time value of money discounted cash flow analysis

DrakeDRAKE UNIVERSITY

MBA

Time Value of MoneyDiscounted Cash Flow Analysis


Fin 200Time Value of Money

A dollar received today is not worth the same amount as a dollar to be received in the future WHY?

You should receive Interest on the dollar received today if it is

invested.


Fin 200A 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 $100 today has a Future Value of $105or

The $105 next year has a Present Value of $100


Fin 200Using 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


Fin 200

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


Fin 200On the time line

Time 0 1 2

Cash -$100 $105 $110.25 Flow

Beginning of year 1

End of Year 1Beginning of

Year 2

End of Year 2


Fin 200Generalizing the Formula

110.25 = (100)(1+.05)2

This can be written more generally: Let t = The number of periods = 2 i = The interest rate per period =.05 PV = The Present Value = $100 FV = The Future Value = $110.25

FV = PV(1+i)n

($110.25) = ($100)(1 + 0.05)2

This works for any combination of n, i, and PV


Fin 200

Future Value Interest Factor

FV = PV(1+i)n (1+i)n is called the Future Value Interest Factor (FVIFi,n)

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+i)n = PV(FVIFi,n)


Fin 200Calculation Methods

FV = PV(1+i)n

Tables using the Future Value Interest Factor (FVIF)

Regular Calculator

Financial Calculator

Spreadsheet


Fin 200Using the tables

FVIF5%,2 = 1.1025

Plugging it into our equation

FV = PV(FVIFi,n)

FV = $100(1.1025) = $110.25


Fin 200Using a Regular Calculator

Calculate the FVIF using the yx key(1+.05)2=1.1025

Proceed as BeforePlugging it into our equation

FV = PV(FVIFi,n)

FV = $100(1.1025) = $110.25


Fin 200Financial Calculator

Financial Calculators have 5 TVM keysN = Number of Periods = 2I = interest rate per period =5PV = Present Value = -$100FV = Future Value =?PMT = Payment per period = 0

After entering the portions of the problem you know, the calculator will provide the answer



Example

On an HP-10B calculator you would enter:

2 N 5 I -100 PV 0 PMT FV

and the screen shows 110.25


Fin 200Spreadsheet Example

Excel has a FV command Excel command =FV(rate,nper,pmt,pv,type) =FV(0.05,2,0,100,0) =$110.25

notes: The inputs needed are basically the same as

on the financial calculator Type refers to whether the payment is at the

beginning (type =1) or end (type=0) of the year


Fin 200Practice 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+i)n = PV(FVIFi,n)

FV = $3,000(1+0.04)5=$3,000(1.216652)FV = $3,649.9587

FVIF0.04,5 = (1+0.04)5 = 1.216652


Fin 200Compounding Interest

Assume that 100 years ago your ancestors invested $5 at 6%. In the first year there would have been $0.30 in interest.If you took out the interest each year you would have received a total of $0.30(100) or $30 in interestHow much would the $5 be worth if the interest reinvests?

5(1.06)100 = $1,696.51


Fin 200Compounding Interest

Leaving the interest in the account allows you to earn interest upon the interest. The impact of the interest compounds or increases over time.The more periods interest is allowed to accumulate, the greater the impact of the compounding will be.


Fin 200Compounding at

Different Rates of Interest

$4,338.587%

$1,696.516%

$657.515%

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 20 40 60 80 100Years

Va

lue

($

)


Fin 200Calculating Present Value

We just showed that FV=PV(1+i)n

This can be rearranged to find PV given FV, i and n.Divide both sides by (1+i)n

n

n

n i)(1

i)PV(1

i)(1

FV

which leaves PV = FV/(1+i)n


Fin 200Example

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+i)n

$110.25

(1+0.05)2PV = = $100


Fin 200

Present Value Interest Factor

PV = FV/(1+i)n 1/(1+i)nis called the Present Value Interest Factor (PVIFi,n)

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+i)n = FV(PVIFi,n)



PV = FV/(1+i)n

Tables using the Present Value Interest Factor (PVIF)

Regular Calculator

Financial Calculator

Spreadsheet


Fin 200Using the tables

PVIF5%,2 = .9070


PV = FV(PVIFi,n)

PV = $110.25(0.9070) = $100.00


Fin 200Using a Regular Calculator

Calculate the PVIF using the yx key(1/(1+.05))2=.9070

Make sure to divide first then square Proceed as Before


PV = FV(PVIFi,n)PV = $110.25(0.9070) = $100.00



Financial Calculators have 5 TVM keysN = Number of Periods = 2I = interest rate per period =5PV = Present Value = ?FV = Future Value =$110.25PMT = Payment per period = 0

After entering the portions of the problem you know, the calculator will provide the answer



Example

On an HP-10B calculator you would enter:

2 N 5 I 110.25 FV 0 PMT PV

and the screen shows -$100.00


Fin 200Spreadsheet Example

Excel has a PV command Excel command =PV(rate,nper,pmt,fv,type) =FV(0.05,2,0,110.25,0) =-$100.00

notes: The inputs needed are basically the same as

on the financial calculator Type refers to whether the payment is at the

beginning (type =1) or end (type=0) of the year


Fin 200Example

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?


Fin 200Put the problem on a time

line

Age 25 35 45 55 65Years 0 10 20 30 40

PV = 1,000,000/(1+.10)40=1,000,000(.02209493) PV = $22,094.93

$1,000,000

PV?

PVIF40,10% = 1/(1.1)40 = 0.02209493


Fin 200

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)


Fin 200Example 2

You decide to attend law school after completing your MBA. You believe that you will need $100,000 when you start Law School in three years. How much would you need in the bank today at 7% to have enough for tuition?

$100,000/(1.07)3 = $81,629.7878

PVIF7%,3 =.8163$100,000(.8163)

=$81,630


Fin 200PV and FV Practice Problem

You hope to buy a new car when you graduate in two years, you believe the car will cost $25,000. If you can earn 9% each year, how much would you need to put in the bank today to be able to buy the car in two years?

PV = $25,000/1.092

PV = $21,041.99

FV or PV? Number of Periods? Interest Rate?


Fin 200Solving for the interest rate

PV = FVt/(1+i)n or PV(1+i)n=FV

Rearrange the above equationFV/PV = (1+i)tn

(FV/PV)1/n = 1+i(FV/PV)1/n-1 = i


Fin 200An Example

What interest rate would you need to double your investment of $1,000 over the next five years?

2,000 = 1,000(1+i)5

2,000/1,000 =2 = (1+i)5

2(1/5)= [(1+i)5](1/5)=1+i1.1468 – 1=.1468


Fin 200Rule of 72 – A shortcut

How long does it take for a sum of money to double in value from compounding at a given rate?If the interest rate is between 5% and 20% then the sum will double in approximately 72/r%If you are earning 8% interest your money would double in approximately 72/8 = 9 years


Fin 200An Introduction to determining

the “Correct” Interest Rate

So far we have just assumed a level of interest rate for our problems.How should the correct interest rate be determined?Interest rates are also linked to the level of risk (we will see this in detail later in the semester). Generally, greater risk results in greater return.


Fin 200Opportunity Cost

An opportunity cost represents the cost of the best foregone alternative.When calculating Time Value problems the correct rate combines the idea of risk and return and opportunity cost. Opportunity Cost Rate – the rate of return on the best available alternative investment of equal risk.


Fin 200Solving for the number of

periods

FV = PV(1+i)n

Rearrange FV/PV = (1+i)n

Take the natural log of both sidesln(FV/PV) = n(ln(1+i))n = ln(FV/PV)/(ln(1+i))


Fin 200Questions

1. What happens to the PV of a future sum as the level of interest rate (discount rate) increases (or decreases)?

2. What happens to the FV as the interest rate increases (or decreases)?

3. What happens to the PV of a future sum if the number of periods increases (or decreases)?

4. What happens to the FV of a current sum if the number of periods increases (or decreases)?


Fin 200Annuities

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


Fin 200Future 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


Fin 200FV 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, i, payment, and t

4

1t

t4.06)(1100FV

n

1t

t-ni)(1PMTFV


Fin 200FVIF 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 FVIFAi,n

FVannuity=PMT(FVIFAi,n)

n

1t

tnAnnuity i)(1PMTFV


Fin 200FVIFA

The FVIFA can be approximated by

FVIFA = [(1+i)n-1]/i=[FVIFi,n-1]/i



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

=437.4616 Regular calculator -Approximate FVIFA

FVIFA = [(1+i)t-1]/i 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


Fin 200Practice 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(FVIFAi)(1PMTFV ni,

n

1t

tnAnnuity


Fin 200Put the problem on a time

line

Age 35 36 37 64 65Years 0 1 2 29 30

2,000 2,000 2,000 2,000

FVIFA30,8% = [(1+0.08)30-1]/0.08 =113.28

2$226,566.428)2,000(113.$FVAnnuity


Fin 200Alternative Solution

Methods

Financial Calculator30 N 8 I 0 PV -2000 PMT FV = $226,533.42

SpreadsheetExcel command

=FV(rate,nper,pmt,pv,type)Excel command =FV(.08,30,0,-2000,0)=$226,566.42


Fin 200Practice Problem 2

Assume you want to have $1,000,000 for retirement at age 65. If you deposit the same amount each year and are 20 years old today how much will you need to deposit each year if you earn 9%?

1,000,000 = PMT(FVIFA45,9%)1,000,000 = PMT(525.8587345)

$1,901.6514What if you wait until you are 30 to start saving?

1,000,000 = PMT(FVIFA35,9%)PMT = $4,635.83


Fin 200Present 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


Fin 200PV 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 i, payment, and t

n

1t

tAnnuity i)][1/(1PMTPV

4

1t

t.06)][1/(1100PV


Fin 200PVIF of an Annuity PVIFAr,t

Just like for the PV of a single sum there is a present value interest factor of an annuity

n

1t


This is the PVIFAi,n

PVannuity=PMT(PVIFAi,n)


Fin 200PVIFA

The PVIFA can be approximated by:

i

PVIF1

PVIFA

ni,

ii)(1

11

ni,

n



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

Regular calculator -Approximate FVIFAPVIFA = [(1/i)-1/i(1+i)n] 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


Fin 200Example: Solving for the

Required Annuity Payment

Your grandfather has retired, he currently has $2,000,000 saved to finance his retirement. How much could he spend each of the next 20 years if his deposits earn 7% annually?

2,000,000 = PMT(PVIFA20,7%)2,000,000 = PMT(10.594)

188,785.85


Fin 200Annuity 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?


Fin 200Future Value of an Annuity Due

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)1=106.00

100(1+.06)2=112.36

100(1+.06)3=119.10100(1+.06)4=126.25

FV = 463.7093


Fin 200FV of Annuity Due

Compare the annuity due to a regular annuity with the same number of payments and interest rate. There is one more period of compounding for each payment, Therefore:

FVAnnuity Due = FVAnnuity(1+i)


Fin 200Present Value of an Annuity Due

The PV of the annuity due is the sum of the PV of each of its payments

Time0 1 2 3 4

100 100 100 100

100/(1+.06)0=100100/(1+.06)1=94.3396

100/(1+.06)2=88.9996100/(1+.06)3=83.9619

PV = 367.3011


Fin 200PV of Annuity Due

PVAnnuity Due There is one less period of discounting for each payment, Therefore

PVAnnuity Due = PVAnnuity(1+i)


Fin 200Which 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


Fin 200So what option should the Shue

Family choose?

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?


Fin 200

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+.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


Fin 200Wrong 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?


Fin 200Present Value of an Annuity


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


Fin 200What is the right rate?

Remember the correct rate is based upon the opportunity cost.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 at the time of the winnings yielded 5.02%)


Fin 200Intuition

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….


Fin 200Perpetuity

A perpetuity is a constant cash flow that is received forever. The PV of a perpetuity would be:

1t

tPerpuity i)][1/(1PMTPV


Fin 200Perpetuity

However the formula can be simplified:

1t

tPerpuity i)][1/(1PMTPV

/i1i)][1/(11t

t

i

PMTPVPerpuity


Fin 200Amortization of a Loan

You want to borrow 1,000 and pay it off over three years. Assume that you are charged 6% each year. How much will your payment be?

$1,000 = PV PMT =????$1,000 = PMT (PVIFA6%,3) =

$1,000 = PMT(2.67)PMT = $374.11


Fin 200Amortization

You pay a total of $374.11(3) = $1,122.33A portion of each payment represents interest charges, the portion of the payment that is interest changes with each paymentYou can find the amount of interest by multiplying the balance at the beginning of the period by the interest rate. At the beginning of the loan, the balance is $1,000 so there is $1,000(.06) = 60 in interest.


Fin 200Amortization

The remainder of the payment pays off principal.

$374.11 - $60=$314.11The remaining principal at the end of the period will then be

$1,000 – $314.11 = $685.89The process then repeats itself every period until the original balance of the loan is paid off.


Fin 200Amortization

Beginning Ending

Year Balance Payment Interest Principal Balance1 1,000 374.11 60.00 314.11 685.89

2 685.89 374.11 41.15 332.96 352.93

3 352.93 374.11 21.18 352.93 0.00


Fin 200Credit Card Debt

Assume that you currently have a $5,183.66 balance on your credit card, and it charges you 18% interest every year (1.5% in interest each month).The Credit Card company require you to make a minimum monthly payment of $80 each month, how long do you think it would take to pay off the balance?


Fin 200Credit Card Problem

Your PV is $5,183.66You pay $80 each month and have a monthly interest rate of 1.5%.You are solving for the number of periods it would take to pay off the debt, (in other words how many months of paying $80 each month has a PV of $5,183.66


Fin 200Amortization Credit Card

Debt

Beginning Ending

Month Balance Payment Interest Principal Balance

1 $5,183.66 80 $77.75 $2.25 $5,181.41

2 $5,181.41 80 $77.72 $2.28 $5,179.13

240 $78.86 80 $1.19 $78.81 0.05


Fin 200Uneven 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=112.36

100(1+.06)3=119.10 FV = 643.4616


Fin 200FV 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.

n

1t

tntsCF'Uneven i)(1CFFV


Fin 200PV 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:

n

1t

ttsUnevenCF' i)][1/(1CFPV


Fin 200A Second Example

Ivan “Pudge” Rodriquez signed a contract reported to be worth $40 Million to play baseball over the next four years for the Detroit Tigers. The contract pays Pudge $7M this year, $8 M next year, $11M in each of the following years plus $3M extra the last year if the team does not retain him for another year. What is the PV of his contract?


Fin 200PV of Playing Baseball

Given an interest rate of 5% Pudge’s contract is only worth $34.9 MillionWith an interest rate of 10% Pudge’s contract is only worth $30 MillionWhich is the best way to value the contract?


Fin 200Quick Review

FV of a Single Sum FV = PV(1+i)n

PV of a Single Sum PV = FV/(1+i)n

FV and PV of annuities and uneven cash flows are just repeated applications of the above two equations

n

1t

tntsCF'Uneven i)(1CFFV

n

1t

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

n

1t


n

1t

tnAnnuity i)(1PMTFV


Fin 200Semiannual 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


Fin 200Semiannual 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


Fin 200Effective 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


Fin 200Inflation

We have ignored the impact of inflation It is possible to adjust the interest rate for the impact of inflationAssume you have $100 today and after investing it for one year you have $116.60.What return did you receive?

16.6%


Fin 200Inflation

Assume that inflation was 6% over the same time as your investment, How much did your purchasing power increase?

(1+r)(106) = 116.6r = .10 = 10%


Fin 200Real Interest Rate

Since your purchasing power did not change – your real return was zero (therefore the real rate of interest is zero)


Fin 200Purchasing Power Example

Jared eats a $5 subway sandwich for lunch every day, he has budgeted $100 each month ($100/5 = 20 sandwiches).If he puts $100 away to spend in one year in an account earning 16.6% and the price of sandwiches increases by 5%, how many sandwiches can he buy each month in one year?

116.6/5.25 = 22.21 vs. 116.6 /5 = 23.32 without the price increase


Fin 200Generally

Rate)Inflation (1

Rate)Interest Nom(1 Investment Value Real

RateInflation 1

RateInt Nom1 rateint real1


Fin 200The Fisher Effect

Let R = The nominal rate of interestr = the real rate of interest

h = the inflation rate

The Fisher Effect States:1+R = (1+r)(1+h)

OrR = r + h + (rh)

Which interest rate is more important to investors?


Fin 200For Next Time

Try the practice problems – let me know if you would like to see any of them in class.

time value of money discounted cash flow analysis

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