Creating an IRFunctionCurve Object

Fitting IRFunctionCurve Object Using a Function Handle

You can use the constructor IRFunctionCurve with a MATLAB® function handle to define an interest-rate curve. For more information on defining a function handle, see the MATLAB Programming Fundamentals documentation.

Example

This example uses a FunctionHandle argument with a value @(t) t.^2 to construct an interest-rate curve:

rr = IRFunctionCurve('Zero',today,@(t) t.^2)
rr = 

  Properties:
    FunctionHandle: @(t)t.^2
              Type: 'Zero'
            Settle: 733600
       Compounding: 2
             Basis: 0

Fitting IRFunctionCurve Object Using Nelson-Siegel Method

Use the method, fitNelsonSiegel, for the Nelson-Siegel model that fits the empirical form of the yield curve with a prespecified functional form of the spot rates which is a function of the time to maturity of the bonds.

The Nelson-Siegel model represents a dynamic three-factor model: level, slope, and curvature. However, the Nelson-Siegel factors are unobserved, or latent, which allows for measurement error, and the associated loadings have economic restrictions (forward rates are always positive, and the discount factor approaches zero as maturity increases). For more information, see "Zero-coupon yield curves: technical documentation," BIS Papers, Bank for International Settlements, Number 25, October, 2005.

Example

This example uses IRFunctionCurve to model the default-free term structure of interest rates in the United Kingdom.

Load the data:

load ukdata20080430

Convert 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');

The IRFunctionCurve object provides the capability to fit a Nelson-Siegel curve to observed market data with the fitNelsonSiegel method. The fitting is done by calling the Optimization Toolbox™ function lsqnonlin. This method has required inputs of Type, Settle, and a matrix of instrument data.

NSModel = IRFunctionCurve.fitNelsonSiegel('Zero',CurveSettle,...
Instruments,'Compounding',-1,'InstrumentPeriod',InstrumentPeriod);

Plot the Nelson-Siegel interest-rate curve for forward rates:

PlottingDates = CurveSettle+20:30:CurveSettle+365*25;
TimeToMaturity = yearfrac(CurveSettle,PlottingDates);
NSForwardRates = getForwardRates(NSModel, PlottingDates);
plot(TimeToMaturity,NSForwardRates)
title('Nelson Siegel model of UK instantaneous nominal forward curve')

Fitting IRFunctionCurve Object Using Svensson Method

Use the method, fitSvensson, for the Svensson model to improve the flexibility of the curves and the fit for a Nelson-Siegel model. In 1994, Svensson extended Nelson and Siegel's function by adding a further term that allows for a second "hump." The extra precision is achieved at the cost of adding two more parameters, β3 and τ2, which have to be estimated.

Example

In this example of using the fitSvensson method, an IRFitOptions structure, previously defined using the IRFitOptions constructor, is used. Thus, you must specify FitType, InitialGuess, UpperBound, LowerBound, and the OptOptions optimization parameters for lsqnonlin.

Load the data:

load ukdata20080430

Convert 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');

Define OptOptions for the IRFitOptions constructor:

OptOptions = optimoptions('lsqnonlin','MaxFunEvals',1000);
fIRFitOptions = IRFitOptions([5.82 -2.55 -.87 0.45 3.9 0.44],...
'FitType','durationweightedprice','OptOptions',OptOptions,...
'LowerBound',[0 -Inf -Inf -Inf 0 0],'UpperBound',[Inf Inf Inf Inf Inf Inf]);

Fit the interest-rate curve using a Svensson model:

SvenssonModel = IRFunctionCurve.fitSvensson('Zero',CurveSettle,...
Instruments,'IRFitOptions', fIRFitOptions, 'Compounding',-1,...
'InstrumentPeriod',InstrumentPeriod)
Local minimum possible.

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

SvenssonModel = 

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

The status message, output from lsqnonlin, indicates that the optimization to find parameters for the Svensson equation terminated successfully.

Plot the Svensson interest-rate curve for forward rates:

PlottingDates = CurveSettle+20:30:CurveSettle+365*25;
TimeToMaturity = yearfrac(CurveSettle,PlottingDates);
SvenssonForwardRates = getForwardRates(SvenssonModel, PlottingDates);
plot(TimeToMaturity,SvenssonForwardRates)
title('Svensson model of UK instantaneous nominal forward curve')

Fitting IRFunctionCurve Object Using Smoothing Spline Method

Use the method, fitSmoothingSpline, to model the term structure with a spline, specifically, the term structure represents the forward curve with a cubic spline.

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

Example

The IRFunctionCurve object is used to fit a smoothing spline representation of the forward curve with a penalty function. Required inputs are Type, Settle, the matrix of Instruments, and Lambdafun, a function handle containing the penalty function

Load the data:

load ukdata20080430

Convert 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 parameters for Lambdafun:

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);

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

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

Using fitFunction to Create Custom Fitting Function

When using an IRFunctionCurve object, you can create a custom fitting function with the fitFunction method. To use fitFunction, you must define a FunctionHandle. In addition, you must also use the constructor IRFitOptions to define IRFitOptionsObj to support an InitialGuess for the parameters of the curve function.

Example

The following example demonstrates the use of fitFunction with a FunctionHandle and an IRFitOptionsObj:

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];
CurveSettle = datenum('30-Apr-2008');

Define the FunctionHandle:

functionHandle = @(t,theta) polyval(theta,t);

Define the OptOptions for IRFitOptions:

OptOptions = optimoptions('lsqnonlin','display','iter');

Define fitFunction:

CustomModel = IRFunctionCurve.fitFunction('Zero', CurveSettle, ...
functionHandle,Instruments, IRFitOptions([.05 .05 .05],'FitType','price',...
'OptOptions',OptOptions));
                                         Norm of      First-order 
 Iteration  Func-count     f(x)          step          optimality   CG-iterations
     0          4         38036.7                      4.92e+04
     1          8         38036.7             10       4.92e+04            0
     2         12         38036.7            2.5       4.92e+04            0
     3         16         38036.7          0.625       4.92e+04            0
     4         20         38036.7        0.15625       4.92e+04            0
     5         24         30741.5      0.0390625       1.72e+05            0
     6         28         30741.5       0.078125       1.72e+05            0
     7         32         30741.5      0.0195312       1.72e+05            0
     8         36         28713.6     0.00488281       2.33e+05            0
     9         40         20323.3     0.00976562       9.47e+05            0
    10         44         20323.3      0.0195312       9.47e+05            0
    11         48         20323.3     0.00488281       9.47e+05            0
    12         52         20323.3      0.0012207       9.47e+05            0
    13         56         19698.8    0.000305176       1.08e+06            0
    14         60           17493    0.000610352          7e+06            0
    15         64           17493      0.0012207          7e+06            0
    16         68           17493    0.000305176          7e+06            0
    17         72         15455.1    7.62939e-05       2.25e+07            0
    18         76         15455.1    0.000177499       2.25e+07            0
    19         80         13317.1     3.8147e-05       3.18e+07            0
    20         84         12865.3    7.62939e-05       7.83e+07            0
    21         88         11779.8    7.62939e-05       7.58e+06            0
    22         92         11747.6    0.000152588       1.45e+05            0
    23         96         11720.9    0.000305176       2.33e+05            0
    24        100         11667.2    0.000610352       1.48e+05            0
    25        104         11558.6      0.0012207       3.55e+05            0
    26        108         11335.5     0.00244141       1.57e+05            0
    27        112         10863.8     0.00488281       6.36e+05            0
    28        116         9797.14     0.00976562       2.53e+05            0
    29        120         6882.83      0.0195312       9.18e+05            0
    30        124         6882.83      0.0373993       9.18e+05            0
    31        128         3218.45     0.00934981       1.96e+06            0
    32        132         612.703      0.0186996       3.01e+06            0
    33        136         13.0998      0.0253882       3.05e+06            0
    34        140       0.0762922     0.00154002       5.05e+04            0
    35        144       0.0731652    3.61102e-06           29.9            0
    36        148       0.0731652    6.32335e-08          0.063            0

Local minimum possible.

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

Plot the custom function that is defined using fitFunction:

Yields = bndyield(CleanPrice,CouponRate,Settle(1),Maturity);
scatter(Maturity,Yields);
PlottingPoints = min(Maturity):30:max(Maturity);
hold on;
plot(PlottingPoints, getParYields(CustomModel, PlottingPoints),'r');
datetick
legend('Market Yields','Fitted Yield Curve')
title('Custom Function fit to Market Data')

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