Your monthly payments

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Your monthly payments

• No arbitrage pricing.

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Key concepts

• Real investment
• Financial investment

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Interest rate defined

• Premium for current delivery

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Basic principle

• Firms maximize value
• Owners maximize utility
• Separately

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Justification

• Real investment with positive NPV shifts
  consumption opportunities outward.
• Financial investment satisfies the owners time
  preferences.

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A typical bond
Note Always start with the time line.
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Definitions

• Coupon -- the amount paid periodically
• Coupon rate -- the coupon times annual payments
  divided by 1000

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Two parts of a bond

• Principal paid at maturity.
• A repeated constant flow -- an annuity

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Strips

• U.S. Treasury bonds
• Stripped coupon is an annuity
• Stripped principal is a payment of 1000 at
  maturity and nothing until then.
• Stripped principal is also called a pure discount
  bond, a zero-coupon bond, or a zero, for short.

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No arbitrage condition

• Price of bond price of zero-coupon bond
  price of stripped coupon.
• Otherwise, a money machine, one way or the other.
• Riskless increase in wealth

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Pie theory

• The bond is the whole pie.
• The strip is one piece, the zero is the other.
• Together, you get the whole pie.
• No arbitrage pricing requires that the values of
  the pieces add up to the value of the whole pie.

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Yogi Berra on finance

• Cut my pizza in four slices, please. Im not
  hungry enough for six.

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Why use interest rates?

• In addition to prices?
• Answer Coherence

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Example discount bonds

• A zero pays 1000 at maturity.
• Price (value) is the PV of that 1000 cash flow,
  using the market rate specific to the asset and
  maturity.

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Example continued

• Ten-year maturity price is 426.30576
• Five-year maturity price is 652.92095
• Similar or different?
• They have the SAME discount rate (interest rate)
  r .089 (i.e. 8.9)

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Calculations

• 652.92095 1000 / (1.089)5
• Note is spreadsheet notation for raising to a
  power
• 426.30576 1000 / (1.089)10

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More realistically

• For the ten-year discount bond, the price is
  422.41081 (not 426.30576).
• The ten-year rate is (1000/422.41081)1/10 - 1
  .09.
• The 1/10 power is the tenth root.
• It solves the equation 422.41081
  1000/(1r)10

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Annuity

• Interest rate per period, r.
• Size of cash flows, C.
• Maturity T.
• If Tinfinity, its called a perpetuity.

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Market value of a perpetuity
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Value of a perpetuity is C(1/r)

• In spreadsheet notation, is the sign for
  multiplication.
• Present Value of Perpetuity Factor, PVPF(r)
  1/r
• It assumes that C 1.
• For any other C, multiply PVPF(r) by C.

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Finished here 1/12/06
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Value of an annuity

• C (1/r)1-1/(1r)T
• Present value of annuity factor
• PVAF(r,T) (1/r)1-1/(1r)T
• or ATr

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Explanation

• Annuity
• difference in perpetuities.
• One starts at time 1,
• the other starts at time T 1.
• Value difference in values (no arbitrage).

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Explanation
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Values

• Value of the perpetuity starting at 1 is 1/r
• in time zero dollars
• Value of the perpetuity starting at T 1 is
  1/r
• in time T dollars,
• or (1/r)1/(1r)T in time zero dollars.
• Difference is PVAF(r,T) (1/r)1-1/(1r)T

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Compounding

• 12 is not 12 ?
• when it is compounded.

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E.A.R. Equivalent Annual rate
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Example which is better?

• Wells Fargo 8.3 compounded daily
• World Savings 8.65 uncompounded

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Solution

• Compare the equivalent annual rates
• World Savings EAR .0865
• Wells Fargo (1.083/365)365 -1 .0865314

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Exam (sub) question

• The interest rate is 6, compounded monthly.
• You set aside 100 at the end of each month for
  10 years.
• How much money do you have at the end?

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Answer
Interest per period is .5 or .005.
Present value is PVAF(120,.005)100 9007.3451
Future value is 9007.3451(1.005)120 16387.934