drake drake university universite d’auvergne time value of money discounted cash flow analysis mba...

DrakeDRAKE UNIVERSITY

UNIVERSITED’AUVERGNE

Time Value of MoneyDiscounted Cash Flow Analysis

MBA 220


DrakeDrake University

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



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 orThe $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 ofFirst 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 1End of Year 1Beginning 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 MethodsFV = PV(1+r)t

Tables using the Future Value Interest Factor (FVIF)

Regular Calculator

Financial Calculator

Spreadsheet



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 BeforePlugging it into our equation

FV = PV(FVIFrr,t)

FV = $100(1.1025) = $110.25



Financial Calculator

Financial Calculators have 5 TVM keysN = Number of Periods = 2

I = interest rate per period =5PV = Present Value = $100PMT = 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) =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

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)

FV = $3,000(1+.04)5=$3,000(1.216652)FV = $3,649.9587

FVIF0.4,5 = (1+0.04)5 = 1.216652



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

t

t

t r)(1

r)PV(1

r)(1

FV



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 = 0.907029Either way original equation can be

rewritten:PV = FV/(1+r)t = FV(PVIFr,t)



Calculating PV of a Single Sum

Tables - Look up the PVIFPVIF5%,2 = 0.9070 PV = 110.25(0.9070) =100.00

Regular calculator -Calculate PVIFPVIF =1/ (1+r)t PV = 110.25(0.9070) = 100.00

Financial Calculator2 N 5 I - 110.25 FV 0 PMT PV = 100.00

SpreadsheetExcel command =PV(rate,nper,pmt,fv,type)Excel command =PV(.05,2,0,110.25,0)=100.00



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 needPV = $1,000,000/(1+.10)30 = $57,308.55

If you are 45 you would needPV = $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=100.00100(1+.06)1=106.00100(1+.06)2=112.36100(1+.06)3=119.10

FV = 437.4616



FV of An Annuity

This could also be writtenFV=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

4

1t

t4.06)(1100FV

t

1j

jtr)(1PMTFV



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)

t

1j

jtAnnuity r)(1PMTFV



Calculation Methods

Tables - Look up the FVIFAFVIFA6%,4 = 4.374616 FV = 100(4.374616) =437.4616

Regular calculator -Approximate FVIFAFVIFA = [(1+r)t-1]/r FV = 100(4.374616) =437.4616

Financial Calculator4 N 6 I 0 PV -100 PMT FV = 437.4616

SpreadsheetExcel 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?

)PMT(FVIFAr)(1PMTFV tr,

t

1j

jtAnnuity

226,566.4228)2,000(113.$FVAnnuity



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 100100 100 100

100/(1+.06)1=94.3396

100/(1+.06)2=88.9996

100/(1+.06)3=83.9619100/(1+.06)4=79.2094

PV = 346.5105



PV of An Annuity

This could also be writtenPV=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

t

1j

jAnnuity r)][1/(1PMTPV

4

1j

j.06)][1/(1100PV



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

t

1j

jAnnuity r)][1/(1CPV

This is the PVIFAr,t

PVannuity=PMT(PVIFAr,t)



Calculation Methods

Tables - Look up the PVIFAPVIFA6%,4 = 3.465105 FV = 100(3.465105) =346.5105

Regular calculator -Approximate FVIFAPVIFA = [(1/r)-1/r(1+r)t] FV = 100(3.465105) =346.5105

Financial Calculator4 N 6 I 0 FV -100 PMT PV = 346.5105

SpreadsheetExcel 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)1=3.559M

3.6M/(1+.03)2=3.456M

3.6M/(1+.03)3=3.355M3.6M/(1+.03)29=1.555MPV =$ 74,024,333

3.6M/(1+.03)0=3.6M



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)1=3.459M

3.6M/(1+.06)2=3.263M

3.6M/(1+.06)3=3.078M3.6M/(1+.06)29=676,708PV =$ 53,499,310

3.6M/(1+.06)0=3.6M



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 $41,060,370If 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….



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, ThereforePVAnnuity 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=212.00100(1+.06)2=112.36100(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.

t

1j

j-tjsCF'Uneven r)(1CFV



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:

t

1j

jjsCF'Uneven r)][1/(1CPV



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

t

1j

j-tjsCF'Uneven i)(1CFV

t

1j

jjsCF'Uneven r)][1/(1CPV

t

1j

jAnnuity r)][1/(1CPV

t

1j

j-tAnnuity r)(1CFV



Perpetuity

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

1j

jPerpetuity r)][1/(1CPV

1j

jr)][1/(1r1

rCPVPerpetuity



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:

n

1t

ttsCF'Uneven r)][1/(1CFPV



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:

1j

jj0Perpetuity Growing r)/(1g)(1CFPV

1j

j

j

01j

jj0 r)(1

g)(1CFr)/(1g)(1CF

1j0j

j

0 gr

g)(1CF

r)(1

g)(1CF



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 months

This increases the FV compared to just 6% yearly



Semiannual CompoundingAn Example

You deposit $100 in an account that pays a 6% annual rate (the periodic rate) and interest compounds semiannually

Time0 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

drake drake university universite d’auvergne time value of money discounted cash flow analysis mba...

Documents