fitSvensson

Fit Svensson model to bond market data

Syntax

outCurve = fitSvensson(Settle,Instruments,CleanPrice)

outCurve = fitSvensson(___,Name=Value)

Description

outCurve = fitSvensson(Settle,Instruments,CleanPrice) fits a Svensson model to bond data.

After creating a parametercurve object for outCurve, you can use the associated object functions discountfactors, zerorates, and forwardrates.

outCurve = fitSvensson(___,Name=Value) fits a Svensson model to bond data using optional name-value arguments for Basis, x0, lb, and ub.

Examples

Fit Svensson Model to Bond Market Data

Define the bond data and use fininstrument to create FixedBond instrument objects.

 Settle = datetime(2017,9,15);
  Maturity = [datetime(2019,9,15);datetime(2021,9,15);...
      datetime(2023,9,15);datetime(2026,9,7);...
      datetime(2035,9,15);datetime(2047,9,15)];
  
  CleanPrice = [100.1;100.1;100.8;96.6;103.3;96.3];
  CouponRate = [0.0400;0.0425;0.0450;0.0400;0.0500;0.0425];
 
nInst = numel(CouponRate);

Bonds = fininstrument("FixedBond",'Maturity',Maturity,'CouponRate',CouponRate);

Use fitSvensson to create a parametercurve object.

SvenModel = fitSvensson(Settle,Bonds,CleanPrice)

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

<stopping criteria details>

SvenModel = 
  parametercurve with properties:

              Type: "zero"
            Settle: 15-Sep-2017
       Compounding: -1
             Basis: 0
    FunctionHandle: @(t)fitF(Params,t)
        Parameters: [3.2984e-08 0.0197 0.0624 0.1391 1.3563 11.7741]

Input Arguments

Settle — Settlement date

datetime scalar | string scalar | date character vector

Settlement date, specified as a scalar datetime, string, or date character vector.

To support existing code, fitSvensson also accepts serial date numbers as inputs, but they are not recommended.

Instruments — Bond instrument objects

scalar | array

Bond instrument objects, specified as a scalar bond object or an array of bond instrument objects using FixedBond.

Data Types: object

CleanPrice — Observed market prices

vector

Observed market prices, specified as a vector.

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: NSModel = fitSvensson(Settle,Bonds,CleanPrice,Basis=4)

Basis — Day count basis

0 (actual/actual) (default) | integer from 0 to 13

Since R2024a

Day count basis, specified as Basis and a scalar or a NINST-by-1 matrix.

• 0 — actual/actual

• 1 — 30/360 (SIA)

• 2 — actual/360

• 3 — actual/365

• 4 — 30/360 (PSA)

• 5 — 30/360 (ISDA)

• 6 — 30/360 (European)

• 7 — actual/365 (Japanese)

• 8 — actual/actual (ICMA)

• 9 — actual/360 (ICMA)

• 10 — actual/365 (ICMA)

• 11 — 30/360E (ICMA)

• 12 — actual/365 (ISDA)

• 13 — BUS/252

For more information, see Basis.

Data Types: double

x0 — Initial estimate

[4/10 -1/10 -1/10 1/10 1 5] (default) | vector

Since R2024a

Initial estimate, specified as x0 and a vector.

Data Types: double

lb — Lower bound

[0 -Inf -Inf -Inf 0 0] (default) | vector

Since R2024a

Lower bound, specified as lb and a vector.

Data Types: double

ub — Upper bound

[Inf Inf Inf Inf Inf Inf] (default) | vector

Since R2024a

Upper bound, specified as ub and a vector.

Data Types: double

Output Arguments

outCurve — Fitted Svensson model

object

Fitted Svensson model, returned as a parametercurve object.

Algorithms

A similar model to the Nelson-Siegel is the Svensson model, which adds two additional parameters to account for greater flexibility in the term structure. This model proposes that the forward rate can be modeled with the following form:

$$f={\beta }_0+{\beta }_1{e}^{\frac{-m}{{\tau }_1}}+{\beta }_2{e}^{\frac{-m}{{\tau }_1}}\frac{m}{{\tau }_1}+{\beta }_3{e}^{\frac{-m}{{\tau }_2}}\frac{m}{{\tau }_2}$$

As above, this can be integrated to derive an equation for the zero curve:

$$s={\beta }_0+{\beta }_1(1-e^{\frac{-m}{{\tau }_1}})(-\frac{{\tau }_1}{m})+{\beta }_2((1-e^{\frac{-m}{{\tau }_1}})\frac{{\tau }_1}{m}-e^{\frac{-m}{{\tau }_1}})+{\beta }_3((1-e^{\frac{-m}{{\tau }_2}})\frac{{\tau }_2}{m}-e^{\frac{-m}{{\tau }_2}})$$

References

[1] Nelson, C.R., Siegel, A.F. “Parsimonious modelling of yield curves.” Journal of Business. Vol. 60, 1987, pp 473–89.

[2] Svensson, L.E.O. “Estimating and interpreting forward interest rates: Sweden 1992-4.” International Monetary Fund, IMF Working Paper, 1994/114.

[3] Fisher, M., Nychka, D., Zervos, D. “Fitting the term structure of interest rates with smoothing splines.” Board of Governors of the Federal Reserve System, Federal Reserve Board Working Paper 1995-1.

[4] Anderson, N., Sleath, J. “New estimates of the UK real and nominal yield curves.” Bank of England Quarterly Bulletin, November, 1999, pp 384–92.

[5] Waggoner, D. “Spline Methods for Extracting Interest Rate Curves from Coupon Bond Prices.” Federal Reserve Board Working Paper 1997–10.

[6] “Zero-coupon yield curves: technical documentation.” BIS Papers No. 25, October 2005.

[7] Bolder, D.J., Gusba, S. “Exponentials, Polynomials, and Fourier Series: More Yield Curve Modelling at the Bank of Canada.” Working Papers 2002–29, Bank of Canada.

[8] Bolder, D.J., Streliski, D. “Yield Curve Modelling at the Bank of Canada.” Technical Reports 84, 1999, Bank of Canada.

Version History

Introduced in R2020a

R2024a: Added Optional Name-Value Arguments for Basis, x0, lb, and ub

fitSvensson supports optional name-value arguments for Basis, x0, lb, and ub.

R2022b: Serial date numbers not recommended

Although fitSvensson supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.