Annuities and No Arbitrage Pricing

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Annuities and 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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equation of the budget constraint
Time one cash flow
status quo
Time zero cash flow
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Financing possibilities, not physical investment
Time one cash flow
With- drawal
deposit
Time zero cash flow
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An investment opportunity that increases value.
Time one cash flow
NPV
Time zero cash flow
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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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Why use interest rates

• Instead of just prices
• Coherence

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Example pure discount bond

• Definition A pure discount bond pays 1000 at
  maturity and has no interest payments before
  then.
• Price is the PV of that 1000 cash flow, using the
  market rate specific to the asset.

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Example continued

• Ten-year discount bond price is 426.30576
• Five-year discount bond price is 652.92095
• Are they similar or different?
• Similar because they have the SAME 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 - 1
  .09The .1 power is the tenth root.
• The longer bond has a higher interest rate. Why?
• Because more time means more risk.

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A typical bond
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Definitions

• Coupon -- the amount paid periodically
• Coupon rate -- the coupon times annual payments
  divided by 1000
• Same as for mortgage payments

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Pure discount bonds on 1/9/02
Matures Ask Ask yield
Feb 04 9820 1.27
Feb 05 9613 1.75
Feb 06 9312 2.23
Feb 07 8915 2.73
Feb 08 8517 3.09
Feb 09 8015 3.46
Feb10 7630 3.73
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No arbitrage principle

• Market prices must admit no profitable, risk-free
  arbitrage.
• No money pumps.
• Otherwise, acquisitive investors would exploit
  the arbitrage indefinitely.

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Example

• Coupons sell for 450
• Principal sells for 500
• The bond MUST sell for 950.
• Otherwise, an arbitrage opportunity exists.
• For instance, if the bond sells for 920
• Buy the bond, sell the stripped components.
  Profit 30 per bond, indefinitely.
• Similarly, if the bond sells for 980

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Two parts of a bond

• Pure discount bond
• A repeated constant flow -- an annuity

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Stripped coupons and principal

• Treasury notes (and some agency bonds)
• Coupons (assembled) sold separately, an annuity.
• Stripped principal is a pure discount bond.

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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

• Start with 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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Justification
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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 ArT

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Explanation

• Value of annuity
• difference in values of perpetuities.
• One starts at time 1,
• the other starts at time T 1.

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Explanation
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Values

• P.V. of Perp at 0 1/r
• P.V. of Perp at T (1/r) 1/(1r)T
• Value of annuity difference (1/r)1-1/(1r)T
  

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Compounding

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

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Compounding 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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When to cut a tree

• Application of continuous compounding
• A tree growing in value.
• The land cannot be reused.
• Discounting continuously.
• What is the optimum time to cut the tree?
• The time that maximizes NPV.

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Numerical example

• Cost of planting 100
• Value of tree -10025t
• Interest rate .05
• Maximize (-10025t)exp(-.05t)
• Check second order conditions
• First order condition .05 25/(-10025t)
• t 24 value 500

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Example continued

• Present value of the tree 500exp(-.0524)
  150.5971.
• Greater than cost of 100.
• NPV 50.5971
• Market value of a partly grown tree at time t lt
  24 is 150.5971exp(.05t)
• For t gt 24 it is -10025t

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Example Cost of College

• Annual cost 25000
• Paid when?
• Make a table of cash flows

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Timing

• Obviously simplified

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Present value at time zero

• 2525PVAF(.06,3)
• 91.825298

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Spreadsheet confirmation
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Saving for college

• Start saving 16 years before matriculation.
• How much each year?
• Make a table.

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The college savings problem
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Solution outlined

• Find PV of target sum, that is, take 91.825 and
  discount back to time 0.
• Divide by (1.06)16
• PV of savings CCPVAF(.06,16)
• Equate and solve for C.

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Numerical Solution

• PV of target sum 36.146687
• PV of savings CC10.105895
• C 3.2547298

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Balance C 1.06 previous balance
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Alternative solution outlined

• Need 91.825 at time 16.
• FV of savings (1.06)16 (CCPVAF(.06,16))
• Equate and solve for C.

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Numerical Solution

• Future target sum 91.825
• FV of savings (1.06)16(CC10.105895)
• 91.825 C((1.06)16)(110.105895)
• C 3.2547298

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Review 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 in two steps

• Step 1. Find PDV of the annuity.
• .005 per month
• 120 months
• PVAF 90.073451
• PVAF100 9007.3451
• Step 2. Translate to money of time 120.
• (1.005)1209007.3451 16387.934