FIXEDINCOME SECURITIES Chapter 6 Beyond Duration Outline Accounting
![FIXED-INCOME SECURITIES Chapter 6 Beyond Duration FIXED-INCOME SECURITIES Chapter 6 Beyond Duration](https://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-1.jpg)
FIXED-INCOME SECURITIES Chapter 6 Beyond Duration
![Outline • Accounting for Larger Changes in Yield • Accounting for a Non Flat Outline • Accounting for Larger Changes in Yield • Accounting for a Non Flat](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-2.jpg)
Outline • Accounting for Larger Changes in Yield • Accounting for a Non Flat Yield Curve • Accounting for Non Parallel Shits
![Beyond Duration Limits of Duration • Duration hedging is – Relatively simple – Built Beyond Duration Limits of Duration • Duration hedging is – Relatively simple – Built](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-3.jpg)
Beyond Duration Limits of Duration • Duration hedging is – Relatively simple – Built on very restrictive assumptions • Assumption 1: small changes in yield – The value of the portfolio could be approximated by its first order Taylor expansion – OK when changes in yield are small, not OK otherwise – This is why the hedge portfolio should be re-adjusted reasonably often • Assumption 2: the yield curve is flat at the origin – In particular we suppose that all bonds have the same yield rate – In other words, the interest rate risk is simply considered as a risk on the general level of interest rates • Assumption 3: the yield curve is flat at each point in time – In other words, we have assumed that the yield curve is only affected only by a parallel shift
![Accounting for Larger Changes in Yield Duration and Interest Rate Risk Accounting for Larger Changes in Yield Duration and Interest Rate Risk](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-4.jpg)
Accounting for Larger Changes in Yield Duration and Interest Rate Risk
![Accounting for Larger Changes in Yield Hedging Error • Let us consider a 10 Accounting for Larger Changes in Yield Hedging Error • Let us consider a 10](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-5.jpg)
Accounting for Larger Changes in Yield Hedging Error • Let us consider a 10 year maturity bond, with a 6% annual coupon rate, a 7. 36 modified duration, and which sells at par • What happens if – Case 1: yield increases from 6% to 6. 01% (small increase) – Case 2: yield increases from 6% to 8% (large increase) • Case 1: – Discount future cash-flows with new yield and obtain $99. 267 – Absolute change : - 0. 733 = (99. 267 -100) – Use modified duration and find that change in price is -100 x 7. 36 x 0. 001= - $0. 736 – Very good approximation • Case 2: – Discount future cash-flows with new yield and obtain $86. 58 – Absolute change : - 13. 42 = (86. 58 -100) – Use modified duration and find that change in price is -100 x 7. 36 x 0. 02= - $14. 72 – Lousy approximation
![Accounting for Larger Changes in Yield Convexity • Relationship between price and yield is Accounting for Larger Changes in Yield Convexity • Relationship between price and yield is](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-6.jpg)
Accounting for Larger Changes in Yield Convexity • Relationship between price and yield is convex: • Taylor approximation: • Relative change • Conv is relative convexity, i. e. , the second derivative of value with respect to yield divided by value
![Accounting for Larger Changes in Yield Convexity and $ Convexity • (Relative) convexity is Accounting for Larger Changes in Yield Convexity and $ Convexity • (Relative) convexity is](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-7.jpg)
Accounting for Larger Changes in Yield Convexity and $ Convexity • (Relative) convexity is • $Convexity = V’’(y) = Conv x V(y) • Example (back to previous) – 10 year maturity bond, with a 6% annual coupon rate, a 7. 36 modified duration, a 6974 $ convexity and which sells at par – Case 2: yields go from 6% to 8% • Second order approximation to change in price – Find: -14. 72 + (6974. (0. 02)²/2) = -$13. 33 – Exact solution is -$13. 42 and first order approximation is -$14. 72
![Accounting for Larger Changes in Yield Properties of Convexity • Convexity is always positive Accounting for Larger Changes in Yield Properties of Convexity • Convexity is always positive](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-8.jpg)
Accounting for Larger Changes in Yield Properties of Convexity • Convexity is always positive • For a given maturity and yield, convexity increases as coupon rate – Decreases • For a given coupon rate and yield, convexity increases as maturity – Increases • For a given maturity and coupon rate, convexity increases as yield rate – Decreases
![Accounting for Larger Changes in Yield Properties of Convexity Accounting for Larger Changes in Yield Properties of Convexity](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-9.jpg)
Accounting for Larger Changes in Yield Properties of Convexity
![Accounting for Larger Changes in Yield Properties of Convexity - Linearity • Convexity of Accounting for Larger Changes in Yield Properties of Convexity - Linearity • Convexity of](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-10.jpg)
Accounting for Larger Changes in Yield Properties of Convexity - Linearity • Convexity of a portfolio of n bonds where wi is the weight of bond i in the portfolio, and: • This is true if and only if all bonds have same yield, i. e. , if yield curve is flat
![Accounting for Larger Changes in Yield Duration-Convexity Hedging • Principle: immunize the value of Accounting for Larger Changes in Yield Duration-Convexity Hedging • Principle: immunize the value of](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-11.jpg)
Accounting for Larger Changes in Yield Duration-Convexity Hedging • Principle: immunize the value of a bond portfolio with respect to changes in yield – Denote by P the value of the portfolio – Denote by H 1 and H 2 the value of two hedging instruments – Needs two hedging instrument because want to hedge one risk factor (still assume a flat yield curve) up to the second order • Changes in value – Portfolio – Hedging instruments
![Accounting for Larger Changes in Yield Duration-Convexity Hedging • Strategy: hold q 1 and Accounting for Larger Changes in Yield Duration-Convexity Hedging • Strategy: hold q 1 and](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-12.jpg)
Accounting for Larger Changes in Yield Duration-Convexity Hedging • Strategy: hold q 1 and q 2 units of the first and second hedging instrument respectively such that
![Duration-Convexity Hedging Duration-Convexity Hedging](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-13.jpg)
Duration-Convexity Hedging
![Solution • Solution (under the assumption of unique dy – parallel shifts). Find q Solution • Solution (under the assumption of unique dy – parallel shifts). Find q](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-14.jpg)
Solution • Solution (under the assumption of unique dy – parallel shifts). Find q 1 and q 2 that solve the linear system in two unknown: – Or (under the assumption of a unique y – flat yield curve)
![Accounting for a Non Flat Yield Curve Allowing for a Term Structure • Problem Accounting for a Non Flat Yield Curve Allowing for a Term Structure • Problem](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-15.jpg)
Accounting for a Non Flat Yield Curve Allowing for a Term Structure • Problem with the previous method: we have assumed a unique yield for all instrument, i. e. , we have assumed a flat yield curve • We now relax this simplifying assumption and consider 3 potentially different yields y, y 1, y 2 • On the other hand, we maintain the assumption of parallel shifts, i. e. , we assume dy = dy 1 = dy 2 • We are still looking for q 1 and q 2 such that
![Accounting for a Non Flat Yield Curve • Solution (under the assumption of unique Accounting for a Non Flat Yield Curve • Solution (under the assumption of unique](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-16.jpg)
Accounting for a Non Flat Yield Curve • Solution (under the assumption of unique dy – parallel shifts) – Or (relaxing the assumption of a flat yield curve)
![Accounting for a Non Flat Yield Curve Time for an Example! • Portfolio at Accounting for a Non Flat Yield Curve Time for an Example! • Portfolio at](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-17.jpg)
Accounting for a Non Flat Yield Curve Time for an Example! • Portfolio at date t – – Price P = $ 32863. 5 Yield y = 5. 143% Modified duration Sens = 6. 76 Convexity Conv =85. 329 • Hedging instrument 1 – – Price H 1 = $ 97. 962 Yield y 1 = 5. 232 % Modified duration Sens 1 = 8. 813 Convexity Conv 1 = 99. 081 • Hedging instrument 2: – – Price H 2 = $ 108. 039 Yield y 2 = 4. 097% Modified duration Sens 2 = 2. 704 Convexity Conv 2 = 10. 168
![Accounting for a Non Flat Yield Curve Time for an Example! • Optimal quantities Accounting for a Non Flat Yield Curve Time for an Example! • Optimal quantities](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-18.jpg)
Accounting for a Non Flat Yield Curve Time for an Example! • Optimal quantities q 1 and q 2 of each hedging instrument are given by – Or q 1 = -305 and q 2 = 140 • If you hold the portfolio, you should sell 305 units of H 1 and buy 140 units of H 2
![Accounting for Non Parallel Shifts Accounting for Changes in Shape of the TS • Accounting for Non Parallel Shifts Accounting for Changes in Shape of the TS •](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-19.jpg)
Accounting for Non Parallel Shifts Accounting for Changes in Shape of the TS • Bad news is: not only the yield curve is not flat, but also it changes shape! • Afore mentioned methods do not allow to account for such deformations – Additional risk factors – One has to regroup different risk factors to reduce the dimensionality of the problem: e. g. , a short, medium and long maturity factors • Systematic approach: factor analysis on historical data has shed some light on the dynamics of the yield curve • 3 factors account for more than 90% of the variations – Level factor – Slope factor – Curvature factor
![Accounting for Non Parallel Shifts Accounting for Non Parallel Shits • To properly account Accounting for Non Parallel Shifts Accounting for Non Parallel Shits • To properly account](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-20.jpg)
Accounting for Non Parallel Shifts Accounting for Non Parallel Shits • To properly account for the changes in the yield curve, one has to get back to pure discount rates • Or, using continuously compounded rates
![Accounting for Non Parallel Shifts Nelson Siegel Model • The challenge is that we Accounting for Non Parallel Shifts Nelson Siegel Model • The challenge is that we](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-21.jpg)
Accounting for Non Parallel Shifts Nelson Siegel Model • The challenge is that we are now facing m risk factors • Reduce the dimensionality of the problem by writing discount rates as a function of 3 parameters • One classic model is Nelson et Siegel’s – – – with R(0, ): pure discount rate with maturity 0 : level factor 1 : rotation factor 2 : curvature factor : fixed scaling parameter • Hedging principle: immunize the portfolio with respect to changes in the value of the 3 parameters
![Accounting for Non Parallel Shifts Nelson Siegel Model • Mechanics of the model: changes Accounting for Non Parallel Shifts Nelson Siegel Model • Mechanics of the model: changes](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-22.jpg)
Accounting for Non Parallel Shifts Nelson Siegel Model • Mechanics of the model: changes in beta parameters imply changes in discount rates, which in turn imply changes in prices • One may easily compute the sensitivity (partial derivative) of R(0, ) with respect to each parameter beta (see next slide) • Very consistent with factor analysis of interest rates in the sense that they can be regarded as level, slope and curvature factors, respectively
![Accounting for Non Parallel Shifts Nelson Siegel Accounting for Non Parallel Shifts Nelson Siegel](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-23.jpg)
Accounting for Non Parallel Shifts Nelson Siegel
![Accounting for Non Parallel Shifts Nelson Siegel Model • Let us consider at date Accounting for Non Parallel Shifts Nelson Siegel Model • Let us consider at date](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-24.jpg)
Accounting for Non Parallel Shifts Nelson Siegel Model • Let us consider at date t=0 a bond with price P delivering the future cash-flows Fi • The price is given by • Sensitivities of the bond price with respect to each beta parameter are
![Accounting for Non Parallel Shifts Example • At date t=0, parameters are estimated (fitted) Accounting for Non Parallel Shifts Example • At date t=0, parameters are estimated (fitted)](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-25.jpg)
Accounting for Non Parallel Shifts Example • At date t=0, parameters are estimated (fitted) to be Beta 0 8% Beta 1 -3% Beta 2 -1% Scale parameter 3 • Sensitivities of 3 bonds with respect to each beta parameter, as well as that of the portfolio invested in the 3 bonds, are
![Accounting for Non Parallel Shifts Hedging with Nelson Siegel • Principle: immunize the value Accounting for Non Parallel Shifts Hedging with Nelson Siegel • Principle: immunize the value](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-26.jpg)
Accounting for Non Parallel Shifts Hedging with Nelson Siegel • Principle: immunize the value of a bond portfolio with respect to changes in parameters of the model – Denote by P the value of the portfolio – Denote by H 1, H 2 and H 3 the value of three hedging instruments – Needs 3 hedging instruments because want to hedge 3 risk factors (up to the first order) – Can also impose dollar neutrality constraint q 0 H 0 + q 1 H 1 + q 2 H 2 + q 3 H 3 + q 4 H 4 = - P (need a 4 th instrument for that) • Formally, look for q 1, q 2 and q 3 such that
![Beyond Duration General Comments • Whatever the method used, duration, modified duration, convexity and Beyond Duration General Comments • Whatever the method used, duration, modified duration, convexity and](http://slidetodoc.com/presentation_image_h/dd8517c2adeed193cef5cc8ac3a53632/image-27.jpg)
Beyond Duration General Comments • Whatever the method used, duration, modified duration, convexity and sensitivity to Nelson and Siegel parameters are time-varying quantities – Given that their value directly impact the quantities of hedging instruments, hedging strategies are dynamic strategies – Re-balancement should occur to adjust the hedging portfolio so that it reflects the current market conditions • In the context of Nelson and Siegel model, one may elect to partially hedge the portfolio with respect to some beta parameters – This is a way to speculate on changes in some factors; it is known as « semi-hedging » strategies – For example, a portfolio bond holder who anticipates a decrease in interest rates may choose to hedge with respect to parameters beta 1 and beta 2 (slope and curvature factors) while remaining voluntarily exposed to a change in the beta 0 parameter (level factor)
- Slides: 27