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## Chapter 6 Beyond Duration

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**FIXED-INCOME SECURITIES**Chapter 6 Beyond Duration**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 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 YieldDuration and Interest**Rate Risk**Accounting for Larger Changes in YieldHedging 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 -100x7.36x0.001= - $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 -100x7.36x0.02= - $14.72 • Lousy approximation**Accounting for Larger Changes in YieldConvexity**• 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 YieldConvexity and $**Convexity • $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 • (Relative) convexity is**Accounting for Larger Changes in YieldProperties 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 YieldProperties of**Convexity**Accounting for Larger Changes in YieldProperties of**Convexity - Linearity • Convexity of a portfolio of n bonds where wiis 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 YieldDuration-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 H1 and H2 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 YieldDuration-Convexity**Hedging • Strategy: hold q1 and q2 units of the first and second hedging instrument respectively such that**Solution**• Or (under the assumption of a unique y – flat yield curve) • Solution (under the assumption of unique dy – parallel shifts). Find q1 and q2 that solve the linear system in two unknown:**Accounting for a Non Flat Yield CurveAllowing 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, y1, y2 • On the other hand, we maintain the assumption of parallel shifts, i.e., we assume dy= dy1 = dy2 • We are still looking for q1 and q2 such that**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 CurveTime 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 H1 = $ 97.962 • Yield y1 = 5.232 % • Modified duration Sens1 = 8.813 • Convexity Conv1 = 99.081 • Hedging instrument 2: • Price H2 = $ 108.039 • Yield y2 = 4.097% • Modified duration Sens2 = 2.704 • Convexity Conv2 = 10.168**Accounting for a Non Flat Yield CurveTime for an Example!**• Optimal quantities q1 and q2 of each hedging instrument are given by • Or q1 = -305 and q2 = 140 • If you hold the portfolio, you should sell 305 units of H1 and buy 140 units of H2**Accounting for Non Parallel ShiftsAccounting 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 ShiftsAccounting 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 ShiftsNelson 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 ShiftsNelson 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 ShiftsNelson 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**Beta 0**Beta 1 Beta 2 Scale parameter 8% -3% -1% 3 Accounting for Non Parallel ShiftsExample • At date t=0, parameters are estimated (fitted) to be • 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 ShiftsHedging 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 H1, H2 and H3 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 q0H0 + q1H1 + q2H2 + q3H3 + q4H4 = - P (need a 4th instrument for that) • Formally, look for q1, q2and q3such that**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)