1 / 48

Annuities and No Arbitrage Pricing

Annuities and No Arbitrage Pricing. Key concepts. Real investment Financial investment. Interest rate defined. Premium for current delivery. equation of the budget constraint:. Time one cash flow. = status quo. Time zero cash flow. Financing possibilities, not physical investment.

iolana
Download Presentation

Annuities and No Arbitrage Pricing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Annuities and No Arbitrage Pricing

  2. Key concepts • Real investment • Financial investment

  3. Interest rate defined • Premium for current delivery

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

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

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

  7. Basic principle • Firms maximize value • Owners maximize utility • Separately

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

  9. Why use interest rates • Instead of just prices • Coherence

  10. 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.

  11. 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%)

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

  13. 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 = .09The .1 power is the tenth root. • The longer bond has a higher interest rate. Why? • Because more time means more risk.

  14. A typical bond

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

  16. Pure discount bonds on 1/9/02

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

  18. 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 …

  19. Two parts of a bond • Pure discount bond • A repeated constant flow -- an annuity

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

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

  22. Market value of a perpetuity • Start with a perpetuity.

  23. 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.

  24. Justification

  25. 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 ArT

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

  27. Explanation

  28. 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 ]

  29. Compounding • 12% is not 12% … ? • … when it is compounded.

  30. Compounding: E.A.R. Equivalent Annual rate

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

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

  33. 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.

  34. 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

  35. 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

  36. Example: Cost of College • Annual cost = 25000 • Paid when? • Make a table of cash flows

  37. Timing • Obviously simplified

  38. Present value at time zero • 25+25*PVAF(.06,3) • =91.825298

  39. Spreadsheet confirmation

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

  41. The college savings problem

  42. 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.

  43. Numerical Solution • PV of target sum = 36.146687 • PV of savings = C+C*10.105895 • C = 3.2547298

  44. Balance = C + 1.06 previous balance

  45. 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.

  46. 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

  47. 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?

  48. 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

More Related