MBA_621_Zietlow_Chapter_3

1
Chapter 3
Present Value
Professor John ZietlowMBA 621
Spring 2006
2
Chapter 3 Overview

• 3.1 The Theory of Present Value
• Borrowing, Lending and Consumption Opportunities
• How Financial Markets Improve Welfare
• 3.2 Future Value of a Lump Sum Amount
• The Concept of Future Value
• The Equation for Future Value
• A Graphic View of Future Value
• 3.3 Present Value of a Lump Sum Amount
• The Concept of Present Value
• The Equation for Present Value
• A Graphic View of Present Value
• 3.4 Future Value of Cash Flow Streams
• Finding the Future Value of a Mixed Stream
• Types of Annuities
• Finding the Future Value of an Ordinary Annuity
• Finding the Future Value of an Annuity Due
• Comparison of an Ordinary Annuity with an Annuity
  Due

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Chapter 3 Overview

• 3.5 Present Value of Cash Flow Streams
• Finding Present Value of a Mixed Stream
• Finding the Present Value of an Ordinary Annuity
• Finding the Present Value of an Annuity Due
• Finding the Present Value of a Perpetuity
• Finding the Present Value of a Growing Perpetuity
• 3.6 Special Applications of Time Value
• Compounding More Frequently than Annually
• Nominal and Effective Annual Rates of Interest
• Deposits Needed to Accumulate a Future Sum
• Loan Amortization
• Appendix Additional Special Applications of
  Time Value
• Interest or Growth Rates
• Determining the Number of Time Periods

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The Focus on Present Value

• Chapter describes how to account for time value
  of money in financial decision-making
• Begins with finding future value of sum invested
  today
• Finance uses compound rather than simple interest
• Compound interest causes sum to grow very large
  with time
• Focus on present value value of future CF
  measured today
• Permits comparing values of CFs received at
  different times
• Present value concepts calculations pervade
  finance
• Managers use PV to evaluate capital investments
• Investors use PV to value securities
• Begin with simplest cases--single cash flows
• Then study complex, multiple CF streams
• A time line can be used to show CFs graphically

5
Using Timelines To Demonstrate Future Value And
Present Value

• Time period 0 is today others represent future
  period-ends
• Unless stated otherwise, period means year
  (end-of-year)
• So t1 is end of year 1 t5 is end of year five
• Negative values represent cash outflows
• Positive values represent cash inflows
• FV uses compounding to find terminal value of CFs
  
• What is value in 5 years of 1 invested at 6
  annual interest?
• PV uses discounting to find todays value of
  future CFs
• What is todays value of 1 to be received in 5
  years at a 6 discount rate?
• FV and PV can be computed in several ways
• Using financial calculators or computer
  spreadsheets
• Using tables with present future value factors
  (PVIF, FVIF)

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Timeline Illustration of Future Value and Present
Value
Compounding
FutureValue
-10,000 3,000 5,000 4,000
3,000 2,000
0 1 2
3 4 5
End of Year
Present Value
Discounting
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Future Value Concepts Terms

• Basic Terminology of Future Value
• Interest rate (r) is the annual rate of interest
  paid on the principal amount. Also called
  compound annual interest
• Present Value (PV) in this setting is the initial
  investment amount (principal) on which interest
  is paid. In other cases, PV is the discounted
  present value of a future sum or sums
• Future Value (FV) is computed by applying annual
  interest to a principal amount over a specified
  period of time
• Number of compounding periods (n) is the number
  of years the principal will earn interest
• Basic formula for the end-of-period n future
  value of a sum invested today at interest rate r
  is
• FVn PV x (1 r)n

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Time Line for 78.35 Invested for Five Years at
5 Interest
FV5 100
PV 78.35
0 1 2
3 4 5
End of Year
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Demonstrating Simple (One CF) Future Value
Computations

• Compute the FV of a 50 sum deposited at 4
  interest at the end of years 1, 2, 3, and 4
• FV end of year 1 50 x (1 0.04) 52
• FV end of year 2 52 x (1.04) 50 x (1.04)2
  54.08
• FV end of year 3 54.08 x (1.04) 50 x
  (1.04)3 56.24
• FV end of year 4 56.24 x (1.04) 50 x
  (1.04)4 58.49
• With compound interest, you earn interest on
  interest, so FV can reach large amounts
  relatively quickly
• FV end of year 9 50 x (1.04)9 71.16
• FV end of year 15 50 x (1.04)15 90.05
• FV end of year 30 50 x (1.04)30 162.17

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Simple Future Value Computations (Continued)

• Find the FV of 3,000 invested at 3.25 interest
  for 3 years
• FV3 PV x (1 r)n 3,000 (1.0325)3
  3,302.11
• Find the FV of 735.5 invested at 6.35 for 5
  years
• FV5 735.5 x (1.0635)5 1000.62
• Find the FV of 100 invested at 6 for 15 months
  Hint 15 months can be specified as 1.25 years
  
• FV1.25 100 x (1.06)1.25 107.55
• Find the FV of 5,000 invested at 6.74 for 8
  years, 3 months Hint express 3 months as
  3/120.25 year
• FV8.25 5,000 (1.0674)8.25 8,536.86
• At high interest rates, FV builds up very fast !

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The Power Of Compound Interest Future Value Of
1 Invested At Different Interest Rates
30.00
20
25.00
15
20.00
Future Value of One Dollar ()
15.00
10.00
10
5.00
5
0
1.00
0 2 4 6 8 10 12 14 16 18 20
22 24
Periods
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Computing Future Values Algebraically And Using
FVIF Tables

• You deposit 1,000 today at 3 interest.
• How much will you have in 5 years?
• Could solve this using basic FV formula
• FVn PV x (1r)n 1,000 x (1.03)5 1159.27
• Or could use future value interest factor formula
  and table
• FV5 PV x FVIFr,n 1,000 x FVIF3,5
• Look up FVIF6,5 in Future Value Interest Factor
  table
• FVIF3,5 1.159
• FV5 1,000 x 1.159 1,159

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Format Of A Future Value Interest Factor (FVIF)
Table
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Computing Future Values Using Excel
You deposit 1,000 today at 3 interest. How much
will you have in 5 years?
Excel Function FV (interest, periods, pmt,
PV) FV (.03, 5, 1000)
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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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The Logic Of Present Value

• Assume you can buy an investment that will pay
  1,000 one year from now
• Also assume you can earn 3.15 on equally risky
  investments
• What should you pay for this opportunity?
• Answer Find how much must be invested today at
  3.15 to have 1,000 in one year
• PV x (1 0.0315) 1,000
• Solving for PV gives

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Calculating The PV Of A Single Amount

• The present value of a future amount can be found
  mathematically by using this formula
• Find the present value of 500 to be received in
  7 years, assuming a discount rate of 6.
• Substitute FV7 500, n 7, and r .06 into PV
  formula

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Present Value of 500 to be Received in 7 Years
at a 6 Discount Rate
0 1 2 3 4
5 6 7
FV7 500
End of Year
PV 332.53
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Format Of A Present Value Factor (PVF) Table
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Calculating Present Value Of A Single Amount
Using A Spreadsheet

• Example How much must you deposit today in
  order to have 500 in 7 years if you can earn 6
  interest on your deposit?

Excel Function PV (interest, periods, pmt,
FV) PV (.06, 7, 500)
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The Power Of High Discount Rates Present Value
Of 1 Invested At Different Interest Rates
1.00
0
0.75
Present Value of One Dollar ()
0.5
5
10
0.25
15
20
0 2 4 6 8 10 12 14 16 18 20
22 24
Periods
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Finding The Future Value Of Cash Flow Streams
(Multiple Cash Flows)

• Two basic types of cash flows streams are
  observed
• A mixed stream has uneven cash flows (no pattern)
• An annuity has equal annual cash flows
• Either type can represent cash inflows (receipts)
  or cash outflows (payments)
• FV of a stream equals sum of FVs of individual
  cash flows
• Basic formula for the FV of a stream (FVMn),
  where CFt equals a cash flow at end of year t
• FVMn CF1 ? (1 r)n-1 CF2 ? (1 r)n-2
  CFn ? (1 r)n-n

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Finding The FV Of A Mixed Stream

• Find the end of year 5 future value of the
  following cash flows, which are invested at 5.5
  annual interest (r5.5, n5)
• End of year 1 3,500 (invested for four years)
• End of year 2 3,800 (invested for three years)
• End of year 3 2,000 (invested for two years)
• End of year 4 3,000(invested for one year)
• End of year 5 2,500 (invested for 0 year)
• Use FVMn formula to calculate terminal (future)
  value
• FVMn CF1 ? (1 r)n-1 CF2 ? (1 r)n-2
  CFn ? (1 r)n-n
• 3,500 (1.055)4 3,800 (1.055)3
  2,000(1.055)2
• 3,000(1.055) 2,500 (1.00)
• 4,335.89 4,462.12 2,226.05 3,165
  2,500
• 16,689.06

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Future Value, at the end of 5 Years of a Mixed
Cash Flow Stream Invested at 5.5
FV5 16,689.06
4,335.89
4,462.12
2,226.06
3,165.00
2,500.00
3,500 3,800
2,000 3,000 2,500
0 1 2
3 4 5
End of Year
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The Future Value of An Annuity

• Annuities are extremely important in finance
• Virtually all bond interest payments structured
  as annuities
• Many capital investment projects have
  annuity-like cash flows
• Two types of annuities ordinary annuity
  annuity due
• Ordinary annuity payments occur at end of period
• Annuity due payments occur at beginning of
  period
• FV of annuity due always higher than FV of
  ordinary annuity
• Since CF invested at beginning--rather than
  end--of period, all CFs earn one more periods
  interest
• Unless otherwise stated, will assume an ordinary
  annuity
• Much more commonly observed in actual finance
  practice

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Finding the Future Value Of An Ordinary Annuity

• FV of annuity (FVA) can be found as with FVM
• Find FV of individual amounts, then sum FVs
• Demonstrated with timeline (next slide)
• Since an annuity has equal payments,
• CF1 CF2 CFn PMT, can simplify FVM
  formula
• Express FV of annuity as the product of the
  payment amount (PMT) times the sum of the FV
  factors
• Summation term to the right of PMT is the future
  value interest factor of an annuity (FVIFAr,n)

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Calculating The Future Value of An Ordinary
Annuity

• How much will your deposits grow to if you
  deposit 1,000 at the end of each year at 4.3
  interest for 5 years.
• Can show computation of FVA as sum of individual
  FVs
• FVA 1,000 (1.043)4 1,000 (1.043)3 1,000
  (1.043)2
• 1,000 (1.043) 1,000 (1.0)
• 1,000 (1.1834) 1,000 (1.1346) 1,000
  (1.0878)
• 1.000 (1.043) 1,000 (1.0) 5,448.8
• Or can multiply payment times sum of FV factors
• FVA 1,000 (1.1834 1.1346 1.0878 1.043
  1.0)
• 1,000 (5.4488) 5,448.8

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Future Value, at the end of 5 Years of an Annuity
Investing 1,000 per year at 4.3
FV5 5,448.8
1,183.4
1,134.6
1,087.8
1,043.0
1,000.0
1,000 1,000
1,000 1,000 1,000
0 1 2
3 4 5
End of Year
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Finding The Future Value Of An Ordinary Annuity
Using A Spreadsheet

• How much will your deposits grow to at the end
  of five years if you deposit 1,000 at the end of
  each year at 4.3 interest for 5 years?

Excel Function FV (interest, periods, pmt,
PV) FV (.043, 5,1000 )
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Cash Flows Of An Ordinary Annuity Versus An
Annuity Due
Comparison of ordinary Annuity and Annuity Due
Cash Flows (1,000, 5 Years)
Annual Cash Flows
End of yeara Annuity A
(ordinary) Annuity B (annuity due)

aThe ends of years 0, 1,2, 3, 4 and 5 are
equivalent to the beginnings of years 1, 2, 3, 4,
5, and 6 respectively
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Calculating The Future Value Of An Annuity Due

• Equation for the FV of an ordinary annuity can
  be converted
• into an expression for the future value of
  an annuity due,
• FVAn (annuity due), by merely multiplying it
  by (1 r)

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Future Value, at the end of 5 Years of an Annuity
Due Investing 1,000 per year at 4.3
FV5 5,683.1
1,234.30
1,183.41
1,134.60
1,087.80
1,043.00
1,000 1,000 1,000 1,000
1,000
0 1 2
3 4 5
End of Year
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Finding The Future Value Of An Annuity Due Using
A Spreadsheet

• How much will your deposits grow to at the end
  of five years
• if you deposit 1,000 at the beginning of each
  year at 4.3
• interest for 5 years?

Excel Function FV (interest, periods, pmt,
PV) FV (.043, 5, 1,000 ) 5,448.89(1.043)
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The Present Value Of A Mixed Stream

• Continuing to let CFt represent the cash flow at
  the end of year t, the present value of an n-year
  mixed stream of cash flows, PVMn, can be
  expressed as

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Calculating The PV Of A Mixed Stream

• Assume you must find the PV of the following
  year-end cash flows, if the discount rate is 6
• End of year 1 1,500,000 End of year 2
  3,000,000
• End of year 3 2,000,000 End of year 4
  5,000,000
• Plug year-end cash flows into PVM formula, with
  k9

PVM4 1,500,000 (0.9434) 3,000,000 (0.8899)
2,000,000 (0.8396) 5,000,000 (0.7921)
9,724,500
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Present Value of a 5-Year Mixed Stream Discounted
at 9
0 1 2
3 4
1,500,000 3,000,000
2,000,000 5,000,000
End of Year
1,415,100
2,669,700
1,679,200
3,960,500
PV5 9,724,500
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Finding The PV Of An Ordinary Annuity

• Since, for an annuity, PMT CF1 CF2
  CFn, the PVMn formula can be modified to compute
  the present value of an n-year annuity, PVAn.
• The rightmost term is the formula for the present
  value interest factor for an annuity, PVIFAr,n
• If PMT 1,250, n 6 years, and r 5, find
  PVA6
• PVA6 PMT x PVIFA5,6 1,250 x 5.0757
  6,344.625

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Present Value of a 6-Year Mixed Stream Discounted
at 5
0 1 2
3 4 5
6
1,250 1,250 1,250
1,250 1,250 1,250
End of Year
1,190.476
1,133.787
1,079.797
1,028.378
979.407
932.769
PV5 6,344.6
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Calculating The PV Of A Perpetuity

• Frequently need to calculate the PV of a
  perpetuity--a stream of equal annual cash flows
  that lasts forever
• Most common finance example valuing preferred
  stock
• Can modify the basic PVAn formula for n ?
  (infinity)
• The summation term reduces to 1/r, so PVA?
  simplifies to
• PVA? PMT x 1/r
• Assume a preferred stock pays 1.5/share, and the
  appropriate discount rate is r 0.07. Find
  stocks PV
• PVA? PMT x 1/r 1.5 x
  (14.286) 21.43

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Compounding More Frequently Than Annually

• Can compute interest with semi-annual, quarterly,
  monthly (or more frequent) compounding periods
• Semi-annual interest computed twice per year
• Quarterly interest computed four times per year
• To change basic FV formula to m compounding
  periods
• Divide interest rate r by m and
• Multiply number of years n by m
• Basic FV formula becomes

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Demonstrating Compounding More Frequently Than
Annually

• Find FV at end of 2 years of 125,000 deposited
  at 5.13 percent interest
• For semiannual compounding, m equals 2
• For quarterly compounding, m equals 4

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Continuous Compounding

• In the extreme case, interest paid can be
  compounded continuously
• In this case, m approaches infinity, and the
  exponential function e (where e 2.7183) is
  used
• The FV formula for continuous compounding
  becomes
• FVn PV x (rxn)
• Use this to find value at the end of two years of
  100 invested at 8 annual interest, compounded
  continuously
• FVn 100 x (e0.08x2) 100 (2.71830.16)
  117.35

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A Basic Result The More Frequent The Compounding
Period, The Larger The FV

• FV of 100 at end of 2 years, invested at 8
  annual interest, compounded at the following
  intervals
• Annually FV 100 (1.08)2 116.64
• Semi-annually FV 100 (1.04)4 116.99
• Quarterly FV 100 (1.02)8 117.17
• Monthly FV 100 (1.0067)24 117.30
• Continuously FV 100 (e 0.16) 117.35

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The Nominal (Stated) Annual Rate Versus The
Effective (True) Annual Interest Rate

• Nominal, or stated, rate is the contractual
  annual rate charged by a lender or promised by a
  borrower
• Does not reflect compounding frequency
• Effective rate the annual rate actually paid or
  earned
• Does reflect compounding frequency
• Can make substantial difference at high interest
  rates. Credit cards often charge 1.5 per month
• Looks like 12 months/year x 1.5/month 18 per
  year
• Actual rate (1.015)12 1.1956-1 0.1956
  19.56 per year

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Effective Rates Are Always Greater Than Or Equal
To Nominal Rates

• For annual compounding, effective nominal
• For semi-annual compounding
• For quarterly compounding

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Special Applications Of Time Value Deposits
Needed To Accumulate A Future Sum

• Frequently need to determine the annual deposit
  needed to accumulate a fixed sum of money so many
  years since
• This is closely related to the process of finding
  the future value of an ordinary annuity
• Can find the annual deposit required to
  accumulate FVAn dollars, given a specified
  interest rate, r, and a certain number of years,
  n by solving this equation for PMT

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Calculating Deposits Needed To Accumulate A
Future Sum

• Suppose a person wishes to buy a house 5 years
  from nowand estimates an initial down payment of
  35,000 will berequired at that time.
• She wishes to make equal annual end-of-year
  deposits in an account paying annual interest of
  4 percent, so she must determine what size
  annuity will result in a lump sum equal to
  35,000 at the end of year 5.
• Find the annual deposit required to accumulate
  FVAn dollars, given an interest rate, r, and a
  certain number of years, n by solving equation
  PMT

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A Loan Amortization Table
Loan Amortization Schedule (6,000 Principal, 10
Interest 4 Year Repayment Period
Payments
End of year
Beginning-of-year principal(2)
End-of-year principal(2) (4)(5)
Interest.10 x (2)(3)
Loan Payment(1)
Principal(1) (3)(4)
aDue to rounding, a slight difference (.40)
exists between beginning-of-year 4 principal (in
column 2) and the year-4 principal payment (in
column 4)
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Determining 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, assume you invested 1,000 in a
  mutual fund in 1997 which grew as shown in the
  table below.
• What compound growth rate did this investment
  achieve?

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 Growth Rates (Continued)

• This chart shows that 1,000 is the present
  value, the future
• value is 5,525, and the number of periods is
  6.
• Want to find the rate, r, that would cause
  1,000 to grow to
• 5,525 over a six-year compounding period.
• Use FV formula FV PV x (1r)n
  5,5251,000 x (1r)6
• Simplify rearrange (1r)6 5,525 ? 1,000
  5.525
• Find sixth root of 5.525 (Take yx, where
  x0.16667), subtract 1
• Find r 0.3296, so growth rate 32.96

Excel Function Rate(periods, pmt, PV,
FV) Rate(6, ,1000, 5525)
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Much Of Finance Involves Finding Future And
(Especially) Present Values

• Central To All Financial Valuation Techniques
• Techniques Used By Investors Firms Alike
• Chapter 4 Bond Stock Valuation
• Chapters 7-9 Capital Budgeting