We will create a table variable to store the data created by the SWAPCURVE function.

DECLARE @SWAPCURVE as TABLE (

mat_date datetime,

df float,

rsource char(1),

zero_cpn float,

cczero float

)

The @SWAPCURVE table contains a unique row for each date. Each row identifies the discount factor (df), the continuously compounded zero coupon rate (cczero), and the zero coupon rate (zero_cpn). It is fairly straightforward to convert from one to the other (see the previous blog entry) and for purposes of this article we will only use the discount factors. In addition, the source for each row is identified in the rsource column: cash rates (C), futures (F), swaps (S), or interpolated rates (I). The source of the rates is very important for deconstructing the rates.

Run the following SQL to populate the @SWAPCURVE table.

Before doing any calculations, we need to establish the dates that are being used. This curve was generated on 2016-09-09. The *spot* date is calculated as 2 business days from the processing date. The spot date is essential in the calculation of all maturity dates as they are quoted with respect to the spot date. In talking about a rate for a 10-year swap, for example, the maturity date is calculated as 10 years from the spot date; not 10 years from the processing date.

DECLARE @TODAY as date = CAST('2016-09-09' as datetime)

DECLARE @SPOT as date = wct.BUSINESSDATE(@TODAY,'D',2,'M','')

Since cash and swap instruments are quoted with respect to the spot date, their associated discount factors are calculated from the spot date to the maturity date. To obtain the discount factor back to the processing date (the @TODAY variable) we need to, effectively, discount the discount factor for the instrument from the spot date back to @TODAY. This requires that we know the discount factor for the spot date.

DECLARE @df_spot as float =(SELECT df FROM @SWAPCURVE WHERE mat_date = @SPOT)

Remember that in calculating the discount factors we use the cash rates until the start of the futures strip, the futures rates until the start of swaps and then we 'bootstrap' the swaps for each coupon date as well as calculating an interpolated swap rate for each coupon date.

At the transition from the cash part of the curve to the futures strip we calculate an interpolated discount factor from the discount factors that surround the start of the futures strip. The discount factor beyond the strip start date is thrown away.

The discount factors from the futures strip are calculated contiguously until the first swap maturity date and any after that date are thrown away. An interpolated discount factor is calculated for each swap coupon date prior to the first swap maturity date.

In many ways it is simpler to work backward from the end of the curve to the beginning of the curve. Remember that the swaps part the curve relied on a bootstrapping process which used this formula:

Since we know all of the discount factors we can reconstruct the swap part of the curve using this formula:

The following SQL implements this formula and puts the results into the #swaps temporary table. We use the BONDCF table-valued function and BUSINESSDATE scalar function to calculate all the coupon dates associated with the swaps so that we pick up the correct deltas and discount factors. We use the YEARFRAC function to calculate the deltas.

--Recover the swap rates

SELECT

*,

ROUND((@df_spot-df)/((df*delta)-(@df_spot*-LAG(bootstrap,1,NULL) OVER (ORDER BY q.mat_date))),10) as [Swap Rate]

INTO

#swaps

FROM (

SELECT

*,

SUM(delta * df) OVER (ORDER BY p.mat_date) as bootstrap

FROM (

SELECT

n.mat_date,

S.rsource,

S.df,

wct.YEARFRAC(@SPOT,n.mat_date,2) - wct.YEARFRAC(@SPOT,LAG(n.mat_date,1,@SPOT) OVER (ORDER BY n.mat_date),2) as delta

FROM (

SELECT

wct.BUSINESSDATE(k.date_pmt,'M',0,'M','') as mat_date

FROM

(SELECT max(mat_date) as maxDate FROM @SWAPCURVE)n(mdate)

CROSS APPLY

wct.BONDCF(@SPOT,n.mdate,0,0,NULL,2,NULL,NULL,NULL,NULL)k

)n

INNER JOIN

@SWAPCURVE S

ON

n.mat_date = S.mat_date

WHERE

n.mat_date > @SPOT

)p

)q

The #swaps table should contain the following results

We can use the following formula to calculate the rates associated with the futures contracts.

We need to calculate the *futures strip start date* to establish a starting point for the calculation. For yield curve construction we know that the Eurodollar futures contracts are delivered on the 3^{rd} Wednesday of the month and that the strip is in 3-month increments. By using the FIRSTWEEKDAY and EDATE function to determine the strip start date, it becomes quite straightforward to calculate the associated rates using the formula. The following SQL puts the results in the #futures temporary table.

--Recover the futures rates

SELECT

*

,ROUND(((LAG(df,1,NULL) OVER (ORDER BY mat_date)/df)-1)/wct.YEARFRAC(LAG(mat_date,1,NULL) OVER (ORDER BY mat_date),mat_date,2),10) as [Futures Rate]

INTO

#futures

FROM (

SELECT

mat_date,

df,

rsource

FROM (

SELECT

--Figure out what the start date is for the futures strip

DATEADD(d,14,wct.FIRSTWEEKDAY(wct.EDATE(MIN(mat_date),-3), 4))

FROM

@SWAPCURVE

WHERE

rsource = 'F')n(date_start)

INNER JOIN

@SWAPCURVE S

ON

n.date_start = S.mat_date

UNION ALL

SELECT

mat_date,

df,

rsource

FROM

@SWAPCURVE S

WHERE

rsource = 'F'

)n

This is what the #futures table should look like.

We can use the following formula to recover the cash rates.

T_{n} in this formulation is the time, in years, from the spot date to the maturity date of the cash instrument. The following SQL uses the YEARFRAC function to calculate the time values in implementing the formula and puts the results in the #cash temporary table.

--Recover the cash rates

SELECT

mat_date,

df,

rsource,

ROUND((@df_spot/df - 1) / wct.YEARFRAC(@SPOT,mat_date,2), 10) as [Cash Rates]

INTO

#cash

FROM

@SWAPCURVE

WHERE

rsource = 'C'

AND mat_date > @SPOT

This what the #cash table should look like.

Notice that in our previous SQL we excluded all the maturity dates less than or equal to @SPOT. If the maturity data is less than or equal to the spot date, then the calculation of the rate value does not have to be adjusted by the discount factor for the spot date. It requires a different formulation.

The calculation for the *tom/next* maturity date (the first entry in the @SWAPCURVE table) is straightforward. This SQL adds the tom/next rate to the #cash temporary table.

INSERT INTO

#cash

SELECT

mat_date,

df,

rsource,

ROUND(((1/df)-1)/wct.YEARFRAC(@TODAY,mat_date,2),10)

FROM

@SWAPCURVE

WHERE

mat_date < @SPOT

The calculation for the spot rate is similarly straightforward but relies on the tom/next rate calculated in the previous step. This SQL adds the spot rate to the #cash temporary table.

INSERT INTO

#cash

SELECT

@SPOT,

S.df,

S.rsource,

ROUND(((n.df/s.df)-1)/wct.YEARFRAC(n.mat_date,@SPOT,2),10)

FROM

(SELECT TOP 1 mat_date, df FROM #cash ORDER BY mat_date asc)n

INNER JOIN

@SWAPCURVE S

ON

S.mat_date = @SPOT

Not quite. We still need to account for the cash rate that was used for the interpolated rate for the start of the futures strip. Recall that the strip start date was 2016-12-21. The interpolated rate for the strip start date was calculated using the cash rates. We have already calculated the cash rates out to 3 months. Given that we know the interpolated discount factor for the strip start date and the discount factor for the 3-month cash rate, can we infer the 6-month cash rate?

When we transition from the cash part of the curve to the futures part of the curve we use *discount factor interpolation* (see the DFINTERP function) to calculate the discount factor for the strip start date. The formula for discount factor interpolation is:

Where t is time in years and t_{n-1} ≤ t ≤ t_{n} and a = (t_{1}-t) / (t_{2} – t_{1}). Simple re-arrangement of terms gives us the following equation.

The following SQL implements the re-arranged equation and stores the result in the #cash temporary table. We use the TENOR2DATE function to calculate the maturity dates, the FIRSTWEEKDAY and EDATE functions to calculate the strip start date and the YEARFRAC function to calculate all the time values.

--Calculating the cash rate used to start the futures strip

DECLARE @date_3m as datetime = wct.TENOR2DATE('3M',@TODAY,@SPOT,'')

DECLARE @date_6m as datetime = wct.TENOR2DATE('6M',@TODAY,@SPOT,'')

DECLARE @date_futures_start as datetime =(SELECT DATEADD(d,14,wct.FIRSTWEEKDAY(wct.EDATE(MIN(mat_date),-3), 4)) FROM @SWAPCURVE WHERE rsource = 'F')

DECLARE @t as float = wct.YEARFRAC(@TODAY,@date_futures_start,3)

DECLARE @tnminus1 as float = wct.YEARFRAC(@TODAY,@date_3m,3)

DECLARE @tn as float = wct.YEARFRAC(@TODAY,@date_6m,3)

DECLARE @alpha as float =(@t-@tnminus1)/(@tn-@tnminus1)

DECLARE @dft as float =(SELECT df FROM @SWAPCURVE WHERE mat_date = @date_futures_start)

DECLARE @dfnminus1 as float =(SELECT df FROM @SWAPCURVE WHERE mat_date = @date_3m)

DECLARE @dfn as float = POWER(@dft*POWER(@dfnminus1,@t*(@alpha-1)/@tnminus1),@tn/(@t*@alpha))

INSERT INTO

#cash

SELECT

@date_6m,

@dfn,

'C',

ROUND(((@df_spot / @dfn) - 1) / wct.YEARFRAC(@SPOT,@date_6m,2), 10)

It's worth noting that the 6-month cash maturity date coincides with the first coupon date from the swap part of the curve. However, other than in the interpolation of the discount factor for the start of the futures strip, this rate is not used. The 6-month rate used in the bootstrapping process is calculated using discount factor interpolation and the futures discount factors.

The cash part of the curve:

SELECT

mat_date,

[Cash Rates]

FROM

#cash

ORDER BY

mat_date

This produces the following result.

The futures part of the curve:

SELECT

DATEADD(d,14,wct.FIRSTWEEKDAY(wct.EDATE(mat_date,-3), 4)) as [Contract Date],

100 *(1-[Futures Rate]) as [Futures Price]

FROM

#futures

WHERE

rsource <> 'I'

This produces the following result.

The swaps part of the curve:

SELECT

mat_date,

[Swap Rate]

FROM

#swaps

WHERE

rsource <> 'I'

This produces the following result.

The benchmark spot rate or zero-coupon rates can be calculated from the discount factors. When we talk about spot rates in this context we are not talking about the cash rate for the spot date (which we used in many of our calculations) but rather the yield on a zero coupon bond implied by the discount factors. This calculation should use the same compounding assumption as the swaps curve, which in our example is semi-annually.

SELECT

mat_date,

df,

[Swap Rate],

ROUND(2*(POWER(@df_spot/df,1e+00/(2*DATEDIFF(year,@SPOT,mat_date)))-1),10) as [Benchmark Spot Rate]

FROM

#SWAPS

WHERE

rsource = 'S'

This produces the following result.

By deconstructing the swap curve we have gained more insight into how cash rates, swap rates, and futures prices are converted into discount factors and continuously compounded zero coupon rates. This article has hopefully helped in understanding the link between the inputs and the outputs to those calculations. To find out more about how to build the yield curve, see our article on Constructing a Yield Curve in SQL Server.

If you are interested in taking advantage of the powerful tools in the XLeratorDB library of functions, you should download the free 15-day free trial today.

If you are not a SQL Server user you should try out the XLeratorDLL library of functions, which also has a free 15-day trial. Download that and see how easy it is to incorporate this functionality into your .NET code.

Got questions? Send us an e-mail at support@westclintech.com or directly to me at cflock@westclintech.com

· TENOR2DATE - Convert an alphanumeric expression ('tenor') to a swaps or money market maturity date

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