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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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= status quo Time zero cash flow Time one cash flow equation of the budget constraint:

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Time zero cash flow Time one cash flow Financing possibilities, not physical investment deposit With- drawal

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Time zero cash flow Time one cash flow An investment opportunity that increases value. NPV

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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 owner’s 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 =.09 The.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 MaturesAskAsk yield Feb 0498:201.27 Feb 0596:131.75 Feb 0693:122.23 Feb 0789:152.73 Feb 0885:173.09 Feb 0980:153.46 Feb1076:303.73

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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 T=infinity, it’s 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/(1+r)^T] Present value of annuity factor PVAF(r,T) = (1/r)[1-1/(1+r)^T] or A r T

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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/(1+r)^T Value of annuity = difference = (1/r)[1- 1/(1+r)^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 -100+25t Interest rate.05 Maximize (-100+25t)exp(-.05t) Check second order conditions First order condition.05 = 25/(-100+25t) t = 24 value = 500

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Example continued Present value of the tree = 500*exp(-.05*24) = 150.5971. Greater than cost of 100. NPV = 50.5971 Market value of a partly grown tree at time t < 24 is 150.5971*exp(.05*t) For t > 24 it is -100+25*t

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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 25+25*PVAF(.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 =C+C*PVAF(.06,16) Equate and solve for C.

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Numerical Solution PV of target sum = 36.146687 PV of savings = C+C*10.105895 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 *(C+C*PVAF(.06,16)) Equate and solve for C.

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Numerical Solution Future target sum = 91.825 FV of savings = (1.06)^16*(C+C*10.105895) 91.825 = C*((1.06)^16)*(1+10.105895) 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 PVAF*100 = 9007.3451 Step 2. Translate to money of time 120. [(1.005)^120]*9007.3451 = 16387.934

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