![Page 1: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/1.jpg)
Principal Components Analysis in Yield-CurveModeling
Carlos F. Tolmasky
April 4, 2007
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 2: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/2.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 3: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/3.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 4: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/4.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 5: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/5.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 6: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/6.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.
Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 7: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/7.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 8: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/8.jpg)
Term Structure Models
Black-Scholes models 1 underlying.
What if we need more? spread, basket options.
Need correlation structure of the market.
What if the market is naturally a curve?
Interest rates.Commodities.
Does it make sense to model each underlying individually?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 9: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/9.jpg)
Front Month Crude
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 10: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/10.jpg)
Crude Curve
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 11: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/11.jpg)
Yield Curve
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 12: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/12.jpg)
Japanese Yield Curve
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 13: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/13.jpg)
Crude Curve through time
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 14: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/14.jpg)
Natural Gas Curve through time
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 15: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/15.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 16: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/16.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.
What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 17: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/17.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 18: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/18.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.
1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 19: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/19.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?
Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 20: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/20.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 21: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/21.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..
Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 22: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/22.jpg)
Term Structure Models
Historically, different approaches:
Black’s model:
Each possible underlying is lognormal.What if we need to use more than one rate?
1-Factor models (Vasicek, Ho-Lee)
Model the short rate, derive the rest of thecurve from it.1-factor not rich enough, how do we add factors?Adding factors not obvious.
HJM
Forget Black-Scholes..Model the whole curve.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 23: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/23.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 24: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/24.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 25: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/25.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 26: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/26.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 27: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/27.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 28: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/28.jpg)
HJM
How?? ∞-many points.
However correlation is high.
Maybe the moves ”live” in a lower dimensional space.
Instead of
dFi
Fi= σidWi i = 1, ..., n
with Wi ,Wj correlated do
dFi
Fi=
k∑i=1
σj ,idWj k < n (hopefully)
But, how do we choose the σj ,i ??
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 29: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/29.jpg)
PCA
Technique to reduce dimensionality.
If X is the matrix containing our data, we look for w so thatarg max‖w‖=1 Var(wTX )
Then we do the same in the subspace orthogonal to w .
It is equivalent to diagonalizing the covariance matrix.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 30: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/30.jpg)
PCA
Technique to reduce dimensionality.
If X is the matrix containing our data, we look for w so thatarg max‖w‖=1 Var(wTX )
Then we do the same in the subspace orthogonal to w .
It is equivalent to diagonalizing the covariance matrix.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 31: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/31.jpg)
PCA
Technique to reduce dimensionality.
If X is the matrix containing our data, we look for w so thatarg max‖w‖=1 Var(wTX )
Then we do the same in the subspace orthogonal to w .
It is equivalent to diagonalizing the covariance matrix.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 32: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/32.jpg)
PCA
Technique to reduce dimensionality.
If X is the matrix containing our data, we look for w so thatarg max‖w‖=1 Var(wTX )
Then we do the same in the subspace orthogonal to w .
It is equivalent to diagonalizing the covariance matrix.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 33: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/33.jpg)
PCA
Technique to reduce dimensionality.
If X is the matrix containing our data, we look for w so thatarg max‖w‖=1 Var(wTX )
Then we do the same in the subspace orthogonal to w .
It is equivalent to diagonalizing the covariance matrix.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 34: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/34.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 35: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/35.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 36: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/36.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 37: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/37.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 38: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/38.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.
Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 39: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/39.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 40: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/40.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 41: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/41.jpg)
Litterman-Scheikman (1991)
Looked at the treasury yield curve.
Found that just a few eigenvectors are the important ones.
Three of them explain most of the moves.
Level-Slope-Curvature
Very Intuitive.Curve trades.
Cortazar-Schwartz (2004) found the same in copper
Loads (or lots?) of other people report the same kind ofresults in many other markets.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 42: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/42.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 43: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/43.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.
Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 44: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/44.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 45: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/45.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.
Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 46: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/46.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.
Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 47: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/47.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 48: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/48.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.
Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 49: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/49.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.
Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 50: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/50.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.
Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 51: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/51.jpg)
Predictive Power
Recently, some work has been done on this.
Monch (2006)
Studies innovations in level-slope-curvature wrt macrovariables.Positive answer for curvature.
Diebold-Li (2006)
Use autoregressive models for each component.Study forecast power at short and long horizons.Report encouraging results at long horizons.
Chantziara-Skiadopoulos (2005).
Study predictive power in oil.Results are weak.Also look at spillover effects among crude (WTI and IPE),heating oil and gasoline.Some spillover effects found.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 52: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/52.jpg)
Table: Correlation Matrix for Changes of the First 12 Crude Oil Futures Prices
1.000 0.992 0.980 0.966 0.951 0.936 0.922 0.08 0.892 0.877 0.860 0.8480.992 1.000 0.996 0.988 0.978 0.966 0.954 0.941 0.927 0.913 0.898 0.8860.980 0.996 1.000 0.997 0.991 0.982 0.973 0.963 0.951 0.939 0.925 0.9140.966 0.988 0.997 1.000 0.998 0.993 0.986 0.978 0.968 0.958 0.946 0.9360.951 0.978 0.991 0.998 1.000 0.998 0.994 0.989 0.981 0.972 0.963 0.9540.936 0.966 0.982 0.993 0.998 1.000 0.999 0.995 0.90 0.983 0.975 0.9670.922 0.954 0.973 0.986 0.994 0.999 1.000 0.999 0.996 0.991 0.984 0.9780.08 0.941 0.963 0.978 0.989 0.995 0.999 1.000 0.999 0.996 0.991 0.9850.892 0.927 0.951 0.968 0.981 0.90 0.996 0.999 1.000 0.999 0.995 0.9910.877 0.913 0.939 0.958 0.972 0.983 0.991 0.996 0.999 1.000 0.998 0.9960.860 0.898 0.925 0.946 0.963 0.975 0.984 0.991 0.995 0.998 1.000 0.9980.848 0.886 0.914 0.936 0.954 0.967 0.978 0.985 0.991 0.996 0.998 1.000
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 53: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/53.jpg)
First four eigenvectors for oil
2 4 6 8 10 12
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Contract
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 54: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/54.jpg)
First four eigenvectors for oil
2 4 6 8 10 12
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Contract
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 55: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/55.jpg)
First four eigenvectors for oil
2 4 6 8 10 12
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Contract
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 56: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/56.jpg)
First four eigenvectors for oil
2 4 6 8 10 12
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Contract
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 57: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/57.jpg)
Forzani-T (2003)
Why is the result ”market-invariant”?
Because all the correlation matrices are very similar.
They all look like ρ|i−j | with ρ close to 1.
Proved that the eigenvectors of those matrices converge tocos(nx) when ρ→ 1 .
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 58: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/58.jpg)
Forzani-T (2003)
Why is the result ”market-invariant”?
Because all the correlation matrices are very similar.
They all look like ρ|i−j | with ρ close to 1.
Proved that the eigenvectors of those matrices converge tocos(nx) when ρ→ 1 .
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 59: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/59.jpg)
Forzani-T (2003)
Why is the result ”market-invariant”?
Because all the correlation matrices are very similar.
They all look like ρ|i−j | with ρ close to 1.
Proved that the eigenvectors of those matrices converge tocos(nx) when ρ→ 1 .
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 60: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/60.jpg)
Forzani-T (2003)
Why is the result ”market-invariant”?
Because all the correlation matrices are very similar.
They all look like ρ|i−j | with ρ close to 1.
Proved that the eigenvectors of those matrices converge tocos(nx) when ρ→ 1 .
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 61: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/61.jpg)
Forzani-T (2003)
Correlation matrix:
1 ρTn ρ2T
n ... ... ρn Tn
ρTn 1 ρ
Tn ... ... ρ(n−1)T
n
... ... ... ... ... ...
... ... ... ... ... ...
ρ(n−1)Tn ρ(n−2)T
n ρ(n−3)Tn ... 1 ρT
n
ρn Tn ρ(n−1)T
n ρ(n−2)Tn ... ρ
Tn 1
or, as an operator:
Kρf (x) =
∫ T
0ρ|y−x |f (y)dy . (1)
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 62: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/62.jpg)
Lekkos (2000)
A big part of the correlation structure is given by:
R(t,T1)T1 = R(t,T0)T0 + f (t,T0,T1)(T1 − T0)
So, it is an artifact.
Even if we generate independent forwards we find structure inthe correlation matrix of the zeros.
Looked at the PCAs of fwds in various markets, found nothinginteresting.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 63: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/63.jpg)
Lekkos (2000)
A big part of the correlation structure is given by:
R(t,T1)T1 = R(t,T0)T0 + f (t,T0,T1)(T1 − T0)
So, it is an artifact.
Even if we generate independent forwards we find structure inthe correlation matrix of the zeros.
Looked at the PCAs of fwds in various markets, found nothinginteresting.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 64: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/64.jpg)
Lekkos (2000)
A big part of the correlation structure is given by:
R(t,T1)T1 = R(t,T0)T0 + f (t,T0,T1)(T1 − T0)
So, it is an artifact.
Even if we generate independent forwards we find structure inthe correlation matrix of the zeros.
Looked at the PCAs of fwds in various markets, found nothinginteresting.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 65: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/65.jpg)
Lekkos (2000)
A big part of the correlation structure is given by:
R(t,T1)T1 = R(t,T0)T0 + f (t,T0,T1)(T1 − T0)
So, it is an artifact.
Even if we generate independent forwards we find structure inthe correlation matrix of the zeros.
Looked at the PCAs of fwds in various markets, found nothinginteresting.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 66: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/66.jpg)
Alexander-Lvov (2003)
They study different fitting techniques for the yield curve.
Found that this choice is crucial to the correlation structureobtained.
Could Lekkos’ critique be just a matter of the choice of thefitting technique?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 67: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/67.jpg)
Alexander-Lvov (2003)
They study different fitting techniques for the yield curve.
Found that this choice is crucial to the correlation structureobtained.
Could Lekkos’ critique be just a matter of the choice of thefitting technique?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 68: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/68.jpg)
Alexander-Lvov (2003)
They study different fitting techniques for the yield curve.
Found that this choice is crucial to the correlation structureobtained.
Could Lekkos’ critique be just a matter of the choice of thefitting technique?
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 69: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/69.jpg)
Lord, Pessler (2005)
They ask the question:
Can we characterize ”level-slope-curvature”?
They look at sign changes in the eigenvectors.
”Level” means no sign changes.
This is solved by Perron’s theorem.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 70: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/70.jpg)
Lord, Pessler (2005)
They ask the question:
Can we characterize ”level-slope-curvature”?
They look at sign changes in the eigenvectors.
”Level” means no sign changes.
This is solved by Perron’s theorem.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 71: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/71.jpg)
Lord, Pessler (2005)
They ask the question:
Can we characterize ”level-slope-curvature”?
They look at sign changes in the eigenvectors.
”Level” means no sign changes.
This is solved by Perron’s theorem.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 72: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/72.jpg)
Lord, Pessler (2005)
They ask the question:
Can we characterize ”level-slope-curvature”?
They look at sign changes in the eigenvectors.
”Level” means no sign changes.
This is solved by Perron’s theorem.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 73: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/73.jpg)
Lord, Pessler (2005)
They ask the question:
Can we characterize ”level-slope-curvature”?
They look at sign changes in the eigenvectors.
”Level” means no sign changes.
This is solved by Perron’s theorem.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 74: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/74.jpg)
Lord, Pessler (2005)
Perron’s Theorem:
Let A be an N × N matrix, all of whose elements are strictlypositive. Then A has a positive eigenvalue of algebraic multiplicityequal to 1, which is strictly greater in modulus than all othereigenvalues of A. Furthermore, the unique (up to multiplication bya non-zero constant) associated eigenvector may be chosen so thatall its components are strictly positive.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 75: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/75.jpg)
Lord-Pessler (2005)
A square matrix A is said to be totally positive (TP) when forall p-uples n,m and p ≤ N, the matrix formed by theelements ani ,mj has nonnegative determinant.
If that condition is valid only for p ≤ k < N then A is calledTPk .
If those dets are strictly positive they are called strictly totallypositive (STP).
This is all classical stuff in matrix theory.
In 1937 Gantmacher and Kreın proved a theorem for STmatrices.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 76: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/76.jpg)
Lord-Pessler (2005)
A square matrix A is said to be totally positive (TP) when forall p-uples n,m and p ≤ N, the matrix formed by theelements ani ,mj has nonnegative determinant.
If that condition is valid only for p ≤ k < N then A is calledTPk .
If those dets are strictly positive they are called strictly totallypositive (STP).
This is all classical stuff in matrix theory.
In 1937 Gantmacher and Kreın proved a theorem for STmatrices.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 77: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/77.jpg)
Lord-Pessler (2005)
A square matrix A is said to be totally positive (TP) when forall p-uples n,m and p ≤ N, the matrix formed by theelements ani ,mj has nonnegative determinant.
If that condition is valid only for p ≤ k < N then A is calledTPk .
If those dets are strictly positive they are called strictly totallypositive (STP).
This is all classical stuff in matrix theory.
In 1937 Gantmacher and Kreın proved a theorem for STmatrices.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 78: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/78.jpg)
Lord-Pessler (2005)
A square matrix A is said to be totally positive (TP) when forall p-uples n,m and p ≤ N, the matrix formed by theelements ani ,mj has nonnegative determinant.
If that condition is valid only for p ≤ k < N then A is calledTPk .
If those dets are strictly positive they are called strictly totallypositive (STP).
This is all classical stuff in matrix theory.
In 1937 Gantmacher and Kreın proved a theorem for STmatrices.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 79: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/79.jpg)
Lord-Pessler (2005)
A square matrix A is said to be totally positive (TP) when forall p-uples n,m and p ≤ N, the matrix formed by theelements ani ,mj has nonnegative determinant.
If that condition is valid only for p ≤ k < N then A is calledTPk .
If those dets are strictly positive they are called strictly totallypositive (STP).
This is all classical stuff in matrix theory.
In 1937 Gantmacher and Kreın proved a theorem for STmatrices.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 80: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/80.jpg)
Lord-Pessler (2005)
Sign-change pattern in STPk matrices
Assume Σ is an N × N positive definite symmetric matrix (i.e. avalid covariance matrix) that is STPk . Then we haveλ1 > λ2 > ... > λk > λk+1 ≥ ...λN > 0, i.e. at least the first keigenvalues are simple. Moreover denoting the jth eigenvector byxj , we have that xj crosses the zero j − 1 times for j = 1, ..., k.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 81: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/81.jpg)
Lord-Pessler (2005)
Therefore STP3 ⇒ ”level-slope-curvature”.
Condition can be relaxed.
Definition: A matrix is called oscillatory if it is TPk and somepower of it is STPk .
Sufficient condition can be relaxed to being oscillatory oforder 3 (actually to having a power which is).
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 82: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/82.jpg)
Lord-Pessler (2005)
Therefore STP3 ⇒ ”level-slope-curvature”.
Condition can be relaxed.
Definition: A matrix is called oscillatory if it is TPk and somepower of it is STPk .
Sufficient condition can be relaxed to being oscillatory oforder 3 (actually to having a power which is).
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 83: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/83.jpg)
Lord-Pessler (2005)
Therefore STP3 ⇒ ”level-slope-curvature”.
Condition can be relaxed.
Definition: A matrix is called oscillatory if it is TPk and somepower of it is STPk .
Sufficient condition can be relaxed to being oscillatory oforder 3 (actually to having a power which is).
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 84: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/84.jpg)
Lord-Pessler (2005)
Therefore STP3 ⇒ ”level-slope-curvature”.
Condition can be relaxed.
Definition: A matrix is called oscillatory if it is TPk and somepower of it is STPk .
Sufficient condition can be relaxed to being oscillatory oforder 3 (actually to having a power which is).
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 85: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/85.jpg)
Lord-Pessler (2005). Schoenmakers-Coffey (2000)
The matrices in Forzani-T have constant diagonal elements
Actually that is not true in reality. The diagonals increase insize.
In modeling correlations Schoenmakers-Coffey proposed afamily of matrices that takes this fact into account.
Lord-Pessler show that these matrices are oscillatory.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 86: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/86.jpg)
Lord-Pessler (2005). Schoenmakers-Coffey (2000)
The matrices in Forzani-T have constant diagonal elements
Actually that is not true in reality. The diagonals increase insize.
In modeling correlations Schoenmakers-Coffey proposed afamily of matrices that takes this fact into account.
Lord-Pessler show that these matrices are oscillatory.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 87: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/87.jpg)
Lord-Pessler (2005). Schoenmakers-Coffey (2000)
The matrices in Forzani-T have constant diagonal elements
Actually that is not true in reality. The diagonals increase insize.
In modeling correlations Schoenmakers-Coffey proposed afamily of matrices that takes this fact into account.
Lord-Pessler show that these matrices are oscillatory.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 88: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/88.jpg)
Lord-Pessler (2005). Schoenmakers-Coffey (2000)
The matrices in Forzani-T have constant diagonal elements
Actually that is not true in reality. The diagonals increase insize.
In modeling correlations Schoenmakers-Coffey proposed afamily of matrices that takes this fact into account.
Lord-Pessler show that these matrices are oscillatory.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 89: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/89.jpg)
Lord-Pessler (2005). Conjecture
Sufficient conditions for a correlation matrix to satisfy”level-slope-curvature” are:
ρi ,j+1 ≤ ρi ,j for j ≥ i .
ρi ,j−1 ≤ ρi ,j for j ≤ i .
ρi ,i+j ≤ ρi+1,i+j+1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 90: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/90.jpg)
Lord-Pessler (2005). Conjecture
Sufficient conditions for a correlation matrix to satisfy”level-slope-curvature” are:
ρi ,j+1 ≤ ρi ,j for j ≥ i .
ρi ,j−1 ≤ ρi ,j for j ≤ i .
ρi ,i+j ≤ ρi+1,i+j+1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
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Lord-Pessler (2005). Conjecture
Sufficient conditions for a correlation matrix to satisfy”level-slope-curvature” are:
ρi ,j+1 ≤ ρi ,j for j ≥ i .
ρi ,j−1 ≤ ρi ,j for j ≤ i .
ρi ,i+j ≤ ρi+1,i+j+1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 92: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/92.jpg)
Lord-Pessler (2005). Conjecture
Sufficient conditions for a correlation matrix to satisfy”level-slope-curvature” are:
ρi ,j+1 ≤ ρi ,j for j ≥ i .
ρi ,j−1 ≤ ρi ,j for j ≤ i .
ρi ,i+j ≤ ρi+1,i+j+1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 93: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/93.jpg)
Lord-Pessler (2005). Conjecture
Sufficient conditions for a correlation matrix to satisfy”level-slope-curvature” are:
ρi ,j+1 ≤ ρi ,j for j ≥ i .
ρi ,j−1 ≤ ρi ,j for j ≤ i .
ρi ,i+j ≤ ρi+1,i+j+1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 94: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/94.jpg)
Extensions: Multi-Curve, Seasonality. Hindanov-T (2002)
Sometimes we need to mix up different markets.
Example: Oil
Not just timespreads, bflies but also cracks.
In that case we could price any structure in a muti-curvemarket.
We can model something like this by assuming a constantcorrelation intercurve and a different, also constant,correlation intracurve.
Depending on how high is the intercurve correlation we willget ”separation” vectors of different orders.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 95: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/95.jpg)
Extensions: Multi-Curve, Seasonality. Hindanov-T (2002)
Sometimes we need to mix up different markets.
Example: Oil
Not just timespreads, bflies but also cracks.
In that case we could price any structure in a muti-curvemarket.
We can model something like this by assuming a constantcorrelation intercurve and a different, also constant,correlation intracurve.
Depending on how high is the intercurve correlation we willget ”separation” vectors of different orders.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 96: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/96.jpg)
Extensions: Multi-Curve, Seasonality. Hindanov-T (2002)
Sometimes we need to mix up different markets.
Example: Oil
Not just timespreads, bflies but also cracks.
In that case we could price any structure in a muti-curvemarket.
We can model something like this by assuming a constantcorrelation intercurve and a different, also constant,correlation intracurve.
Depending on how high is the intercurve correlation we willget ”separation” vectors of different orders.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 97: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/97.jpg)
Extensions: Multi-Curve, Seasonality. Hindanov-T (2002)
Sometimes we need to mix up different markets.
Example: Oil
Not just timespreads, bflies but also cracks.
In that case we could price any structure in a muti-curvemarket.
We can model something like this by assuming a constantcorrelation intercurve and a different, also constant,correlation intracurve.
Depending on how high is the intercurve correlation we willget ”separation” vectors of different orders.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 98: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/98.jpg)
Extensions: Multi-Curve, Seasonality. Hindanov-T (2002)
Sometimes we need to mix up different markets.
Example: Oil
Not just timespreads, bflies but also cracks.
In that case we could price any structure in a muti-curvemarket.
We can model something like this by assuming a constantcorrelation intercurve and a different, also constant,correlation intracurve.
Depending on how high is the intercurve correlation we willget ”separation” vectors of different orders.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 99: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/99.jpg)
PCA of crude and heating oil together
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0
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0
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0
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0
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0
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1 2 3 4 5 6 7 8 9 10
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0
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Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 100: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/100.jpg)
Model for multiple curves
Let µ and λ be the intercurve and intracurve correlations.
Then the correlation matrix C is given by:(Cρ µCρ
µCρ Cρ
)where
1 ρ ρ2 ... ... ρn
ρ 1 ρ ... ... ρn−1
... ... ... ... ... ...
... ... ... ... ... ...ρn−1 ρn−2 ρn−3 ... 1 ρρn ρn−1 ρn−2 ... ρ 1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 101: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/101.jpg)
Model for multiple curves
Let µ and λ be the intercurve and intracurve correlations.
Then the correlation matrix C is given by:(Cρ µCρ
µCρ Cρ
)where
1 ρ ρ2 ... ... ρn
ρ 1 ρ ... ... ρn−1
... ... ... ... ... ...
... ... ... ... ... ...ρn−1 ρn−2 ρn−3 ... 1 ρρn ρn−1 ρn−2 ... ρ 1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 102: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/102.jpg)
Model for multiple curves
Let µ and λ be the intercurve and intracurve correlations.
Then the correlation matrix C is given by:
(Cρ µCρ
µCρ Cρ
)where
1 ρ ρ2 ... ... ρn
ρ 1 ρ ... ... ρn−1
... ... ... ... ... ...
... ... ... ... ... ...ρn−1 ρn−2 ρn−3 ... 1 ρρn ρn−1 ρn−2 ... ρ 1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 103: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/103.jpg)
Model for multiple curves
Let µ and λ be the intercurve and intracurve correlations.
Then the correlation matrix C is given by:(Cρ µCρ
µCρ Cρ
)
where
1 ρ ρ2 ... ... ρn
ρ 1 ρ ... ... ρn−1
... ... ... ... ... ...
... ... ... ... ... ...ρn−1 ρn−2 ρn−3 ... 1 ρρn ρn−1 ρn−2 ... ρ 1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 104: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/104.jpg)
Model for multiple curves
Let µ and λ be the intercurve and intracurve correlations.
Then the correlation matrix C is given by:(Cρ µCρ
µCρ Cρ
)where
1 ρ ρ2 ... ... ρn
ρ 1 ρ ... ... ρn−1
... ... ... ... ... ...
... ... ... ... ... ...ρn−1 ρn−2 ρn−3 ... 1 ρρn ρn−1 ρn−2 ... ρ 1
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 105: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/105.jpg)
Model for multiple curves
If v1, ..., vn are the eigenvectors of Cρ with eigenvalues λ1, ..., λn.
Then the eigenvectors of C are of the form (vk , vk) and (vk ,−vk)with 1 ≤ k ≤ n and
eigenvalues λk(1 + µ) and λk(1− µ).
So, depending on the size of the intercurve correlation we will getdifferent order of importance between common frequencies andseparating frequencies.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 106: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/106.jpg)
Model for multiple curves
If v1, ..., vn are the eigenvectors of Cρ with eigenvalues λ1, ..., λn.
Then the eigenvectors of C are of the form (vk , vk) and (vk ,−vk)with 1 ≤ k ≤ n and
eigenvalues λk(1 + µ) and λk(1− µ).
So, depending on the size of the intercurve correlation we will getdifferent order of importance between common frequencies andseparating frequencies.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 107: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/107.jpg)
Model for multiple curves
If v1, ..., vn are the eigenvectors of Cρ with eigenvalues λ1, ..., λn.
Then the eigenvectors of C are of the form (vk , vk) and (vk ,−vk)with 1 ≤ k ≤ n and
eigenvalues λk(1 + µ) and λk(1− µ).
So, depending on the size of the intercurve correlation we will getdifferent order of importance between common frequencies andseparating frequencies.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 108: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/108.jpg)
Model for multiple curves
If v1, ..., vn are the eigenvectors of Cρ with eigenvalues λ1, ..., λn.
Then the eigenvectors of C are of the form (vk , vk) and (vk ,−vk)with 1 ≤ k ≤ n and
eigenvalues λk(1 + µ) and λk(1− µ).
So, depending on the size of the intercurve correlation we will getdifferent order of importance between common frequencies andseparating frequencies.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 109: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/109.jpg)
Model for multiple curves
If v1, ..., vn are the eigenvectors of Cρ with eigenvalues λ1, ..., λn.
Then the eigenvectors of C are of the form (vk , vk) and (vk ,−vk)with 1 ≤ k ≤ n and
eigenvalues λk(1 + µ) and λk(1− µ).
So, depending on the size of the intercurve correlation we will getdifferent order of importance between common frequencies andseparating frequencies.
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling
![Page 110: Principal Components Analysis in Yield-Curve Modeling](https://reader035.vdocuments.us/reader035/viewer/2022071600/613d1963736caf36b7594d3b/html5/thumbnails/110.jpg)
Seasonality in the Eigenvalues (o=heating oil, x=crude)
0.94
0.95
0.96
0.97
0.98
0.99
1
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
00.0050.01
0.0150.02
0.0250.03
0.0350.04
0.0450.05
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
Carlos F. Tolmasky Principal Components Analysis in Yield-Curve Modeling