Yield Curve Estimation

1.Product specifications

1.1. General

ZERO+ is a software application for estimating the zero-coupon yield curve of fixed income securities. The application is a response to the market needs for more accurate calculators of term structure as the bond market and fixed income funds have grown significantly and therefore competition has sharpened. ZERO+ has been developed as a mean for investors and dealers to make sound decisions in trading and managing the risk of their portfolios. With ZERO+ we offer a tool for advanced analysis and profitable active trading.

1.2. Estimation Methodology

It is explicit, that the more accurate the yield estimation, the more advantage the user will have. As a new approach for the calculation of zero-coupon yield curve of a set of fixed income securities, such as bonds, futures, swaps etc., ZERO+ provides a high degree of precision due to the smoothing spline method used in the basic algorithm.

Traditionally there have been two different approaches to the estimation of the term structure of interest rates. The first approach, Nelson and Siegel (1987), defines a functional form to fit the yield curve. The other approach is a spline-based method, pioneered by McCulloch (1971, 1975) and extended by Adams and Van Deventer (1994), among others. These papers have proposed interpolating and least-squares spline methods, which may not produce a smooth function. Fisher, Nyckha and Zervos (1995) and Käppi (1997) have extended the literature by proposing a smoothing spline method.

The smoothing spline method solves the well-known problem of stiff curves and high pricing errors derived from the functional approaches. This is why it has received a lot of attention lately. Our method uses a new smoothing norm, the square of the discontinuity jump in the third derivatives at the interior knot points, and it locates the internal knot points by the size of the fitting errors. The smoothing splines try to combine the two different approximation objectives, to fit a smooth curve to the data and to simultaneously keep the pricing errors as small as possible. ZERO+ offers this possibility to the user.

Optionally, ZERO+ provides also the Nelson-Siegel methodology for the estimation of yield curves. ZERO+ employs the extended Nelson-Siegel function because the standard method would bring along several shortcomings. For example, the standard Nelson-Siegel method is a stiff function, which does not fit steep inverted humped yield curves. Therefore the term structure is incorrect and the pricing errors are high.

Even though the extended Nelson-Siegel estimation function gives good results, such as small pricing errors, there are certain features that one should understand. For example, although the yield curve might be upward sloping at the estimation time period, say from 1 month to 10 years, it suddenly may start to slope down at longer maturities. Further, none of the function parameters has sound econometric interpretations. If used in simulation models, this result causes problems to the simulation runs. On the other side, the simplicity of the function makes the simulation of the curve faster.

1.3. Yield Curve calculation


Input information consists of bond and other fixed income instruments quotations, which can be either entered manually or fetched from real-time market data sources such as Reuters, or Bloomberg. There are two advantages with respect to the ease of inserting the input data into ZERO+ : firstly, all the necessary information is quoted by the data providers, so that the user does not have to prepare derived information, for example by doing intermediate computations; secondly, the user can enter multiple bond series which also can have different coupon characteristics, maturities and risk classes.

In general, the following information for each instrument constitutes the required input for ZERO+:

  1. Settlement date
  2. Maturity date
  3. Annualized coupon rate
  4. Yield- or price-based quotation
  5. Frequency of coupon payments
  6. Daycount basis


The main purpose of ZERO+ is to estimate the zero-coupon yield curve of the instruments that the user entered into the calculation. This is the key information in the relationship between time to maturity and yield to maturity and which is used for instrument pricing. There are as many sets of output curves as there are input sets. Also, there is virtually no limit on the number of the instruments used for the calculation. In this way, ZERO+ overcomes another problem with the traditional estimating methods, the limitation of the number of coupon paying instruments that they consider. The intended use of the multiple curve feature is to facilitate the comparison of portfolios from different markets and risk classes. For example this can be utilized to obtain both the bid and ask zero-coupon curves, or to estimate yield spreads between different government benchmark instruments.

As the time-to-maturity - yield-to-maturity relationship can also be represented by the forward rates, ZERO+ provides the user with the forward yield curve option. Actually, the application goes still one step further; it calculates an implied spot yield curve which enables the user to take a future position on his portfolio. The implied spot yield curve shows the future spot yield curve one period of time ahead from the present moment. In other words, there can be two ways of projecting the future yields: first, one can show the yield level as constant-maturity one-year rate at various future dates (that we call the forward yield curve); second, one can construct the yield level as an implied spot curve at one future date in one time period ahead (and, that we call the implied spot curve). The one year forward curve shows by how much the spot curve needs to change over the next year to make all bonds earn the same holding period return. There can be two ways to present this adjustment: by the (traditional) forward curve and by the implied spot curve as defined above.

Another feature that ZERO+ provides is the pricing error estimation and yield error estimation which allow the user to spot the differences between the market quotation and the estimated value and which therefore can be very useful for trading and hedging purposes.


Market conventions are not a restriction for running the application. The system is able to handle the standard (non-optional or convertible) straight bonds from different markets. ZERO+ is flexible enough to take care of the conventions in the area of accrued interest calculations. The computation is possible also for bonds with odd first coupon payment. Option characteristics or other abnormalities are not taken care of. The range of instruments handled by ZERO+ includes: standard bonds, bonds with odd first coupon period, swaps, eurodollar futures and other futures quoted similarly to eurodollar ones, as well as FRAs quoted like the ones in London.

1.4. Price calculation

ZERO+ pricing module makes use of the yield calculation results to price standard bonds, swaps and forward-start swaps from different markets and maturities.


  • Fixed and float payment features - frequency, daycount basis and float premium
  • Settlement date
  • Maturity date
  • Par value of the instrument
  • Forward start date (for forward-start swaps)


Pricing module calculates the instrument price as par yield figure. The user may choose between cubic splining or Nelson-Siegel results as ZERO+ reports the values according to both methodologies.

1.5. Technical requirements

ZERO+ has been implemented as a Microsoft Excel application. In addition to visible interface there is a calculation module which has been implemented as a separate DLL. This library is written using Fortran 90 compiler and sophisticated mathematical analysis libraries. ZERO+ requires Excel version 7.0 or 8.0. The technical side of ZERO+ is very convenient for the user because it can be run in stand-alone workstations and does not require any investment in additional hardware or software. This makes a considerable deduction in the potential costs of the package.

2.What the user might like to know

2.1. Who will use ZERO+ and for what purposes?

ZERO+ is intended to be a reliable ‘gauge’ in the hands of a bond and bond futures trader, fixed income derivatives dealer, portfolio manager and the like.

Time-to-maturity - yield-to-maturity relationship of zero-coupon bonds is the key concept in the fixed-income market. As this market has grown and become more competitive, the need to use sophisticated technology has increased as well. ZERO+ provides zero-coupon yield values for pricing and trading of fixed income instruments.

Also, ZERO+ detects mispricing and arbitrage opportunities for those who prefer active funds management versus passive management approaches. Portfolio managers will find the tool very handy as it will assist them in setting up and pricing the securities of their portfolios.

The estimations of ZERO+ will assist in the risk management process by providing accurate basis for estimating hedging values. Bond futures contract is one of the most actively traded financial futures contracts in the world. However, the pricing of bond futures has been little studied. For example, if bond futures or derivatives on them are used to hedge a certain value of exposure, the manager should know how much of the securities should be used and what the real value of these securities is going to be. ZERO+ calculator provides a good estimate of term structure of fixed income derivative instruments, which is the basis of pricing models in use. Mispricing can cause serious problems, if one wants to use the future contracts as a hedging instrument, because it effects negatively the effectiveness of the hedge.

Finally, Value-at-Risk applications need good estimates of the yield curve to map the cash flows from fixed income securities, in order to estimate the risks in the portfolios. ZERO+ can be effectively used as an estimation tool for risk management applications, such as VAR systems.

2.2. What are the advantages of the application?

  1. A precise pricing technology, able to realize two objectives:
    1. Fitting a smooth curve to the data
    2. Minimum pricing errors
  2. Nelson-Siegel is optional
  3. Easy to integrate into existing systems
  4. Fed with real-time data
  5. Simple user interface
  6. Fast, as complicated calculations run in Fortran 90
  7. Relatively inexpensive, as
  8. Product is narrow and focused
  9. Product is almost maintenance-free
  10. No additional hardware or software is needed
  11. Open to tailor-made features

Furthermore, the buyer of ZERO+ is entitled to free-of-charge updated versions of the application.

3. End note

For all our products, full demonstrations along with case studies are available on an individually tailored basis. For more information please contact:

CD Financial Technology
Mikonkatu 8

E-mail: sales@cdgroup.fi
Tel: +358 9 612 3322