1. Your monthly payments No arbitrage pricing.

2. Key concepts  Real investment  Financial investment

3. Interest rate defined  Premium for current delivery

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

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

6. A typical bond Note: Always start with the time line.

7. Definitions  Coupon -- the amount paid periodically  Coupon rate -- the coupon times annual payments divided by 1000

8. Two parts of a bond  Principal paid at maturity.  A repeated constant flow -- an annuity

9. 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.

10. 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

11. 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.

12. Yogi Berra on finance  Cut my pizza in four slices, please. I’m not hungry enough for six.

13. Why use interest rates?  In addition to prices?  Answer: Coherence

14. 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.

15. 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%)

16. Calculations  652.92095 = 1000 / (1+.089) 5  Note: ^ is spreadsheet notation for raising to a power  426.30576 = 1000 / (1+.089) 10

17. 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

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

19. Market value of a perpetuity

20. 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.

21. Finished here 1/12/06

22. 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 T r

23. Explanation  Annuity =  difference in perpetuities.  One starts at time 1,  the other starts at time T + 1.  Value = difference in values (no arbitrage).

24. Explanation

25. 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 ]

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

27. E.A.R. Equivalent Annual rate

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

29. Solution  Compare the equivalent annual rates  World Savings: EAR =.0865  Wells Fargo: (1+.083/365) 365 -1 =.0865314

30. 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?

31. Answer t=012…120 CF0100 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