Yield curve explained: what are the implications?

The yield curve is a central instrument of financial market analysis — it provides information on interest rate expectations, economic developments and influences financing decisions by companies, states and investors. But what exactly does the curve show — and how can it be calculated reliably?

This article provides a structured overview of methods, models and challenges in determining yield curves — from yield structure to zero bonds to bootstrapping and interpolation processes.

To determine the yield curve

The interest rate structure describes how interest rates change as a function of the maturity of a financial instrument. It is illustrated by the yield curve, which provides investors and economists with important information about future economic developments.

Normal vs. inverse yield curve

Long-term interest rates are generally higher than short-term interest rates, as investors expect a higher return on a longer capital commitment. This shape is known as a normal yield curve. However, there are also periods in which short-term interest rates exceed long-term interest rates — a phenomenon known as an inverse yield curve. An inverse interest rate structure often indicates an expected economic weakening or falling interest rates in the future.

Relevance of the interest rate structure for the economy and financial market

The interest rate structure plays a decisive role in the financing decisions of companies, states and households:
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▪️ Companies rely on long-term interest rates to plan investments.
▪️ Governments use the interest rate structure to strategically place bonds.
▪️ Banks are adjusting their long-term loan interest rates accordingly.
▪️ Central banks analyse the interest rate structure in order to steer their monetary policy in a targeted manner.

While short-term interest rates are heavily influenced by monetary policy measures, long-term interest rates are more dependent on inflation expectations and economic growth.

Valuation basis for financial instruments

The yield curve serves as a central basis for valuing financial instruments such as bonds, lease liabilities or derivatives. It significantly influences the present value of future cash flows and thus the pricing of these instruments.

Yield curve vs. zero curve

Although the terms are often used interchangeably, they refer to different concepts:
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▪️ The yield curve shows how the effective returns (yield-to-maturity, YTM) of bonds with different maturities are developing.
▪️ The zero curve shows the effective interest rates of zero-coupon bonds over different maturities.

Calculation of the yield curve

The return is the interest rate at which the present value of all future payments is equal to the current market price:

Since this equation cannot be solved directly after yTM, the calculation in practice is usually carried out using algorithmic approximation methods.

Limits of the yield curve

One disadvantage of the yield curve is that it only provides an average interest rate per bond. Especially when interest rates vary significantly over different terms, this can lead to distortions, as coupon payments and repayment amount are combined.

The zero curve

The zero curve shows the effective interest rates of zero-coupon bonds. Since they only have one payment at the end of the term, an exact interest rate can be determined for each term — without distortion through coupons.

Bootstrapping to determine the interest rate structure

Since there are rarely enough zero-coupon bonds in practice, Bootstrapping resorted to. Coupon bonds are gradually converted into synthetic zero-coupon bonds. example:

First, r1 from a well-known one-year zero-coupon bond is used. R₂ is then algorithmically determined for the two-year term — and so on. Each interest rate builds on the previous one, hence the name “bootstrapping.”

Data basis: government bonds as a benchmark

Government bonds are usually used to create the yield curve, as they are considered a risk-free benchmark within a country.

The procedure:
1. Determine yield-term combinations for all available bonds.
2. Establish a functional relationship — with the help of mathematical models.

Methods for interpolating the interest rate structure

Two established methods are used particularly frequently:

Regression model (Deutsche Bundesbank)

Since 1997, has been using the Svensson model, an extension of the Nelson Seal feature. Here, estimated parameters are adjusted to the observed returns using regression.

Spline interpolation (US FED)

The US Federal Reserve uses a Quasi-cubic Hermite Spline (HS) interpolation. Returns are rounded to defined bases, averaged and linked by cubic functions.

Data providers such as Bloomberg or LSEG (formerly Refinitiv) also use similar methods to generate marketable yield curves.