1. Bond Price Volatility Zvi Wiener Based on Chapter 4 in Fabozzi
Bond Markets, Analysis and Strategies
Fall-02

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?
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Fabozzi Ch 4

3. Money Manager value Original plan New plan Market shock 0 t D
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4. 8% Coupon Bond
Zero Coupon Bond
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5. Price-Yield for option-free bonds
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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:
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7. Derivatives
F(x) x
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8. Properties of derivatives
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9. Zero-coupon example
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10. Example y=10%, y=0.5% T P0 P1 P 1 90.90 90.09 -0.45%% % %
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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.
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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?
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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?
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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?
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15. What affects price volatility?
Linkage
Credit considerations
Time to maturity
Coupon rate
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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.
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17. Duration and IR sensitivity
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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?
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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.
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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.
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21. Duration
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22. Duration
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23. Duration
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24. Measuring Price Change
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25. The Yield to Maturity
The yield to maturity of a fixed coupon bond y is given by
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26. Macaulay Duration Definition of duration, assuming t=0. Zvi Wiener
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27. 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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28. Meaning of Duration r $
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29. Parallel shift r upward move Current TS Downward move T Zvi Wiener
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30. Comparison of two bonds
Coupon bond with duration
Price (at 5% for 6m.) is $
If IR increase by 1bp
(to 5.01%), its price will fall to $ , or
0.359% decline.
Zero-coupon bond with equal duration must have years to maturity.
At 5% semiannual its price is
($1,000/ )=$
If IR increase to 5.01%, the price becomes:
($1,000/ )=$831.66
0.359% decline.
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Fabozzi Ch 4

31. Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity
0 3m 6m 1yr 3yr 5yr 10yr 30yr
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32. Example A bond with 30-yr to maturity Coupon 8%; paid semiannually
YTM = 9%
P0 = $897.26
D = Yrs
if YTM = 9.1%, what will be the price?
P/P = - y D*
P = -(y D*)P = -$9.36
P = $ $9.36 = $887.90
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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.
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34. What Determines Duration?
Holding the coupon rate constant, duration not always increases with ttm.
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35. Convexity r $
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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
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37. Example If the yield changes to 7% the price change is Zvi Wiener
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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 If the yield decreases by 1 bp, the price will increase to What is the modified duration of this bond?
A. 5.0
B. -5.0
C. 4.5
D. -4.5
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39. FRM-98, Question 17
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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 B C D
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41. FRM-98, Question 22
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42. Portfolio Duration
Similar to a single bond but the cashflow is determined by all Fixed Income securities held in the portfolio.
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43. 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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44. Duration of a portfolio
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45. ALM Duration Does NOT work! Wrong units of measurement
Division by a small number
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46. Duration Gap A - L = C, assets - liabilities = capital Zvi Wiener
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47. ALM Duration A similar problem with measuring yield Zvi Wiener
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48. Do not think of duration as a measure of time!
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49. Principal component duration Partial duration
Key rate duration
Principal component duration
Partial duration
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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?
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51. 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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52. Home Assignment Chapter 4
Ch. 4: Questions 1, 2, 3, 4, 15.
Calculate duration of a consul (perpetual bond).
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53. End Ch. 4
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54. 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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55. Callable bond
The buyer of a callable bond has written an option to the issuer to call the bond back.
Rationally this should be done when …
Interest rate fall and the debt issuer can refinance at a lower rate.
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Fabozzi Ch 4

56. Puttable bond
The buyer of a such a bond can request the loan to be returned.
The rational strategy is to exercise this option when interest rates are high enough to provide an interesting alternative.
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57. Embedded Call Option r regular bond strike callable bond Zvi Wiener
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58. Embedded Put Option r puttable bond regular bond Zvi Wiener
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59. Convertible Bond Stock Payoff Convertible Bond Straight Bond Stock
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60. Timing of exercise European American Bermudian Lock up time Zvi Wiener
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61. Macaulay Duration
Modified duration
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62. Bond Price Change
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63. Duration of a coupon bond
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64. Exercise Find the duration and convexity of a consol (perpetual bond).
Answer: (1+y)/y.
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65. FRM-98, Question 29
A and B are perpetual bonds. A has 4% coupon, and B has 8% coupon. Assume that both bonds are trading at the same yield, what can be said about duration of these bonds?
A. The duration of A is greater than of B
B. The duration of A is less than of B
C. They have the same duration
D. None of the above
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Fabozzi Ch 4

66. FRM-97, Question 24
Which of the following is NOT a property of bond duration?
A. For zero-coupon bonds Macaulay duration of the bond equals to time to maturity.
B. Duration is usually inversely related to the coupon of a bond.
C. Duration is usually higher for higher yields to maturity.
D. Duration is higher as the number of years to maturity for a bond selling at par or above increases.
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Fabozzi Ch 4

67. FRM-99, Question 75
You have a large short position in two bonds with similar credit risk. Bond A is priced at par yielding 6% with 20 years to maturity. Bond B has 20 years to maturity, coupon 6.5% and yield of 6%. Which bond contributes more to the risk of the portfolio?
A. Bond A
B. Bond B
C. A and B have similar risk
D. None of the above
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Fabozzi Ch 4

68. Portfolio Duration and Convexity
Portfolio weights
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69. 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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70. 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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71. 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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72. FRM-98, Question 18 Note that $100 means notional amount
and can be misunderstood.
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73. Useful formulas
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74. 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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75. 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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Fabozzi Ch 4

76. 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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