HFT 4464

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HFT 4464

• Chapter 5
• Time Value of Money

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Chapter 5Introduction

• This chapter introduces the topic of financial
  mathematics also known as the time value of
  money.
• This is a foundation topic relevant to many
  finance decisions for a hospitality firm
• Capital budgeting decisions
• Cost of capital estimation
• Pricing a bond issuance

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Organization of Chapter

• Financial math will be presented in the context
  of personal finance. Later chapters will apply
  financial math to various finance applications
  for a hospitality firm.
• Financial math topics covered include
• Computation of future values, present values,
  annuity payments, interest rates, etc.
• Perpetuities and non-constants cash flows
• Effective annual rates and compounding periods
  other than annual

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Future Value of a Lump Sum

• The future value in 2 years of 1,000 earning 5
  annually is an example of computing the future
  value of a lump sum. We can compute this in any
  one of three ways
• Using a calculator programmed for financial math
• Solve the mathematical equation
• Using financial math tables (Table 5.1) p. 91

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Solve for the Future Value

• The general equation for future value is
• FVn PV x (1i)n
• Computing the future value in the example
• FV2 1,000 x (15)2 1,102.50

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Present Value of a Lump Sum

• How much do you need to invest today so you can
  make a single payment of 30,000 in 18 years if
  the interest rate is 8? This is an example of
  the present value of a lump sum.
• Again we can solve it using a programmed
  calculator, solving the math or using Table 5.2.
  p.93

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Solve for the Present Value

• The general equation for present value is
• Computing the present value in the example

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Annuities

• Two or more periodic payments
• All payments are equal in size.
• Periods between each payment are equal in length.

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PV Versus FV of an Annuity

• The value of an annuity can be expressed as an
  equivalent lump sum value.
• The PV of an annuity is the lump sum value of an
  annuity at a point in time earlier than the
  payments.
• The FV of an annuity is the lump sum value of an
  annuity at a point in time later than the
  payments.

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Ordinary Annuity Versus Annuity Due

• The PV of an ordinary annuity is located one
  period before the first annuity payment. (Paid at
  the end of each period)
• The PV of an annuity due is located on the same
  date as the first annuity payment. (Paid at the
  beginning of each period)
• The FV of an ordinary annuity is located on the
  same date as the last annuity payment. (Last
  annuity payment is at the end of the last period)
• The FV of an annuity due is located one period
  after the last annuity payment. ( Last annuity
  payment is at the beginning of the last period)

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Future Value of an Annuity

• Suppose you plan to deposit 1,000 annually into
  an account at the end of each of the next 5
  years. If the account pays 12 annually, what is
  the value of the account at the end of 5 years?
  This is a future value of an annuity example.
• We can solve this problem using a programmed
  calculator, solving the math, or using Table 5.4.
  (p. 96)

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Solve for the Future Value of an Annuity

• The general equation for a FV of an annuity is
• The FV of the annuity in the example is

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Present Value of an Annuity

• You plan to withdraw 1,000 annually from an
  account at the end of each of the next 5 years.
  If the account pays 12 annually, what must you
  deposit in the account today? This is an example
  of a present value of an annuity.
• We can solve this problem using a programmed
  calculator, solving the math, or using Table 5.5.
  ( p. 102)

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Solve for the Present Value of an Annuity

• The general equation for PV of an annuity is
• The PV of the annuity in the example is

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PerpetuityAn Infinite Annuity

• A perpetuity is essentially an infinite annuity.
• An example is an investment which costs you
  1,000 today and promises to return to you 100
  at the end of each forever!
• What is your rate of return or the interest rate?

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The Present Value of a Perpetuity

• Another investment pays 90 at the end of each
  year forever. If 10 is the relevant interest
  rate, what is the value of this investment to you
  today? We need to solve for the present value of
  the perpetuity.

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Present Value of a Deferred Annuity

• There are 3 different PV of annuity computations
• The payments on an ordinary annuity begin one
  period after the PV.
• The payments on an annuity due begin on the same
  date as the PV.
• The payments on a deferred annuity begin 2 or
  more periods after the PV. Thus it is called a
  deferred annuity since the payments are deferred
  more than one period from the present.

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Computing the PV of a Deferred Annuity

• An investment promises to pay 100 annually
  beginning at the end of 5 years and continuing
  until the end of 10 years. What is the value of
  this investment today at a 7 interest rate?
  Because the payments are deferred 5 years, this
  is a PV of deferred annuity problem.
• 1st stepCompute the PV of an ordinary annuity.

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Computing the PV of a Deferred Annuity

• 2nd step Discount the PV of the ordinary
  annuity through deferral period.

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General Formula for PV of a Deferred Annuity

• PMT amount of the perpetuity payment
• i interest rate
• n the number of perpetuity payments
• m the deferral period minus 1

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PV of a Series of Non-Constant Cash Flows

• The PV of a series of non-constant cash flows is
  just the sum of the individual PV equations for
  each cash flow.
• Where the Cfis are a series of non-constant cash
  flows from year 1 to year n.

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PV of a Series of Non-Constant Cash Flows

• Suppose some new kitchen equipment for your
  restaurant is expected to save you 1,000 in 1
  year, 750 in 2 years, and 500 in 3 years. What
  is the PV of these cost savings today if 10 is
  the relevant interest rate?

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Compounding Periods Other Than Annual

• Future value of a lump sum.
• inom nominal annual interest rate
• m number of compounding periods per year
• n number of years

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Compounding Periods Other Than Annual

• A 1,000 investment earns 6 annually compounded
  monthly for 2 years.

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Compounding Periods Other Than Annual

• PV of a lump sum uses a similar adjustment to the
  basic equation for non-annual compounding.
• inom nominal annual interest rate
• m number of compounding periods per year
• n number of years

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Compounding Periods Other Than Annual

• Annuity computations require the annuity period
  and the compounding period to be the same.
• For example, suppose a car loan for 12,000
  required 20 equal monthly payments and uses a 12
  annual rate compounded monthly.
• The annuity payments and the compounding periods
  are both monthly.
• The interest rate needs to be expressed as a
  monthly rate
• I 12 / 12 1

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Compounding Periods Other Than Annual

• The car loan payment can be computed with the
  following equation
• And the car loan payment 664.98.

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Effective Annual Rate

• An effective annual rate is an annual compounding
  rate. When compounding periods are not annual,
  the rate can still be expressed as an effective
  annual rate using the following
• inom nominal annual rate
• m number of compounding periods in 1 year

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Effective Annual Rate

• A bank offers a certificate of deposit rate of 6
  annually compounded monthly. What is the
  equivalent effective annual rate?

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Amortized Loans

• Amortized loans are paid off in equal payments
  over a set period of time and can be viewed as
  the PV of an ordinary annuity.
• An amortization schedule follows for a 120,000
  mortgage to be paid of with 360 monthly payments
  of 965.55 each over 30 years. The interest rate
  is 9 annually compounded monthly or 0.75 per
  month.

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Loan Amortization Schedule
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Loan Amortization Schedule

• A loan amortization schedule shows
• The amount of each payment apportioned to pay
  interest. The amount paid towards interest
  declines since the principal balance is
  declining.
• The amount of each payment apportioned to pay
  principal balance. The amount paid towards
  principal balance increases as the interest
  amount declines.
• The remaining balance after each payment.

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Summary

• FV PV of a lump sum
• FV PV of an annuity and PV of a perpetuity
• PV of a series of non-constant cash flows
• Compounding other than annual and effective
  annual rates
• Loan amortization schedule

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Homework Problems
1,2,3,4,13 and Problem Sheet
on the website