Hi all,
The traditional yield curve depicts the relationship between yields and the time to maturity of debt securities that pay periodic interest payments (coupons). It is widely used for pricing bonds and other fixed-income securities that pay periodic interest. When buying and selling bonds, investors include their expectations of future inflation, real interest rates and their assessment of risks, which corresponds to future cash flows discounting.
The zero-coupon yield curve represents the yields of zero-coupon bonds at various maturity dates. It is used for pricing bonds and other financial instruments that do not pay periodic interest, such as zero-coupon bonds, forward rate agreements, or that pay periodic amounts or coupons like interest rate swaps.
Building the (zero-coupon) yield curve serves as a foundational step in many financial modeling endeavors. To achieve this, diverse methodologies, including bootstrapping, are utilized to extrapolate continuous yield curves from an obviously only discrete data points set.
Before delving deeper into methodologies, let’s first consider the market data utilized to construct the yield curve.
Curve building - Market Data in Static Data
In LexiFi Apropos, different types of interest rate instruments are used to construct the yield curve. These instruments can be found in Static Data => Interest Rate Market Data, the most used ones being:
- Overnight rates: interest rates for very short-term borrowing or lending, typically overnight. They are often used as a benchmark for other short-term interest rates.
- Deposit rates: these rates apply to deposits with maturities ranging from two days to several months. They represent the interest rates offered on short-term deposits by banks or financial institutions.
- Futures: in the context of curve building, futures contracts are used to estimate future interest rates. By deducting the futures rate from the contract’s price, you can derive an implied interest rate, which can then be used to construct the yield curve.
- Forward Rate Agreements (FRAs): FRAs are contracts that allow parties to lock in an interest rate for a future period, starting at a specified future date. FRAs are often used by market participants to hedge against fluctuations in future interest rates. In the context of yield curve building, FRAs provide valuable information about market expectations for short-term interest rates over specific future periods. By observing the prices of FRAs with different start and end dates, you can infer the market’s expectations for short-term interest rates at those points in time. FRASs and futures differ among other things in collateralization. FRAs, typically traded over-the-counter (OTC), do not usually require collateral and are customized contracts negotiated between parties, resulting in higher credit risk. In contrast, futures are exchange-traded, standardized contracts that require collateral in the form of initial and variation margin, which reduces credit risk by involving a clearinghouse.
- Swap Rates: Swap rates are crucial for estimating longer-term interest rates on the yield curve. Interest rate swaps involve the exchange of cash flows between parties based on a specified notional amount. These cash flows typically represent fixed and floating interest payments. By observing the market prices of interest rate swaps with various maturities, you can derive swap rates, which represent the market’s expectations for future interest rates over the corresponding periods. These swap rates provide key data points for constructing the longer end of the yield curve.
All these instruments and some others contribute to the overall shape and structure of the yield curve by providing data points for different maturity periods. Incorporating information from these instruments allows for the construction of a yield curve that accurately reflects market expectations for interest rates across various time horizons. In LexiFi, the yield curve is built using a typical bootstrapping procedure on these instruments.
In LexiFi Apropos, these data points are either derived from LexiFi data or an external provider. LexiFi Apropos integrates with market data providers to access real-time or historical yield curve data.
The curve building section allows to use the Market Data defined in the Interest Rate Market Data section to build the curve. A default configuration exists in LexiFi Apropos and can of course be adapted to users’ needs.
Parameters for curve building include:
- Curve Type: these are the parameters for the bootstrap. Curve type allows users to choose the size of the affine part of the instantaneous forward rate interpolation.
As mentioned above, bootstrapping in yield curve construction is a technique used to derive the yields for a series of fixed-income securities with different maturities, starting from the observed market prices or yields or spreads of certain benchmark securities. The process involves iteratively estimating the yields for each maturity point on the curve, beginning with the shortest maturity and moving towards longer maturities. Bootstrapping is particularly useful because it allows for the creation of a yield curve using only observable market prices or yields, without the need for external models . It ensures consistency between observed market prices and derived yields and is widely used in fixed-income markets for pricing and risk management purposes.
- Curve Instrument: allows users to choose the adequate instrument for building the curve.
- The Rules section: allows to specify rules per currency. In this case, the rules are either Interest rate, that allows to choose one of the preceding cases, or FX.
- FX Curve Rules: specify the convention when building the curve using FX instruments (FX forwards and/or Cross Currency Swaps). Applicable when using Curves from FX in a pricing.
- Default Curve Tags: Multiple curves per item type are now supported by adding tags to yield curve market data items. We implement a tagged curve (defined in Market Data transformation) with an associated priority list extension.
💡To further explore curve building methodologies in LexiFi, you can visit the blog article on our website Curve Building.
Once the curve is built, users can use the Curve Inspector feature to visualize and analyze the curve. Users may visualize the curve in different forms (Discount factors, Continous Spot Rates,…). The primary function of the curve inspector is to provide a graphical representation of the yield curve, allowing users to visualize the relationship between interest rates and their corresponding maturities. Curve inspector is thus a great tool to analyze and visualize yield curves, including curve spreads, shifts, and shapes. Users may be able to plot multiple curves, identify trends or anomalies.
💡More details on the Curve Inspector are available on the Features section of the website under Curve Inspector.
Funding-adjusted yield curve
The funding-adjusted yield curve incorporates explicitly issuer risk into the yield curve. This type of yield curve adjusts for the credit risk associated with different issuers of fixed-income securities. Unlike traditional yield curves, which primarily reflect the relationship between interest rates and time to maturity, funding-adjusted yield curves consider the creditworthiness of the issuer.
To construct a funding-adjusted yield curve, market data related to the credit spreads of various issuers is utilized in addition to the yields of government bonds or other risk-free securities. Credit spreads represent the additional yield investors demand for holding bonds issued by entities with credit risk compared to risk-free securities. By incorporating these credit spreads into the yield curve analysis, the funding-adjusted yield curve provides a more comprehensive view of the term structure of interest rates, taking into account the varying levels of credit risk across different maturities. This adjusted curve is particularly useful (and often needed) for assessing the cost of funding for issuers and investors, as well as for pricing fixed-income securities with credit risk.
LexiFi Apropos offers the possibility to build a funding-adjusted yield curve either by:
- Uploading funding grids directly into the Market Data when available
- Using Credit Default Swaps (CDS) or Credit Swaps Options as proxys. CDS spreads are used as a proxy for the credit spread or risk premium demanded by investors for holding bonds issued by a particular issuer. By adding the observed CDS spread to the risk-free rate, the zero-coupon rate can be adjusted to reflect the additional compensation required by investors for bearing the credit risk of the issuer.
The risk-adjusted zero coupon rates can be found by clicking on the Extra Results of the Pricing section:
