1. BONDS HAVE DIFFERENT INTEREST SENSITIVITIES

2. BOND PRICE-YIELD RELATIONSHIP

3. MEASURES OF INTEREST SENSITIVITY èPrice Value of a Basis Point (PVBP) Change in bond price given a one basis point change in yield èYield Value of a 32nd (YV32) Change in yield given a price change of 1/32 èDuration Macaulay: Price elasticity w.r.t. (1+y/2) Modified: % price change given yield change

4. MACAULAY DURATION

5. MACAULAY DURATION: CENTER OF GRAVITY INTERPRETATION

6. MACAULAY DURATION: CLOSED FORM SOLUTION

7. MACAULAY DURATION THREE EXAMPLE BONDS (AT 10% YLD)

8. MACAULAY DURATION: SPECIAL CASES

9. PROPERTIES OF DURATION èDURATION: èDecreases with coupon rate èDecreases with yield èUsually increases maturity (but watch out for long-term deep discount bonds)

10. MODIFIED DURATION Used more in practice because of easier interpretation Calculate as shown, using closed form solution for D MAC or use built-in Excel function

11. CALCULATING DURATION èA. Program Closed-Form Solution into a spreadsheet èB. Use Excel functions DURATION and MDURATION (in Analysis ToolPak) èC. Approximate: D MOD  (P - - P + )/[2(P 0 )(  y)]

12. DURATION OF A PORTFOLIO

13. APPLICATION: HEDGING For our portfolio, given a change in yield,  P/P  - D MOD (p)  y To hedge the portfolio against yield changes, set D MOD (p) = 0 This implies: W A = - W B [D MOD (A)/ D MOD (B)]

14. APPROXIMATING BOND PRICE CHANGES But what if  P/P  - D MOD (p)  y isn’t a very good approximation? A better approximation comes from carrying the Taylor’s Series one more term:

15. CONVEXITY

16. WHAT DOES CONVEXITY MEASURE? èNote that convexity is a second derivative measure èIt tells us about the degree of curvature in the price-yield relationship

17. APPROXIMATING CONVEXITY A closed form solution exists, but convexity can be approximated quite accurately by measuring the change in the change in price, given a change in yield (including between coupon dates)

18. CONVEXITY PROPERTIES èAs yield increases, convexity decreases èFor a given yield and maturity, as coupon rate increases, convexity decreases èe.g., zeros have greatest convexity among bonds of given yield and maturity èFor a given yield and modified duration, as coupon rate increases, convexity increases èe.g., zeros have least convexity among bonds of given yield and D MOD

19. IMMUNIZATION

20. IMMUNIZATION CAVEATS èThe immunization technique protects us only against a small, one-time shift in rates èneed to rebalance after yield shifts occur èFor portfolios, immunization only works for parallel yield curve shifts èin fact, the yield curve often flattens or steepens èWhy not just use zeros to immunize?