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fitSmithWilson

Fit the Smith-Wilson model to observed bond prices

Since R2024a

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Syntax

outCurve = fitSmithWilson(Settle,Instruments,CleanPrice,UltimateForwardRate,LastLiquidPoint)
outCurve = fitSmithWilson(___Name=Value)

Description

example

outCurve = fitSmithWilson(Settle,Instruments,CleanPrice,UltimateForwardRate,LastLiquidPoint) fits a Smith-Wilson curve to observed bond prices and returns a parametercurve object.. After creating a parametercurve object for outCurve, you can use the associated object functions discountfactors, zerorates, and forwardrates.

example

outCurve = fitSmithWilson(___Name=Value) specifies options using one or more name-value arguments in addition to the input arguments in the previous syntax.

Examples

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Open Live Script

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

Settle = datetime("15-Jun-2023");
Maturity = Settle + calyears([2 4 6 9 18 30]');
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];
Bonds = fininstrument("FixedBond",Maturity=Maturity,CouponRate=CouponRate)
Bonds=6×1 FixedBond array with properties:
    CouponRate
    Period
    Basis
    EndMonthRule
    Principal
    DaycountAdjustedCashFlow
    BusinessDayConvention
    Holidays
    IssueDate
    FirstCouponDate
    LastCouponDate
    StartDate
    Maturity
    Name

Use fitSmithWilson to create a parametercurve object. 

UltimateForwardRate = .042;
LastLiquidPoint = 20;
SW = fitSmithWilson(Settle,Bonds,CleanPrice,UltimateForwardRate,LastLiquidPoint,Basis=4)
SW = 
  parametercurve with properties:

              Type: "discount"
            Settle: 15-Jun-2023
       Compounding: -1
             Basis: 4
    FunctionHandle: @(t)W(u_cf,t(:)')'*CF'*zeta+exp(-ultFwdInt*t(:))
        Parameters: 0.1317

You can use the parametercurve object (SW) to compute discountfactors, zerorates, and forwardrates.

outDiscountFactors=discountfactors(SW,[datetime("15-Jun-2024");datetime("15-Jun-2027")])
outDiscountFactors = 2×1

    0.9627
    0.8456


outZeroRates = zerorates(SW,[datetime("15-Jun-2024");datetime("15-Jun-2027")])
outZeroRates = 2×1

    0.0380
    0.0419


outForwardRates = forwardrates(SW,datetime("15-Jun-2024"),datetime("15-Jun-2027"))
outForwardRates = 0.0432
Open Live Script

Define the bond market data and use bndyield to compute the raw yield to maturity. 

Settle = datetime("15-Jun-2023");
Maturity = Settle + calyears([2 4 6 9 18 30]');
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];

YTM = bndyield(CleanPrice,CouponRate,Settle,Maturity)
YTM = 6×1

    0.0395
    0.0422
    0.0435
    0.0446
    0.0473
    0.0448


scatter(Maturity,YTM)

Use fininstrument to create a FixedBond instrument object using the bond data. 

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

Use fitSmithWilson to create a parametercurve object. 

UltimateForwardRate = .033;
LastLiquidPoint = 20;
SW = fitSmithWilson(Settle,Bonds,CleanPrice,UltimateForwardRate,LastLiquidPoint)
SW = 
  parametercurve with properties:

              Type: "discount"
            Settle: 15-Jun-2023
       Compounding: -1
             Basis: 3
    FunctionHandle: @(t)W(u_cf,t(:)')'*CF'*zeta+exp(-UltimateForwardRate*t(:))
        Parameters: 0.2514

Use fitNelsonSiegel to create a parametercurve object. 

NS = fitNelsonSiegel(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>

NS = 
  parametercurve with properties:

              Type: "zero"
            Settle: 15-Jun-2023
       Compounding: -1
             Basis: 0
    FunctionHandle: @(t)fitF(Params,t)
        Parameters: [6.7176e-08 0.0363 0.0904 16.5171]

Plot a comparison of the fit of a Smith-Wilson model to a Nelson-Siegel model for the bond market data.

hold on
PlottingTimes = calmonths(6:6:12*30)';
PlottingDates = Settle + PlottingTimes;
PY_SW = zero2pyld(zerorates(SW,PlottingDates),PlottingDates,Settle,'InputCompounding',-1,'InputBasis',3,'OutputCompounding',2);
PY_NS = zero2pyld(zerorates(NS,PlottingDates),PlottingDates,Settle,'InputCompounding',-1,'InputBasis',3,'OutputCompounding',2);
plot(PlottingDates,PY_SW,'r')
plot(PlottingDates,PY_NS,'g')
legend(["Raw YTM","Smith-Wilson model","Nelson-Siegel model"])

Figure contains an axes object. The axes object contains 3 objects of type scatter, line. These objects represent Raw YTM, Smith-Wilson model, Nelson-Siegel model.

Input Arguments

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

Data Types: datetime | string | char

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

Data Types: object

Observed market prices, specified as a vector.

Data Types: double

Long-term constant forward rate, specified as a numeric.

Data Types: double

Point on the yield curve beyond which there is insufficient liquidity to value cash flows accurately, specified as a numeric.

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: outCurve = fitSmithWilson(Settle,Bonds,CleanPrice,UltimateForwardRate,LastLiquidPoint,Basis=4,alpha=0.05)

Day count basis, specified as Basis and a scalar integer.

  • 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

Alpha value represents level of smoothness or flexibility of the yield curve, specified as alpha and a scalar numeric between 0 and 1.

The alpha value controls the tradeoff between fitting the observed market data and ensuring a smooth and plausible yield curve. By adjusting the alpha value, you can calibrate the Smith-Wilson model to different market conditions and risk preferences. A higher alpha value results in a smoother yield curve, while a lower alpha value allows for more flexibility and responsiveness to market movements.

Note

If you specify alpha, then fitSmithWilson ignores the input argument for LastLiquidPoint.

Data Types: double

Output Arguments

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Fitted Smith-Wilson model, returned as a parametercurve object.

More About

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Smith-Wilson Model

The Smith-Wilson model is a technique used to estimate the yield curve for discounting future cash flows.

The Smith-Wilson model is designed to handle long-dated liabilities and to accurately reflect the term structure of interest rates, particularly at the long end of the yield curve where market data is sparse. It uses a combination of market data and an ultimate forward rate (UFR) to extrapolate the yield curve to longer maturities.

Given the discount factor for tenor t, the Smith-Wilson equation to fit N prices for zero-coupon bonds with tenors u1,...uN, is

P(t):=e−wt+∑j=1NςjW(t,uj),t≥0,where w:=log(1+UFR),W(t,uj):=e−w(t+uj)(α(t∧uj)−e−α(t∨uj)sinh(α(t∧uj)))

where:

  • UFR is the ultimate forward rate.

  • ɑ is a parameter determining the rate of convergence to the UFR.

References

[1] Lagerås A., and M. Lindholm. "Issues with Smith-Wilson Method." Insurance: Mathematics and Economics. Vol. 71, 2016, pp. 93–102.

[2] Smith, A., and T. Wilson. "Fitting Yield Curves with Long Term Constraints." Research report, Bacon and Woodrow, 2000.

Version History

Introduced in R2024a

See Also

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