Chapter 4
The Time Value of Money
1
Learning Outcomes
Chapter 4
Identify various types of cash flow patterns
Compute the future value and the present value of
different cash flow streams
Compute the return on an investment and how long
it takes to reach a financial goal
Explain the difference between the Annual
Percentage Rate and the Effective Annual Rate
Describe an amortized loan. Compute amortized loan
payments and the amount that must be paid at a
specific point during the life of the loan
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Time Value of Money
The principles and computations used to
revalue cash payoffs from different times so
they are stated in dollars of the same time
period.
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Cash Flow Time Lines
Graphical representations used to show timing of cash flows
PV = Present Value – the beginning amount that can be
invested. PV also represents the current value of some future
amount.
FV = Future Value – the value to which an amount invested
today will grow at the end of n periods.
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Types of Cash Flow Patterns
Lump Sum Amount – a single payment (received or
made) that occurs either today or at some date in
the future.
Annuity – Multiple payments of the same amount
over equal time periods.
Uneven Cash Flows – Multiple payments of different
amounts over a period of time.
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Future Value
Compounding – To compute the future value of an
amount we push forward the current amount by
adding interest for each period in which the money
can earn interest in the future.
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Future Value of a Lump-Sum Amount
FVn
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Four Ways to Solve Time Value of Money
Problems
Use Cash Flow Time Line
Use Equations
Use Financial Calculator
Use Electronic Spreadsheet
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Time Line Solution
The Future Value of $700 invested at 10% per
year for 3 years
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Equation Solution
FV3
= $700(1.10)3
= $700(1.33100)
= $931.70
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Financial Calculator Solution
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Spreadsheet Solution - MS Excel
1.
Set up a table that contains
the data used to solve the
problem
2.
Click fx and choose function
3.
Click the cells containing the
appropriate data to calculate
the answer
Rate (B2)
Number of periods (B1)
Present value (B3)
Payment (B4) (not used)
Type (B5) (not used)
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Future Value of an Annuity
Annuity - A series of payments of equal
amounts at fixed intervals for a specified
number of periods.
Ordinary (deferred) Annuity - An annuity
whose payments occur at the end of each
period.
Annuity Due - An annuity whose payments
occur at the beginning of each period.
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What’s the FV of a 3-year Ordinary
Annuity of $400 at 5%?
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Equation Solution:
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Financial Calculator Solution
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Find the FV of an Annuity Due
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Equation Solution
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Financial Calculator Solution
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Find the FV of an Uneven Cash Flow
Stream – FVCFn
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Equation Solution
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Present Value
Present value is the value today of a future cash flow
or series of cash flows.
Discounting is the process of finding the present
value of a future cash flow or series of future cash
flows; it is the reverse of compounding.
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Present Value of a Lump-Sum Amount
PV
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PV of a Lump-Sum Amount
Financial Calculator
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Present Value of an Annuity (Ordinary)
PVAn = the present value of an annuity with
n payments.
Each payment is discounted, and the sum of
the discounted payments is the present
value of the annuity.
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What is the PV of $400 due in 3 years if r
= 5%?
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Equation Solution
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Financial Calculator Solution
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Present Value of an Annuity Due
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Equation Solution
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Financial Calculator Solution
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Perpetuities
Streams of equal payments that are expected to go
on forever
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Uneven Cash Flow Streams
A series of cash flows in which the amount
varies from one period to the next:
 Payment (PMT) designates constant cash flows—
that is, an annuity stream.
 Cash flow (CF) designates cash flows in general,
both constant cash flows and uneven cash flows.
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PV of an Uneven Cash Flow Stream –
PVCFn
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Equation Solution
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Financial Calculator Solution
Input in “CF” register:




CF0
CF1
CF2
CF3
=
=
=
=
0
400
300
250
Enter I = 5, then press NPV button to get
NPV = -869.02.
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Solving for Interest Rates (r)
You pay $78.35 for an investment that promises
to pay you $100 per year for the next five years.
What annual rate of return will you earn on this
investment?
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Financial Calculator Solution
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Solving for Time (n)
A security costing $68.30 will provide a return of
10% per year and you want to keep the
investment until it grows to a value of $100. How
long will it take the investment to grow to $100?
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Financial Calculator Solution
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Semiannual and Other Compounding
Periods
Annual compounding is the process of
determining the future value of a cash flow or
series of cash flows when interest is added
once a year.
Semiannual compounding is the process of
determining the future value of a cash flow or
series of cash flows when interest is added
twice a year.
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The FV of a lump sum be larger if we
compound more often, holding the stated
r constant? Why?
If compounding is more frequent than once a
year—for example, semi-annually, quarterly,
or daily—interest is earned on interest—that
is, compounded—more often.
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Distinguishing Between Different
Interest Rates
rSIMPLE = Simple (Quoted) Rate
used to compute the interest paid per period
rEAR = Effective Annual Rate
the annual rate of interest actually being earned
APR = Annual Percentage Rate = rSIMPLE periodic
rate X the number of periods per year
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Comparison of Different Types of Interest
Rates
rSIMPLE: Written into contracts, quoted by
banks and brokers. Not used in calculations
or shown on time lines.
rPER: Used in calculations, shown on time
lines.
rEAR: Used to compare returns on
investments with different payments per
year.
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Simple (Quoted) Rate
rSIMPLE is stated in contracts
Periods per year (m) must also be given
Examples:
 8%, compounded quarterly
 8%, compounded daily (365 days)
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Periodic Rate
Periodic rate = rPER = rSIMPLE/m, where m is
number of compounding periods per year. m
= 4 for quarterly, 12 for monthly, and 360 or
365 for daily compounding.
Examples:
 8% quarterly: rPER = 8/4 = 2%
 8% daily (365): rPER = 8/365 = 0.021918%
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Effective Annual Rate
The annual rate that causes PV to grow to the same
FV as under multi-period compounding.
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How do we find rEAR for a simple rate of
10%, compounded semi-annually?
m
rEAR

rSIMPLE 
= 1 +


m 
- 1
2
0.10 

= 1 +
 - 1.0
2 

= (1.05) - 1.0 = 0.1025 = 10.25%
2
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Amortized Loans
Amortized Loan - A loan that is repaid in
equal payments over its life
Amortization tables are widely used for home
mortgages, auto loans, business loans,
retirement plans, and so forth to determine
how much of each payment represents
principal repayment and how much
represents interest
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An amortization schedule for a $15,000,
8 percent loan that requires three equal
annual payments.
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Financial Calculator Solution
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Amortization Schedule
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