Principles of Managerial Finance 9th Edition

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Principles of Managerial Finance9th Edition

• Chapter 5

Time Value of Money
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Learning Objectives

• Discuss the role of time value in finance and the
  use of computational aids used to simplify its
  application.
• Understand the concept of future value, its
  calculation for a single amount, and the effects
  of compounding interest more frequently than
  annually.
• Find the future value of an ordinary annuity and
  an annuity due and compare these two types of
  annuities.
• Understand the concept of present value, its
  calculation for a single amount, and its
  relationship to future value.

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Learning Objectives

• Calculate the present value of a mixed stream of
  cash flows, an annuity, a mixed stream with an
  embedded annuity, and a perpetuity.
• Describe the procedures involved in
• determining deposits to accumulate a future sum,
• loan amortization, and
• finding interest or growth rates

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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.

Question? Would it be better for a company to
invest 100,000 in a product that would return a
total of 200,000 in one year, or one that would
return 500,000 after two years?
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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.

Answer! It depends on the interest rate!
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Basic Concepts

• Future Value compounding or growth over time
• Present Value discounting to todays 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 Spreadsheets

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Computational Aids
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Computational Aids
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Computational Aids
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Computational Aids
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Simple Interest
With simple interest, you dont 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
• k 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(1k)n PV(FVIFk,n)
• PV0 FVn1/(1k)n FV(PVIFk,n)
• FVAn A (1k)n - 1 A(FVIFAk,n)
• k
• PVA0 A 1 - 1/(1k)n A(PVIFAk,n)
• k

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Future Value Example
Algebraically and Using FVIF Tables
You deposit 2,000 today at 6 interest. How much
will you have in 5 years?
2,000 x (1.06)5 2,000 x FVIF6,5 2,000 x
1.3382 2,676.40
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Future Value Example
Using Excel
You deposit 2,000 today at 6 interest. How much
will you have in 5 years?
Excel Function FV (interest, periods, pmt,
PV) FV (.06, 5, , 2000)
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Future Value Example
A Graphic View of Future Value
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Compounding 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 More Frequently than Annually

• For example, what would be the difference in
  future value if I deposit 100 for 5 years and
  earn 12 annual interest compounded (a) annually,
  (b) semiannually, (c) quarterly, an (d) monthly?

Annually 100 x (1 .12)5 176.23 Semiannual
ly 100 x (1 .06)10 179.09 Quarterly 100
x (1 .03)20 180.61 Monthly 100 x (1
.01)60 181.67
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Compounding More Frequently than Annually
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Continuous Compounding

• With continuous compounding the number of
  compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
  becomes

FVn (continuous compounding) PV x (ekxn) where
e has a value of 2.7183.

• 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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Continuous Compounding

• With continuous compounding the number of
  compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
  becomes

FVn (continuous compounding) PV x (ekxn) where
e has a value of 2.7183.
FVn 100 x (2.7183).12x5 182.22
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Nominal Effective Rates

• 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 gt nominal rate
  whenever compounding occurs more than once per
  year

EAR (1 k/m) m -1
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Nominal Effective Rates

• For example, what is the effective rate of
  interest on your credit card if the nominal rate
  is 18 per year, compounded monthly?

EAR (1 .18/12) 12 -1 EAR 19.56
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Present Value

• 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.
• It is also known as discounting, the reverse of
  compounding.
• The discount rate is often also referred to as
  the opportunity cost, the discount rate, the
  required return, and the cost of capital.

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Present Value Example
Algebraically and Using PVIF Tables
How much must you deposit today in order to have
2,000 in 5 years if you can earn 6 interest on
your deposit?
2,000 x 1/(1.06)5 2,000 x PVIF6,5
2,000 x 0.74758 1,494.52
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Present Value Example
Using Excel
How much must you deposit today in order to have
2,000 in 5 years if you can earn 6 interest on
your deposit?
Excel Function PV (interest, periods, pmt,
FV) PV (.06, 5, , 2000)
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Present Value Example
A Graphic 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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Annuities
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Future Value of an Ordinary Annuity
Using the FVIFA Tables

• Annuity Equal Annual Series of Cash Flows
• Example How much will your deposits grow to if
  you deposit 100 at the end of each year at 5
  interest for three years.

FVA 100(FVIFA,5,3) 315.25
Year 1 100 deposited at end of
year 100.00 Year 2 100 x .05 5.00 100
100 205.00 Year 3 205 x .05 10.25
205 100 315.25
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Future Value of an Ordinary Annuity
Using Excel

• Annuity Equal Annual Series of Cash Flows
• Example How much will your deposits grow to if
  you deposit 100 at the end of each year at 5
  interest for three years.

Excel Function FV (interest, periods, pmt,
PV) FV (.06, 5,100, )
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Future Value of an Annuity Due
Using the FVIFA Tables

• Annuity Equal Annual Series of Cash Flows
• Example How much will your deposits grow to if
  you deposit 100 at the beginning of each year at
  5 interest for three years.

FVA 100(FVIFA,5,3)(1k) 330.96
FVA 100(3.152)(1.05) 330.96
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Future Value of an Annuity Due
Using Excel

• Annuity Equal Annual Series of Cash Flows
• Example How much will your deposits grow to if
  you deposit 100 at the beginning of each year at
  5 interest for three years.

Excel Function FV (interest, periods, pmt,
PV) FV (.06, 5,100, ) 315.25(1.05)
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Present Value of an Ordinary Annuity
Using PVIFA Tables

• Annuity Equal Annual Series of Cash Flows
• Example How much could you borrow if you could
  afford annual payments of 2,000 (which includes
  both principal and interest) at the end of each
  year for three years at 10 interest?

PVA 2,000(PVIFA,10,3) 4,973.70
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Present Value of an Ordinary Annuity
Using Excel

• Annuity Equal Annual Series of Cash Flows
• Example How much could you borrow if you could
  afford annual payments of 2,000 (which includes
  both principal and interest) at the end of each
  year for three years at 10 interest?

Excel Function PV (interest, periods, pmt,
FV) PV (.10, 3, 2000, )
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Present Value of a Mixed Stream
Using Tables

• A mixed stream of cash flows reflects no
  particular pattern
• Find the present value of the following mixed
  stream assuming a required return of 9.

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Present Value of a Mixed Stream
Using EXCEL

• A mixed stream of cash flows reflects no
  particular pattern
• Find the present value of the following mixed
  stream assuming a required return of 9.

Excel Function NPV (interest, cells containing
CFs) NPV (.09,B3B7)
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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.

PV Annuity/k

• 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?

PV 1,000/.08 12,500
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Loan Amortization
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Determining 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.
• For example, you invested 1,000 in a mutual fund
  in 1994 which grew as shown in the table below?

It is first important to note that although there
are 7 years show, there are only 6 time periods
between the initial deposit and the final value.
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Determining 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.
• For example, you invested 1,000 in a mutual fund
  in 1994 which grew as shown in the table below?

Thus, 1,000 is the present value, 5,525 is the
future value, and 6 is the number of periods.
Using Excel, we get
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Determining 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.
• For example, you invested 1,000 in a mutual fund
  in 1994 which grew as shown in the table below?

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Determining 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.
• For example, you invested 1,000 in a mutual fund
  in 1994 which grew as shown in the table below?

Excel Function Rate(periods, pmt, PV,
FV) Rate(6, ,1000, 5525)