Time Value of Money Discounted Cash Flow Analysis MBA 220

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 Million Paid in 30 yearly payments of $3,666,666 $60 Million

Time Value of Money 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

How much will you have in one year? A Simple Example 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 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? 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

On the time line Time 0 1 2 Cash -$100 $105 110.25 Flow Beginning of year 1 End of Year 1 Beginning of Year 2 End of Year 2

Generalizing the Formula 110.25 = (100)(1+.05)2 This can be written more generally: Let t = The number of periods = 2 r = The interest rate per 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 FV = PV(1+r)t (1+r)t is called the Future Value Interest Factor (FVIFr,t) FVIF’s can be found in tables or calculated Interest Rate 4.0 4.5 5.0 5.5 Periods 1 2 3 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 Tables using the Future Value Interest Factor (FVIF) Regular Calculator Financial Calculator Spreadsheet

Plugging it into our equation Using the tables FVIF5%,2 = 1.1025 Plugging it into our equation FV = PV(FVIFr,t) FV = $100(1.1025) = $110.25

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

Financial Calculator Example On an HP-10B calculator you would enter: 2 N 5 I -100 PV 0 PMT FV and the screen shows 110.25

Spreadsheet Example Excel has a FV command =FV(rate,nper,pmt,pv,type) =110.25 note: Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year

FV = PV(1+r)t = PV(FVIFr,t) Practice Problem 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) FVIF0.4,5 = (1+0.04)5 = 1.216652 FV = $3,000(1+.04)5=$3,000(1.216652) FV = $3,649.9587

Calculating Present Value 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 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 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 2 3 0.907029 OR 1/(1+.05)2 = 0.907029 Either way original equation can be rewritten: PV = FV/(1+r)t = FV(PVIFr,t)

Calculating PV of a Single Sum Tables - Look up the PVIF PVIF5%,2 = 0.9070 PV = 110.25(0.9070) =100.00 Regular calculator -Calculate PVIF 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

How much would you need in the bank today if you were 25? Example Assume 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

What if you are currently 35? Or 45? If you are 35 you would need PV = $1,000,000/(1+.10)30 = $57,308.55 If you are 45 you would need PV = $1,000,000/(1+.10)20 = $148,643.63 This process is called discounting (it is the opposite of compounding)

Annuities 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 100 100 100 100

Future Value of an Annuity 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 4 100 100 100 100 FV of CF 100(1+.06)0=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 This could also be written or for any n, r, payment, and t

FVIF of an Annuity (FVIFAr,t) 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)

Calculation Methods Tables - Look up the FVIFA FVIFA6%,4 = 4.374616 FV = 100(4.374616) =437.4616 Regular calculator -Approximate FVIFA 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 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 The PV of the annuity is the sum of the PV of each of its payments Time 0 1 2 3 4 100 100 100 100 100/(1+.06)1=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 This could also be written or for any r, payment, and t

PVIF of an Annuity PVIFAr,t 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)

Calculation Methods Tables - Look up the PVIFA PVIFA6%,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

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

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?

Present Value of an Annuity Due The PV of the annuity due is the sum of the PV of each of its payments Time 0 1 2 3 29 3.6M 3.6M 3.6M 3.6M 3.6M 3.6M/(1+.03)0=3.6M 3.6M/(1+.03)1=3.559M 3.6M/(1+.03)2=3.456M 3.6M/(1+.03)3=3.355M 3.6M/(1+.03)29=1.555M PV =$ 74,024,333

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?

Present Value of an Annuity Due Time 0 1 2 3 29 3.6M 3.6M 3.6M 3.6M 3.6M 3.6M/(1+.06)0=3.6M 3.6M/(1+.06)1=3.459M 3.6M/(1+.06)2=3.263M 3.6M/(1+.06)3=3.078M 3.6M/(1+.06)29=676,708 PV =$ 53,499,310

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%)

Intuition 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 $3.6Million for $41,060,370 and used the rest of the $60 million for something else….

FV an PV of Annuity Due 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)

Uneven Cash Flow Streams What if you receive a stream of payments that are not constant? For example: Time 0 1 2 3 4 100 100 200 200 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 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 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:

Quick Review 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 Cash flows continue forever instead of over a finite period of time.

Growing Perpetuity 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 However, in this case the cash flows grow at a constant rate which implies CF1 = CF0(1+g) CF2 = CF1(1+g) = [CF0(1+g)](1+g) CF3 =CF2(1+g) = CF0(1+g)3 CFt = CF0(1+g)t

Growing Perpetuity The PV is then Given as:

Semiannual Compounding 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 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)(1+.03)=100(1.03)2=106.09

Effective Annual Rate 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 per period Our example EAR = (1+.06/2)2-1=1.032-1=.0609