2.2.7. Brazilian Yield Curve Exponential Interpolation and Extrapolation
Interpolation is a mathematical used to, from limits points (in edges), create intermediate points. Its not necessary use interpolations when the searched point its on any limit (inferior or superior), simply using the point coincident, although the calculated result by interpolation its the same.
Exists a lot of types of interpolation: linear, polynomial, exponential etc. Choose the interpolation which must be used depends from the type of data which its being analysed. Its a point of coherence with the variable behavioring represented by data or by points.
The exponetial interpolation has aplication in finances in function of policies of capitalization composed aplied to interest and other cumulative indicators (profit,inflation etc).
In particular case of interests rates, the exponteial interpolation between two points represents the course which an financial aplication has from the first point to the second considering a continuous movement,so, without "degree" between points. This can be reached considering the interests rate inlaid betwenn these limits points like a constant throughout the time.
Yet for interest rates case, the exponential interpolation its the interpolation with bigger economic purpose. Although this, the exponetial interpolation can present discontinuities when its analysed the speed of changing of interests rates or the first deriative of this curve. This discontinuities can appear only over the points which create a interest curve and represent a changing of rates between two consecutive periods.
Extrapolation functions of interests rates are most hard to be shaped and demands an extra number of hypothesis to be constructed.
2.2.7.1. Function MX.INTERPOLEX
Access:
- Menu - Insert | Function | Metrixus
 -Padrão | Metrixus ToolBar
Description:
Returns the interest rates exponetial interpoled or extrapoled to the period indicated from the interests curve given. In case of extrapolations, that is, periods beyond the last curve expiration informed, the rate returned its negative to signalize the extrapolation.
Are necessary at least 3 points in the curve to calculate the extrapolated interests rates, accomplished through the differences in base points between the last 3 expirations and through determinating a new inclination to the interests rates. This inlcination its attenuated until the date informed ( flat curve) by a curve with a constant second derivative and the first derivative equals 0 on the flat date (inclination tending zero to the flat date).
One interests curve can't have more than one information for the same point in time (in example one information of future interests and other of swap's). If its occur, will prevail the bigger rate.
Call: MX.INTERPOLEX (Workdays, Yield Curve, Years Flat)
Argument |
Type |
Description |
Workdays |
integer |
Workdays for which interest rate is calculated. This number must be bigger than 1.
|
Yield Curve |
range |
Range (n rows and 2 columns) describing Brazilian yield curve (rates base 252) on the first column and workdays in the second column. This data are used to calculate interests rates for any expiration date, including extrapolation for long and after periods for the last date indicated. Must have at least 2 differents points in one yield curve.
|
Years Flat |
double |
Optional. Number of years (base 252) after the last rate informed where the yield curve flats. The extrapoled rates are informed negative and are base in the difference in base points of the 3 lasts rates informed. The extrapolation its effected by a curve with a constant second derivative and with first derivative equals 0 in the flat date. The default value is 0, being the curve considered flat after the last date.
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The points on the yield curve informed does not need to be sorted in time, but must be disposed in a range with two columns only.
Cells containing text or empty for the yield curve are ignored by the function. Data less than 1 for the parameter Workdays also be ignored!
Important:
The interests rates informed in the parameter Yield Curve must be the anual rates with base 252. The rates must be in the first column and the workdays in the second.
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Important:
To repeated points in time, will prevail that with bigger rate!
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The result for a yield curve informed with n points is:
- Interpolation: considering Pi and Pi+1 as points in the yield curve, Di and Di+1 as workdays corresponding to this points and Dint to the date of the searched point, being Dint between Di and Di+1, the interests rate Pint interpolated for the searched point wil be given by:
- Interpolation:
Important:
If the searched point is a expiration date before the first date of the yield curve, the rate returned will be equal to the interest rate of the first date!
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- Extrapolation: considering Pn, Pn-1 and Pn-2 as the 3 lasts points in the yield curve, Dn, Dn-1 and Dn-2 as the workdays corresponding to this points, Dext and the date on the point searched and AFLAT the years after the last point given in the yield curve, being Dext bigger than Dn, the interests rate Pext extrapoleed to the point searched will be given using:
- Extrapolation Dext >=DFLAT:
- Extrapolation Dext <=DFLAT:
Onde:
- Days flat:
- Inclination of the 2 lasts stretches of the yield curve:
- Factor of convergence:
- Initial inclination for extrapolation:
- Rate flat:
Important:
In case of extrapolations, the rates informed are negative only to intensify the idea of extrapolation of interest rates and this sign does not have any economic sense. Are necessary at least 3 points to allows extrapolations of interests rates or the function will return ERROR.
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Important:
This algorithm is proprietary of Élin Duxus.
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Example using the exponential interpolation:
Exponetial interpolation of yield curve - parameters:
- Workdays: 120
- Years flat: 1,0
- Curve: A2:B11 (11 points not sorted)
| 18,75% | 20 |
| 20,65% | 220 |
| 20,70% | 300 |
| 20,40% | 140 |
| 19,15% | 40 |
| 19,75% | 80 |
| 20,71% | 350 |
| 20,60% | 180 |
| 19,40% | 60 |
| 20,00% | 100 |
= MX.INTERPOLEX( 120; A2:B11; 1,0)
Results:
Yield curve interpolated for many points. To construct this graphic the interpolated rates was calculated for all workdays until the curve reachs one trajectory flat (252 days or 1 year after the last date informed in the parameter Curve). See one line to the first date, indicating that does not have interpolations on this period.
Example using extrapolation:
Extrapolation to the yield curve invert - parameters:
- Workdays 450
- Years flat 0,75 (9 meses após último vencimento)
- 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 |
= MX.INTERPOLEX( 450; A2:B11; 0,75)
Results:
The negative result informed does not have economic sense, its used only to intensify that it is an extrapolation of interests rates, effected following the proprietary model of Élind Duxus.
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