1. 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:
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2. Derivatives
F(x) x
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3. Properties of derivatives
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4. Zero-coupon example
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5. Example y=10%, y=0.5% T P0 P1 P 1 90.90 90.09 -0.45%% % %
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6. 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.
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7. Modified Duration
The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity.
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8. Duration
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9. Duration
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10. Duration
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11. Measuring Price Change
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12. Macaulay Duration
A weighted sum of times to maturities of each coupon.
What is the duration of a zero coupon bond?
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13. Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity
0 3m 6m 1yr 3yr 5yr 10yr 30yr
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14. 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.
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15. Convexity r $
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16. Example 10 year zero coupon bond with a semiannual yield of 6%
The duration is 10 years, the modified duration is:
The convexity is
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17. Example If the yield changes to 7% the price change is Zvi Wiener
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18. Bond Price Derivatives
D* - modified duration, dollar duration is the negative of the first derivative:
Dollar convexity = the second derivative,
C - convexity.
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19. Duration of a portfolio
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20. ALM Duration Does NOT work! Wrong units of measurement
Division by a small number
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21. Duration Gap A - L = C, assets - liabilities = capital Zvi Wiener
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22. ALM Duration A similar problem with measuring yield Zvi Wiener
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23. Do not think of duration as a measure of time!
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24. Principal component duration Partial duration
Key rate duration
Principal component duration
Partial duration
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25. 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?
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26. Home Assignment What is the duration of a floater?
What is the duration of an inverse floater?
How coupon payments affect duration?
Why modified duration is better than Macaulay duration?
How duration can be used for hedging?
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27. Understanding of Duration/Convexity
What happens with duration when a coupon is paid?
How does convexity of a callable bond depend on interest rate?
How does convexity of a puttable bond depend on interest rate?
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28. Embedded Put Option r puttable bond regular bond Zvi Wiener
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29. Convertible Bond Stock Payoff Convertible Bond Straight Bond Stock
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30. Macaulay Duration
Modified duration
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31. Bond Price Change
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32. Duration of a coupon bond
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33. Portfolio Duration and Convexity
Portfolio weights
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34. Example
Construct a portfolio of two bonds: A and B to match the value and duration of a 10-years, 6% coupon bond with value $100 and modified duration of 7.44 years.
A. 1 year zero bond - price $94.26
B. 30 year zero - price $16.97
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35. Barbel portfolio consists of very short and very long bonds.
Modified duration
Barbel portfolio consists of very short and very long bonds.
Bullet portfolio consists of bonds with similar maturities.
Which of them has higher convexity?
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36. FRM-98, Question 18
A portfolio consists of two positions. One is long $100 of a two year bond priced at 101 with a duration of 1.7; the other position is short $50 of a five year bond priced at 99 with a duration of What is the duration of the portfolio?
A. 0.68
B. 0.61 C D
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37. Useful formulas
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38. Volatilities of IR/bond prices
Price volatility in % End 99 End 96
Euro 30d
Euro 180d
Euro 360d
Swap 2Y
Swap 5Y
Swap 10Y
Zero 2Y
Zero 5Y
Zero 10Y
Zero 30Y
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39. Duration approximation
What duration makes bond as volatile as FX?
What duration makes bond as volatile as stocks?
A 10 year bond has yearly price volatility of 8% which is similar to major FX.
30-year bonds have volatility similar to equities (20%).
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40. Volatilities of yields
Yield volatility in %, 99 (y/y) (y)
Euro 30d
Euro 180d
Euro 360d
Swap 2Y
Swap 5Y
Swap 10Y
Zero 2Y
Zero 5Y
Zero 10Y
Zero 30Y
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