NPV and the

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Chapter 4

• NPV and the
• Time Value of Money

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Chapter Outline

• 4.1 The Timeline
• 4.2 Solving Single Sum problems
• 4.3 Valuing a Stream of Cash Flows
• 4.4 The Net Present Value of a Stream of Cash
  Flows
• 4.5 Annuities and Perpetuities
• 5.2 Discount Rates and Loans
• Amortization Schedule

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Learning Objectives

• How to construct a cash flow timeline
• Calculate Present Value and Future Value of a
  Single Sum
• Value a series of cash flows and compute the net
  present value (NPV)
• Uneven Cash Flows
• Calculate Present Value and Future Value of
  multiple cash flows
• Annuities and Perpetuities
• Calculate loan payments and construct loan
  amortization schedules

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Types of Time Value Problems

• Single Sum One dollar amount
• Uneven Cash Flows A series of unequal cash flows
• Net Present Value (NPV)
• Annuity A series of equal cash flows for a
  specified number of periods at a constant
  interest rate
• Perpetuity An infinite series of equal payments

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Basic Terms

• Present Value (PV) Value as of today
• Future Value (FV) Value at some point in the
  future
• Interest rate (r) rate of return (FV) or
  discount rate (PV)
• Cost of capital
• Required return
• Number of Periods (n) - usually measured in
  months or years
• Payment (C) used for annuity problems

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4.1 The Timeline

• A series of cash flows lasting several periods is
  defined as a stream of cash flows. We can
  represent a stream of cash flows on a timeline, a
  linear representation of the timing of the
  expected cash flows.

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Constructing a Timeline 
Date 0 represents the present. Date 1 is one year
later
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Distinguishing Cash Inflows from Outflows
The first cash flow at date 0 (today) is
represented as negative because it is an outflow
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Example 4.1 Constructing a Timeline

• Suppose you must pay tuition of 10,000 per year
  for the next four years. Your tuition payments
  must be made in equal installments of 5,000 each
  every 6 months. What is the timeline of your
  tuition payments?

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Example 4.1 Constructing a Timeline

• Assuming today is the start of the first
  semester, your first payment occurs at date 0
  (today). The remaining payments occur at 6-month
  intervals. Using one-half year (6 months) as the
  period length, we can construct a timeline as
  follows

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4.2 Valuing Cash Flows at Different Points in Time

• Three Important Rules Central to Financial
  Decision Making
• Rule 1 Comparing and Combining Values
• Rule 2 Compounding Future Value
• Rule 3 Discounting Present Value

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Rule 1 Comparing and Combining Values

• A dollar today and a dollar in one year are not
  equivalent. Having money now is more valuable
  than having money in the future if you have the
  money today you can earn interest on it.

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Rule 2 Compounding

• The process of moving forward along the timeline
  to determine a cash flows value in the future
  (its future value) is known as compounding.

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Rule 2 Compounding

• We can apply this rule repeatedly. Suppose we
  want to know how much the 1,000 is worth in two
  years time. If the interest rate for year 2 is
  also 10, then

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Rule 2 Compounding

• This effect of earning interest on both the
  original principal plus the accumulated interest,
  so that you are earning interest on interest,
  is known as compound interest.
• Compounding the cash flow a third time, assuming
  the competitive market interest rate is fixed at
  10, we get

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Figure 4.1 The Compounding of Interest over Time
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Future Value of a Single SumText Book Formula
(Eq. 4.1)
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Rule 3 Discounting

• The process of finding the equivalent value today
  of a future cash flow is known as discounting.  

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Present Value of a Single Sum Text Book Formula
(Eq. 4.2)
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Single Sum Formulas Simplified

• No need to have C in both formulas. So we will
  replace it to simplify the equation and,
  hopefully, avoid using the wrong formula
• Future Value Formula
• FV PV x (1 r) n
• Present Value Formula
• PV FV / (1 r) n

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Future Value Example

• Suppose you purchased a house ten years ago for
  125,000 and houses in your neighborhood have
  increased in value by an average of 10 per year.
  What should your house be worth today?
• FV PV(1 r) n
• FV 125,000 (1.10) 10 324,218

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Example 4.2 Personal Finance Present Value of a
Single Sum

• You are considering investing in a savings bond
  that will pay 15,000 in ten years. If the
  competitive market interest rate is fixed at 6
  per year, what is the bond worth today?
• PV FV / (1 r) n
• PV 15,000 / (1.06) 10 8,375.92

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4.3 Valuing a Stream of Cash Flows

• Previous examples were easy to evaluate because
  there was one cash flow. What do we need to do if
  there are multiple cash flow?
• Equal Cash Flows Annuity or Perpetuity
• Unequal/Uneven Cash Flows

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4.3 Valuing a Stream of Cash Flows
? 1000 2000 3000

• Uneven cash flows are when there are different
  cash flow streams each year
• Treat each cash flow as a Single Sum problem and
  add the PV amounts together.

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4.3 Valuing a Stream of Cash Flows

• What is the present value of the preceding cash
  flow stream using a 12 discount rate?
• PV FV / (1 r) n
• Yr 1 1,000 / (1.12)1 893
• Yr 2 2,000 / (1.12)2 1,594
• Yr 3 3,000 / (1.12)3 2,135
• 4,622

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4.4 The Net Present Value of a Stream of Cash
Flows

• We can represent investment decisions on a
  timeline as negative cash flows.
• The NPV of an investment opportunity is also the
  present value of the stream of cash flows of the
  opportunity
• If the NPV is positive, the benefits exceed the
  costs and we should make the investment.

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4.4 The Net Present Value of a Stream of Cash
Flows

• You have been offered the following investment
  opportunity If you invest 1 million today, you
  will receive the cash flow stream below. If you
  could otherwise earn 12 per year on your money,
  should you undertake the investment opportunity?
• Yr 1 250,000
• Yr 2 150,000
• Yr 3 200,000
• Yr 4 300,000
• Yr 5 400,000
• Yr 6 500,000

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4.5 Annuities and Perpetuities

• Annuity A series of equal cash flows for a
  specified number of periods at a constant
  interest rate
• Ordinary annuity cash flows occur at the end of
  each period
• Annuity due cash flows occur at the beginning of
  each period (not covered in class)
• Perpetuity An infinite series of equal payments

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Examples of Annuities and Perpetuities

• Annuity
• If you buy a bond, you will receive equal coupon
  interest payments over the life of the bond.
• If you borrow money to buy a house or a car, you
  will pay a stream of equal payments.
• Amortization Schedule
• Perpetuity
• If you buy preferred stock, you will receive
  equal dividend payments forever.
• Assuming infinite life of corporation

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Annuity Formulas in Text
(Eq. 4.5)
(Eq. 4.6)
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Annuities Formulas - Simplified

• This presentation utilizes a simplified version
  of the formulas that eliminates the 1/r component

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Interest Rates and Time Periods

• Keep in mind that payments (C) are not always
  annual. Thus the interest rates (r) and number of
  periods (N) need to be adjusted if there is more
  than 1 payment per year. For example, here are
  interest rates and number of periods for a 5-year
  loan at 6 interest
• Annual Payments N 5 r 6.0
• Semi-Annual Payments N 10 r 3.0
• Quarterly Payments N 20 r 1.5
• Monthly Payments N 60 r 0.5

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Present Value Annuity Example

• You can afford to pay 632 per month towards a
  new car. If you can get a loan that charges 1
  interest per month and has a 40-month term, how
  much can you borrow?

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Present Value Annuity Example

• You can afford to pay 632 per month towards a
  new car. If you can get a loan that charges 1
  interest per month and has a 40-month term, how
  much can you borrow?

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Example 4.8 Personal Finance Retirement Savings
Plan Annuity

• Ellen is 35 years old, and she has decided it is
  time to plan seriously for her retirement. At the
  end of each year until she is 65, she will save
  10,000 in a retirement account. If the account
  earns 10 per year, how much will Ellen have
  saved at age 65?

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Example 4.8 Personal Finance Retirement Savings
Plan Annuity

• Ellen is 35 years old, and she has decided it is
  time to plan seriously for her retirement. At the
  end of each year until she is 65, she will save
  10,000 in a retirement account. If the account
  earns 10 per year, how much will Ellen have
  saved at age 65?

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Problems

• With 10,000 to invest and assuming a 10
  interest rate, how much will you have at the end
  of 5 years?
• Suppose you need 15,000 in 3 years. If you can
  earn 6 annually, how much do you need to invest
  today?
• Which of the following is better?
• Investing 500 at a 7 annual interest rate for 5
  years.
• Investing 500 today and receiving 750 in 5
  years.

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Problems

• An investment will provide you with 100 at the
  end of each year for the next 10 years. What is
  the present value of that annuity if the discount
  rate is 8 annually?
• 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?
• You are saving for a new house, and you put
  10,000 per year in an account paying 8. How
  much will you have at the end of 3 years?

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Present Value of a Perpetuity
(Eq. 4.4)

• PV is the value of the perpetuity today
• C is the recurring payment
• r is the required interest rate (not dividend
  rate)
• If we know any two variables, we can solve for
  the third

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Example 4.6 Personal Finance Endowing a
Perpetuity

• You want to endow an annual graduation party at
  your alma mater that is budgeted to cost 30,000
  per year forever. If the university can earn 8
  per year on its investments and the first party
  is in one years time, how much will you need to
  donate to endow the party?

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Example 4.6 Personal Finance Endowing a
Perpetuity

• You want to endow an annual graduation party at
  your alma mater that is budgeted to cost 30,000
  per year forever. If the university can earn 8
  per year on its investments and the first party
  is in one years time, how much will you need to
  donate to endow the party?

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Perpetuity Example Preferred Stock

• There is a preferred stock with a par value of
  25 and a dividend rate of 5 per year. If
  preferred stock with a similar risk profile is
  yielding 4, what price should the stock be
  trading at?

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Loan Payments and Amortization Schedules

• One of the most useful applications of the
  annuity formula is calculating the payment on a
  loan. To do so, we revise the annuity formula to
  solve for Payment (C)

(Eq. 4.8)
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Example 4.10Computing a Loan Payment

• Your firm plans to buy a warehouse for 100,000.
  The bank offers you a 30-year loan with equal
  annual payments and an interest rate of 8 per
  year. The bank requires that your firm pay 20 of
  the purchase price as a down payment, so you can
  borrow only 80,000. What is the annual loan
  payment?

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Example 4.10Computing a Loan Payment

• Eq. 4.8 calculates the payment as follows

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Loan Payments and Amortization Schedules

• In Example 4.10, our firm will need to pay
  7,106.19 each year to fully repay the loan over
  30 years.
• To understand how the loan will be repaid, we
  must create an Amortization Schedule for the loan.

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Amortization Schedule

• An Amortization Schedule contains five columns,
  as illustrated below
• The payment is typically monthly, quarterly, or
  annually, so we may need to adjust the interest
  rate and number of periods accordingly.

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Amortization ScheduleEqual Payments

• Payment
• Stays the Same
• Interest Expense
• Declines Over Time
• Principal Repayment
• Increases Over Time
• Balance of Loan
• Fully repaid by maturity

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Amortization ScheduleEqual Payments

• Example 4.10
• Purchase Warehouse for 100,000
• 20,000 Down Payment 80,000 Loan
• 30-year Term
• 8 Interest Rate
• Annual End of Year Payments

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Amortization Schedule

• Beginning Balance is Loan Amount
• Calculate Payment

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Amortization Schedule

• Calculate Interest
• Principal x Rate x Time (80,000)(8)(1 yr)
• Principal Payment - Interest
• Balance Prior Balance - Principal

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Amortization Schedule
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Amortization ExampleEqual Payments

• Create an amortization schedule for the first
  month based on the following terms
• Purchase Building for 1,000,000
• 200,000 Down Payment, Balance Borrowed
• 5 yr. Term
• 9 Interest Rate
• Monthly Payments

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Calculating Payment in Excel

• Excel can easily calculate the payment for a loan