSITE D’AUVERGNE Drake University 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 = $105 + 105(. 05)= $105(1+. 05) = $110. 25 Or substituting $100(1+. 05) for $105 = [$100(1+. 05)](1+. 05) = $100(1+. 05)2 =$110. 25

UNIVERSITE D’AUVERGNE On the time line Drake University Time 0 Cash -$100 Flow Beginning of year 1 1 2 $105 110. 25 End of Year 1 Beginning of Year 2 End of Year 2

UNIVERSITE D’AUVERGNE Generalizing the Formula Drake University 110. 25 = (100)(1+. 05)2 This can be written more generally: Let t = The number of periods = 2 r = The interest rate period =. 05 PV = The Present Value = $100 FV = The Future Value = $110. 25 FV = PV(1+r)t ($110. 25) = ($100)(1 + 0. 05)2 This works for any combination of t, r, and PV

Future Value Interest Factor UNIVERSITE D’AUVERGNE Drake University FV = PV(1+r)t is called the Future Value Interest Factor (FVIFr, t) FVIF’s can be found in tables or calculated Interest Rate Periods 1 2 3 4. 0 4. 5 5. 0 5. 5 1. 1025 OR (1+. 05)2 = 1. 1025 Either way original equation can be rewritten: FV = PV(1+r)t = PV(FVIFr, t)

Calculation Methods FV = PV(1+r)t UNIVERSITE D’AUVERGNE Tables using the Future Value Interest Factor (FVIF) Regular Calculator Financial Calculator Spreadsheet Drake University

Using the tables UNIVERSITE D’AUVERGNE Drake University FVIF 5%, 2 = 1. 1025 Plugging it into our equation FV = PV(FVIFr, t) FV = $100(1. 1025) = $110. 25

Using a Regular Calculator UNIVERSITE D’AUVERGNE Drake University Calculate the FVIF using the yx key (1+. 05)2=1. 1025 Proceed as Before Plugging it into our equation FV = PV(FVIFrr, t) FV = $100(1. 1025) = $110. 25

Financial Calculator UNIVERSITE D’AUVERGNE Drake University Financial Calculators have 5 TVM keys N = Number of Periods = 2 I = interest rate period =5 PV = Present Value = $100 PMT = Payment period = 0 FV = Future Value =? After entering the portions of the problem that you know, the calculator will provide the answer

UNIVERSITE D’AUVERGNE Financial Calculator Example Drake University On an HP-10 B calculator you would enter: 2 N 5 I -100 PV 0 PMT FV and the screen shows 110. 25

Spreadsheet Example UNIVERSITE D’AUVERGNE Drake University Excel has a FV command =FV(rate, nper, pmt, pv, type) =FV(0. 05, 2, 0, 100, 0) =110. 25 note: Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year

Practice Problem UNIVERSITE D’AUVERGNE Drake University If 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) FVIF 0. 4, 5 = (1+0. 04)5 = 1. 216652 FV = $3, 000(1+. 04)5=$3, 000(1. 216652) FV = $3, 649. 9587

Calculating Present Value UNIVERSITE D’AUVERGNE Drake University We just showed that FV=PV(1+r)t This can be rearranged to find PV given FV, i and n. Divide both sides by (1+r)t which leaves PV = FV/(1+r)t

Example UNIVERSITE D’AUVERGNE Drake University If you wanted to have $110. 25 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)t PV = $110. 25/(1+0. 05)2 = $100. 00

Present Value Interest Factor UNIVERSITE D’AUVERGNE Drake University PV = FV/(1+r)t 1/(1+r)t is called the Present Value Interest Factor (PVIFr, t) PVIF’s can be found in tables or calculated Interest Rate 4. 0 4. 5 5. 0 5. 5 Periods 0 1 0. 907029 2 3 OR 1/(1+. 05)2 = 0. 907029 Either way original equation can be rewritten: PV = FV/(1+r)t = FV(PVIFr, t)

UNIVERSITE D’AUVERGNE Calculating PV of a Single Sum Tables - Look up the PVIF 5%, 2 = 0. 9070 PV = 110. 25(0. 9070) =100. 00 Regular calculator -Calculate PVIF =1/ (1+r)t PV = 110. 25(0. 9070) = 100. 00 Financial Calculator 2 N 5 I - 110. 25 FV 0 PMT PV = 100. 00 Spreadsheet Excel command =PV(rate, nper, pmt, fv, type) Excel command =PV(. 05, 2, 0, 110. 25, 0)=100. 00 Drake University

Example UNIVERSITE D’AUVERGNE Drake University Assume you want to have $1, 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/(1+. 10)40=$22, 094. 93

What if you are currently 35? Or 45? UNIVERSITE D’AUVERGNE Drake University If you are 35 you would need PV = $1, 000/(1+. 10)30 = $57, 308. 55 If you are 45 you would need PV = $1, 000/(1+. 10)20 = $148, 643. 63 This process is called discounting (it is the opposite of compounding)

UNIVERSITE D’AUVERGNE Annuities Drake University Annuity: 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. Time 0 1 2 3 4 CF’s 100 100

Future Value of an Annuity UNIVERSITE D’AUVERGNE Drake University The FV of the annuity is the sum of the FV of each of its payments. Assume 6% a year Time 0 1 2 3 4 100 100 FV of 0 CF 100(1+. 06) =100. 00 100(1+. 06)1=106. 00 100(1+. 06)2=112. 36 100(1+. 06)3=119. 10 FV = 437. 4616

FV of An Annuity UNIVERSITE D’AUVERGNE Drake University This could also be written FV=100(1+. 06)0 +100(1+. 06)1 +100(1+. 06)2+ 100(1+. 06)3 FV=100[(1+. 06)0 +(1+. 06)1 +(1+. 06)2+(1+. 06)3] or for any n, r, payment, and t

FVIF of an Annuity (FVIFAr, t) UNIVERSITE D’AUVERGNE Drake University Just like for the FV of a single sum there is a future value interest factor of an annuity This is the FVIFAr, t FVannuity=PMT(FVIFAr, t)

UNIVERSITE D’AUVERGNE Calculation Methods Drake University Tables - Look up the FVIFA 6%, 4 = 4. 374616 FV = 100(4. 374616) =437. 4616 Regular calculator -Approximate FVIFA = [(1+r)t-1]/r FV = 100(4. 374616) =437. 4616 Financial Calculator 4 N 6 I 0 PV -100 PMT FV = 437. 4616 Spreadsheet Excel command =FV(rate, nper, pmt, pv, type) Excel command =FV(. 06, 4, 100, 0, 0)=437. 4616

Practice Problem UNIVERSITE D’AUVERGNE Drake University Your 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?

Present Value of an Annuity UNIVERSITE D’AUVERGNE Drake University The PV of the annuity is the sum of the PV of each of its payments Time 0 1 2 3 4 100 100 1 100/(1+. 06) =94. 3396 100/(1+. 06)2=88. 9996 100/(1+. 06)3=83. 9619 100/(1+. 06)4=79. 2094 PV = 346. 5105

PV of An Annuity UNIVERSITE D’AUVERGNE Drake University This could also be written PV=100/(1+. 06)1+100/(1+. 06)2+100/(1+. 06)3+100/(1+. 06)4 PV=100[1/(1+. 06)1+1/(1+. 06)2+1/(1+. 06)3+1/(1+. 06)4] or for any r, payment, and t

PVIF of an Annuity PVIFAr, t UNIVERSITE D’AUVERGNE Drake University Just like for the PV of a single sum there is a future value interest factor of an annuity This is the PVIFAr, t PVannuity=PMT(PVIFAr, t)

UNIVERSITE D’AUVERGNE Calculation Methods Drake University Tables - Look up the PVIFA 6%, 4 = 3. 465105 FV = 100(3. 465105) =346. 5105 Regular calculator -Approximate FVIFA PVIFA = [(1/r)-1/r(1+r)t] FV = 100(3. 465105) =346. 5105 Financial Calculator 4 N 6 I 0 FV -100 PMT PV = 346. 5105 Spreadsheet Excel command =PV(rate, nper, pmt, fv, type) Excel command =PV(. 06, 4, 100, 0, 0)=346. 5105

UNIVERSITE D’AUVERGNE Annuity Due Drake University The payment comes at the beginning of the period instead of the end of the period. Time CF’s 0 CF’s Annuity Due 100 2 3 4 100 Annuity 1 100 100 100 How does this change the calculation methods?

So what about the Shue Family? UNIVERSITE D’AUVERGNE Drake University The PV of the 30 equal payments of $3, 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?

Present Value of an Annuity Due UNIVERSITE D’AUVERGNE Drake University The PV of the annuity due is the sum of the PV of each of its payments Time 0 3. 6 M/(1+. 03)0=3. 6 M/(1+. 03)1=3. 559 M 3. 6 M/(1+. 03)2=3. 456 M 3. 6 M/(1+. 03)3=3. 355 M 3. 6 M/(1+. 03)29=1. 555 M PV =$ 74, 024, 333 1 2 3. 6 M 29

Wrong Choice? UNIVERSITE D’AUVERGNE Drake University 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?

Present Value of an Annuity Due UNIVERSITE D’AUVERGNE Drake University Time 3. 6 M 0 3. 6 M/(1+. 06)0=3. 6 M/(1+. 06)1=3. 459 M 3. 6 M/(1+. 06)2=3. 263 M 3. 6 M/(1+. 06)3=3. 078 M 3. 6 M/(1+. 06)29=676, 708 PV =$ 53, 499, 310 1 2 3. 6 M 3 3. 6 M 29 3. 6 M

What is the right rate? UNIVERSITE D’AUVERGNE Drake University 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%)

Intuition UNIVERSITE D’AUVERGNE Drake University Over 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, 370 If you can guarantee a 9% return you could buy an annuity that made 30 equal payments of $3. 6 Million for $41, 060, 370 and used the rest of the $60 million for something else….

FV an PV of Annuity Due UNIVERSITE D’AUVERGNE Drake University FVAnnuity 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, Therefore PVAnnuity Due = PVAnnuity(1+r)

UNIVERSITE D’AUVERGNE Uneven Cash Flow Streams Drake University What if you receive a stream of payments that are not constant? For example: Time 0 1 100 2 100 3 200 4 200 FV of CF 200(1+. 06)0=200. 00 200(1+. 06)1=212. 00 100(1+. 06)2=112. 36 100(1+. 06)3=119. 10 FV = 643. 4616

FV of An Uneven CF Stream UNIVERSITE D’AUVERGNE Drake University 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.

PV of an Uneven CF Streams UNIVERSITE D’AUVERGNE Drake University 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:

UNIVERSITE D’AUVERGNE Quick Review Drake University FV of a Single Sum FV = PV(1+r)t PV of a Single Sum PV = FV/(1+r)t FV and PV of annuities and uneven cash flows are just repeated applications of the above two equations

Perpetuity UNIVERSITE D’AUVERGNE Drake University Cash flows continue forever instead of over a finite period of time.

Growing Perpetuity UNIVERSITE D’AUVERGNE Drake University What 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:

Growing Perpetuity UNIVERSITE D’AUVERGNE Drake University However, in this case the cash flows grow at a constant rate which implies CF 1 = CF 0(1+g) CF 2 = CF 1(1+g) = [CF 0(1+g)](1+g) CF 3 =CF 2(1+g) = CF 0(1+g)3 CFt = CF 0(1+g)t

Growing Perpetuity UNIVERSITE D’AUVERGNE Drake University The PV is then Given as:

Semiannual Compounding UNIVERSITE D’AUVERGNE Drake University Often interest compounds at a different rate than the periodic rate. For example: 6% yearly compounded semiannual This implies that you receive 3% interest each six months This increases the FV compared to just 6% yearly

Semiannual Compounding An Example UNIVERSITE D’AUVERGNE Drake University You deposit $100 in an account that pays a 6% annual rate (the periodic rate) and interest compounds semiannually Time 0 1/2 1 3% 3% -100 106. 09 FV=100(1+. 03)=100(1. 03)2=106. 09

Effective Annual Rate UNIVERSITE D’AUVERGNE Drake University The effective Annual Rate is the annual rate that would provide the same annual return as the more often compounding EAR = (1+inom/m)m-1 m= # of times compounding period Our example EAR = (1+. 06/2)2 -1=1. 032 -1=. 0609