Time Value of Money

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• Chapter 2
• Time Value of Money

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TIME LINES

• Does money have time value? Why?
• A dollar today is worth more than a dollar a
  year from now
• A time line specifies the amount, sign, and
  timing of a series of cash flows

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FUTURE VALUE OF A SINGLE DEPOSIT

• FVn PV(1 i)n
• If you invest 5,000 today in an account earning
  8 interest, how much will it grow to in nine
  years?
• Algebraic solution?
• Calculator solution?

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• How much of the future value is invested
  principal?
• How much of the future value is earned
  interest?
• What is another name for finding future value?

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PRESENT VALUE OF A FUTURE RECEIPT

• PV FVn/(1 i)n
• How much must you invest today into an account
  earning 6 interest to have 1,000 five years
  from now?
• Algebraic solution?

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• Calculator solution?
• What is another name for finding present value?

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SOLVING FOR INTEREST RATE OR TIME

• Suppose a security will cost 78.35 today, and
  will return 100 after 5 years. What is the
  annual return?
• What is the algebraic solution?

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• How long would it take an initial investment of
  50 to double if you are earning 10?
• Do either of these problems have algebraic
  solutions? No How are they solved?

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FUTURE VALUE OF AN ANNUITY

• What is an annuity?
• A series of payments that are equal in size,
  equally-spaced over time, and eventually
  end
• How does an annuity due differ from an ordinary
  annuity?
• With an annuity due, pmts begin today (i.e., are
  at the start of each period)

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Future Value of An (Ordinary) Annuity

• If you deposit 1,000 at the end of each year
  into an account earning 10, how much will you
  have in 5 years?

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Future Value of An Annuity Due

• How much will you have in 5 years in the prior
  problem if the deposits are made at the start of
  each year?

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• Which type of annuity produces a greater future
  value? Annuity due
• Why? Because each deposit has a longer time to
  earn interest.

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Present Value of an Ordinary Annuity

• How much must you deposit today in an account
  earning 12 to be able to withdraw 2,000 at the
  end of each year for the next three years?

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Present Value of An Annuity Due

• How much must you deposit today in the previous
  problem if withdrawals are made at the start of
  each year?

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• Which type of annuity produces the greater
  present value? annuity due
• Why? Because you receive the CFs sooner worth
  more

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PERPETUITY

• What are the characteristics of a perpetuity?
  Same as an ordinary annuity, but the payments
  never end
• How much must you deposit today in an account
  earning 8 to be able to withdraw 1,000 at the
  end of each year forever?

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FINDING PRESENT VALUE FOR UNEVEN CASH FLOW STREAMS

• What is the present value of the following cash
  flows at 7 interest?
• CF1 200 CF2 100
• CF3 100 CF4 100

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SEMIANNUAL AND OTHER COMPOUNDING PERIODS

• If you invest 1,000 in an account earning 8
  interest compounded semiannually, how big will
  the account be in two years?
• How big will it be in ten years?

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• What would it be worth in ten years if the
  interest is compounded quarterly?
• Why is this answer greater? The account is
  earning interest on interest more frequently.

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Nominal vs Effective Rate

• EFF ( or EAR) (1 iNOM/m)m 1.0
• If you are earning 8 compounded quarterly, what
  is the equivalent in terms of an annually
  compounded rate?
• What is the periodic rate?

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• The nominal, quoted, or APR rate ignores
  accelerated compounding (i.e., it assumes your
  account grows only by 8 per year in this
  problem)
• The effective rate (EAR, or EFF) incorporates the
  effect of accelerated compounding (i.e., it
  assumes you account will grow by 8.24 per year)

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AMORTIZED LOANS

• What is the payment structure of an amortized
  loan? The loan is repaid in equal periodic
  payments that are each part repayment of
  principal and part interest expense

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• Assume you borrow 10,000 at 10 to be repaid in
  three equal annual end-of-year installments.
  What is the size of each annual payment?

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• What is the amortization table for this loan?
• Annual Repay of Loan
• N Pmt IntExp Prin. Balance
• 1 4021 1000 3021 6979
• 2 4021
• 3 4021 366 3656 0

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• Why do we need to find interest expense for each
  year? Because for businesses or homeowners the
  interest portion of each payment is tax deductible