In Parts 1, 2 and 3 of this series, we outlined the pros and cons of the Nelson-Siegel approach to yield curve smoothing versus the spline based approach that dominates non-financial applications like computer graphics and computer animation. In Part 2, we provided a worked example of the Nelson-Siegel approach. In Part 3, we showed how the maximum smoothness forward rate approach can be used to improve on the Nelson-Siegel approach both in terms of accuracy in fitting observable bond prices and in terms of the smoothness of the forward rates produced. In today’s post, we show with some simple examples that the Nelson-Siegel approach’s forward rate function is too simple to provide accurate mark to market valuations for realistically shaped forward rate curves.
In Part 2, we discussed how the Nelson-Siegel zero coupon yield function
can be fitted to observable market data using 3 Russian Federation bonds. We also noted that the continuous forward rate function can be derived from the yield function based on this formula from Chapter 8 of Advanced Financial Risk Management (van Deventer, Imai and Mesler, John Wiley & Sons, 2004):
When we evaluate this forward rate function for the Nelson-Siegel yield curve, we find that continuous forward rates follow a very simple functional form:
By contrast, the maximum smoothness forward rate method uses quartic polynomials pasted together to fit observable bond prices with maximum accuracy and forward rate smoothness, given the number of line segments that the analyst has chosen to use and the constraints that the user has imposed on the left and right hand sides (the shortest and longest maturities) of the forward rate curve:
In the example from posts 2 and 3 in this series, we used three Russian Federation bonds as input to the smoothing process. We used two line segments in fitting the maximum smoothness forward rate function to this data. Because the number of bond prices used was so few, both methods produce very similar forward rate functions that have a high degree of smoothness:
The forward rate curve for the maximum smoothness method flattens at the right hand side of the curve because we imposed that constraint in the smoothing process. This graph may lead some to the erroneous conclusion that the two methods are reasonably equivalent, but nothing could be farther from the truth.
Since we had only three bonds as inputs to the smoothing process, the data won’t reveal any twists or bends in the yield curve. Consider a more complex set of yields from the Federal Reserve Board’s H15 Statistical Release for the U.S. Dollar swap curve on December 31, 2008:
The smoothing in this graph is simply the interpolation built into standard spreadsheet software, but it does show clearly that there can be a series of bends and twists in yields even for a capital market with as much depth as the U.S. Dollar market.
The maximum smoothness forward rate approach is ALWAYS able to fit actual data exactly (when there are no data errors). If there are N distinct data points, using “maximum smoothness” with N+1 line segments achieves this objective. If the number of data points is large or if this is a production run for risk management purposes, this code is embedded in a very sophisticated enterprise wide risk management software system.
Given that the maximum smoothness approach fits the data exactly, how does the Nelson-Siegel approach do using this simple forward rate function for the entire maturity spectrum?
A few simple examples will show that its performance is not sufficiently accurate to meet normal best practice standards for risk management or for Financial Accounting Standard 157 valuation purposes. For these examples, we assume that we are a close relative of God and that we are given the “true” continuous forward rates for maturities of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 years. The “0” maturity, of course, is actually 1 day. We then use the non-linear optimization routine in standard spreadsheet software to maximize the goodness of fit for the Nelson-Siegel approach by setting alpha, beta, gamma and delta to the “best” levels. Of course for the maximum smoothness approach, we fit the observed points exactly.
Consider example 1 and its best fitting Nelson-Siegel parameters. In minimizing the sum of squared errors, we have multiplied the errors by one million to insure that the spreadsheet optimization routine is not affected by rounding error.
We can then graph the actual forward rates versus those implied by Nelson-Siegel. Note that we have connected the fitted forward rates at maturities 0, 1, …, 10 with straight line segments. It is the even annual maturity data that one should focus on:
The virtue of the Nelson-Siegel approach is the simplicity of the forward rate function. Unfortunately, the function is extraordinarily inaccurate even on this simple example which shows bends than are much less severe than those graphed above for December 31, 2008.
Example 2 shows another common forward rate curve shape. Again, we optimize the Nelson-Siegel parameters to maximize the goodness of fit, but we still cannot match the perfect fit of the maximum smoothness forward yield curve approach.
Again, when we graph the results, we see that the Nelson-Siegel fit is better than example 1 but still far from exact.
These “fails” to fit lead to pricing errors. When the fitted NS curve is used to price the bonds used as inputs to the smoothing process, the calculated bond prices will not be the same as the actual prices. This means that risk measurement will be incorrect and that there will be the appearance of arbitrage opportunities when in fact there are none.
Example 3 is another variation on the same theme:
The graph again shows a poor fit between the Nelson-Siegel forward rate function and the actual data points:
In short, the Nelson-Siegel approach can lead to very serious mispricings. After the failures of the copula approach and value at risk during the credit crisis, the use of a model that is known in advance to be inaccurate is nearly impossible to defend to management, the board of directors and regulators.
We think that the implications of this four part series on Nelson-Siegel versus spline technologies are clear:
- Spline approaches in general and the maximum smoothness forward rate approach in particular are superior to the Nelson-Siegel approach in two critical dimensions: (a) accuracy in fitting observable bond prices and other market data, and (b) smoothness and reasonableness of the forward rate curves, given that the forward curve must fit observable data.
- Central bank government yield curves derived using the Nelson-Siegel approach are potentially very inconsistent with actual bond prices used as input to the smoothing process. A careful analyst should reject use of this Nelson-Siegel smoothed data as inputs to risk calculations.
- Nelson-Siegel smoothing is not accurate enough for FAS 157 valuations and key risk reports given the magnitude of errors that can result and the tendency of users to accept results at face value without checking whether the fit to a given set of data was good or not. This lack of “self-assessment” was at the heart of the copula driven meltdown of the CDO market in the current credit crisis.
Some analysts might ask “What about the case where the data is bad?” That’s an entirely different issue that shouldn’t obscure the main point of this series: even when the data is good, the Nelson-Siegel approach fails to fit it with 100% accuracy.
Comments and suggestions are welcomed at email@example.com.
Sean Klein and Donald R. van Deventer
Honolulu, August 18, 2009