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A yield curve is a plot of yields for various bonds (often government bonds) versus the bonds' maturities. We also often plot swap and other LIBOR rates to get the (related) swap curve.

Description

A yield curve is a plot of yields for various bonds (often government bonds) versus the bonds' maturities.

Yield curves often use benchmark tenors (times to maturity) so that the curves are directly comparable across time. In the US, these benchmark tenor bonds are known as "constant-maturity Treasuries" (CMTs) and are inferred from the yields on bonds near those benchmark tenors.

When we plot yield curves, we typically plot an overnight rate and then rates for 1-, 3-, 6-, and 12-months as well as 2-, 5-, 10-, 20-, and 30-years. The axis of tenors is often not plotted to scale.

Why People Look At Yield Curves

We plot yield curves because the curve tells us about the market's willingness to lock up money for various periods of time. Interest rates express the cost people demand to rent their money out for those tenors. Thus we can compare these terms to see if people seem concerned about lending money to short-term borrowers or eager to lock their money up in a safer investment than, say, equities for a long period of time.

A yield curve which is upward-sloping suggests economic growth ahead since people sell long-term bonds and invest their money in risky ventures and are willing to lend money to short-term borrowers (who tend to be more affected by crises).

For the opposite reasons, an inverted (downward-sloping) yield curve tends to presage looming economic distress.

We often see the front-end of the curve elevated when the central bank or monetary authority is trying to slow economic growth and lower when those same people are trying to increase economic growth.

Finally, we may see a humped curve (with higher yields in the 2Y to 10Y tenor range) when countries try to prop up medium-term yields to attract capital or reduce capital flight. This may occur because a country has promised but not delivered on economic reforms and can be seen as a currency defense.

Building a Yield Curve

We often build a yield curve from discount rates; however, that is complicated by most bonds beyond 1Y tenors being coupon bonds. Thus we may need to bootstrap (impute) discount rates starting from discount bonds up to 1Y.

Modeling or Fitting Yield Curves

To model or fit a yield curve, people use a number of approaches. Some use cubic splines while others use models such as Nelson and Siegel (1987) which has constant, decaying term, and decaying humped terms or the Svensson (1994) model which adds a second decaying humped term. Diebold and Li (2006) discuss using these models to predict yield curve movements.

Another approach is to use an arbitrage-free model such as Heath, Jarrow, and Morton (1992).

Common Yield Curve Movements

Litterman and Scheinkman (1991) looked at a principle components analysis of yield curve movements to see which sorts of movements were the most common. They found that the first principal component (which explain the most movement) is the overall level of the curve; the second principal component was the slope of the curve; and, the third principal component was the curvature of the curve. These correspond closely to the Nelson-Siegel factors.

Related Curves

The yield curve produced by (non-inflation-indexed) government bonds will differ from the curve for inflation-indexed government bonds. Those curves will differ for curves produced from LIBOR and swap rates (which are not considered "risk-free") or commercial paper and corporate bonds.

We also often plot swap and other LIBOR rates to get the (related) swap curve.

Further Reference

For more on yield curves, one could read the short overview article by Brousseau (2002). For more depth, the following texts are excellent:

  • Brigo and Mercurio, Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit, 2nd Ed., 2006.
  • Rebonato, Bond Pricing and Yield Curve Modeling: A Structural Approach, 2018.