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.
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.
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) =
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]* =