1. Chapter 6 Beyond Duration
FIXED-INCOME SECURITIES
Chapter 6
Beyond Duration

2. Outline Accounting for Larger Changes in Yield
Accounting for a Non Flat Yield Curve
Accounting for Non Parallel Shits

3. 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

4. Accounting for Larger Changes in Yield Duration and Interest Rate Risk

5. 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 : = ( )
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 : = ( )
-100x7.36x0.02= - $14.72
Lousy approximation

6. 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

7. Accounting for Larger Changes in Yield Convexity 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: (6974.(0.02)²/2) = -$13.33
Exact solution is -$13.42 and first order approximation is -$14.72
(Relative) convexity is

8. 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

9. Accounting for Larger Changes in Yield Properties of Convexity

10. 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

11. 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 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

12. Accounting for Larger Changes in Yield Duration-Convexity Hedging
Strategy: hold q1 and q2 units of the first and second hedging instrument respectively such that

13. Duration-Convexity Hedging

14. Solution
Solution (under the assumption of unique dy – parallel shifts). Find q1 and q2 that solve the linear system in two unknown:
Or (under the assumption of a unique y – flat yield curve)

15. 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, 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

16. 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)

17. Accounting for a Non Flat Yield Curve Time for an Example!
Portfolio at date t
Price P = $
Yield y = 5.143%
Modified duration Sens = 6.76
Convexity Conv =85.329
Hedging instrument 1
Price H1 = $
Yield y1 = %
Modified duration Sens1 = 8.813
Convexity Conv1 =
Hedging instrument 2:
Price H2 = $
Yield y2 = 4.097%
Modified duration Sens2 = 2.704
Convexity Conv2 =

18. Accounting for a Non Flat Yield Curve Time 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

19. 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

20. 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

21. 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

22. 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

23. Accounting for Non Parallel Shifts Nelson Siegel

24. 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

25. Accounting for Non Parallel Shifts Example
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
Beta 0
Beta 1
Beta 2
Scale parameter 8%
-3%
-1% 3

26. 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 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, q2 and q3 such that

27. 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)