drake drake university mba time value of money discounted cash flow analysis
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DrakeDRAKE UNIVERSITY
MBA
Time Value of MoneyDiscounted Cash Flow Analysis
DrakeDRAKE UNIVERSITY
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.
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
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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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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)
DrakeDRAKE UNIVERSITY
Fin 200Calculation Methods
FV = PV(1+i)n
Tables using the Future Value Interest Factor (FVIF)
Regular Calculator
Financial Calculator
Spreadsheet
DrakeDRAKE UNIVERSITY
Fin 200Using the tables
FVIF5%,2 = 1.1025
Plugging it into our equation
FV = PV(FVIFi,n)
FV = $100(1.1025) = $110.25
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
Fin 200Financial 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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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.
DrakeDRAKE UNIVERSITY
Fin 200Compounding at
Different Rates of Interest
$4,338.587%
$1,696.516%
$657.515%
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 20 40 60 80 100Years
Va
lue
($
)
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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)
DrakeDRAKE UNIVERSITY
Fin 200Calculation Methods
PV = FV/(1+i)n
Tables using the Present Value Interest Factor (PVIF)
Regular Calculator
Financial Calculator
Spreadsheet
DrakeDRAKE UNIVERSITY
Fin 200Using the tables
PVIF5%,2 = .9070
Plugging it into our equation
PV = FV(PVIFi,n)
PV = $110.25(0.9070) = $100.00
DrakeDRAKE UNIVERSITY
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
Plugging it into our equation
PV = FV(PVIFi,n)PV = $110.25(0.9070) = $100.00
DrakeDRAKE UNIVERSITY
Fin 200Financial Calculator
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
DrakeDRAKE UNIVERSITY
Fin 200Financial Calculator
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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?
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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)
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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?
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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.
DrakeDRAKE UNIVERSITY
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.
DrakeDRAKE UNIVERSITY
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))
DrakeDRAKE UNIVERSITY
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)?
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
Fin 200FVIFA
The FVIFA can be approximated by
FVIFA = [(1+i)n-1]/i=[FVIFi,n-1]/i
DrakeDRAKE UNIVERSITY
Fin 200Calculation Methods
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
tAnnuity i)][1/(1PMTPV
This is the PVIFAi,n
PVannuity=PMT(PVIFAi,n)
DrakeDRAKE UNIVERSITY
Fin 200PVIFA
The PVIFA can be approximated by:
i
PVIF1
PVIFA
ni,
ii)(1
11
ni,
n
DrakeDRAKE UNIVERSITY
Fin 200Calculation Methods
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
DrakeDRAKE UNIVERSITY
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
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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?
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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
DrakeDRAKE UNIVERSITY
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)
DrakeDRAKE UNIVERSITY
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
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Fin 200PV of Annuity Due
PVAnnuity Due There is one less period of discounting for each payment, Therefore
PVAnnuity Due = PVAnnuity(1+i)
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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
$60 Million
DrakeDRAKE UNIVERSITY
Fin 200So what option should the Shue
Family choose?
Lets assume their local banker tells them they can earn 3% interest each year on a savings account. Using that as the interest rate what is the PV of the 30 payments?
DrakeDRAKE UNIVERSITY
Fin 200
Present Value of an Annuity Due
The PV of the annuity is the sum of the PV of each of its payments
Time 0 1 2 3 29
3.6M 3.6M 3.6M 3.6M 3.6M
3.6M/(1+.03)1=3.559M
3.6M/(1+.03)2=3.456M
3.6M/(1+.03)3=3.355M3.6M/(1+.03)29=1.555MPV =$ 74,024,333
3.6M/(1+.03)0=3.6M
DrakeDRAKE UNIVERSITY
Fin 200Wrong Choice?
It would cost $74,024,333 to generate the same annuity payments each year, the Shue’s took the $60 Million instead of the 30 payments, did they made a mistake?Not necessarily, it depends upon the interest rate used to find the PV.The rate should be based upon the risk associated with the investment. What if we used 6% instead?
DrakeDRAKE UNIVERSITY
Fin 200Present Value of an Annuity
The PV of the annuity is the sum of the PV of each of its payments
Time 0 1 2 3 29
3.6M 3.6M 3.6M 3.6M 3.6M
3.6M/(1+.06)1=3.459M
3.6M/(1+.06)2=3.263M
3.6M/(1+.06)3=3.078M3.6M/(1+.06)29=676,708PV =$ 53,499,310
3.6M/(1+.06)0=3.6M
DrakeDRAKE UNIVERSITY
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%)
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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….
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
Fin 200Perpetuity
However the formula can be simplified:
1t
tPerpuity i)][1/(1PMTPV
/i1i)][1/(11t
t
i
PMTPVPerpuity
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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.
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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.
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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
DrakeDRAKE UNIVERSITY
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?
DrakeDRAKE UNIVERSITY
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
DrakeDRAKE UNIVERSITY
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
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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
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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
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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
DrakeDRAKE UNIVERSITY
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?
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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?
DrakeDRAKE UNIVERSITY
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
tAnnuity i)][1/(1PMTPV
n
1t
tnAnnuity i)(1PMTFV
DrakeDRAKE UNIVERSITY
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
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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
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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
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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%
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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%
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Fin 200Real Interest Rate
Since your purchasing power did not change – your real return was zero (therefore the real rate of interest is zero)
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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
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Fin 200Generally
Rate)Inflation (1
Rate)Interest Nom(1 Investment Value Real
RateInflation 1
RateInt Nom1 rateint real1
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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?
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Fin 200For Next Time
Try the practice problems – let me know if you would like to see any of them in class.