Tài chính doanh nghiệp - Chapter 4: Time value of money
Compounding Interest
More Frequently Than Annually (cont.)
• A General Equation for Compounding More
Frequently than Annually
– Recalculate the example for the Fred Moreno
example assuming (1) semiannual compounding and
(2) quarterly compounding.
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Chapter 4
Time Value
of Money
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-2
Learning Goals
1. Discuss the role of time value in finance, the
use of computational aids, and the basic
patterns of cash flow.
2. Understand the concept of future value and
present value, their calculation for single
amounts, and the relationship between them.
3. Find the future value and the present value of
both an ordinary annuity and an annuity due,
and the present value of a perpetuity.
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Learning Goals (cont.)
4. Calculate both the future value and the present
value of a mixed stream of cash flows.
5. Understand the effect that compounding
interest more frequently than annually has on
future value and the effective annual rate
of interest.
6. Describe the procedures involved in (1)
determining deposits needed to accumulate to
a future sum, (2) loan amortization, (3) finding
interest or growth rates, and (4) finding an
unknown number of periods.
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Question
Would it be better for a company to invest $100,000 in a
product that would return a total of $200,000 after one
year, or one that would return $220,000 after two years?
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits
that are spread out over time.
• Time value of money allows comparison of cash
flows from different periods.
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Answer
It depends on the interest rate!
The Role of Time Value
in Finance (cont.)
• Most financial decisions involve costs &
benefits that are spread out over time.
• Time value of money allows comparison
of cash flows from different periods.
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Basic Concepts
• Future Value: compounding or growth
over time
• Present Value: discounting to
today’s value
• Single cash flows & series of cash flows
can be considered
• Time lines are used to illustrate
these relationships
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Computational Aids
• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
• Use Electronic Spreadsheets
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Computational Aids (cont.)
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Computational Aids (cont.)
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Computational Aids (cont.)
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Computational Aids (cont.)
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Advantages Electronic Spreadsheets
• Spreadsheets go far beyond the computational abilities
of calculators.
• Spreadsheets have the ability to program logical
decisions.
• Spreadsheets display not only the calculated values of
solutions but also the input conditions on which solutions
are based.
• Spreadsheets encourage teamwork.
• Spreadsheets enhance learning.
• Spreadsheets communicate as well as calculate.
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Basic Patterns of Cash Flow
• The cash inflows and outflows of a firm can be described by its
general pattern.
• The three basic patterns include a single amount, an annuity, or a
mixed stream:
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Simple Interest
• With simple interest, you don’t earn interest
on interest.
• Year 1: 5% of $100 = $5 + $100 = $105
• Year 2: 5% of $100 = $5 + $105 = $110
• Year 3: 5% of $100 = $5 + $110 = $115
• Year 4: 5% of $100 = $5 + $115 = $120
• Year 5: 5% of $100 = $5 + $120 = $125
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Compound Interest
• With compound interest, a depositor earns interest
on interest!
• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00
• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25
• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76
• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55
• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63
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Time Value Terms
• PV0 = present value or beginning amount
• i = interest rate
• FVn = future value at end of “n” periods
• n = number of compounding periods
• A = an annuity (series of equal payments
or receipts)
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Four Basic Models
• FVn = PV0(1+i)
n = PV x (FVIFi,n)
• PV0 = FVn[1/(1+i)
n] = FV x (PVIFi,n)
• FVAn = A (1+i)
n - 1 = A x (FVIFAi,n)
i
• PVA0 = A 1 - [1/(1+i)
n] = A x (PVIFAi,n)
i
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Future Value of a Single Amount
• Future Value techniques typically measure cash flows
at the end of a project’s life.
• Future value is cash you will receive at a given
future date.
• The future value technique uses compounding to find
the future value of each cash flow at the end of an
investment’s life and then sums these values to find the
investment’s future value.
• We speak of compound interest to indicate that the
amount of interest earned on a given deposit has
become part of the principal at the end of the period.
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$100 x (1.08)1 = $100 x FVIF8%,1
$100 x 1.08 = $108
Future Value of a Single Amount:
Using FVIF Tables
• If Fred Moreno places $100 in a savings
account paying 8% interest compounded
annually, how much will he have in the
account at the end of one year?
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FV5 = $800 X (1 + 0.06)
5 = $800 X (1.338) = $1,070.40
Future Value of a Single Amount:
The Equation for Future Value
• Jane Farber places $800 in a savings
account paying 6% interest compounded
annually. She wants to know how much
money will be in the account at the end of
five years.
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Future Value of a Single Amount:
Using a Financial Calculator
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Future Value of a Single Amount:
Using Spreadsheets
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Future Value of a Single Amount:
A Graphical View of Future Value
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Present Value of a Single Amount
• Present value is the current dollar value of a future
amount of money.
• It is based on the idea that a dollar today is worth more
than a dollar tomorrow.
• It is the amount today that must be invested at a given
rate to reach a future amount.
• Calculating present value is also known as discounting.
• The discount rate is often also referred to as the
opportunity cost, the discount rate, the required
return, or the cost of capital.
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$300 x [1/(1.06)1] = $300 x PVIF6%,1
$300 x 0.9434 = $283.02
Present Value of a Single Amount:
Using PVIF Tables
• Paul Shorter has an opportunity to receive
$300 one year from now. If he can earn
6% on his investments, what is the most
he should pay now for this opportunity?
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PV = $1,700/(1 + 0.08)8 = $1,700/1.851 = $918.42
Present Value of a Single Amount:
The Equation for Future Value
• Pam Valenti wishes to find the present
value of $1,700 that will be received
8 years from now. Pam’s opportunity
cost is 8%.
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Present Value of a Single Amount:
Using a Financial Calculator
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Present Value of a Single Amount:
Using Spreadsheets
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Present Value of a Single Amount:
A Graphical View of Present Value
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Annuities
• Annuities are equally-spaced cash flows of
equal size.
• Annuities can be either inflows or outflows.
• An ordinary (deferred) annuity has cash flows
that occur at the end of each period.
• An annuity due has cash flows that occur at the
beginning of each period.
• An annuity due will always be greater than an
otherwise equivalent ordinary annuity because
interest will compound for an additional period.
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Types of Annuities
Note that the amount of both annuities total $5,000.
• Fran Abrams is choosing which of two annuities
to receive. Both are 5-year $1,000 annuities;
annuity A is an ordinary annuity, and annuity B
is an annuity due. Fran has listed the cash
flows for both annuities as shown in Table 4.1
on the following slide.
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Future Value of an Ordinary Annuity
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Future Value
of an Ordinary Annuity (cont.)
• Fran Abrams wishes to determine how much money she
will have at the end of 5 years if he chooses annuity A,
the ordinary annuity and it earns 7% annually. Annuity a
is depicted graphically below:
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Future Value of an Ordinary Annuity:
Using the FVIFA Tables
FVA = $1,000 (FVIFA,7%,5)
= $1,000 (5.751)
= $5,751
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Future Value of an Ordinary Annuity:
Using a Financial Calculator
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Future Value of an Ordinary Annuity:
Using Spreadsheets
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FVA = $1,000(FVIFA,7%,5)(1+.07)
= $1,000 (5.751) (1.07)
= $6,154
Future Value of an Annuity Due:
Using the FVIFA Tables
• Fran Abrams now wishes to calculate the
future value of an annuity due for annuity
B in Table 4.1. Recall that annuity B was
a 5 period annuity with the first annuity
beginning immediately.
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Future Value of an Annuity Due:
Using a Financial Calculator
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Future Value of an Annuity Due:
Using Spreadsheets
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Present Value of an Ordinary Annuity
• Braden Company, a small producer of plastic toys,
wants to determine the most it should pay to purchase a
particular annuity. The annuity consists of cash flows of
$700 at the end of each year for 5 years. The required
return is 8%.
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Present Value of an Ordinary Annuity:
The Long Method
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Present Value of an Ordinary Annuity:
Using PVIFA Tables
PVA = $700 (PVIFA,8%,5)
= $700 (3.993)
= $2,795.10
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Present Value of an Ordinary Annuity:
Using a Financial Calculator
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Present Value of an Ordinary Annuity:
Using Spreadsheets
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PVA = $700 (PVIFA,8%,5) (1.08)
= $700 (3.993) (1.08)
= $3,018.40
Present Value of an Annuity Due:
Using PVIFA Tables
• In the earlier example, we found that the value
of Braden Company’s $700, 5 year ordinary
annuity discounted at 8% to be about $2,795. If
we now assume that the cash flows occur at the
beginning of the year, we can find the PV of the
annuity due.
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Present Value of an Annuity Due:
Using a Financial Calculator
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Present Value of an Annuity Due:
Using Spreadsheets
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PV = Annuity/Interest Rate
PV = $1,000/.08 = $12,500
Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or cash
flow stream continues forever.
• For example, how much would I have to deposit
today in order to withdraw $1,000 each year
forever if I can earn 8% on my deposit?
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Future Value of a Mixed Stream
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Future Value of a Mixed Stream (cont.)
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Future Value of a Mixed Stream:
Using Excel
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Present Value of a Mixed Stream
• Frey Company, a shoe manufacturer, has
been offered an opportunity to receive the
following mixed stream of cash flows over
the next 5 years.
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Present Value of a Mixed Stream
• If the firm must earn at least 9% on its
investments, what is the most it should pay for
this opportunity?
• This situation is depicted on the following
time line.
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Present Value of a Mixed Stream
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Present Value of a Mixed Stream:
Using Excel
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Compounding Interest
More Frequently Than Annually
• Compounding more frequently than once a year
results in a higher effective interest rate
because you are earning on interest on interest
more frequently.
• As a result, the effective interest rate is greater
than the nominal (annual) interest rate.
• Furthermore, the effective rate of interest will
increase the more frequently interest is
compounded.
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Compounding Interest
More Frequently Than Annually (cont.)
• Fred Moreno has found an institution that will pay him
8% annual interest, compounded quarterly. If he leaves
the money in the account for 24 months (2 years), he
will be paid 2% interest compounded over eight periods.
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Compounding Interest
More Frequently Than Annually (cont.)
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Compounding Interest
More Frequently Than Annually (cont.)
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Compounding Interest
More Frequently Than Annually (cont.)
• A General Equation for Compounding
More Frequently than Annually
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Compounding Interest
More Frequently Than Annually (cont.)
• A General Equation for Compounding More
Frequently than Annually
– Recalculate the example for the Fred Moreno
example assuming (1) semiannual compounding and
(2) quarterly compounding.
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Compounding Interest More Frequently
Than Annually: Using a Financial Calculator
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Compounding Interest More Frequently
Than Annually: Using a Spreadsheet
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FVn (continuous compounding) = PV x (e
kxn)
where “e” has a value of 2.7183.
Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation
thus becomes:
• Continuing with the previous example, find the Future
value of the $100 deposit after 5 years if interest is
compounded continuously.
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FVn (continuous compounding) = PV x (e
kxn)
where “e” has a value of 2.7183.
FVn = 100 x (2.7183).08x2 = $117.35
Continuous Compounding (cont.)
• With continuous compounding the number
of compounding periods per year
approaches infinity.
• Through the use of calculus, the equation
thus becomes:
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Continuous Compounding:
Using a Financial Calculator
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Continuous Compounding:
Using a Spreadsheet
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EAR = (1 + i/m) m - 1
Nominal & Effective
Annual Rates of Interest
• The nominal interest rate is the stated or
contractual rate of interest charged by a lender
or promised by a borrower.
• The effective interest rate is the rate actually
paid or earned.
• In general, the effective rate > nominal rate
whenever compounding occurs more than once
per year
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Nominal & Effective
Annual Rates of Interest (cont.)
• Fred Moreno wishes to find the effective annual rate
associated with an 8% nominal annual rate (I = .08)
when interest is compounded (1) annually (m=1); (2)
semiannually (m=2); and (3) quarterly (m=4).
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Special Applications of Time Value: Deposits
Needed to Accumulate to a Future Sum
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PMT = $30,000/5.637 = $5,321.98
Special Applications of Time Value: Deposits
Needed to Accumulate to a Future Sum (cont.)
• Suppose you want to buy a house 5 years from
now and you estimate that the down payment
needed will be $30,000. How much would you
need to deposit at the end of each year for the
next 5 years to accumulate $30,000 if you can
earn 6% on your deposits?
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Special Applications of Time Value: Deposits
Needed to Accumulate to a Future Sum (cont.)
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Special Applications of Time Value: Deposits
Needed to Accumulate to a Future Sum (cont.)
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Special Applications of Time Value:
Loan Amortization
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Special Applications of Time Value:
Loan Amortization (cont.)
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Ray Noble wishes to find the rate of interest or growth
reflected in the stream of cash flows he received from a
real estate investment over the period from 2002 through
2006 as shown in the table on the following slide.
Special Applications of Time Value:
Interest or Growth Rates
• At times, it may be desirable to determine
the compound interest rate or growth rate
implied by a series of cash flows.
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PVIFi,5yrs = PV/FV = ($1,250/$1,520) = 0.822
PVIFi,5yrs = approximately 5%
Special Applications of Time Value:
Interest or Growth Rates (cont.)
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Special Applications of Time Value:
Interest or Growth Rates (cont.)
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Special Applications of Time Value:
Interest or Growth Rates (cont.)
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Ann Bates wishes to determine the number of years it
will take for her initial $1,000 deposit, earning 8% annual
interest, to grow to equal $2,500. Simply stated, at an
8% annual rate of interest, how many years, n, will it
take for Ann’s $1,000 (PVn) to grow to $2,500 (FVn)?
Special Applications of Time Value:
Finding an Unknown Number of Periods
• At times, it may be desirable to determine the
number of time periods needed to generate a
given amount of cash flow from an initial
amount.
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PVIF8%,n = PV/FV = ($1,000/$2,500) = .400
PVIF8%,n = approximately 12 years
Special Applications of Time Value:
Finding an Unknown Number of Periods (cont.)
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Special Applications of Time Value:
Finding an Unknown Number of Periods (cont.)
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Special Applications of Time Value:
Finding an Unknown Number of Periods (cont.)
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