Fixed Income > YCM 2001 - Interpolation Techniques

Interpolation Techniques

Copyright © 1996-2006Investment Analytics

Copyright © 1996-2006 Investment Analytics Interpolation Techniques Slide: 2

Interpolation Techniques

Why interpolate?Straight line interpolationCubic spline interpolationBasis spline interpolation


Why Interpolate

StructuringProject security cash flowsNeed forward rates on coupon dates

ValuationNeed spot rates on coupon dates

In either case coupon dates may not coincide with dates for which zero-coupon yields are known.


Interpolation MethodsStraight LinePolynomial

Single high order polynomialUnstable between points and at ends

Splined polynomialLow order polynomials linked together

Basis SplinesRepresent discount function as weighted sum of other functions


Straight Line Interpolation –Pros and ConsSimple to estimate intermediate points on curveNot accurate for undulating curvesGives different results on discount factorsProduces discontinuous forward rate curve


Linear Interpolation

Intermediate values lie on a straight line between the nearest data points.

Ri = R1 + (R2 - R1 ) x (Ti - T1) / (T2 - T1)

T1

Ti

T2

R1

Ri

R2


Linear Interpolation: Rates or Discount Factors?

If interest rates lie on a straight line, discount factors do notExample:

Using Rates Using DF’sR1 = R2 = 5.00% D1 = 0.9877T1 = 90 D2 = 0.9756T2 = 180 Di = 0.9836Ti = 120Ri = 5.00% Ri = 4.99%


Linear and Exponential InterpolationLinear interpolation on continuously compounded interest rates is equivalent to exponential interpolation on discount factors

D e D eR R R

T TT T

D D D

R T R T

i

i

i

TT

TT

i i

1

1 2

1

2 1

1

1

2

1 1 2 2

1 2

21

= == − +

=−−

⇒ =

− −

−

,( )

( )

α α

α

α α


Cubic Spline Interpolation

A different cubic polynomial is fitted between each pair of data pointsThe polynomials are twice differentiableEnsures that:

The slope of the curve is smoothThe rate of change of the slope is smoothThe curves “join” at the end points


Cubic Spline Curve Fitting

Ri(t) = ai(t-ti)3 + bi (t-ti)2 + ci (t-ti) + di

Ri+1(t)

Ri-1(t)

5.00%

5.50%

6.00%

6.50%

7.00%

7.50%

8.00%

8.50%

0 200 400 600 800 1000 1200 1400 1600 1800 2000


Natural Splines

End Curve ConditionsConditions of the two ends of the yield curve must be specified for a solution.

Natural SplineSecond derivative (rate of change of the slope of the yield curve) equal to zero at both ends.

Slope of curve is constant at the endsYou typically only care about points in the belly of the curve


Cubic Splines – Pros & Cons

Smooth curve -twice differentiable at every data pointCan be used on both rates and DF’s Works for undulating curvesProduces continuous forward rate curveNot so easy to calculateCan suffer from oscillation


Lab: Building Yield Curves with Cubic Splines

Excel workbook; Yield Curve Modeling.xlsWorksheet: Cubic Spline CurveBuild 3m forward rate curve using:

Linearly interpolated DFsLinearly interpolated spot ratesCubic Spline interpolated spot rates

See Notes & Solution


Solution: Cubic SplineForward Curves

5.00%

5.50%

6.00%

6.50%

7.00%

7.50%

8.00%

8.50%

0 500 1000 1500 2000Days

Linear Interp on DF

Linear Interp on R

Cspline Interp on DF


Basis SplinesAnother widely used interpolation methodUsed for modeling discount functionTypically combined with regression analysis


Regression: More Payment Dates than Bonds

This is the usual case, as bond coupon dates fall on different days in the year.Have to represent discount factors by a function

Insufficient bonds to estimate model parametersSingular matrix

Use regression to determine parameters of the discount function

Then calculate discount factors on any chosen date


Representing the Discount Function by Basis Splines

Represent DF’s by function d(t):

Bond prices can be expressed as the sum of discounted cash flows:

Determine values of weights to fit bond prices to market data.

)()(1

tftd l

L

ll∑

=

= αl = 1...L: the number of basis spline functions f.α: weights applied to each function

)(11

tfCP l

L

ll

n

jiji ∑∑

==

= αC: Bond cash flowsP: Bond price


Estimating the Discount Function

Rearrange bond price equation:Sum of discounted cash flows

Sum of weighted cashflow x spline function

Spline functions are defined over the whole periodAs long as we have more bonds than weighting factors α,regression can be used.

)(11

tfCP l

L

ll

n

jiji ∑∑

==

= α

)(11

tfCP l

n

jij

L

lli ∑∑

==

= α


Basis Splines & Knot PointsBasis Splines

The discount function is a weighted average of a number of overlapping B-Splines.Cubic B-Spline functions usually selected.Individual spline functions are not linked.

Knot PointsEach spline function is non-zero over a well-defined interval.The start and end points of the splines are called “knot points”.


Basis Spline CurvesThe discount function is the weighted sum of individual splines

Knot Points

0.00E+00

2.00E-05

4.00E-05

6.00E-05

8.00E-05

1.00E-04

1.20E-04

1.40E-04

1.60E-04

1.80E-04

2.00E-04

0 500 1000 1500 2000 2500 3000


Selection of Knot Points

Results can be sensitive to placing of knot points

Unless there is an even distribution of bonds.

Important to have an equal number of bonds with maturities between each knot point.

Reduces estimation error.


Building a Zero Coupon Curve from Treasury Bonds

Often have more payment dates than bondsNo unique set of discount factors that will price all bonds

Use Regression AnalysisDetermine Least Squares Estimates of Discount Factors

Minimize the square of the difference between the observed bond prices and those based on estimated discount factors.

Discount factors must be linked by a functional formCubic splines have problems due to correlation.Basis splines are independent but watch “knot points”.


Lab: Building a Yield Curve with Basis Splines

Worksheet: Basis SplinesBuild yield curve using bond dataMethod:

Basis Splines & Regression

See Notes & Solution


Spot and Forward Rate Curves

5.0%

5.5%

6.0%

6.5%

7.0%

7.5%

0 500 1000 1500 2000 2500 3000

Spot

Forw ard


Confidence Intervals

3%

4%

4%

5%

5%

6%

6%

7%

7%

8%

0 500 1000 1500 2000 2500 3000

Spot Rate

Upper 95%

Low er 95%


Interpolation Methods: Summary

Straight Line InterpolationInaccurate.Leads to discontinuous forward rates.

Cubic SplinesBetter than linear interpolation.Due to smoothness condition points on the yield curve are linkedtogether.Linking causes multicollinearity.Accuracy of and one discount factor cannot be determined.

Basis SplinesFunctions go to zero at defined points.Need to use a weighted combination of several B-Splines.

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