A FORECASTING MODEL FOR INTEREST RATES

A COMPARISON BETWEEN NELSON SIEGEL SVENSSON MODEL AND PCA APPROACH.

PIERPAOLO CASSESE  

ASSICURAZIONI GENERALI FINANCE DEPARTMENT

 9TH JULY 2012

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Summary

o Nelson Siegel Model

o An extention of Nelson Siegel Model: Nelson Siegel Svensson Model

o Calibration Model

o Principal Component Analysis

o Case Studies

o Concluding Remarks

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Nelson Siegel Model

Nelson Siegel Model (1987) is an exponential model which is

frequently adopted to estimate the Yield Curve. Foremost it

considers the instantaneous intensity of Interest instead of

Yield Curve directly. As shown beyond, this model could be

explained indipendently by the time t, since it is built

considering just one time period. This statement recognises

the no dynamic structure to Nelson Siegel Model.

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Nelson Siegel Model

Nelson Siegel Model (henceforth NS) is used commonly for modelling the instantaneous

interest intensity, beginning from period t and computed as follows:

d (t,s) = β0 + β1 + β2

(1)

Where must be s ≥ t and βk (k = 1, 2) e τk(k = 1) are the real parameters in order to

satisfy (2) :

β0 > 0, β1 > - β0, τ1 > 0

(2)

Basically, NS’s function is directly extrapolated by the main factors shown ahead. It is

painted with a monotone and exponential shape, indipendent from time t or s, while it’s

correlated to their difference (t-s).

Its structure is composed mainly by 3 factors:

1. β0 which consists in the long term interest rates factor.

2. β1 is the parameter that shows the spread among spot rate r and the value of Long

Term interest rates β0

3. β2 is the measure ables to represent the central part of the Yield Curve (medium

term interest rates factor)

4. t1 gives the contribution to the shape of the Yield Curve, even if it is caught likewise

the velocity with which the interest rates will converge to Long Term Interest level β0

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Nelson Siegel ModelMain Factors

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Example of Italian Swap Curve gained throughout NS Model…

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An estention of Nelson Siegel:Nelson Siegel Svensson Model

Nelson Siegel previously analysed, in some cases, losses its

affinity to the real Yield Curve it isn’t able to fit each real Yield

Curve maturities present in that phase on the Money Market.

For filling this error, two new parameters have been inserted

in the basic NS Model by Svensson. Adding the third

parameter, seems that the Yield Curve applying the new

Model, managed to fit better than with NS for all points of

Yield Curve. The formula should help to understand this better

:

d (t,s) = β0 + β1 + β2 + β3 (3)

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Nelson Siegel Svensson ModelMain Factors

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Example of Italian Swap Curve gained throughout NSS Model…

Nelson Siegel & Nelson Siegel SvenssonHow do these models calibrate?

Although NS and NSS belong to the math environment, to gain the

output of them is required by using an econometrician support, in

detail the non linear least square approach, with which to obtain the

main factors of the models. The technique used frequently is to set

the latest time series of interest rate, at t-1 time point, applying a non

linear regression to the Main maturities. Each key factor1, is indeed

just valued beforehand, attributing to these a hypothetical value.

After regression, the output will be merged in a vector and its

parameters procured as follows:

Min [4]

RMSE = [5]

[1] qNSS = [ b0, b1, b2, b3, t1, t2 ]

[2] The minimizing procedure is realized guessing that the difference vectors between rmkt – rNSS are the standard errors computed with the NLS estimation. When we calculate these, than we should keep in mind that how much smaller are these values, so much close will be the Yield Curve realized with NSS compared to the real market Curve.

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Nelson Siegel & Nelson Siegel SvenssonHow do these models calibrate?

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Principal Component Analysis

• Principal Component Analysis is piece of Multivariate Statistic Methodology in order to

studying the maximum variation regarding a time series enough deep in term of time

period.

• Principal components fix like goal that to aggregate a whole dataset and express this in a

few factors, usually indicated in three key factors. The PC are orthogonal linear

combinations of the original risk factors performed following a linear transformation of the

initial parameters.

• You need to aggregate all the time series of interest rates in a new matrix indicated with Z.

• The result that you have to perform will be a new Matrix Y where its values will be

uncorrelated among them.

Y=LZ (6)

Where L is the orthogonal matrix of eigenvectors of the variance and covariance matrix of Z.

Z = r (t) - r (t-k) (7)

Z represents the difference between the Interest Rates at t and t-k, with k which will be

defined as the deep of time series.

Principal Components classification: How to extrapolate PCs?

• First procedure to follow is to compute the differences between the value of interest rates (X).….

• Second step is to compute the Variance-Covariance Matrix of X

S = (8)

• PC Methodology tightly requires to maximize the goal function :

L= wTSw-λ(wTw-1) (9)

where λ is a Lagrange’s multiplier and w are the weights.

• If we maximize L goal function to w with the scope to find the combination of W in order to gain

the maximum variance expressed to the S matrix.

S-λI=0 (10)

Where I is the Identity Matrix.

When will be found the optimal combination in order to become max the L function, you need to

order the maximum explained variance of PC, by ranking from highest to smallest the variance of

these.

Thus the numbers of PC will be able to explain the total variance included in the range 80% - 90%,

than you could choice those PC defined optimal to define the main risk factors

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Meaning of PCs..

• Recently in the financial environment is always more frequent to analyze the interest rate risk factors adopting the PC technique..

• The sense attributed to them is that to put near to the variability of mains risk factors delineated in the components of the Yield Curve:

How do we read the Principal Components in Finance?

• PC1 is commonly identified with the

term of “Shift” of the interest rates

• PC2 is known better with the

synonym of “hump”, because it describes the shape of the Yield Curve.

• PC3 classified as the “Twist” for the

Yield Curve because it reveals the curvature of its.

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Scenario built adopting PC’s Strategy..

Difference realized between Scenario Curve and Real Swap Curve 22Th February

A comparison between Scenario vs Real Interest Swap Curve

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A brief comparison among Swap Curve created both NSS and PCA..

To test the two models described within

this presentation, We are going to

compare the result got following the

two previuos methodologies shown

ahead. The length of Times series is

nearly to 13 years, from January 1999

to February 2012 with a kind of dataset

got daily.

The study has been realized in order to

choose the better way to rear a good

scenario or a forecast, to foresee some

possible scenarios for the Italian Swap

Curve for the date 22th February 2012.

Seems that the PC’s Approach is not

only more flexible than NSS because

doesn’t require to consider just one

series of Interest Rates Swap values,

but it needs to also be good of the

whole series of these and, in add, to fit

almost perfectly to the real Swap

Curve.

Case Study

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This article is aimed the task to introduce two models to estimate the Interest rate curve generally speaking. The main

goal of this presentation has been that to supply a method and also to make some scenarios of Interest rates which is

oriented, for example, to the Swap Curve as has been done in this work, but also other kind of financial products as

derivative contracts belonging to the family of Interest Rates.

Thus to make a forecast for example for a new Swap Curve, you might realize this, applying either Nelson Siegel

Svensson Model or PCA in order to stay with your scenario close to the real market value of the Swap Curve.

In detail, has been possible to notice that both models performed very good toward the reality relevated on the Market,

but with some differences.

For PC model, the precision indicated in terms of standard error has been lower than NSS, so more precision means

more reliability and affinity to the real world.

Nelson Siegel Svensson since being a valid technique to create a forecast for the analyst or for investor, it doesn’t

perform perfectly as does PC, especially on the medium part of Swap Curve in terms of Maturities. Indeed isn’t able to

capture all points of Swap Curve hence isn’t allowed to fit good the real path signed by the Swap Curve indicated by the

Money Market.

This declaration plays to PC’s favour when you must make a choice what is the better Model to dedicate for computing

the possible Scenario for the future Swap Curve.

In conclusion, seems that the way to follow is that to tend to the PCA methodology for determining the ideal scenario

closer to the real world. This conclusion emerges after it is though to esclude the NSS Model since it doesn’t consider for

its analysis the deepness of the time series of data but just one of this, restricted to one day, not going beyond. Anyway

the NSS Model is known as a no dynamic model for this specific reason.

Concluding Remarks

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The study is undertaken beginning by the

length of time series of the differences of

interest rates. This is thought as a whole of

dataset that manages to supply a pool of

information widely large than other criteria.

It

considers the differences of the interest

rates, which they are always understood as

stationary variables, easier than the original

dataset both to manage and to elaborate. At

the end of analysis, the pool of information

that you had before, will be summarized in

two or three key factors, hence compared to

before, you will lose the original information.

This methodology allows to admit that,

given its technicalities, is requested a solid

background so you can valuate it.

The NSS can’t fit good all points of Yield Curve

compared to each maturities.

foremost this weakness point of NSS is visible

on the medium term of maturities, in the

range 5Y – 10Y, while for the rest of backet of

maturities, the fitting seems good but no

perfectly as well as we saw for the PC’s

Methodology.

NSS as underlined before, captures just one

time period, in fact isn’t totally reliable as way

of thinking. Its style adopted to make the

scenario should be strengthened to become

market consistent.

Therefore compared to PCA, it’s easy to catch

highest differences in terms of interest rates

which they transform in a minor similarity to

the value of the real Yield Curve.

Why do you choise PC’s Methodology instead NSS?

PCA Nelson Siegel Svensson

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