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Bond Price Volatility

Bond Price Volatility. Zvi Wiener Based on Chapter 4 in Fabozzi Bond Markets, Analysis and Strategies. You Open a Bank!. You have 1,000 customers. Typical CD is for 1-3 months with $1,000. You pay 5% on these CDs. A local business needs a $1M loan for 1 yr.

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Bond Price Volatility

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  1. Bond Price Volatility Zvi Wiener Based on Chapter 4 in Fabozzi Bond Markets, Analysis and Strategies http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

  2. You Open a Bank! • You have 1,000 customers. • Typical CD is for 1-3 months with $1,000. • You pay 5% on these CDs. • A local business needs a $1M loan for 1 yr. • The business is ready to pay 7% annually. • What are your major sources of risk? • How you can measure and manage it? Fabozzi Ch 4

  3. Original plan New plan Market shock Money Manager value 0 t D Fabozzi Ch 4

  4. 8% Coupon Bond Zero Coupon Bond Fabozzi Ch 4

  5. Price-Yield for option-free bonds price yield Fabozzi Ch 4

  6. Taylor Expansion • To measure the price response to a small change in risk factor we use the Taylor expansion. • Initial value y0, new value y1, change y: Fabozzi Ch 4

  7. Derivatives F(x) x Fabozzi Ch 4

  8. Properties of derivatives Fabozzi Ch 4

  9. Zero-coupon example Fabozzi Ch 4

  10. Example y=10%, y=0.5% T P0 P1 P 1 90.90 90.09 -0.45% 2 82.64 81.16 -1.79% 10 38.55 35.22 -8.65% Fabozzi Ch 4

  11. Property 1 • Prices of option-free bonds move in OPPOSITE direction from the change in yield. • The price change (in %) is NOT the same for different bonds. Fabozzi Ch 4

  12. Property 2 • For a given bond a small increase or decrease in yield leads very similar (but opposite) changes in prices. • What does this means mathematically? Fabozzi Ch 4

  13. Property 3 • For a given bond a large increase or decrease in yield leads to different (and opposite) changes in prices. • What does this means mathematically? Fabozzi Ch 4

  14. Property 4 • For a given bond a large change in yield the percentage price increase is greater than the percentage decrease. • What does this means mathematically? Fabozzi Ch 4

  15. What affects price volatility? • Linkage • Credit considerations • Time to maturity • Coupon rate Fabozzi Ch 4

  16. Bond Price Volatility • Consider only IR as a risk factor • Longer TTM means higher volatility • Lower coupons means higher volatility • Floaters have a very low price volatility • Price is also affected by coupon payments • Price value of a Basis Point (PVBP)= price change resulting from a change of 0.01% in the yield. Fabozzi Ch 4

  17. Duration and IR sensitivity Fabozzi Ch 4

  18. 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? Fabozzi Ch 4

  19. Duration • The bond price volatility is proportional to the bond’s duration. • Thus duration is a natural measure of interest rate risk exposure. Fabozzi Ch 4

  20. Modified Duration • The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity. Fabozzi Ch 4

  21. Duration Fabozzi Ch 4

  22. Duration Fabozzi Ch 4

  23. Duration Fabozzi Ch 4

  24. Measuring Price Change Fabozzi Ch 4

  25. The Yield to Maturity • The yield to maturity of a fixed coupon bond y is given by Fabozzi Ch 4

  26. Macaulay Duration • Definition of duration, assuming t=0. Fabozzi Ch 4

  27. Macaulay Duration • What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon. Fabozzi Ch 4

  28. $ r Meaning of Duration Fabozzi Ch 4

  29. upward move Current TS Downward move Parallel shift r T Fabozzi Ch 4

  30. 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 Fabozzi Ch 4

  31. Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity 0 3m 6m 1yr 3yr 5yr 10yr 30yr Fabozzi Ch 4

  32. 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 Fabozzi Ch 4

  33. 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. Fabozzi Ch 4

  34. What Determines Duration? • Holding the coupon rate constant, duration not always increases with ttm. Fabozzi Ch 4

  35. $ r Convexity Fabozzi Ch 4

  36. Example • 10 year zero coupon bond with a semiannual yield of 6% The duration is 10 years, the modified duration is: The convexity is Fabozzi Ch 4

  37. Example If the yield changes to 7% the price change is Fabozzi Ch 4

  38. FRM-98, Question 17 • A bond is trading at a price of 100 with a yield of 8%. If the yield increases by 1 bp, the price of the bond will decrease to 99.95. If the yield decreases by 1 bp, the price will increase to 100.04. What is the modified duration of this bond? • A. 5.0 • B. -5.0 • C. 4.5 • D. -4.5 Fabozzi Ch 4

  39. FRM-98, Question 17 Fabozzi Ch 4

  40. FRM-98, Question 22 • What is the price of a 10 bp increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50? • A. -0.705 • B. -0.700 • C. -0.698 • D. -0.690 Fabozzi Ch 4

  41. FRM-98, Question 22 Fabozzi Ch 4

  42. Portfolio Duration • Similar to a single bond but the cashflow is determined by all Fixed Income securities held in the portfolio. Fabozzi Ch 4

  43. Bond Price Derivatives • D* - modified duration, dollar duration is the negative of the first derivative: Dollar convexity = the second derivative, C - convexity. Fabozzi Ch 4

  44. Duration of a portfolio Fabozzi Ch 4

  45. ALM Duration • Does NOT work! • Wrong units of measurement • Division by a small number Fabozzi Ch 4

  46. Duration Gap • A - L = C, assets - liabilities = capital Fabozzi Ch 4

  47. ALM Duration • A similar problem with measuring yield Fabozzi Ch 4

  48. Do not think of duration as a measure of time! Fabozzi Ch 4

  49. Key rate duration • Principal component duration • Partial duration Fabozzi Ch 4

  50. Very good question! • Cashflow: • Libor in one year from now • Libor in two years form now • Libor in three years from now (no principal) • What is the duration? Fabozzi Ch 4

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