1 / 97

Financial Engineering

Financial Engineering. Interest Rates and Fixed Income Securities Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049. Bonds. A bond is a contract, paid up-front that yields a known amount at a known date (maturity). The bond may pay a dividend (coupon) at fixed times during the life.

virgo
Download Presentation

Financial Engineering

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. Financial Engineering Interest Rates and Fixed Income Securities Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 ContTimeFin - 6

  2. Bonds A bond is a contract, paid up-front that yields a known amount at a known date (maturity). The bond may pay a dividend (coupon) at fixed times during the life. Additional options: callable, puttable, indexed, prepayment options, etc. Credit risk, recovery ratio, rating. ContTimeFin - 6

  3. Term Structure of IR r long term IR short term IR spot rate time to maturity ContTimeFin - 6

  4. Known IR V - value of a contract. r(t) - short term interest rate. If there is no risk and no coupons then dV = rVdt V(t) = V(T)e-rt if there is a continuous dividend stream dV+cVdt = rVdt ContTimeFin - 6

  5. Known IR If r is not constant, but not risky r(t) dV = r(t)Vdt If there is a continuous dividend stream dV+c(t)Vdt = r(t)Vdt ContTimeFin - 6

  6. Known IR Assume that there are zero coupon bonds for all possible ttm (time to maturity). Denote the price of these bonds by V(t,T). ContTimeFin - 6

  7. Known IR ContTimeFin - 6

  8. Yield ContTimeFin - 6

  9. increasing humped decreasing Typical yield curves yield time to maturity ContTimeFin - 6

  10. Typical yield curves • increasing - the most typical. • decreasing - short rates are high but expected to fall. • humped - short rates are expected to fall soon. ContTimeFin - 6

  11. Term Structure Explanations Expectation hypothesis states F0=E(PT) this hypothesis is be true if all market participants were risk neutral. ContTimeFin - 6

  12. Term Structure Explanations Normal Backwardation (Keynes), commodities are used by hedgers to reduce risk. In order to induce speculators to take the opposite positions, the producers must offer a higher return. Thus speculators enter the long side and have the expected profit of E(PT) – F0 > 0 ContTimeFin - 6

  13. Term Structure Explanations Contango is similar to the normal backwardation, but the natural hedgers are the purchasers of a commodity, rather than suppliers. Since speculators must be paid for taking risk, the opposite relation holds: E(PT) – F0 < 0 ContTimeFin - 6

  14. 8% Coupon Bond Zero Coupon Bond ContTimeFin - 6

  15. Duration F. Macaulay (1938) Better measurement than time to maturity. Weighted average of all coupons with the corresponding time to payment. Bond Price = Sum[ CFt/(1+y)t ] suggested weight of each coupon: wt = CFt/(1+y)t /Bond Price What is the sum of all wt? ContTimeFin - 6

  16. Macaulay Duration A weighted sum of times to maturities of each coupon. What is the duration of a zero coupon bond? ContTimeFin - 6

  17. Macaulay Duration ContTimeFin - 6

  18. Macaulay Duration ContTimeFin - 6

  19. Duration Sensitivity to IR changes: • Long term bonds are more sensitive. • Lower coupon bonds are more sensitive. • The sensitivity depends on levels of IR. ContTimeFin - 6

  20. Duration The bond price volatility is proportional to the bond’s duration. Thus duration is a natural measure of interest rate risk exposure. ContTimeFin - 6

  21. Modified Duration The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity. ContTimeFin - 6

  22. Coupon bond with duration 1.8853 Price (at 5% for 6m.) is $964.5405 If IR increase by 1bp (to 5.01%), its price will fall to $964.1942, or 0.359% decline. Zero-coupon bond with equal duration must have 1.8853 years to maturity. At 5% semiannual its price is ($1,000/1.053.7706)=$831.9623 If IR increase to 5.01%, the price becomes: ($1,000/1.05013.7706)=$831.66 0.359% decline. Comparison of two bonds ContTimeFin - 6

  23. Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity 0 3m 6m 1yr 3yr 5yr 10yr 30yr ContTimeFin - 6

  24. Example A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P0 = $897.26 D = 11.37 Yrs if YTM = 9.1%, what will be the price? ContTimeFin - 6

  25. Example A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P0 = $897.26 D = 11.37 Yrs if YTM = 9.1%, what will be the price? P/P = - y D* P = -(y D*)P = -$9.36 P = $897.26 - $9.36 = $887.90 ContTimeFin - 6

  26. What Determines Duration? • Duration of a zero-coupon bond equals maturity. • Holding ttm constant, duration is higher when coupons are lower. • Holding other factors constant, duration is higher when ytm is lower. • Duration of a perpetuity is (1+y)/y. ContTimeFin - 6

  27. What Determines Duration? • Holding the coupon rate constant, duration not always increases with ttm. ContTimeFin - 6

  28. Duration ContTimeFin - 6

  29. ContTimeFin - 6

  30. ContTimeFin - 6

  31. ContTimeFin - 6

  32. Modern Approach Duration can be regarded as the discount-rate elasticity of the bond price ContTimeFin - 6

  33. Modern Approach Duration can be used to measure the price volatility of a bond: ContTimeFin - 6

  34. Modern Approach What are the natural bounds on duration? Can duration be bigger than maturity? Can duration be negative? How to measure duration of a portfolio? ContTimeFin - 6

  35. Duration: Modern Approach ContTimeFin - 6

  36. Duration of a Portfolio ContTimeFin - 6

  37. Duration of a Portfolio ContTimeFin - 6

  38. Modern Approach to Duration Simon Benninga, Financial Modelling, the MIT press, Cambridge, MA, ISBN 0-262-02437-3, $45 MIT Press tel: 800-356-0343 http://mitpress.mit.edu/book-home.tcl?isbn=0262024373 see also my advanced lecture notes on duration Convexity is a similar measurement but with second derivative. ContTimeFin - 6

  39. Financial Modellingby Simon Benninga • Implementation in Excel • Duration Patterns • Duration of a bond with uneven payments • Calculating YTM for uneven periods • Nonflat term structure and duration • Immunization strategies • Cheapest to deliver option and Duration ContTimeFin - 6

  40. Passive Bond Management Passive management takes bond prices as fairly set and seeks to control only the risk of the fixed-income portfolio. • Indexing strategy • attempts to replicate a bond index • Immunization • used to tailor the risk to specific needs (insurance companies, pension funds) ContTimeFin - 6

  41. Bond-Index Funds Similar to stock indexing. Major indices: Lehman Brothers, Merill Lynch, Salomon Brothers. Include: government, corporate, mortgage-backed, Yankee bonds (dollar denominated, SEC registered bonds of foreign issuers, sold in the US). ContTimeFin - 6

  42. Bond-Index Funds Properties: many issues not all are liquid replacement of maturing issues Tracking error is a good measurement of performance. According to Salomon Bros. With $100M one can track the index within 4bp. tracking error per month. ContTimeFin - 6

  43. Cellular approach ContTimeFin - 6

  44. Immunization Immunization techniques refer to strategies used by investors to shield their overall financial status from exposure to interest rate fluctuations. ContTimeFin - 6

  45. Net Worth Immunization Banks and thrifts have a natural mismatch between assets and liabilities. Liabilities are primarily short-term deposits (low duration), assets are typically loans or mortgages (higher duration). When will banks lose money, when IR increase or decline? ContTimeFin - 6

  46. Gap Management ARM are used to reduce duration of bank portfolios. Other derivative securities can be used. Capital requirement on duration (exposure). Basic idea: to match duration of assets and liabilities. ContTimeFin - 6

  47. Target Date Immunization Important for pension funds and insurances. Price risk and reinvestment risk. What is the correlation between them? ContTimeFin - 6

  48. Target Date Immunization Accumulated value Original plan 0 t* t ContTimeFin - 6

  49. Target Date Immunization Accumulated value IR increased at t* 0 t* t ContTimeFin - 6

  50. Target Date Immunization Accumulated value 0 t* D t ContTimeFin - 6

More Related