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## Interest rate curve construction (IRCC)Part 1: Core elements of IRCC

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Use of appropriate interest rates and term structure is core to dealing with financial instruments. Whether it is about getting the right valuations or risk measures, using the interest rate term structure plays a key role. The use of IRCC depends on the type of financial instruments, end use and availability of market data. Interest rate term structures are used for forecasting the forward rates as well as for discounting purposes.

In this article, we are going to introduce the core elements of IRCC including the interest rate (IR) instruments that are used in construction of different type of IR curves while explaining the nuances associated with par and spot yield curves. We will also go through a simplistic approach for IR curve bootstrapping and the desired end results for the generation of a zero coupon yield curve (ZCYC),  discount factors (DFs) or forward rates.

##### Introduction to IRCC and compositional aspects

Seamless discounting of future cash flows (CFs) using correct IR remains one of the key building blocks on which the whole ensemble of mark-to-market (MTM) valuation for market traded products rests. Depending on the reference financial market there are a range of IR products (both cash and derivative ones) that get traded in the short (<1Y time-to-maturity (TTM)), medium (generally from 1Y to 3-4Y TTM) and long (>3-4Y TTM) term maturity buckets.

The purpose for constructing an IR curve is to identify the appropriate traded product for different maturities to define tenors and rates. Coverage of IR product types in this article is limited in nature for ease of explanation of the underlying concepts, with explanations being provided around sovereign bonds and vanilla interest rate swaps (IRS). By default, the construction of an IR yield curve (YC) necessitates logically placing these IR products in an ascending order of TTM while accounting for trading and settlement (T&S) conventions.

There are essentially two broad classes of IR YCs that we observe in the marketplace. The first class comprises IR YCs where only cash products are being used in the construction. The most commonly observed type in this class are sovereign YCs where treasury bills (T-bills) are used at the short-end of the maturity spectrum for the YC construction, while sovereign bonds are used for constructing the medium-to-long end of the maturity spectrum.

The second class comprises IRS YCs where cash as well as derivative products are being used in curve construction. Herein a money market (MM) observed cash or deposit rate is being used at the short end of the IRS YC. For the reference market, if traded interest rate futures (IRFs) or forward rate agreements (FRAs) exist, typically IRF / FRA are being used for the tenor points between 6M to 2Y TTM. For maturities >2Y, typical market practice is to use swap quotes in the construction of the IRS YC. If there is absence of / inadequacy associated with IRF / FRA quotes, those types of IRC curve construction may be purely carried out based on cash rates for short end and swap quotes for medium / long end.

Use of traded instruments depends on the availability and liquidity of the traded instruments. In the absence of the traded instruments, curves are also constructed on proxies as well as polled rates. This will be explained in a separate article.

##### Deriving the ZCYC from par YC – Bootstrapping

A Zero Coupon Yield Curve (ZCYC)is used for discounting purpose as it eliminates the noise of intermediate coupons and is an accurate reflection of the time value of money for a specific maturity. However, most of the longer tenure traded instruments have an intermediate coupon attached to them. Hence, there is a need to derive ZCYC from par YC.

Money Market (MM) observed IRs are typically quoted on a spot basis on account of a single cash flow on an MM instrument’s maturity date. However, the medium-to-long tenor rates usually possess an intermittent coupon (and associated coupon reinvestment risk at YTM) effect and are quoted on a par basis. The IRYCs as quoted in the interbank market is based on the associated trading practice from a front office (FO) perspective, and in their innate form they are termed par YCs. From a mid-office (MO) / market risk (MR) perspective, it is preferred that the discounting of CFs happen on a bootstrapped ZCYC / spot curve basis.

Given a par YC, bootstrapping is the process of finding the sequential spot rates (in an TTM based ascending order) while following the algebraic transformation in an iterative way. We can explain the approach using a simplified setup with a sovereign YC that comprises T-bills and fixed coupon bearing sovereign bonds.

An example:

Assuming we have a T-bill quote for one year maturity, $$DF_1$$  can be computed using the 1 Y discount yield i.e. $$1\over {(1+Z_1)}^1$$, where $$Z_1$$  is the MM discount yield on the 1Y TTM T-bill). So for a unit notional (say INR 1) 2Y bond (say with annual coupon $$C_2$$ ) trading at par the pricing equation will be as follows, we will try to compute the DF for 2Y tenor point i.e. $$DF_2$$ :

$1 = {C_2\times DF_1 + ((1+C_2)\times DF_2)}$

where

$DF_2 = {1-(C_2\times DF_1) \over 1+C_2}$

In the next step we compute DF3 for the 3Y TTM bond with an annual fixed coupon $$C_3$$ , hence the pricing equation will be as follows:

$1 = {(C_3\times DF_1)+(C_3\times DF_2) +((1+C_3)\times DF_3)}$

where

$(1-(C_3\times (DF_1+DF_2)) = {(1+C_3)\times DF_3}$

and where

$DF_3 = {(1-C_3)\times (DF_1+DF_2)\over 1+C_3 }$

Moving iteratively by mathematical induction we can infer that,

$DF_n = {1-C_i\times {{\sum_{i=1}^{n-1}DF_i}} \over 1+C_i}$

An illustration of bootstrapping of 3M USD Libor based IRS curve is being provided here as an illustration (Click here)

Given the definition of a zero rate (i.e. single CF on maturity), there is a natural mathematical relation between the tenored DFs and the relevant tenor zero rate, for a unit notional (say INR 1 ) “n” year TTM bond with zero coupon rate $$Z_n$$  and DF defined as $$DF_n$$ we can write:

$DF_n = {1\over (1+Z_n)^ n }$

where

${\left(\frac{1}{DF_n}\right)}^{1\over n} -1 = Z_n$

##### Computation of implied forward rates (IFR) using zero rates

Present day estimation of IR at a future point of time happens on the basis of the principal of interest rate / CF compounding, such estimated IR is termed as implied forward rate (IFR). Let us say corresponding to TTM $$T_1$$, we have zero rate $$Z_1$$ and $$DF_1$$, likewise for TTM $$T_2$$ (with $$T_2$$ > $$T_1$$) we have zero rate $$Z_2$$ and $$DF_2$$. We are going to find the estimated IFR say $$Z_{1x2}$$ between $$T_1$$ and $$T_2$$ (i.e. being rate set at $$T_1$$ and applicable till $$T_2$$. For unit notional CF the pricing equation will be:

${(1+Z_1)^T}\times (1+Z_{1x2})^{(T_2-T_1)}={(1+Z_2)}^{T_2}$

where

${\left(\frac{1}{DF_1}\right)}\times {(1+Z_{1x2})^{(T_2-T_1)}} = \left (1\over DF_2\right)$

where

${(1+Z_{1x2})}=\left (DF_1\over DF_2\right)^{1\over (T_2-T_1)}$

and where

$Z_{1x2}=\left (DF_1\over DF_2\right)^{1\over (T_2-T_1)}-1$

So this is how mathematically DFs, spot rates and IFRs are estimated in a simplified setup. We will delve into finer aspects of these estimations in the next article in this IRCC series.