Managerial Finance
MBA 643, Online
Week 1
Basic Definitions
• Present Value – earlier money on a time line
• Future Value – later money on a time line
• Interest rate – “exchange rate” between earlier money and later money
•
•
•
•
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Discount rate
Cost of capital
Opportunity cost of capital
Required return
Time Value of Money
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Time Value of Money
Future Value
Present Values
Discount Rate
Discounted Cash Flow Valuation
Future and Present Values of Multiple Cash
Flows
Valuing Level Cash Flows
Annuities and Perpetuities
The Effect of Compounding
Loan Types and Loan Amortization
Future Values
Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year?
• Interest = 1, 000 × .05 = 50
• Value in one year = principal + interest = 1, 000 + 50 = 1, 050
• Future Value (FV) = 1, 000(1 + .05) = 1, 050
Suppose you leave the money in for another year. How much will you have two years from
now?
FV = 1, 000 × 1.05 × 1.05 = 1, 000 × 1.052 = 1, 102.50
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Time Value of Money
Future Value
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Future Values
General Formula
We have the formula you saw before:
FV = PV(1 + r )t
where
FV future value
PV present value
r period interest rate, expressed as a decimal
t number of periods
The future value interest factor is the factor (1 + r )t
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Time Value of Money
Future Value
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Compounding
The process of reinvesting money and any interest earned for more than one period is called
compounding. Compounding the interest means earning interest on interest.
Consider the previous example:
• FV with simple interest = 1,000 + 50 + 50 = 1,100
• FV with compound interest = 1,102.50
• The extra 2.50 comes from the interest of .05 × 50 = 2.50 earned on the first interest
payment
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Time Value of Money
Future Value
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Compounding
How important is compounding?
The effect of compounding is small for a small number of periods, but increases as the number
of periods increases.
In our example, with two periods we have a compound interest of $2.50 against a simple
interest of $100 = $50 + $50. The compound interest is very small when compared to the
simple interest.
On the other hand, if we invest for 50 periods, the total simple interest amounts to $2, 500.00,
while the compound interest is $7, 967.40.
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Time Value of Money
Future Value
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Future Value as a general formula
In fact the Future Value formula is applicable to more than just money quantities.
For example, suppose your company expects to increase unit sales of widgets by 15% per year
for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do
you expect to sell in 5 years?
FV = 3, 000, 000 × (1 + .15)5 = 6, 034, 072
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Time Value of Money
Future Value
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Time Value of Money
Future Value
Present Values
Discount Rate
Discounted Cash Flow Valuation
Future and Present Values of Multiple Cash
Flows
Valuing Level Cash Flows
Annuities and Perpetuities
The Effect of Compounding
Loan Types and Loan Amortization
Present Value
How much do we have to invest today to have some amount in the future?
FV = PV(1 + r )t
Rearrange to solve for
PV =
FV
(1 + r )t
When we talk about discounting, we mean finding the present value PV of some future
amount.
By the “value” of something, we mean the present value unless we specifically indicate that we
want the future value.
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Time Value of Money
Present Values
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Present Value
Example
You want to begin saving for your daughter’s college education and you estimate that she will
need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do
you need to invest today?
PV =
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150, 000
= $40, 540.34
(1 + .08)17
Time Value of Money
Present Values
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Present Value
Important Relationship I
For a given interest rate – the longer the time period, the lower the present value.
Example:
What is the present value of $500 to be received in 5 years? 10 years? The discount rate is
10%
• 5 years: PV = 500/(1 + .1)5 = 310.46
• 10 years: PV = 500/(1 + .1)10 = 192.77
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Time Value of Money
Present Values
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Present Value
Important Relationship II
For a given time period – the higher the interest rate, the smaller the present value
Example:
What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?
• 10% rate: PV = 500/(1 + .1)5 = 310.46
• 15% rate: PV = 500/(1 + .15)5 = 248.59
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Time Value of Money
Present Values
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Present Value
All the moving parts
The PV formula
PV =
FV
(1 + r )t
has four parts: PV, FV, r , and t
If we know any three, we can solve for the fourth. For example,
r=
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Time Value of Money
FV
PV
1
t
−1
Present Values
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Time Value of Money
Future Value
Present Values
Discount Rate
Discounted Cash Flow Valuation
Future and Present Values of Multiple Cash
Flows
Valuing Level Cash Flows
Annuities and Perpetuities
The Effect of Compounding
Loan Types and Loan Amortization
Discount Rate
The quantity used in the PV formula
1
(1 + r )t
is usually called the discount factor and the rate r the discount rate.
Calculating the present value of a future cash flow to determine its value today is called the
discounted cash flow valuation (DCF).
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Time Value of Money
Discount Rate
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Discount Rate
Examples
You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today.
What is the implied rate of interest?
r=
1, 200
1, 000
1/5
− 1 = .03714 = 3.714%
Suppose you are offered an investment that will allow you to double your money in 6 years.
You have $10,000 to invest. What is the implied rate of interest?
r=
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20, 000
10, 000
1/6
Time Value of Money
− 1 = .122462 = 12.25%
Discount Rate
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Finding the Number of Periods
Start with the basic equation
FV = PV(1 + r )t
and solve for t (remember your logs1 ):
t = log
FV
PV
/ log(1 + r )
You can use the financial keys on the calculator as well; just remember the sign convention.
1
here log means the natural log function. In Excel, this is given by LN
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Time Value of Money
Discount Rate
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Finding the Number of Periods
Example
You want to purchase a new car, and you are willing to pay $20,000. If you can invest at 10%
per year and you currently have $15,000, how long will it be before you have enough money to
pay cash for the car?
20, 000
/ log(1 + .1) = 3.02 years
t = log
15, 000
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Time Value of Money
Discount Rate
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Finding the Number of Periods
Example
Suppose you want to buy a new house. You currently have $15,000, and you figure you need
to have a 10% down payment plus an additional 5% of the loan amount for closing costs.
Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year.
How long will it be before you have enough money for the down payment and closing costs?
How much do you need to have in the future?
• Down payment = .1 × 150, 000 = 15, 000
• Closing costs = .05 × (150, 000 − 15, 000) = 6, 750
• Total needed = 15, 000 + 6, 750 = 21, 750
Compute the number of periods using the formula:
t = log(21, 750/15, 000)/ log(1 + .075) = 5.14 years
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Time Value of Money
Discount Rate
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Using Excel for Time Value of Money Computations
Use the following formulas for TVM calculations:
• FV(rate,nper,pmt,pv) – future value
• PV(rate,nper,pmt,fv) – present value
• RATE(nper,pmt,pv,fv) – discount rate
• NPER(rate,pmt,pv,fv) – number of periods
where
rate interest rate
nper number of periods
pmt equals zero
pv present value
fv future value
The formula icon in Excel is very useful when you can’t remember the exact formula!
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Time Value of Money
Discount Rate
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Time Value of Money
Future Value
Present Values
Discount Rate
Discounted Cash Flow Valuation
Future and Present Values of Multiple Cash
Flows
Valuing Level Cash Flows
Annuities and Perpetuities
The Effect of Compounding
Loan Types and Loan Amortization
Multiple Cash Flows
Future Value
You think you will be able to deposit $4,000 at the end of each of the next three years in a bank
account paying 8 percent interest.
You currently have $7,000 in the account.
How much will you have in three years? How much will you have in four years?
Find the value at year 3 of each cash flow and add them together:
• Today (year 0): FV = 7000 × (1.08)3 = 8, 817.98
• Year 1: FV = 4, 000 × (1.08)2 = 4, 665.60
• Year 2: FV = 4, 000 × (1.08) = 4, 320
• Year 3: FV = 4, 000
Thus, the total value in 3 years is 8, 817.98 + 4, 665.60 + 4, 320 + 4, 000 = 21, 803.58.
The value at year 4 is simply 21, 803.58 × (1.08) = 23, 547.87
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Discounted Cash Flow Valuation
Future and Present Values
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Multiple Cash Flows
Future Value
Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9%
annually, how much will you have in two years?
FV = 500 × (1.09)2 + 600 × (1.09) = 1, 248.05
How much will you have in 5 years if you make no further deposits?
First way:
FV = 500 × (1.09)5 + 600 × (1.09)4 = 1, 616.26
Second way – use the value at year 2:
FV = 1, 248.05 × (1.09)3 = 1, 616.26
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Discounted Cash Flow Valuation
Future and Present Values
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Multiple Cash Flows
Future Value
To calculate the future value of multiple cash flows, we can roll them forward one year at a
time using the interest rate, or, in a more direct way, look at how many years we need to
compound each cashflow and apply the interest rate that many times.
today
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1
2
Discounted Cash Flow Valuation
n
Future and Present Values
time
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Multiple Cash Flows
Present Value
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years, and $3,000
in three years. If you want to earn 10% on your money, how much would you be willing to pay?
• PV = 1000/(1.1)1 = 909.09
• PV = 2000/(1.1)2 = 1, 652.89
• PV = 3000/(1.1)3 = 2, 253.94
• PV = 909.09 + 1, 652.89 + 2, 253.94 = 4, 815.92
or, looking at the PV of cash flows at each period:
• Present Value of at year 3: PV3 = 3, 000
• Present Value of at year 2: PV2 = 3, 000/1.1 + 2, 000 = 4, 727.27
• Present Value of at year 1: PV1 = 4, 727.27/1.1 + 1, 000 = 5, 297.52
• Present Value of cash flows today: PV = 5, 297.52/1.1 = 4, 815.93
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Discounted Cash Flow Valuation
Future and Present Values
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Multiple Cash Flows
Present Value
That is, just like for Future Value, we calculate the present value of multiple cash flows by
rolling them backwards one year at a time using the interest rate, or, in a more direct way,
discount by the amount of years.
today
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1
2
Discounted Cash Flow Valuation
n
Future and Present Values
time
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Multiple Cash Flows
Present Value
You are offered the opportunity to put some money away for retirement. You will receive five
annual payments of $25,000 each beginning in 40 years. How much would you be willing to
invest today if you desire an interest rate of 12%?
• Cash flows in years 0 – 39 are zero.
• Cash flows in years 40 – 44 are $25K.
Thus,
PV = sum of all Present Values = $1, 084.71
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Discounted Cash Flow Valuation
Future and Present Values
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Time Value of Money
Future Value
Present Values
Discount Rate
Discounted Cash Flow Valuation
Future and Present Values of Multiple Cash
Flows
Valuing Level Cash Flows
Annuities and Perpetuities
The Effect of Compounding
Loan Types and Loan Amortization
Annuities and Perpetuities
An annuity is a finite series of equal payments that occur at regular intervals. If the first
payment occurs at the end of the period, it is called an ordinary annuity. If the first payment
occurs at the beginning of the period, it is called an annuity due.
A perpetuity is an infinite series of equal payments.
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Present Value
For an ordinary annuity, instead of having to sum multiple cash-flows, we can use the following
formula
1
1 − (1+r
)t
PV = C ×
r
where
C amount paid every period
t periods
r interest rate
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Future Value
The formula on the previous slide can be reworked to give the future value:
FV = C ×
(1 + r )t − 1
r
where
C amount paid every period
t periods
r interest rate
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Valuing Level Cash Flows
Annuities and Perpetuities
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Perpetuities
Present Value
For perpetuities there is no Future Value computation (since it’s a perpetual), but we can
compute the Present Value as:
C
PV =
r
where
C amount paid every period
r interest rate
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Present Value
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in
equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%,
how much is the sweepstakes actually worth today?
Using the present value formula for an annuity,
PV = 333, 333.33 ×
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1
1 − 1.05
30
.05
Valuing Level Cash Flows
!
= $5, 124, 150.29
Annuities and Perpetuities
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Annuities
Present Value
You are ready to buy a house, and you have $20,000 for a down payment and closing costs.
Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000,
and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your
monthly income.
The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a
30-year fixed rate loan.
• How much money will the bank loan you?
• How much can you offer for the house?
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Present Value
• Bank loan
• Monthly income = 36, 000/12 = 3, 000
• Maximum payment = .28 × 3, 000 = 840
30×12
= 140, 105
• PV = 840 1−1/1.005
.005
• Total Price
• Closing costs = .04 × 140, 105 = 5, 604
• Down payment = 20, 000 − 5, 604 = 14, 396
• Total Price = 140, 105 + 14, 396 = 154, 501
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Finding the Payment Amount
Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year,
compounded monthly (8%/12 = .66667% per month).
If you take a 4 year loan, what is your monthly payment?
20, 000 = C ×
1
1 − 1.0066667
48
.0066667
!
Solving leads to C = $488.26.
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Number of Payments
Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42.
How long before you pay off the loan?
2, 000 = 734.42 × 1 − 1/1.05t /.05
Solving, .136161869 = 1 − 1/1.05t , which leads to 1.157624287 = 1.05t .
Finally,
t = log(1.157624287)/ log(1.05) = 3 years
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Finding the rate
Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for
60 months. What is the monthly interest rate?
10, 000 = 207.58 × 1 − 1/(1 + r )60 /r
We can use the RATE in Excel, use the Solver feature, or even proceed by trial-and-error:
• Choose an interest rate and compute the PV of the payments based on this rate
• Compare the computed PV with the actual loan amount
• If the computed PV > loan amount, then the interest rate is too low
• If the computed PV < loan amount, then the interest rate is too high
• Adjust the rate and repeat the process until the computed PV and
the loan amount are equal
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities
Future Values
Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA at the end of
every year.
If the interest rate is 7.5%, how much will you have in 40 years?
FV = 2, 000(1.07540 − 1)/.075 = $454, 513.04
What if you deposit the $2,000 at the beginning of every year, starting today?
The value of an annuity due is such that
Annuity due value = Ordinary annuity value × (1 + r )
This is because every cash flow for an annuity due has one more year to earn interested when compared
to those of an ordinary annuity.
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Valuing Level Cash Flows
Annuities and Perpetuities
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Annuities and Perpetuities
Summary of Formulas
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Valuing Level Cash Flows
Annuities and Perpetuities
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Example of a Perpetuity
Preferred Stock
Preferred stock (or preference stock) is a type of stock corporations sell to investors. The
investor is typically promised a fixed cash dividend every period forever2 . This dividend must
be paid before any dividend can be paid to regular stockholders.
Because the PV of a perpetuity is PV = Cr , if for example a company wants to raise $10M by
issuing preferred stock, and the required return from investors is 5% annually, then it needs to
pay out dividends in the amount of
C = PV ×r = 10, 000, 000 × 5% = $500, 000
2
there are some less common issues of preferred stock with long maturities that are not perpetual
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Valuing Level Cash Flows
Annuities and Perpetuities
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Growing Annuities
Present Value
Some annuities have payments that grow over time at a fixed rate g. The present value for
such annuities is
t
1 − 1+g
1+r
PV = C ×
r −g
where
C amount paid every period
t periods
r interest rate
g rate at which payments grow
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Valuing Level Cash Flows
Annuities and Perpetuities
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Growing Perpetuity
Present Value
As for annuities, the present value formula for a Perpetuity with growing payments at a rate g
is
C
PV =
r −g
where
C amount paid every period
r interest rate
g rate at which payments grow
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Valuing Level Cash Flows
Annuities and Perpetuities
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Growing Annuities
Example
A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the
annual payment by three-percent each year.
What is the present value at retirement if the discount rate is 10 percent?
1−
PV = 20, 000 ×
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1+.03
1+.10
40
.10 − .03
Valuing Level Cash Flows
= $265, 121.57
Annuities and Perpetuities
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Effective Annual Rate (EAR)
The Effective Annual Rate (EAR) is the actual rate paid (or received) after accounting for
compounding that occurs during the year.
If you want to compare two alternative investments with different compounding periods, you
need to compute the EAR and use that for comparison.
For example, if you are paid 1% every month then in one year $100 would grow to
$100 × (1 + .01)12 = $112.68 = $100 + $12.68
Thus, your EAR is 12.68%.
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The Effect of Compounding
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Annual Percentage Rate (APR)
The Annual Percentage Rate (APR) is the annual rate that is quoted by law. By definition,
APR = period rate × number of periods per year
Consequently, to get the period rate we just rearrange the APR equation:
period rate =
APR
number of periods per year
Note that you should never divide the effective annual rate by the number of periods per year
– it will not give you the period rate.
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The Effect of Compounding
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Annual Percentage Rate (APR)
Examples
What is the APR if the monthly rate is .5%?
.5% × 12 = 6%
What is the APR if the semiannual rate is .5%?
.5% × 2 = 1%
What is the monthly rate if the APR is 12% with monthly compounding?
12%/12 = 1%
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The Effect of Compounding
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Annual Percentage Rate (APR)
Things to Remember
You always need to make sure that the interest rate and the time period match.
• If you are looking at annual periods, you need an annual rate.
• If you are looking at monthly periods, you need a monthly rate.
If you have an APR based on monthly compounding, you have to use monthly periods for
lump sums, or adjust the interest rate appropriately if you have payments other than monthly.
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The Effect of Compounding
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Effective Annual Rate (EAR)
Examples
Suppose you can earn 1% per month on $1 invested today.
• What is the APR?
1% × 12 = 12%
• How much are you effectively earning?
FV = 1 × (1.01)12 = $1.1268
• Rate = (1.1268 − 1)/1 = .1268 = 12.68%
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The Effect of Compounding
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Effective Annual Rate (EAR)
Examples
Suppose if you put it in another account, you earn 3% per quarter.
• What is the APR?
3% × 4 = 12%
• How much are you effectively earning?
FV = 1 × (1.03)4 = 1.1255
• Rate = (1.1255 − 1)/1 = .1255 = 12.55%
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The Effect of Compounding
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EAR and APR
The following formula allows one to convert APR into EAR:
EAR = 1 +
APR
m
m
−1
where
m number of times interest is paid annually
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The Effect of Compounding
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EAR and APR
Decision Making
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other
pays 5.3% with semiannual compounding. Which account should you use?
• First account:
EAR = (1 + .0525/365)365 − 1 = 5.39%
• Second account:
EAR = (1 + .053/2)2 − 1 = 5.37%
Which account should you choose and why?
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The Effect of Compounding
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EAR and APR
Decision Making
Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in
each account in one year?
• First account:
• Daily rate = .0525/365 = .00014383562
• FV = 100 × (1.00014383562)365 = $105.39
• Second account:
• Daily rate = .0539/2 = .0265
• FV = 100 × (1.0265)2 = $105.37
You have more money in the first account.
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The Effect of Compounding
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EAR and APR
To convert EAR into APR,
1
APR = m (1 + EAR) m − 1
where
m number of times interest is paid annually
For example, suppose you want to earn an effective rate of 12% and you are looking at an
account that compounds on a monthly basis. What APR must they pay?
1
APR = 12 × (1 + .12) 12 − 1 = .1139 = 11.39%
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The Effect of Compounding
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Computing Payments with APRs
Suppose you want to buy a new computer system and the store is willing to allow you to make
monthly payments. The entire computer system costs $3,500.
The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding.
What is your monthly payment?
• Monthly rate = .169/12 = .01408333333
• Number of months = 2 × 12 = 24
• 3, 500 = C × 1 − (1/1.01408333333)24 /.01408333333
• C = 172.88
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The Effect of Compounding
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Continuous Compounding
Sometimes investments or loans are figured based on continuous compounding
EAR = exp(q) − 1
where exp is is the exponential function. In Excel, use the EXP function to compute the
formula above.
Example: What is the effective annual rate of 7% compounded continuously?
EAR = exp(.07) − 1 = .0725 = 7.25%
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The Effect of Compounding
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Pure Discount Loans
A pure discount loan is the simplest form of loan. The borrower receives money today and
repays a single lump sum at some time in the future. To get the PV of such a loan, just
compute the discounted value of the single lump sum payment in the future.
Treasury bills (or T-bills) are excellent examples of pure discount loans. A T-bill is a promise
by the government to repay a fixed amount at some time in the future – for example, 3
months or 12 months – without any periodic interest payments.
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Loan Types and Loan Amortization
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Interest-Only Loans
In an interest-only loan, the borrower pays interest each period and repays the entire principal
(the original loan amount) at some point in the future.
For example, consider a 5-year, interest-only loan with a 7% interest rate. The principal
amount is $10,000, and interest is paid annually.
What would the stream of cash flows be?
• Years 1 – 4: Interest payments of .07 × 10, 000 = 700
• Year 5: Interest + principal = 10, 700
This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about
them in greater detail later.
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Loan Types and Loan Amortization
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Amortized Loans
Unlike a pure discount or interest-only loan, when the principal is repaid all at once, an
amortized loan is such that the lender receives part of the loan amount over time. The process
of providing for a loan to be paid off by making regular principal reductions is called
amortizing the loan.
For example, consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the
firm to pay $5,000 in principal each year plus interest for that year. Each payment covers the
interest expense plus reduces principal.
The table on the next slide shows an amortization schedule and payments for such a loan.
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Loan Types and Loan Amortization
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Amortized Loans
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Year
Beginning
Balance
Interest
Payment
Principal
Payment
Total
Payment
Ending
Balance
1
2
3
4
5
6
7
8
9
10
50,000
45,000
40,000
35,000
30,000
25,000
20,000
15,000
10,000
5,000
4,000
3,600
3,200
2,800
2,400
2,000
1,600
1,200
800
400
5,000
5,000
5,000
5,000
5,000
5,000
5,000
5,000
5,000
5,000
9,000
8,600
8,200
7,800
7,400
7,000
6,600
6,200
5,800
5,400
45,000
40,000
35,000
30,000
25,000
20,000
15,000
10,000
5,000
0
Loan Types and Loan Amortization
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