Annuities and No Arbitrage Pricing

Key concepts  Real investment  Financial investment

Interest rate defined  Premium for current delivery

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

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

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

Basic principle  Firms maximize value  Owners maximize utility  Separately

Justification  Real investment with positive NPV shifts consumption opportunities outward.  Financial investment satisfies the owner’s time preferences.

Why use interest rates  Instead of just prices  Coherence

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.

Example continued  Ten-year discount bond: price is  Five-year discount bond: price is  Are they similar or different?  Similar because they have the SAME interest rate r =.089 (i.e. 8.9%)

Calculations  = 1000 / (1+.089)^5  Note: ^ is spreadsheet notation for raising to a power  = 1000 / (1+.089)^10

More realistically  For the ten-year discount bond, the price is (not ).  The ten-year rate is (1000/ )^ =.09 The.1 power is the tenth root.  The longer bond has a higher interest rate. Why?  Because more time means more risk.

A typical bond

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

Pure discount bonds on 1/9/02 MaturesAskAsk yield Feb 0498: Feb 0596: Feb 0693: Feb 0789: Feb 0885: Feb 0980: Feb1076:303.73

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

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 …

Two parts of a bond  Pure discount bond  A repeated constant flow -- an annuity

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

Annuity  Interest rate per period, r.  Size of cash flows, C.  Maturity T.  If T=infinity, it’s called a perpetuity.

Market value of a perpetuity  Start with a perpetuity.

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.

Justification

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

Explanation  Value of annuity =  difference in values of perpetuities.  One starts at time 1,  the other starts at time T + 1.

Explanation

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 ]

Compounding  12% is not 12% … ?  … when it is compounded.

Compounding: E.A.R. Equivalent Annual rate

Example: which is better?  Wells Fargo: 8.3% compounded daily  World Savings: 8.65% uncompounded

Solution  Compare the equivalent annual rates  World Savings: EAR =.0865  Wells Fargo: (1+.083/365) =

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.

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

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

Example: Cost of College  Annual cost =  Paid when?  Make a table of cash flows

Timing  Obviously simplified

Present value at time zero  25+25*PVAF(.06,3)  =

Spreadsheet confirmation

Saving for college  Start saving 16 years before matriculation.  How much each year?  Make a table.

The college savings problem

Solution outlined  Find PV of target sum, that is, take and discount back to time 0.  Divide by (1.06)^16  PV of savings =C+C*PVAF(.06,16)  Equate and solve for C.

Numerical Solution  PV of target sum =  PV of savings = C+C*  C =

Balance = C previous balance

Alternative solution outlined  Need at time 16.  FV of savings =(1.06)^16 *(C+C*PVAF(.06,16))  Equate and solve for C.

Numerical Solution  Future target sum =  FV of savings = (1.06)^16*(C+C* )  = C*((1.06)^16)*( )  C =

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?

Answer in two steps  Step 1. Find PDV of the annuity. .005 per month  120 months  PVAF =  PVAF*100 =  Step 2. Translate to money of time 120.  [(1.005)^120]* =