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Fixed Income Markets - Part 2 Duration and convexity

Fixed Income Markets - Part 2 Duration and convexity. FIN 509: Foundations of Asset Valuation Class session 2 Professor Jonathan M. Karpoff. Sleeping Beauty bond case - Central points. Bond prices are sensitive to changes in interest rates

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Fixed Income Markets - Part 2 Duration and convexity

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  1. Fixed Income Markets - Part 2Duration and convexity FIN 509: Foundations of Asset Valuation Class session 2 Professor Jonathan M. Karpoff

  2. Sleeping Beauty bond case - Central points • Bond prices are sensitive to changes in interest rates • This sensitivity tends to be greater for longer term bonds • But duration is a better measure of term than maturity • Duration for 100-year bond = 14.24 • Duration for 30-year zero = 30 • Duration for 30-year coupon = 12.64 • Sleeping beauty bond has longer maturity but less sensitivity to interest rates than the 30-year zero bond • 30-year coupon and zero bonds have the same maturity but 30-year zero is more sensitive than the 30-year coupon bond Class session 2

  3. Duration and convexity: Outline • I. Macaulay duration • II. Modified duration • III. Examples • IV. The uses and limits of duration • V. Duration intuition • VI. Convexity • VII. Examples with both duration and convexity • VIII. Takeaways Class session 2

  4. I. (Macauly) duration • Weighted average term to maturity • Measure of average maturity of the bond’s promised cash flows • Duration formula: where: • t is measured in years Class session 2

  5. Duration - The expanded equation • Duration is shorter than maturity for all bonds except zero coupon bonds • Duration of a zero-coupon bond is equal to its maturity Class session 2

  6. II. Modified duration (D*m) • Direct measure of price sensitivity to interest rate changes • Can be used to estimate percentage price volatility of a bond Class session 2

  7. Derivation of modified duration • So D*m measures the sensitivity of the % change in bond price to changes in yield Class session 2

  8. III. An example • Compare the price sensitivities of: • Two-year 8% coupon bond with duration of 1.8853 years • Zero-coupon bond with maturity AND duration of 1.8853 years • Semiannual yield = 5% • Suppose yield increases by 1 basis point to 5.01% • Upshot: Equal duration assets are equally sensitive to interest rate movements Class session 2

  9. Another example • Consider a 3-year 10% coupon bond selling at $107.87 to yield 7%. Coupon payments are made annually. Class session 2

  10. Another example – page 2 • Modified duration of this bond: • If yields increase to 7.10%, how does the bond price change? • The percentage price change of this bond is given by: = –2.5661  .0010  100 = –.2566 Class session 2

  11. Another example – page 3 • What is the predicted change in dollar terms? New predicted price: $107.87 – .2768 = $107.5932 Actual dollar price (using PV equation):$107.5966 Good approximation! Class session 2

  12. Summary: Steps for finding the predicted price change • Step 1: Find Macaulay duration of bond. • Step 2: Find modified duration of bond. • Step 3: Recall that when interest rates change, the change in a bond’s price can be related to the change in yield according to the rule: • Find percentage price change of bond • Find predicted dollar price change in bond • Add predicted dollar price change to original price of bond  Predicted new price of bond Class session 2

  13. IV. Why is duration a big deal? • Simple summary statistic of effective average maturity • Measures sensitivity of bond price to interest rate changes • Measure of bond price volatility • Measure of interest-rate risk • Useful in the management of risk • You can match the duration of assets and liabilities • Or hedge the interest rate sensitivity of an investment Class session 2

  14. Qualifiers • First-order approximation • Accurate for small changes in yield • Limitation: Depends on parallel shifts in a flat yield curve • Multifactor duration models try to address this • Strictly applicable only to option-free (e.g., non-convertible) bonds Class session 2

  15. An aside: Corporate bonds and default risk • Most corporate bonds are either “callable” or “convertible” • Callable bonds give the firm the right to repurchase these bonds at a pre-specified price on or after a pre-specified date • Convertible bonds give their holders the right to convert the bonds they hold into common stock of the firm • Bond Rating: An indicator or assessment of the issuer’s ability to meet its interest and principal payments • Moody’s: Aaa; Aa; A; Baa; Ba; Caa; Ca; C (1-3) • S&P: AAA; AA; A; BBB; BB; B; CC; C; CI; D (+/-) Class session 2

  16. V. Check your intuition • How does each of these changes affect duration? • Having no coupon payments. • Decreasing the coupon rate. • Increasing the time to maturity. • Decreasing the yield-to-maturity. Class session 2

  17. Pictorial look at duration* • Cash flows of a seven year 12% bond discounted at 12%. • Shaded area of each box is PV of cash flow • Distance (x-axis) is a measure of time Class session 2

  18. Effects of the coupon • Duration is similar to the distance to the fulcrum (5.1 years) Duration High C, Lower Duration Low C, Higher Duration Class session 2

  19. Example of the coupon effect • Consider the durations of a 5-year and 20-year bond with varying coupon rates (semi-annual coupon payments): Class session 2

  20. Effect of maturity and yield on duration • Duration increases with increased maturity • Effect of yield  yield, weight on earlier payments , fulcrum shifts left  yield, weight on earlier payments , fulcrum shifts right Class session 2

  21. VI. A complication • Notice the convex shape of price-yield relationship • Bond 1 is more convex than Bond 2 • Price falls at a slower rate as yield increases Bond 1 Bond 2 Price A B 5% 10% Yield Class session 2

  22. Convexity • Measures how much a bond’s price-yield curve deviates from a straight line • Second derivative of price with respect to yield divided by bond price • Allows us to improve the duration approximation for bond price changes Class session 2

  23. Predicted percentage price change • Recall approximation using only duration: • The predicted percentage price change accounting for convexity is: Class session 2

  24. VII. Numerical example with convexity • Consider a 20-year 9% coupon bond selling at $134.6722 to yield 6%. Coupon payments are made semiannually. • Dm= 10.98 • The convexity of the bond is 164.106. Class session 2

  25. Numerical example - page 2 • If yields increase instantaneously from 6% to 8%, the percentage price change of this bond is given by: • First approximation (Duration): –10.66  .02  100 = –21.32 • Second approximation (Convexity) 0.5  164.106  (.02)2 100 = +3.28 Total predicted % price change: –21.32 + 3.28 = –18.04% (Actual price change = –18.40%.) Class session 2

  26. Numerical example - page 3 • What if yields fall by 2%? • If yields decrease instantaneously from 6% to 4%, the percentage price change of this bond is given by: • First approximation (Duration): –10.66  –.02  100 = 21.32 • Second approximation (Convexity) 0.5  164.106  (–.02)2 100 = +3.28 Total predicted price change: 21.32 + 3.28 = 24.60% Note that predicted change is NOT SYMMETRIC. Class session 2

  27. VIII. Takeaways: Duration and convexity • Price approximation using only duration: New Bond Price ($) = P + [P (Duration)] • Price approximation using both duration and convexity: New Bond Price ($) = P + [P (Duration)] + [P (Convexity)] Class session 2

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