2.2.7. Duration, Exposure and Convexity for Fixed Income Portfolios
The duration or duration of a bond means the mean time that the owner have to wait to receive the payments. A stock without intermediate payments or 'cupom????' have the same duration as its settlement. Now a stock with 'cupom??' have a smaller duration.
The duration also is a measure of sensibilitty to the alterations in the yield curve, being so used in calcs of hedge operations of assets related to fixed income.
Exists many types of duration, as Macauley (Frederik Macauley), modified and effective. Each type has a distinct purpose.
Macauley duration or simply duration of a stock its equivalent to the mean of limits of each 'cupom???' weightedby its present value (VPi). In case of a portfolio, the same thought can be used, only understanding each stock as a additional set of payments or 'cupons????'.
Thus, for duration in days:
where Pi,day is the daily interest rate for the date di(duration in days) and VFi is the face value of the actual payment.
In anual terms for interest rates (base 252) and considering the duration in days, having:
where Pi,aa is the daily fixed income Pi,dia anual for the date di, yet in days.
The same calc can be used also for portflios in future interest contracts, where each contract is represented by its face value and its settlement.
Its useful determinate also the final exposure equivalente to what can be resumed all the positions of a portfolio, that is, the portfolio its changed by a unique equivalent position of the same final duration. For the calcs of this exposure, can be used the concepts of hedge by modified duration - see section 2.2.8. Hedge by Modified Duration for Fixed Income
Important:
The concept of final exposure its different from the concept of actual value of a portfolio and represents a equivalent position in the duration of the portfolio for the risk of small and parallel oscilations in the yield curve!
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The convexity of a portfolio attached to the interest rates its important to determinate the changes in the actual value of this portfolio in front of a parallel oscilation in the yield curve, but with a relative big amplitude. This cause convexity its the measure of the gradient or of the derivative of the duration and indicates how the duration alternates according to the size of the oscilation in the interests rates.
the convexuty its more important for stocks (or portfolios) with various coupons (or payments) than stocks with only a settlement.
For small oscilations all the chnges in the actual value can be explained by the duration only.
The convexity of a portfolio (or set of payments and coupons) for the anual interests rate and the duration in days its given by:
The variation on the actual value of a portfolio in function of a parallel oscilation in the yield curve can be aprroached by the second order of the Taylor series:
where the index * represents the modified duration:
The calc of the duration of a portfolio also can wrap bought positions (usally with the positive face value) or sold (face value negative). In this case, the simple application of the concept of duration does not have a pratical sense, cause the actual value of the positions appears in the denominator of the equation of duration and the proximity of this value to 0 makes hard to understand the results. Alternally, the opposite position can be understood as a redution in the other side (bought or sold) and, not considering the convexity, the concept of hedge by modified duration - see section 2.2.8. Hedge by Modified Duration for Fixed Income - can be used in the calcs.
A important detail is that the existence of opposite positions in a portfolio can take to two results types: maintenance of duration and reducing face value from main position or maintenance of the face value and reduction of duration in the main position, where main position (buy or sell) its that with the bigger value for multiply the actual value by the ajusted duration. Both results are equivalent from risk sight of small operations (convexitynot considered) and parallel in the yield curve. The result over continuous duration is most commom.
The duration calc of a portfolio its so used in portfolio of fixed income to condense all positions in a duration only and just for effect the total or partial hedge of this portfolios.
2.2.7.1. Function CC.DURATION
Access:
- Menu - Insert | Function | Calculus
 - Toolbar Default | Calculus
Description:
Returns the Macauley duration or duration in workdays of a portfolio directly from the yield curve and assets or payments which compose this portfolio, as well the convexity, modified duration and the face value and actual value of the total exposure duration of the portfolio.
Accepts postions bought (negative value) or sold (positive value) composing the portfolio and returns, in this cases, the duration by the usage of hedge by modified duration for reducing main position (bigger value represented by actual value - buy or sell - multiplied by the respective modified duration ) with continuous duration and with beta - coefficient of correlation between the buying or selling positions - equal to 1.
The calc of the final exposure also uses the concept of hedge by modified duration.
In case of opposite positions in portfolio, the convexity to the ajsuted duration are not calculated and teh fuction returns 0.
For the calc of rates relative to payments of the portfolio, realizes the interpolation by the exponential method. Does not effected extrapolations, being the payment interest rates which occur in after periods to the last point from the yield curve informed equal to the rate in this point. Also does not exist interpolations before the last settlement of the yield curve.
A self yield curve can't have more than one information for the same point in time (in example a information of future interests and other of swap's). If this occur, will prevail the bigger rate.
Important:
The interpolation method used for calcs of duration its the expontial interpolation, without extrapolations!
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The function return its obtained by the matrical format, being necessary, to obtain all output data, the selection of 5 vertical cells and use of CTRL + SHIFT + RETURN after typing the formula.
Call: CC.DURATION ( Yield Curve, Portfolio)
Argument |
Type |
Description |
Yield Curve |
range |
Interval (matrix n rows by 2 columns) containing the interests rates to the year (base 252) in the first column and the workdays until the settlement of each point in the second column. This data are used in the calcs for interest rates for any settlement. May exist at least 2 different points in the yield curve.
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Portfolio |
range |
Interval (matrix n rows by 2 columns) containing the face values of various payments of a portfolio in the first column and the workdays until each payment in the second column. The values bought and sold must be informed with opposite operators (usually positive for bought and negative for sold).
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Note 1: Results must be extracted from Excel selecting the region for the results, formula insertion with the output selected and pressing CTRL+SHIFT+RETURN.
The points in the informed yield curve does not need to be in time order, but need to be disposed in a interval with only two columns. The portfolios payment also does not need to be in time order.
Data containing text or empty for a yield curve as well for the portfolio are ignored by the function.
Important:
The interests rates informed in the parameter of the yield curve must be the anual interest rates with base 252. The rates must be in the first column and the workdays in the second.
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Important:
For repeated points, will prevail that with bigger rate!
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Important:
Positions of portfolio in the buy or sell must have opposite operators!
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The result for a informed yield curve in a portfolio with n positions is:
- Duration: weighted mean of duration of each payment with face value VF. This value its rounded and depends of the existence of opposite positions in the portfolio.
- Without opposite positions:
Where Pint its the interest rate to the year base 252 interpoled corresponding to a payment, considering Pj and Pj+1 with point from the yield curve, Dj and Dj+1 as the number of workdays corresponding to this points and Di the date of this payment, being Di between Dj and Dj+1:
- With opposite positions, uses hedge by modified duration with beta equal to 1 and continuous duration, being the present value of the positions given by VPcompra and the present value of soldpositions given by VPvenda and the modified duration of the shops and sells given, respectively, by Dshop and Dsell and D*shop and D*sell:
Important:
If the payment day is previous to the firt settlement in the yield curve, the interpoled rate will be equal to the interest rate of the first point in the yield curve! If its previous to the last settlement, the interpoled rate will be equal to the last rate in the yield curve (without extrapolations).
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- Final exposure - present value: present value for a position equivalent to teh portfolio, obtained by the concept of hegde by modified duration. Being VP, Ddays and D*days - respectively, the present value, duration and duration ajusted - relative to the principal position and VPopposite and D*days,opposite respectively, the present value and duration ajusted - raletive to the opposite position and where VP and VPoposta have contrary operators:
- Final exposure:
Important:
The values of ajusted duration for the main position (shops or sells) and opposite (sells or shops) are obtained as the same way that for duration without opposite positions, but concentrating all sells and all shops separately.
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- Final exposure - face value: future value of final exposure for the duration date of the portfolio. Considering PD the anual interest rate (base 252) for the final duration , have:
- VF:
- Convexity: convexity of portfolio. This value must be used for calculus of variations in the present value of the face portfolio to a not so small oscilation and parallel in the yield curve. In case of opposite positions, the convexity isn't calculated and return 0:
- Convexity:
(Note: 'Convexidade': Convexity; 'Oposta': Opposite; 'Sem oposta': Without opposite)
- Modified Duration: This value also is used for calculate variations in the present value given an oscilation in the yield curve. In case of opposite positions, return 0:
- Ajusted duration:
Example using bought positions:
Calc of duration for the composed portfolio by pre-fixed titles and future interest contracts. All the positions are bought:
- 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 |
- Portfolio: cells C2:D5. The interval represents 2 stocks bought and a bought position of 100 contracts of future interest DI1. All values are positive (or have the same operator).
| 10,000,000 | 100 |
| 20,000,000 | 220 |
| 10,000,000 | 50 |
= {CC.DURATION( A2:B11; C2:D5)}
Results:
| 143 |
| 36,031,941.36 |
| 40,038,574.73 |
| 0.67595917 |
| 0.47240009 |
The return its matrical and can be returned from 1 to 5 cells, being:
- 1ª cell => Macauley duration
- 2ª cell => final exposure or equivalent present value
- 3ª cell => final exposure face value for duration
- 4ª cell => convexity
- 5ª cell => modified duration
The duration's value (rounded) of the portfolio is 143 days and the portfolio have an equivalent final exposure with present value of 36,031,941.36 and a face value of 40,038,574.73 for the correponding date to the day 143 from yield curve. In other words, the portfolio's risk for parallel and small oscilations of yield curve its equal to that with only one position with face value of 40,038,574.73 for 143 days. For the case of using hedge, the face value of the final exposure must be used.
It's good to observe that the present value of final exposure its different from the present value of stocks of 35,943,357.84 (not calculated). This cause the final exposure its a risk position equivalent for small (not consider convexity) and parallel oscilations in interest rate and not a presente value equivalent position.
In example, considering an oscilation of -0.1% in the anual rate, where all rates decreases 0.1%, the present value of the portfolio goes to 35,960,349.64 (not calculated) representing a increase of 16,991.80. The present value of equivalent exposure increases for 36,048,932.06 (40,038,574.73 present value for a interest rate with oscilation for 143 days) - a increase of 16,990.70.
Once more, for doing hedge by modified duration, what interest is the face value of the exposure and the duration.
Considering now a parallel variation of the yield curve of +1.0% in anual rate, where all rates increases 1.0%, and the present value variation of the portfolio can be simulated. As this oscilation its not a small oscilation, the convexity must be used together with modified duration. Recalculating, so, for an oscilation of +1.0%, having a percentual variation given by:
This is equivalent to say that the portfolio's present value of 35,943,357.84 (not calculated) will fall from 0.046902% or a lost of 168,581.64.
Simulating an oscilation in the yield curve, the present value for a upward shift of 1.0% goes to 35,774,766.98 - a lost of 168,590.65
The difference between values can be unconsidered and its attributed to the oscilations size, once that Taylor series its used and it's from second order (discarding upper orders).
Finishing, for small oscilations, can be used the hedge or changing values by convexity and modified duration concepts. For big oscilations, only the second concept can be used, cause using convexity in hedge its more complex. In both cases, althought, the yield curve oscilations must be parallel.
Example using bought and sold positions - oposite positions:
This example uses the same yield curve from previous example. Will be used bought and sold positions for the portfolio and will be calculated the duration, total exposure and face value by hedge concept. The convexity and modified duration are returned null.
- Portfolio: C2:D5. The interval represent 3 stocks bought and a sold position of 300 contracts of future interests DI1. The bought values are positives and the sold value in future contract has negative value.
| 10,000,000 | 100 |
| 20,000,000 | 220 |
| 10,000,000 | 50 |
| -30,000,000 | 50 |
= {CC.DURATION( A2:B11, C2:D5)}
Results:
| 143 |
| 25,808,357.78 |
| 28,678,162.28 |
| 0 |
| 0 |
The final duration its equal to bought duration (previous example) cause the present value multiplied by ajusted duration of bought position its upper to the equivalent multiplication of the sold position. In other words, the sell of future contracts of this portfolio reduced the exposure (see section 2.2.8. Hedge by Modified Duration for Fixed Income) from buying 40,038,574.73 (face value from total exposure - previous example) for a face value of 28,678,162.28 with the same duration of 143 days.
Considering an oscilation of +0.1% in anual rate, where all rates decreases 0.1%, the portfolio's present value decrase to 6,975,601.22 (not calculated) for 6,693,449.05 representing a loss of 12,152.18. The equivalent exposure's present value decreases to 25,796,203.80 (28,678,162.28 to present value for a interest rate with an oscilation for 143 days) - a fall of 12,153.99.
Once more, with use of contrary positions, its possible to obtain two results with the ssame risk for parallel and small oscilations from yield curve, being more usual the duration calc considering the duration of the main continuous position considering the oposite position a reduction of the main (effected the function).
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