Which yield curve fits a market best?

A yield curve has to be estimated from the prices of the bonds that happen to trade, and central banks use different methods to do it. The choice matters: the same prices can give different curves, and everything priced off the curve inherits the difference.

What we did

We used 953 trading days of Turkish discount bond prices, from 3 January 2006 to 23 October 2009, taken from the Istanbul Stock Exchange's daily bond bulletins. We left out the least traded 5% of bonds and those with under 30 days to run.

We compared four methods widely used by central banks: Nelson-Siegel, its extension by Svensson, the cubic spline and the cubic smoothing spline. As far as we know, this was the first time the smoothing spline had been applied to the Turkish market.

Each method faced two tests. The first was how well it fitted the bonds it was built from. The second, harder test was how well it priced bonds it had not seen: each day we removed the longest maturities, covering the last 120 days of the curve, and asked each method to predict them.

What we found

All four methods produced sensible curves that reflected how the market reacted to events. On the first test, the cubic spline won clearly, with the lowest error on 558 of the 953 days.

On the second test the picture reversed. Because the cubic spline is pure curve-fitting with no economic shape behind it, it broke down badly on some days, and its average error out of sample was more than ten times that of any other method. Svensson and the smoothing spline did best, and were close: head to head, the smoothing spline won on 488 of the 953 days.

Why it matters

The method that fits the past best is not necessarily the one that prices the next bond best. Testing out of sample is the test that matters, and that lesson goes well beyond yield curves.

There is also a trade-off. Nelson-Siegel and Svensson have parameters with an economic meaning, such as the level and slope of the curve, which makes them easier to explain. The smoothing spline gives a smoother curve and better predictions. We found Svensson and the smoothing spline the most successful for the Turkish market.