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

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• 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

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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.