18 minutes reading time (3669 words)

Building in Excel the Implied Risk-Free Discounting Curve when the Collateral is kept in another Currency. Example: Mexico, where no Local OIS Market exists.

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Before the 2007-08 financial crisis, all swap cash flows were discounted with the same yield curve used in "forecasting" the future Libor rates and thus the amounts of the floating cash flows.

At any given time, this yield curve was constructed so that it reproduced the market prices of the swaps traded at that time.

After its construction, the yield curve could be used to calculate the price of non-traded custom swaps and other interest rate derivatives.

This practice has changed since then.

Now two different curves are required to price any collateralized contract, such as a vanilla fixed-to-floating interest rate swap traded over the counter.

One of these curves is referred as the discounting curve (or basis curve) and used exclusively for calculating the present value of future cash flows whose amount is considered known.

The risk profile of that curve must correspond to the risk profile of the collateral used in the collateralization of the swap's flows.

Since the latter is often a margin account holding almost risk-free overnight securities, the discounting curve is built out of the market-quoted rates of OIS (Overnight Index Swaps) that reference the same, or similar, overnight index as that of the securities held in the collateral account.

The second curve is referred as the forecasting curve and used exclusively for calculating the forward rates needed in the calculation of the floating cash flows.

Given an already constructed discount curve, one can easily build the forecasting curve by bootstrapping over the swap rates quoted in the market.

The complete process is described in my post about Exogenously Bootstrapping the Discounting and Forecasting USD Libor curve.

A problem arises when the collateral is held on a currency other than that of the collateralized swap.

This may happen when a contractual clause in the Credit Support Annex (CSA) accompanying the swap contracts requires a USD-denominated collateral account, as is the case in several countries such as Mexico, South Africa etc.

In the sequel, I will focus on the MXN swap market in Mexico, where the collateral accounts of Mexican Peso (MXN) swaps are held in USD currency and earn the overnight USD Fed Funds interest.

Also note that no local OIS market exists in Mexico. 


The Interest Rate Swap Market in Mexico 

The Banco de México publishes daily the Interbank Equilibrium Interest Rate, known as TIIE from Tasa de Interés Interbancaria de Equilibrio, which reflects the average interbank rate polled from Mexican commercial banks.

There exist four tenors that cover 1, 28, 91 and 182 days.

A typical fixed-to-floating MXN swap involves the exchange of the TIIE28 index (the TIIE index with a 28-day tenor) for a fixed rate.

As mentioned already, collateral is held in USD currency earning the USD Fed Funds overnight rate.

A liquid swap market exists with tenors from 3 "months" to 30 "years".

I have surrounded the words "months" and "years" with quotes because of the following Mexican market peculiarity:

Due to the 28-day (4 weeks) tenor of the TIIE28 index, the Mexican "month" in the context of swaps refers to a lunar month consisting of 4 weeks.

Below I will be using the symbol L (from Lunar) for representing a time period of 4 weeks.

So, a swap tenor of 3 "months" corresponds to a tenor equal to 3L, which is a period of 12 weeks (or 48 days).

A swap tenor of one "year" corresponds to a tenor equal to 13L, which is a period of 52 weeks (or 364 days).

Note that the maturity of a one "year" swap will occur at least one calendar day earlier than the end date of a regular one-year period.

The longest quoted tenor of 30 "years" corresponds to a tenor equal to 390L, which is a period of 30*52 weeks (or 30*364 days).

It follows that the maturity of a 30 "year" swap will occur at least 30 calendar days earlier than the end date of a regular 30-year period.

Below is a screenshot of the Bloomberg quotation of MXN vs TIIE28 swaps as of 06 May 2019.

Note the notation Nx1 for a NL (N Lunar months) term. 


Extracting the Implied Discounting Curve in the case of MXN Swaps 

In order to extract the implied discounting curve, I will progress with the following steps:

Step 1

Build the "risk-free", discounting USD yield curve USD-OIS out of the USD OIS market rates because this is the curve associated with the risk profile of the USD-denominated collateral.

The details are described in my post about building discounting and forecasting curves using exogenous bootstrapping.

Diagrammatically, this step is represented as: 


Step 2

Build the forecasting USD 3M Libor yield curve USD-3M out of the USD-OIS yield curve and the USD 3M Libor swap market rates because it will be needed in the next step for the construction of the forecasting USD-1M yield curve.

The details are described in the link mentioned above.

Diagrammatically, this step is represented as: 


Step 3

Build the forecasting USD 1M Libor yield curve USD-1M out of the USD-OIS and USD-3M yield curves and the USD 3M vs 1M tenor basis swap market spreads because it will be needed in the final step for the construction of the implied discounting MXN yield curve.

The details are described in my post about building a forecasting curve out of tenor basis swap spreads.

Diagrammatically, this step is represented as: 


Step 4

Build the discounting MXN yield curve MXN-DISC out of the USD-OIS and USD-1M yield curves, the MXN TIIE28 swap market rates and the USD vs MXN cross currency basis swap market spreads.

Details of this construction are presented further below.

Diagrammatically, this step is represented as: 


Below is the consolidated flowchart of all the four steps.  


Detailed Analysis of the Last Step involving Cross Currency Basis Swaps 

Deriscope can build the discounting curve associated with a currency such as MXN provided the following is known:

  • A.The market spreads of a group of cross currency basis swaps between USD and MXN
  • B.The "risk-free" discounting yield curve for USD
  • C.The forecasting yield curve for USD
  • D.The forecasting yield curve for MXN

The USD vs MXN XCCY basis swaps are floating-to-floating swaps exchanging USD 1M Libor plus spread for MXN TIIE28, which is by the way opposite to the common convention having the spread added on the non-USD index.

Both legs have the same 28-day accrual period, which means the accrual period on the USD leg is slightly less than the one-month tenor of its 1M Libor index.

More details on XCCY swaps and the corresponding yield curve extraction can be found in my post about building a curve out of cross currency basis swaps.

Unfortunately, we cannot use the above recipe due to missing the D part, i.e. the forecasting yield curve for MXN.

But, why is the D part important?

Only because the market data refer to XCCY floating-to-floating basis swaps, which are instruments involving the future TIIE28 fixings. The forecasting yield curve for MXN is then needed only for forecasting these fixings.

This observation brings us to the idea of replacing the XCCY floating-to-floating basis swaps with XCCY floating-to-fixed swaps, which are cross currency swaps exchanging floating USD 1M Libor against a fixed MXN rate. Then we could imply the MXN discounting curve without worrying about the unknown MXN forecasting curve.

The problem is that no such swaps are traded!

On the other hand, such swaps may be synthesized from XCCY USD vs MXN floating-to-floating basis swaps and vanilla MXN floating-to-fixed swaps.

More specifically, the following holds, at least approximately:

Let:

Swap 1: Receive USD 1M Libor plus Spread vs Pay MXN TIIE28

Swap 2: Receive MXN TIIE28 vs Pay MXN Fixed Rate

Synthetic Swap: Receive USD 1M Libor plus Spread vs Pay MXN Fixed Rate

Then:

Synthetic Swap = Swap 1 + Swap 2

Diagrammatically: 


This synthetic construction is only valid in an approximate sense because the floating MXN TIIE28 leg paid in Swap 1 does not exactly cancel out the corresponding leg received in Swap 2, due to differing conventions.

Concretely, Swap 1 settles at T+1 while Swap 2 settles at T+2, a fact that leads to mismatched accrual and payment dates. Also Swap 1 follows the union of the US and Mexican calendars, whereas Swap 2 follows just the Mexican calendar.

Although the effect is minor, one may calculate an appropriate adjustment that can be applied on the fixed rate in Swap 2, so it is at par with a slightly modified floating leg that exactly matches that of Swap 1. 


Building the Implied Discounting MXN Yield Curve in Excel: Steps 1 to 3 

I will now implement in Excel the four steps that lead to the creation of the implied discounting yield curve for MXN by using actual market data from Bloomberg as of May 22, 2019.

The three prerequisite curves in steps 1 to 3 are constructed as described in my post about building the forecasting yield curve implied by tenor basis swaps.

Below are the formulas and data that lead to the creation of the respective objects of type Yield Curve at the top/left of each image: 


The ImpYC XCCY Disc type for Building an Implied Discounting Yield Curve out of XCCY Swaps 

The fourth and last step requires as input all the above curves plus the market rates of the vanilla TIIE28 swaps and the market spreads of the XCCY basis swaps.

Although it could be technically possible, it is not necessary to build in the spreadsheet the set of synthetic XCCY floating-to-fixed swaps.

Deriscope provides a special type called ImpYC XCCY Disc that derives from Yield Curve Input and contains all data needed to build these synthetic swaps, as well as the final implied discounting curve for the MXN currency.

ImpYC XCCY Disc belongs to a family of special types inheriting from ImpYC, itself a sub-type of Yield Curve Input.

All ImpYC sub-types – such as the ImpYC XCCY Disc – represent collections of implied (or synthetic) instruments, i.e. instruments that are not directly traded in the market but can be assumed they exist in a mathematical sense with price quotes derived from proper traded instruments.

I can access the parent type ImpYC in the Type Selector of the Deriscope wizard, as shown below.

The four sub-types are also visible, as well as the description of ImpYC at the Info Area at the bottom. 


If I now select the type ImpYC XCCY Disc appearing at the top of the dropdown list, the wizard displays the Create function of the selected type, as shown below:  


The Info Area at the bottom contains the description of the currently selected ImpYC XCCY Disc type.

Before I go over the meaning of the various input parameters of the Create function, I notice the top key/value pair -Currency= %GBP.

This entry determines the currency of the discounting curve implied by the object under construction here. It is set by the wizard to %GBP only because my computer happens to use the English locale.

Since I am interested in producing an implied discounting curve for MXN, I can click on the pen-on-a-pad sign on the cell with the %GBP entry and change the currency to %MXN. Then the wizard automatically adjusts all remaining entries, with the following result: 


The meaning of the various entries is as follows:

On the outset, there exist two currencies, the USD and MXN, which are referred as source (Src) and target (Tgt) currency respectively.

The key Src Disc Curve= supplies the "riskless" discounting curve associated with the source currency USD.

The key Ccy Swaps Type= specifies the type of the supplied cross currency swaps. Here the associated value is SrcFltTgtFlt, which is an abbreviated form of Source Floating Target Floating meaning that the XCCY swaps involve the exchange of a floating rate of the source currency against a floating rate of the target currency.

The key Src Forc Crv= supplies the forecasting curve associated with the source currency USD. The associated value has been left blank by the wizard and must be filled by me in the spreadsheet later.

The key Tgt Mkt Crv= supplies an object of type Yield Curve created by market rates of vanilla swaps in the target currency MXN.

The key Ccy Swap Curve= supplies an object of type Yield Curve created by XCCY basis swap spreads.

The key Method= specifies the algorithm used to extract the discounting curve of the target currency MXN. The associated value ImplSrcFltTgtFxd is an abbreviated form of Implied Source Floating Target Fixed, which means that the algorithm is based on the synthetic construction of (implied) swaps that involve the exchange of a floating rate of the source currency against a fixed rate of the target currency.

Note that as explained earlier, this algorithm makes use of only the market rates of the vanilla MXN swaps and the market spreads of the XCCY basis swaps. Nevertheless, a full Yield Curve object is required as input to the Ccy Swap Curve= key. This is for technical reasons in order to support a better orthogonalization of the involved data sets and to anticipate more complex requirements of other, not yet implemented, methods.

The last key Adjust Fxd Rates= expects a boolean value that defines whether the fixed rates of the vanilla MXN swaps should be adjusted, as explained in the description of the step 4 above, or not. 


Building the Implied Discounting MXN Yield Curve in Excel: Step 4 (Tgt Mkt Crv) 

The following screenshot shows the creation of the Yield Curve object that contains the MXN swap rates. It is required as input to the Tgt Mkt Crv key:  


The rates are taken from Bloomberg as of May 22, 2019.

The single-element group of deposit instruments created as the handle &YldCrvDep_C20:1.1 in cell C20 is not required by the current algorithm. It is nevertheless provided with the actual Bloomberg MXN deposit rate of 8.5066% for completeness in case one is interested in studying the related MXN curve.

As explained above, our algorithm only needs the MXN swap rates supplied by the handle &YldCrvSwp_F4:1.1 created in cell F4.

Note the values for the keys Fxd Ref Period= and Ibor Index= in cells G7 and G9.

They are respectively %1L and %MXNTiie|1L, where L is used as symbol for the Lunar month period of 4 weeks.

The label MXNTiie points to the Deriscope built-in index for the Mexican TIIE rate.

Combining it with the 1L time period, I form the object %MXNTiie|1L that is considered "trivial" in the sense that its contents are fully defined by the text in its handle name, as explained in my introduction.

The built-in conventions are seen in the wizard if I select the cell containing the handle %MXNTiie|1L as below: 


Building the Implied Discounting MXN Yield Curve in Excel: Step 4 (Ccy Swap Curve) 

The following screenshot shows the creation of the Yield Curve object that contains the MXN swap rates. It is required as input to the Ccy Swap Curve key:  


The Yield Curve &USD-MXN.1 is constructed in cell C4 by using one set of instruments of type cross-currency basis swap corresponding to the object &YldCrvFxb_F4:3.1 entered to the right of the key Market Data= in cell C15.

The shown basis spreads are taken from Bloomberg as of May 22, 2019.

As already mentioned, these spreads are added on the USD Libor leg, contrary to the usual convention of being added on the non-USD leg. This is specified by the entry Spread On Src= TRUE.

Details on the meaning of the various keys can be found on my XCCY basis swaps curve post.

Note though that several entries are not required by the current algorithm, which only needs the basis spreads. In fact, the whole curve construction here is superfluous and is carried out only because the key Ccy Swap Curve expects as value an object of type Yield Curve. Nevertheless, most entries have been filled with reasonable values for completeness and convenience in case one wants to study this particular curve. 


Building the Implied Discounting MXN Yield Curve in Excel: Step 4 (Completion) 

Putting everything together, the final implied discounting curve for MXN is created as the object with handle name &MXN-OIS.1 by the formula =ds(C5:D16) in cell C4, as shown below:  


I can see the contents of the object &MXN-DISC.1 by selecting the cell C4 while the Deriscope taskpane is open:  


Note the curve is constructed without a direct input of any market rates.

All such rates are fed in indirectly through the Yield Curve objects &USD-OIS.1, &USD-1M.1, &MXN-TIIE.1 and &USD-MXN.1.

The dependency upon the various curves is shown schematically below: 


As mentioned above, this curve produces discount factors by relying on a dynamically constructed yield curve created out of synthetic XCCY swaps that exchange floating USD 1M Libor against fixed MXN.

The latter curve can be seen next to the key _Synthesized Curve=.

Clicking on the respective lens sign, I see the following: 


At this point, I may browse through the cash flows of the synthetically created XCCY swaps as explained in this post.

But I would rather click on the lens sign to the right of Market Data= in order to see the definitions (not the cash flows) of the synthetic XCCY swaps. 


What I see above, are the conventions applying on the synthetic XCCY swaps and the curves required in order to build the discounting curve associated with the target currency MXN.

For example, I see on the top that TYPE= Yield Curve Fxb, which tells me that the currently seen object represents a collection of XCCY basis swaps.

Then the Src Index= %USDLibor|1M and Tgt Index= %MXNTiie|1L tell me that the source index USD 1M Libor is exchanged against the target index MXN TIIE28.

But the Tgt Gearing= 1.000E-69 indicates that the multiplier to the target index is practically zero, a fact that makes the XCCY basis swap into a floating-to-fixed XCCY swap!

I also see that both discounting and forecasting curves for the USD currency are provided through the pairs Src Disc Crv= &USD-OIS.1 and Src Forc Crv= &USD-1M.1 respectively.

Note that a forecasting curve for the MXN currency is also provided through the pair Tgt Forc Crv= &MXN-TIIE.1, but is practically not used because of the zero TIIE28 multiplier.

So, the only unknown remains the discounting curve for the MXN currency, which is the curve being finally produced.

The set of synthetic swaps can be seen by clicking on the lens sign appearing at the bottom, next to the Set= key: 


The middle column contains the spreads in the market XCCY USD 1M Libor vs MXN TIIE28 basis swaps.

The last column contains the fixed rates in the market MXN fixed-to-floating swaps.

The title appears as #Spread2 because the synthetic XCCY swaps are structured as floating-to-floating basis swaps, where the MXN legs are structured with a zero index multiplier and a spread set equal to the shown rate.


Comparing the Discount Factors against Bloomberg 

Given the handle name &MXN-DISC.1 I may now use the Deriscope function DF to calculate the discount factor implied by the discounting curve for any maturity.

The following table shows the discount factors calculated by Deriscope and compares them to those returned by the Bloomberg terminal: 


The last column shows the difference between the Deriscope and Bloomberg discount factors. A difference like 0.0001 for a maturity of one year represents roughly a corresponding zero rate difference of one basis point. We observe that the annualized differences are about 1/3 bp for the first 5 years until 2024 and then increase to about 1 bp.

The main reason for these differences is the approximation applied by Bloomberg in its effort to construct the implied discounting curve on the absence of OIS market data.

At least as of December 2017, Bloomberg's approach was to create synthetic floating-to-floating XCCY OIS basis swaps, where the OIS of the collateral currency (eg USD) was exchanged against the OIS of the local currency (eg MXN) plus a spread.

Then the following formula was used: 


In case the XCCY basis spread is paid on the leg of the collateral currency 1 (the MXN case), the above formula must be modified to:  


In my next post, I will also build the forecasting yield curve, use both curves to price various swaps and analyze the impact of the implied discounting and forecasting curves on pricing and risk output.  


Click on YieldCurveBlbgImpliedOIS.xlsx to download the spreadsheet produced with the above steps.

Feel free to contact me if you want to share any thoughts with regard to this product or if you want to request any particular features. Contact info and social media links are available at my web site https://www.deriscope.com

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