Method and system for covariance matrix estimation

A computing apparatus for estimating a covariance with respect to a plurality of bonds, the computing apparatus comprising:

• a processor;
• a display device;
• a memory;
• and
• a communication interface coupled to each of the processor, the display device, and the memory,
• wherein the processor is configured to:
• receive, via the communication interface, first information that relates to historical bond market data;
• use a first algorithm based on an Auto-Regressive-Moving-Average (ARMA) model to calculate a set of regression errors based on a first period of the historical bond market data;
• use a second algorithm based on a logarithmic Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model to calculate an estimated volatility vector based on the first period of the historical bond market data;
• use the calculated set of regression errors and the calculated volatility vector to estimate a time-varying covariance matrix of the set of regression errors with respect to the first period of the historical bond market data;
• combine the estimated time-covariance matrix of the ARMA model regression errors and the calculated volatility vector to estimate a time-varying covariance matrix of the bond returns for the first period of the historical bond market data;
• project the estimated time-varying covariance matrix of the bond returns onto another estimated time-varying covariance matrix of the bond returns for a second period of the historical bond market data, wherein the second period is at least six times longer than the first period, wherein the second period overlaps the first period, and wherein the second period is at most two years;
• use the estimated time-varying covariance matrix of the bond returns that has been projected to calculate a set of predicted hedge ratios;
• and
• display, on a user interface of the display device, a result of the calculating of the predicted hedge ratios that includes a respective graph of a fractional change in a corresponding return of each of a first instrument against which a hedge is to be made and a plurality of candidate instruments to be used for hedging as a function of a prediction date,
• wherein the processor is further configured to use the first algorithm based on the ARMA model to calculate the set of regression errors based on the first period of the historical bond market data by:
• using an ordinary least squares technique with respect to a first subset of the bond market data to estimate respective values of at least two parameters of the first algorithm;
• training the ARMA model based on a second subset of the bond market data that corresponds to a predetermined training interval;
• adjusting the estimated respective values of the at least two parameters based on a result of the training;
• receiving, via the communication interface, third information that relates to bond market data corresponding to a next predetermined time interval that occurs after the predetermined training interval;
• retraining the ARMA model based on the third information;
• and
• readjusting the estimated respective values of the at least two parameters based on a result of the retraining.

A non-transitory computer readable storage medium storing instructions for estimating a covariance with respect to a plurality of bonds, the storage medium comprising executable code which, when executed by at least one processor, causes the at least one processor to:

• receive first information that relates to historical bond market data;
• use a first algorithm based on an Auto-Regressive-Moving-Average (ARMA) model to calculate a set of regression errors based on a first period of the historical bond market data;
• use a second algorithm based on a logarithmic Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model to calculate an estimated volatility vector based on the first period of the historical bond market data;
• use the calculated set of regression errors and the calculated volatility vector to estimate a time-varying covariance matrix of a set of regression errors with respect to the first period of the historical bond market data;
• combine the estimated time-covariance matrix of the ARMA model regression errors and the calculated volatility vector to estimate a time-varying covariance matrix of the bond returns for the first period of the historical bond market data;
• project the estimated time-varying covariance matrix of the bond returns onto another estimated time-varying covariance matrix of the bond returns for a second period of the historical bond market data, wherein the second period is at least six times longer than the first period, wherein the second period overlaps the first period, and wherein the second period is at most two years;
• use the estimated time-varying covariance matrix of the bond returns that has been projected to calculate a set of predicted hedge ratios;
• and
• display, on a user interface, a result of the calculating of the predicted hedge ratios that includes a respective graph of a fractional change in a corresponding return of each of a first instrument against which a hedge is to be made and a plurality of candidate instruments to be used for hedging as a function of a prediction date,
• wherein the executable code causes the at least one processor to perform the use of the first algorithm based on the ARMA model to calculate the set of regression errors based on the first period of the historical bond market data by:
• using an ordinary least squares technique with respect to a first subset of the bond market data to estimate respective values of at least two parameters of the first algorithm;
• training the ARMA model based on a second subset of the bond market data that corresponds to a predetermined training interval;
• adjusting the estimated respective values of the at least two parameters based on a result of the training;
• receiving third information that relates to bond market data corresponding to a next predetermined time interval that occurs after the predetermined training interval;
• retraining the ARMA model based on the third information;
• and
• readjusting the estimated respective values of the at least two parameters based on a result of the retraining.