Fixed Income > Ycm 2001 - Interpolation Techniques

Interpolation Techniques Copyright © 1996-2006 Investment Analytics

Interpolation Techniques „ „ „ „

Why interpolate? Straight line interpolation Cubic spline interpolation Basis spline interpolation

Copyright © 1996-2006 Investment Analytics

Interpolation Techniques

Slide: 2

Why Interpolate „

Structuring „ „

„

Valuation „

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Project security cash flows Need forward rates on coupon dates Need spot rates on coupon dates

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

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Interpolation Techniques

Slide: 3

Interpolation Methods „ „

Straight Line Polynomial „ „

„

Splined polynomial „

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Single high order polynomial Unstable between points and at ends Low order polynomials linked together

Basis Splines „

Represent discount function as weighted sum of other functions

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Interpolation Techniques

Slide: 4

Straight Line Interpolation – Pros and Cons „

„ „ „

Simple to estimate intermediate points on curve Not accurate for undulating curves Gives different results on discount factors Produces discontinuous forward rate curve

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Interpolation Techniques

Slide: 5

Linear Interpolation „

Intermediate values lie on a straight line between the nearest data points. T2 R2

Ti T1

Ri

R1 „

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

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Interpolation Techniques

Slide: 6

Linear Interpolation: Rates or Discount Factors? „

„

If interest rates lie on a straight line, discount factors do not Example: „

Using Rates R1 = R2 = 5.00% T1 = 90 T2 = 180 Ti = 120 Ri = 5.00%

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Using DF’s D1 = 0.9877 D2 = 0.9756 Di = 0.9836 Ri = 4.99%

Interpolation Techniques

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Linear and Exponential Interpolation „

Linear interpolation on continuously compounded interest rates is equivalent to exponential interpolation on discount factors −R T −R T D1 = e

1 1

, D2 = e

2 2

Ri = (1 − α ) R1 + αR2 Ti − T1 α= T2 − T1 (1−α )

⇒ Di = D1 Copyright © 1996-2006 Investment Analytics

Ti T1

α

D2

Ti T2

Interpolation Techniques

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Cubic Spline Interpolation „

„ „

A different cubic polynomial is fitted between each pair of data points The polynomials are twice differentiable Ensures that: „ „ „

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

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Interpolation Techniques

Slide: 9

Cubic Spline Curve Fitting Ri+1(t)

8.50%

8.00%

7.50%

Ri-1(t)

7.00%

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

6.50%

6.00%

5.50%

5.00% 0

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400

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800

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1800

Interpolation Techniques

2000

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Natural Splines „

End Curve Conditions „

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Conditions of the two ends of the yield curve must be specified for a solution.

Natural Spline „

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

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Slope of curve is constant at the ends

You typically only care about points in the belly of the curve

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Interpolation Techniques

Slide: 11

Cubic Splines – Pros & Cons „

„ „ „ „ „

Smooth curve -twice differentiable at every data point Can be used on both rates and DF’s Works for undulating curves Produces continuous forward rate curve Not so easy to calculate Can suffer from oscillation

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Interpolation Techniques

Slide: 12

Lab: Building Yield Curves with Cubic Splines „ „ „

Excel workbook; Yield Curve Modeling.xls Worksheet: Cubic Spline Curve Build 3m forward rate curve using: „ „ „

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Linearly interpolated DFs Linearly interpolated spot rates Cubic Spline interpolated spot rates

See Notes & Solution

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Interpolation Techniques

Slide: 13

Solution: Cubic Spline Forward Curves 8.50% 8.00% 7.50% 7.00% 6.50%

Linear Interp on DF

6.00%

Linear Interp on R 5.50%

Cspline Interp on DF

5.00%

0

500

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Days

Interpolation Techniques

2000

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Basis Splines „ „ „

Another widely used interpolation method Used for modeling discount function Typically combined with regression analysis

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Interpolation Techniques

Slide: 15

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 parameters „

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Singular matrix

Use regression to determine parameters of the discount function „

Then calculate discount factors on any chosen date

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Interpolation Techniques

Slide: 16

Representing the Discount Function by Basis Splines „

Represent DF’s by function d(t): L

d (t ) = ∑ α l f l (t ) l =1

„

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

L

j =1

l =1

Pi = ∑ C ij ∑ α l f l (t ) „

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

C: P:

Bond cash flows Bond price

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

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Interpolation Techniques

Slide: 17

Estimating the Discount Function „

Rearrange bond price equation: „

Sum of discounted cash flows n

L

j =1

l =1

Pi = ∑Cij ∑αl fl (t) „

Sum of weighted cashflow x spline function L

n

l =1

j =1

Pi = ∑ α l ∑ C ij f l (t ) „ „

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

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Interpolation Techniques

Slide: 18

Basis Splines & Knot Points „

Basis Splines „

„ „

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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 Points „

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Each spline function is non-zero over a welldefined interval. The start and end points of the splines are called “knot points”.

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Interpolation Techniques

Slide: 19

Basis Spline Curves „

The discount function is the weighted sum of individual splines 2.00E-04 1.80E-04 1.60E-04 1.40E-04 1.20E-04 1.00E-04 8.00E-05 6.00E-05 4.00E-05 2.00E-05 0.00E+00 0

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Knot Points Copyright © 1996-2006 Investment Analytics

Interpolation Techniques

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Selection of Knot Points „

Results can be sensitive to placing of knot points „

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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.

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Interpolation Techniques

Slide: 21

Building a Zero Coupon Curve from Treasury Bonds „

Often have more payment dates than bonds „

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No unique set of discount factors that will price all bonds

Use Regression Analysis „

Determine Least Squares Estimates of Discount Factors „

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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 form „ „

Cubic splines have problems due to correlation. Basis splines are independent but watch “knot points”.

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Interpolation Techniques

Slide: 22

Lab: Building a Yield Curve with Basis Splines „ „ „

Worksheet: Basis Splines Build yield curve using bond data Method: „

„

Basis Splines & Regression

See Notes & Solution

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Interpolation Techniques

Slide: 23

Spot and Forward Rate Curves 7.5%

7.0%

6.5%

6.0%

5.5%

Spot Forw ard

5.0% 0

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1000

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2000

Interpolation Techniques

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Confidence Intervals 8% 7% 7% 6% 6% 5% 5%

Spot Rate

4%

Upper 95%

4%

Low er 95%

3% 0

500

1000

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2000

Interpolation Techniques

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Interpolation Methods: Summary „

Straight Line Interpolation „ „

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Cubic Splines „ „

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Inaccurate. Leads to discontinuous forward rates. Better than linear interpolation. Due to smoothness condition points on the yield curve are linked together. Linking causes multicollinearity. Accuracy of and one discount factor cannot be determined.

Basis Splines „ „

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

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Interpolation Techniques

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