2.2.8. Hedge by Modified Duration for Fixed Income
The hedge can be definedd as an operation which finds to reduce the risk or esposition in a financial asset without been necessary to negotiate teh asset itself. In particular for the yield curve, the operations of hedge allows to reduce the exposure to oscilations on rates inlayed in the curve. That is, its possible to reduce a active position in interests rates - bought or sold - without effectively the asset attached to interests ( bonds etc.).
Exists various types of assets related to the risk of the yield curve (prefixed stocks, option stocks, sqaps etc) and for each one exist a different way to do the hedge.
Looking for assets which cash-flow does not change by the interest rate level (without options inlayed) and for small and parallel changes in the yeild curve, can be used the concepts of modified duration to effect the hedge.
Changes not parallel to the yield curve causes inefficiency of hedge operations by duration. To increases the inefficiency in this cases, can be used the concept of immunization, to equalize the time based distribution of hedge with active positions (that will result in many hedge operations spread acording to expirations).
Then for bigger changes in the yield curve, the effect of convexity also must be considered in the calculus of actual value oscilations. However, including convexity the hedge operations be more complex.
For stocks with options inlayed, must be used the concept of effective duration.
By duration (Macauley duration) , understood as mean of limits of the interests rates positions weighed by its present value ( (VPi). Or, for a portfolio with n positions in interests rates:
where di its the duration of each stock. If di was measured in days, D must be too.
Modified duration its an adaption of the concept of duration for capitalizatons of interests rates in any limit. To the regime of composed interest rates with anual daily capitalizaton and duration also daily, have:
where Paa is equal to the anual daily interest rate (base 252) and di measured in days for each payment.
The type of asset used to do the hedge is important too. How much bigger is the correlation between the asset of hedge and the asset of teh possition, bigger will be the efficiency of the hedge . When effects teh cross-hedge (different assets), its possible to relate the amount of assets needed for the hedge or with a beta (similar to the used in stocks). This beta represents the relation between he asset variation for the hedge in the position or the coefficient of correlation between this assets.
The future contracts of interests DI1 negotiated in the BM&F are instruments usually used in the operations of hedge for interests rates.
2.2.8.1. Function CC.HEDGE
Access:
- Menu - Insert | Function | Calculus
 - Toolbar Default | Calculus
Description:
Returns the amount of contracts of DI1 from BM&F in expiration date indicated which are necessary to do the hedge of the prefixed position set. The calc is effected for small and parallel changes of the yield curve using the concept of modified duration. The result can be negative or positive, being that the negative operator mean contract selling and the positive, contracts buying.
This function alsso allows to change the duration of a position through establishing a target for duration, being for this, necessary inform the interests rate related to the target duration.
Important:
The operations of hedge by modified duration returned are efficient for small and parallel changes in the yield curve (without modify the curve inclination)!
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Call: CC.HEDGE ( BMF DI1 Yield, Workdays, Face Value, Portfolio Yield, Settlement, Target Duration, Target Yield, Beta)
Argument |
Type |
Description |
BMF DI1 Yield |
double |
BMF DI1 annual yield (base 252) used for hedging.
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Workdays |
integer |
Workdays till BMF DI1 future settlement - expire date - used for hedging.
|
Face value |
long integer |
Face value of portfolio position to be hedged. Mark positive values for long positions and negative value for short positions.
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Portfolio Yield |
double |
Equivalent annual yield (base 252) for portfolio duration or portfolio settlement.
|
Settlement |
integer |
Workdays till portfolio (hedged) settlement.Must be bigger than 1.
|
Target Duration |
integer |
Optional. Workdays of final duration. Default is 0, meaning total hedge.Must be bigger or equal to 0.
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Target Yield |
double |
Optional. Equivalent annual yield (base 252) for target duration. Must be informed if Target Duration is used. Must be a positive number.
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Beta |
double |
Optional. Relationship between BMF DI1 future contracts and portfolio bonds. Default value is 1.
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The result for an operation of hedge is:
- Hedge: number of contracts of BM&F DI1 rounded for hedge. Considering FV as the face value of the position, PP as the anual interests rate for the position duration, DP the duration in position days the anual interest rate of the instrument contract of hedge, Vcto the settlement (duration) of this contract, DA the duration target in days and PA the anual interest rate correspondig to the target duration and also beta for the relation between position assets and the future contract:
- Hedge total:
- Hedge partial – duration reducing:
Where:
Important:
The establishing of a target duration (hedge partial) for a position keep the face value of the position and change only duration (face value continuous). This result is different from the calc of the duration of portfolio composed by partial hedge calculated and by the original position by the function CC.DURATION, cause, as exists opposites positions, the calc of the function CC.DURATION keep the main position and reduce its face value (continuous duration). The risk for small and parallel changes in the yield curve, however, its the same in both cases!
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Important:
To interpret the result of partial hedge with the function CC.DURATION, simulate the total hedge from the part of the position which must be reduced to be equal to the orignal position obtained by the function CC.DURATION!
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Example using total hedge:
Position total Hedge – duration target is zero.
- Contract rate for hedge 18.5%
- Contract settlement 80 (workdays)
- Face value of bought position 100.000.000,00
- Rate for position 19.5% (base 252)
- Position duration 100 (workdays)
= CC.HEDGE( 0.185, 80, 100000000, 0.195, 100)
Results:
To do the total hedge of the position, its necessary sell (negative operator) 1,219 DI1 contracts. Small and parallel changes in the yield curve it will not affect the portfolio with total hedge.
Example using hedge for change duration:
Duration change - hedge partial - of the position with betaa different of 1. In this example, its simulated the duration reducing of private stocks, which can be more sensible to changes in the interest rates and justify the usage of a beta different of 1!
- Future contract rate for hedge 18.5% (base 252)
- Contract settlement 80 (workdays)
- Face value of bought position 100.000.000,00
- Rate for position 19.5% (base 252)
- Position duration 100 (workdays)
- Target duration 80 (dias úteis)
- Interest rates for the period 18.5%
- Relation between the oscilations in interest contract and in the interest rates of the position (beta)1.1
= CC.HEDGE( 0.185, 80, 100000000, 0.195, 100, 80, 0.185, 1.1)
Results:
To do the partial hedge of the position, reducing its duration to the equivalent to the future contract itself (same duration as the contract), its necessary sell 259 DI1 contracts. Parallel changes in the yield curve will impact the portfolio with the same intensity that the interest DI1 contract (final portfolio has the same duration than teh contract).
Also for partial hedge or establishing of target duration, the sell of 259 contracts reduce the duration of th face value for the target duration, being that the face value its continuos, that is, the new position its equivalent to the face value of 100,000,000.00 for a 80 days duration.
Example using partial hedge for changing the duration and comparisson with the function CC.DURATION:
In this example, the rates for each limit was obtained from the yield curve which will be used in the function CC.DURATION.
- Future contract rate for hedge 19.3% (base 252)
- Contract settlement 50 (workdays)
- Face value of bought position 20.000.000,00
- Rate for position 20.58% (efetiva base 252)
- Position duration 220 (dias úteis)
- Target duration 100 (dias úteis)
- Interest rate for the period 20.0%
- Relation between the oscilations in interest contract and in the interest rates of the position (beta)1.1
= CC.HEDGE( 0.193, 50, 200000000, 0.2058, 220, 100, 0.2, 1)
Results:
To reduce the duration of a value with face of 20,000,000.00 from 220 to 100 days, its necessary sell 416 future contracts of DI1.
To sample the comparisson with the function CC.DURATION, have:
- Curve: A2:B11 (11 points not sorted)
| 18.75% | 20 |
| 19.15% | 40 |
| 19.40% | 60 |
| 19.75% | 80 |
| 20.00% | 100 |
| 20.40% | 140 |
| 20.60% | 180 |
| 20.58% | 220 |
| 20.40% | 300 |
| 20.30% | 350 |
- Portfolio: C2:D4 . The interval represent 1 stock bought in a sold position of 416 contracts of future interests DI1. The face value of the sold contracts its negative.
| 20,000,000 | 220 |
| -41,600,000 | 50 |
= {CC.DURATION( A2:B11, C2:D4)}
Results:
| 220 |
| 7,758,197.55 |
| 9,135,144.96 |
| 0 |
| 0 |
By the calculations of the duration, have the keeping of the purchase with face value redution by selling 416 future contracts. This contracts effected a redution of 10,864,855.04 (20,000,000 – 9,135,144.96 = 10,864,855.04) in the face value of the position.
By the result of the function CC.DURATION, was not obtained the expected before: redution of duration for 100 days with the same face value (function CC.HEDGE). This cause, the function CC.DURATION keep the duration continuous and the partial hedge keep the face value continuous. Althought this, the results are equivalents.
To show the equivalence of results, the calc for total hedge for a position of 10,864,855.04 (position anulled by selling future contracts) with a duration of 220 days and a rate of 20.58% (observed directly in the curve) using a future contract with settlement in 50 days and a interpoled rate with base in the curve of the example 19.30%.
- Contract rate for hedge 19.3%
- Contract settlement 50 (workdays)
- Face value of bought position 10,864,855.04
- Rate for position 20.58% (base 252)
- Position duration 220 (workdays)
= CC.HEDGE( 0.193, 50, 10864855.04, 0.2058, 220)
Results:
The result is selling 416 contracts.
This show that the duration of the original position plus the contracts sold its of 220 days for a face valueof 9.135.144,96 (function CC.DURATION). This duration its equal, at risks for small and parallel oscilations of the yield curve informed, for a duration of 100 days for a face value of 20,000,000.00 (partial hedge with target duration).
Its important distinguish the difference between the duration calculated by the function CC.DURATION, which keeps the duration continuous and reduce the face value for using target duration in hedge by the function CC.HEDGE, which keeps the face value but reduces duration. Once more, the two results are equivalent!
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