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## Your monthly payments

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**Your monthly payments**No arbitrage pricing.**Key concepts**• Real investment • Financial investment**Interest rate defined**• Premium for current delivery**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.**A typical bond**Note: Always start with the time line.**Definitions**• Coupon -- the amount paid periodically • Coupon rate -- the coupon times annual payments divided by 1000**Two parts of a bond**• Principal paid at maturity. • A repeated constant flow -- an annuity**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.**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**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.**Yogi Berra on finance**• Cut my pizza in four slices, please. I’m not hungry enough for six.**Why use interest rates?**• In addition to prices? • Answer: Coherence**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.**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%)**Calculations**• 652.92095 = 1000 / (1+.089)5 • Note: ^ is spreadsheet notation for raising to a power • 426.30576 = 1000 / (1+.089)10**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/(1+r)10**Annuity**• Interest rate per period, r. • Size of cash flows, C. • Maturity T. • If T=infinity, it’s called 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 ATr**Explanation**• Annuity = • difference in perpetuities. • One starts at time 1, • the other starts at time T + 1. • Value = difference in values (no arbitrage).**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/(1+r)T] in time zero dollars. • Difference is PVAF(r,T)= (1/r)[1-1/(1+r)T]**Compounding**• 12% is not 12% … ? • … when it is compounded.**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)365 -1 = .0865314**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?**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