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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.

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Financial Engineering

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  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

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