fitSmoothingSpline

Fit smoothing spline to bond market data

Class

@IRFunctionCurve

Syntax

CurveObj = IRFunctionCurve.fitSmoothingSpline(Type,Settle,Instruments,Lambdafun)
CurveObj = IRFunctionCurve.fitSmoothingSpline(Type,Settle,Instruments,Lambdafun,Name,Value)

Arguments

Note

You must have a license for Curve Fitting Toolbox™ software to use the fitSmoothingSpline method.

Type	Type of interest-rate curve for a bond: Forward, Zero, or Discount.
Settle	Scalar for the Settle date of the curve.
Instruments	N-by-4 data matrix for Instruments where the first column is Settle date, the second column is Maturity, the third column is the clean price, and the fourth column is a CouponRate for the bond.
Lambdafun	Penalty function that takes as its input time and returns a penalty value. Use a function handle to support the penalty function. The function handle for the penalty function which takes one numeric input (time-to-maturity) and returns one numeric output (penalty to be applied to the curvature of the spline). For more information on defining a function handle, see the MATLAB® Programming Fundamentals documentation. Note The smoothing spline represents the forward curve. The spline is penalized for curvature by specifying a penalty function. This fit may only be done with a FitType of DurationWeightedPrice.
Knots	(Optional) Vector of knot locations (times-to-maturity); by default, knots is set to be a vector comprised of 0 and the time to maturity of all input instruments. The default is for the spline type to be cubic but you can specify any spline type by explicitly specifying the knots. User-defined knots can be specified using the following command, wherek is the order: augknt(knots,k).
Compounding	(Optional) Scalar that sets the compounding frequency per year for the IRFunctionCurve object: −1 = Continuous compounding (default) 1 = Annual compounding 2 = Semiannual compounding 3 = Compounding three times per year 4 = Quarterly compounding 6 = Bimonthly compounding 12 = Monthly compounding
Basis	(Optional) Day-count basis of the interest-rate curve. A scalar of integers. 0 = actual/actual (default) 1 = 30/360 (SIA) 2 = actual/360 3 = actual/365 4 = 30/360 (BMA) 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.

Instrument Parameters

For each bond Instrument, you can specify the following additional instrument parameters as name-value pairs. For example, InstrumentBasis distinguishes a bond instrument's Basis value from the curve's Basis value.

InstrumentPeriod	(Optional) Coupons per year of the bond. A vector of integers. Allowed values are 0, 1, 2 (default), 3, 4, 6, and 12.
InstrumentBasis	(Optional) Day-count basis of the bond. A vector of integers. 0 = actual/actual (default) 1 = 30/360 (SIA) 2 = actual/360 3 = actual/365 4 = 30/360 (BMA) 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.
InstrumentEndMonthRule	(Optional) End-of-month rule. A vector. This rule applies only when Maturity is an end-of-month date for a month having 30 or fewer days. 0 = ignore rule, meaning that a bond's coupon payment date is always the same numerical day of the month. 1 = set rule on (default), meaning that a bond's coupon payment date is always the last actual day of the month.
InstrumentIssueDate	(Optional) Date when an instrument was issued.
InstrumentFirstCouponDate	(Optional) Date when a bond makes its first coupon payment; used when bond has an irregular first coupon period. When FirstCouponDate and LastCouponDate are both specified, FirstCouponDate takes precedence in determining the coupon payment structure. If you do not specify a FirstCouponDate, the cash flow payment dates are determined from other inputs.
InstrumentLastCouponDate	(Optional) Last coupon date of a bond before the maturity date; used when bond has an irregular last coupon period. In the absence of a specified FirstCouponDate, a specified LastCouponDate determines the coupon structure of the bond. The coupon structure of a bond is truncated at the LastCouponDate, regardless of where it falls, and is followed only by the bond's maturity cash flow date. If you do not specify a LastCouponDate, the cash flow payment dates are determined from other inputs.
InstrumentFace	(Optional) Face or par value. Default = 100.

Note

When using Instrument name-value pairs, you can specify simple interest for a bond by specifying the InstrumentPeriod value as 0. If InstrumentBasis and InstrumentPeriod are not specified for a bond, the following default values are used: Basis is 0 (act/act) and Period is 2.

Description

Fcurve = IRFunctionCurve.fitSmoothingSpline(Type, Settle, Instruments, Lambdafun,Name,Value) fits a smoothing spline to market data for a bond. You must enter the optional arguments for Basis, Compounding, and Knots as comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Examples

Use a Smoothing Spline Function to Fit Market Data For a Bond

This example shows how to use a smoothing spline function to fit market data for a bond.

Settle = repmat(datenum('30-Apr-2008'),[6 1]);
Maturity = [datenum('07-Mar-2009');datenum('07-Mar-2011');...
datenum('07-Mar-2013');datenum('07-Sep-2016');...
datenum('07-Mar-2025');datenum('07-Mar-2036')];

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];
Instruments = [Settle Maturity CleanPrice CouponRate];
PlottingPoints = datenum('07-Mar-2009'):180:datenum('07-Mar-2036');
Yield = bndyield(CleanPrice,CouponRate,Settle,Maturity);

% use the AUGKNT function to construct the knots for a cubic spline at every 5 years
CustomKnots = augknt(0:5:30,4);
SmoothingModel = IRFunctionCurve.fitSmoothingSpline('Zero',datenum('30-Apr-2008'),...
Instruments,@(t) 1000,'knots', CustomKnots);

% create the plot
plot(PlottingPoints, getParYields(SmoothingModel, PlottingPoints),'b')
hold on
scatter(Maturity,Yield,'black')
datetick('x')

Fitting an IRFunctionCurve Object Using the Smoothing Spline Method With the Penalty Function

Use the fitSmoothinSpline method to fit the interest-rate curve and model the Lambdafun penalty function. First, load the data.

load ukdata20080430

Convert the repo rates to be equivalent zero coupon bonds.

RepoCouponRate = repmat(0,size(RepoRates));
RepoPrice = bndprice(RepoRates, RepoCouponRate, RepoSettle, RepoMaturity);

Aggregate the data.

Settle = [RepoSettle;BondSettle];
Maturity = [RepoMaturity;BondMaturity];
CleanPrice = [RepoPrice;BondCleanPrice];
CouponRate = [RepoCouponRate;BondCouponRate];
Instruments = [Settle Maturity CleanPrice CouponRate];
InstrumentPeriod = [repmat(0,6,1);repmat(2,31,1)];
CurveSettle = datenum('30-Apr-2008');

Choose the parameters for the Lambdafun input argument.

L = 9.2;
S = -1;
mu = 1;

Define the Lambdafun penalty function.

lambdafun = @(t) exp(L - (L-S)*exp(-t/mu));
t = 0:.1:25;
y = lambdafun(t);
figure
semilogy(t,y);
title('Penalty Function for VRP Approach')
ylabel('Penalty')
xlabel('Time')

Use the fitSmoothinSpline method to fit the interest-rate curve and model the Lambdafun penalty function.

VRPModel = IRFunctionCurve.fitSmoothingSpline('Forward',CurveSettle,...
Instruments,lambdafun,'Compounding',-1, 'InstrumentPeriod',InstrumentPeriod)

VRPModel = 

			 Type: Forward
		   Settle: 733528 (30-Apr-2008)
	  Compounding: -1
			Basis: 0 (actual/actual)

Plot the smoothing spline interest-rate curve for the forward rates.

PlottingDates = CurveSettle+20:30:CurveSettle+365*25;
TimeToMaturity = yearfrac(CurveSettle,PlottingDates);
VRPForwardRates = getForwardRates(VRPModel, PlottingDates);
figure;plot(TimeToMaturity,VRPForwardRates)
title('Smoothing Spline model of UK instantaneous nominal forward curve')

Algorithms

The term structure can be modeled with a spline — specifically, one way to model the term structure is by representing the forward curve with a cubic spline. To ensure that the spline is sufficiently smooth, a penalty is imposed relating to the curvature (second derivative) of the spline:

where the first term is the difference between the observed price P and the predicted price, $$\widehat{P}$$, (weighted by the bond's duration, D) summed over all bonds in our data set and the second term is the penalty term (where λ is a penalty function and f is the spline).

See [3], [4], [5] below.

There have been different proposals for the specification of the penalty function λ. One approach, advocated by [4], and currently used by the UK Debt Management Office, is a penalty function of the following form:

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.