Lecture 13: Hedging with duration and convexity and review Finance 688: Investment Administration...
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Transcript of Lecture 13: Hedging with duration and convexity and review Finance 688: Investment Administration...
Lecture 13: Hedging with duration and convexity and review
Finance 688: Investment AdministrationProfessor John Chalmers
Read Chapter 12
problems 1-5
Duration and Convexity are risk management tools Basic ideas are applicable to all assets
Often not analytically tractable, make heroic assumptions
Primary uses Asset liability management (managing the firm’s exposure)
Bank managers manage loan portfolio risk
insurance company portfolios, pension fund portfolios
Portfolio selection (in which bonds do we invest)
risk aversion of investors
matching particular liabilities (a retirement plan)
Security selection (how to best implement a trading strategy)
how to best play information about interest rates
e.g. if you know rates are coming down long maturities? MBS?
Convexity helps the Estimates
-40
-20
0
20
40
60
80
0 0.05 0.1 0.15 0.2 0.25
True Price Modif ied Duration (7.22) Duration + Convexity Price
erroryy
P
Py
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PP
P 22
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11
Three portfolios
• Duration increases as coupons decrease
• Convexity increases as coupons decrease
• Suppose your liabilities look like the 5% bond, what can we do to hedge with the other two portfolios? 85.00
90.00
95.00
100.00
105.00
110.00
115.00
120.00
8.00% 9.00% 10.00% 11.00% 12.00%
Portf
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Price 10% 1.43 5% 2.59 Zeros
Ten year 10%, 5% and 0% bonds109.84 8.50% dP dy 1st Der 1st der/P 1st Der 2nd Der 2nd Der/P106.42 9.00% -3.42 0.0050 -684.87 6.24103.14 9.50% -3.28 0.0050 -655.65 6.16 29.22 5844.23 54.92100.00 10.00% -3.14 0.0050 -627.88 6.09 27.77 5554.39 53.8596.99 10.50% -3.01 0.0050 -601.48 6.01 26.40 5280.51 52.8194.11 11.00% -2.88 0.0050 -576.37 5.94 25.11 5021.63 51.7791.35 11.50% -2.76 0.0050 -552.48 5.87 23.88 4776.86 50.76
1.44 Units5% Bond 5% bond Yield Price Chg Yield Chg dP/dy Duration Chg d2P/dy2 Convexity
77.04 111.19 8.50% dP dy 1st Der 1st der/P 1st Der 2nd Der 2nd Der/P74.33 107.29 9.00% -3.91 0.0050 -781.15 7.0371.75 103.56 9.50% -3.73 0.0050 -745.95 6.95 35.21 7041.12 65.6369.28 100.00 10.00% -3.56 0.0050 -712.54 6.88 33.41 6681.06 64.5166.92 96.59 10.50% -3.40 0.0050 -680.84 6.81 31.71 6341.21 63.4164.66 93.34 11.00% -3.25 0.0050 -650.73 6.74 30.10 6020.36 62.3362.51 90.23 11.50% -3.11 0.0050 -622.15 6.67 28.59 5717.34 61.25
2.59 Units0% Bond Zeros Yield Price Chg Yield Chg dP/dy Duration Chg d2P/dy2 Convexity
44.23 114.72 8.50% dP dy 1st Der 1st der/P 1st Der 2nd Der 2nd Der/P42.24 109.56 9.00% -5.15 0.0050 -1030.99 8.9940.35 104.66 9.50% -4.90 0.0050 -980.26 8.95 50.73 10146.60 92.6138.55 100.00 10.00% -4.66 0.0050 -932.24 8.91 48.02 9604.35 91.7736.84 95.57 10.50% -4.43 0.0050 -886.77 8.87 45.47 9093.35 90.9335.22 91.35 11.00% -4.22 0.0050 -843.71 8.83 43.06 8611.68 90.1133.67 87.33 11.50% -4.01 0.0050 -802.92 8.79 40.79 8157.52 89.30
Neutral hedge The objective of a neutral hedge is to desensitize portfolio value
from changes in interest rates. In general, any hedging problem solves for the amount to buy of
various instruments that you can use to hedge. The number of assets required to hedge with will be equal to the number of dimensions on which you wish to hedge.
If D is zero this implies that changes in interest rates will have no impact on the value of your portfolio. This is portfolio immunization. Depends on parallel shift assumption.
Suppose liability is 10% bond. Duration hedge with zero: Remember the duration of portfolio is weighted average of the
duration of the assets in the portfolio
Duration Hedge
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Suppose liability is 10% bond. Duration hedge with zero:
Duration and Convexity Hedge Match the duration of your portfolio along with the
convexity of the portfolios
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Bullet versus Barbell Hedge
Bullet effectively matches duration with assets of maturity similar to the asset that is being hedged. For example hedge a bond with 6 year duration with 6 year zero.
Barbell matches duration with bonds with very different maturities. For example, hedge a 6 year duration bond with a 1 year zero and a 13 year zero.
Bullet hedges will come closer to matching duration and convexity than a barbell hedge. The barbell will have higher convexity, which is fine if rates are a changing.
Summary
• Hedging with duration and convexity• This is useful in many contexts, including the
corporate managers, portfolio managers and business line people.
• Duration and PVBP are the crudest but most often encountered measures of price sensitivity
• The topics on the exam.