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Class Business

Class Business. Upcoming Homework. Duration. A measure of the effective maturity of a bond The weighted average of the times (periods) until each payment is received, with the weights proportional to the present value of the payment Duration is equal to maturity for zero coupon bonds

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Class Business

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  1. Class Business • Upcoming Homework

  2. Duration • A measure of the effective maturity of a bond • The weighted average of the times (periods) until each payment is received, with the weights proportional to the present value of the payment • Duration is equal to maturity for zero coupon bonds • Duration of a perpetuity is (1+r)/r

  3. Duration Formula

  4. Duration FormulaAnother Perspective

  5. Workout Problem-Duration • Calculate the duration of an asset that makes nominal payments of $120 one year from now, $140 two years from now, and $160 three years from now. Assume the YTM is 10%. Calculate the duration of another asset that makes nominal payments of $160 one year from now, $140 two years from now, and $120 three years from now, also with an YTM of 10%. • Spreadsheet

  6. Duration Properties • The longer the term to maturity of a bond, everything else being equal, the greater its duration. • When interest rates rise, everything else being equal, the duration of a coupon bond falls. (convexity) • The higher the coupon rate on the bond, everything else being equal, the shorter the bond’s duration. • Duration is additive: The duration of a portfolio of securities is the weighted average of the durations of the individual securities, with the weights reflecting the proportion of the portfolio invested in each.

  7. Algebraic Duration Relations • Where D* is modified Duration, D* = D/(1+y) • But, using some algebra

  8. Immunization Example • Insurance co must make $19,487 in 7 years, market rates are 10%, PV of payment is $10,000. Using 4 year zero coupon bonds and perpetuities, immunize this obligation against interest rate risk. Duration of Liabilities = 7 years Duration of zero-coupon bonds = 4 Duration of perpetuities = 1.1/.1 = 11 Solve: x*4 + (1-x)*11 = 7 x = 57%, therefore buy $5,700 worth of zero coupon bonds and $4,300 worth of perpetuities

  9. Pricing Error from Convexity Price Pricing Error from Convexity Duration Yield

  10. Correction for Convexity Modify the pricing equation: Convexity is Equal to:

  11. Convexity • How does convexity affect the approximation error of the bond return when we match only the modified duration? • As an investor, do we like convexity? • In general, the higher the coupon rate, the lower the convexity of the bond.

  12. Example • Annual coupon paying bond • matures in 2 years, par=1000, • coupon rate =10%, y=10% • Price=$1000 • Time when cash is received: • t1=1 ($100 is received), t2=2 ($1100 is received) • Find approximate percentage change in bond price using both duration and convexity if yields increase by 100bps

  13. Example • Using D* only (D* = 1.7355) • Using both D* and convexity (4.6583) • Differences can be meaningful

  14. Convexity of a Portfolio • The convexity of a portfolio is the weighted sum of the convexity’s of each bond in the portfolio where weights are the fraction of your investment equity in each • Therefore, we can match convexity of portfolio similar to matching modified duration

  15. Active Bond Management: Swapping Strategies • Substitution • Intermarket • Rate anticipation • Pure yield pickup • Tax • Others

  16. Interest Rate Swaps • Interest rate swap basic characteristics • One party pays fixed and receives variable • Other party pays variable and receives fixed • Principal is notional (not exchanged) • Growth in market • Started in 1980 • Estimated over $60 trillion today • Hedging applications – Banks • Speculative applications – Fixed Income Asset Management

  17. A and B Transform Assets • Assume Portfolio manager A thinks interest rates are going up while manager B thinks interest rates will stay level or go down { Swap 5% 5.2% A B Existing Asset LIBOR+0.8% Existing Asset LIBOR

  18. Swap Example:Financial Institution is Involved 4.95% 5.05% 5.2% A F.I. B LIBOR+0.8% LIBOR LIBOR

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