Systems and methods for compound risk factor sampling with integrated market and credit risk

A system for generating an integrated market and credit loss distribution for the purpose of calculating one or more risk measures associated with a portfolio of instruments by performing a simulation on a computer, wherein said computer comprises:

• at least one processor configured to execute at least a compound risk factor sampling module and a risk measure module;
• at least one memory coupled to the at least one processor;
• and
• at least one database;
• wherein the at least one processor is configured to:
• identify at least a first time horizon for said simulation;
• receive data identifying X, the data identifying X storable in the at least one database, wherein X is a vector of scalar-valued market risk factor processes, each market risk factor process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receiving data identifying Y, the data identifying Y storable in the at least one database, wherein Y is a vector of scalar-valued systemic credit driver processes, each systemic credit driver process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receiving data identifying Z, the data identifying Z storable in the at least one database, wherein Z is a vector of scalar-valued idiosyncratic credit risk factor processes indexed by counterparty name, each idiosyncratic credit risk factor process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receive data comprising one or more co-variance matrices that define the joint evolution of X and Y over said first time horizon, the data comprising the one or more co-variance matrices storable in the at least one database;
• identify a first parameter M wherein M>0, a second parameter S wherein S>1, and a third parameter I wherein I>0;
• wherein M defines a desired number of market risk factor samples, S defines a desired number of systemic credit driver samples for each of M market risk factor samples, and I defines a desired number of idiosyncratic credit risk factor samples for each of S systemic credit driver samples;
• partition said portfolio of instruments into a first sub-portfolio and a second sub-portfolio;
• generate MS conditional loss distributions for said first sub-portfolio by performing acts comprising:
• generating N simulated loss samples for said first sub-portfolio, wherein N =MSI, over said first time horizon by performing acts comprising:
• generating N scenarios for said first sub-portfolio, said N scenarios for said first sub-portfolio defined by N sets of X, Y, and Z values (Xm, Yms,Zmsi) for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I;
• wherein said act of generating N scenarios for said first sub-portfolio comprises:
• for each m from 1 to M, generating a sample, having index m, of a vector of normal random variables represented by Ξ;
• for each m from 1 to M and for each s from 1 to S, generating a random sample, having index ms, of ΔY from a conditional distribution of ΔY derived from the sample of the vector Ξ having index m and from at least one of the one or more co-variance matrices, ΔY being an increment of Y;
• for each m from 1 to M and for each s from 1 to S and for each i from 1 to I, independently generating a random sample, having index msi, of ΔZ, ΔZ being an increment of Z;
• and
• computing said N sets of X, Y, and Z values (Xm,Yms,Zmsi) for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I,
• wherein Xm is calculated as a value of X at the first time horizon based on a previous value of Xm, the at least one function associated with X, and the sample having index m of the vector Ξ,
• wherein Yms is calculated as a value of Y at the first time horizon based on a previous value of Yms, the at least one function associated with Y, and the random sample having index ms of ΔY,
• wherein Zmsi is calculated as a value of Z at the first time horizon based on a previous value of Zmsi, the at least one function associated with Z, and the random sample having index msi of ΔZ, and
• wherein if said first time horizon comprises exactly one time step, said previous value of Xm, Yms, and Zmsi is the start value associated with X, Y and Z respectively, for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I;
• and computing, by the at least one processor, said N simulated loss samples by simulating the first sub-portfolio over the N scenarios over said first time horizon;
• and
• for each (Xm,Yms) for all m from 1 to M and for all s from 1 to S of said N scenarios for said first sub-portfolio, calculate a conditional loss distribution FXmYms for said first time horizon to generate said MS conditional loss distributions for the first sub-portfolio;
• generate MS conditional loss distributions for said second sub-portfolio by performing acts comprising:
• identifying MS scenarios for said second sub-portfolio, wherein said MS scenarios for said second sub-portfolio comprise each of the MS scenarios for said first sub-portfolio defined by MS sets of X and Y values (Xm,Yms) for all m from 1 to M, and for all s from 1 to S obtained for the first sub-portfolio;
• and
• analytically deriving a conditional loss distribution FXmYms for each of said MS scenarios for said second sub-portfolio to generate said MS conditional loss distributions for said second sub-portfolio;
• for each m from 1 to M and each s from 1 to S, convolute the ms-th conditional loss distribution for said first sub-portfolio with the ms-th conditional loss distribution for said second sub-portfolio;
• wherein MS conditional loss distributions for said portfolio are produced;
• produce an unconditional loss distribution {circumflex over (F)} for said first time horizon as a mixture of the MS conditional loss distributions for said portfolio;
• and
• provide the unconditional loss distribution {circumflex over (F)} for said first time horizon for calculating one or more risk measures from said unconditional loss distribution {circumflex over (F)}, said one or more risk measures for use in evaluating risk associated with said portfolio.

A non-transitory computer-readable medium upon which a set of instructions are stored for execution on a computer, said computer comprising at least one processor and at least one memory, said non-transitory computer-readable medium comprising at least one module configured to perform a method for generating an integrated market and credit loss distribution for the purpose of calculating one or more risk measures associated with a portfolio of instruments by performing a simulation, said method comprising:

• identifying at least a first time horizon for said simulation;
• receiving data identifying X, wherein X is a vector of scalar-valued market risk factor processes, each market risk factor process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receiving data identifying Y, wherein Y is a vector of scalar-valued systemic credit driver processes, each systemic credit driver process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receiving data identifying Z, wherein Z is a vector of scalar-valued idiosyncratic credit risk factor processes indexed by counterparty name, each idiosyncratic credit risk factor process defined by a start value, at least one function representing a model, and zero or more parameters for the model;
• receiving data comprising one or more co-variance matrices that define the joint evolution of X and Y over said first time horizon;
• identifying a first parameter M wherein M>0, a second parameter S wherein S>1, and a third parameter I wherein 1>0;
• wherein M defines a desired number of market risk factor samples, S defines a desired number of systemic credit driver samples for each of M market risk factor samples, and I defines a desired number of idiosyncratic credit risk factor samples for each of S systemic credit driver samples;
• partitioning said portfolio of instruments into a first sub-portfolio and a second sub-portfolio;
• generating MS conditional loss distributions for said first sub-portfolio by performing acts comprising:
• generating N simulated loss samples for said first sub-portfolio, wherein N =MSI, over said first time horizon by performing acts comprising:
• generating N scenarios for said first sub-portfolio, said N scenarios for said first sub-portfolio defined by N sets of X, Y, and Z values (Xm,Yms,Zmsi) for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I;
• wherein said act of generating N scenarios for said first sub-portfolio comprises:
• for each m from 1 to M, generating a sample, having index m, of a vector of normal random variables represented by Ξ;
• for each m from 1 to M and for each s from 1 to S, generating a random sample, having index ms, of ΔY from a conditional distribution of ΔY derived from the sample of the vector Ξ having index m and from at least one of the one or more co-variance matrices, ΔY being an increment of Y;
• for each m from 1 to M and for each s from 1 to S and for each i from 1 to I, independently generating a random sample, having index msi, of ΔZ, ΔZ being an increment of Z;
• and
• computing said N sets of X, Y, and Z values (Xm,Yms,Zmsi) for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I,
• wherein Xm is calculated as a value of X at the first time horizon based on a previous value of Xm, the at least one function associated with X, and the sample having index m of the vector Ξ,
• wherein Yms is calculated as a value of Y at the first time horizon based on a previous value of Yms, the at least one function associated with Y, and the random sample having index ms of ΔY,
• wherein Zmsi is calculated as a value of Z at the first time horizon based on a previous value of Zmsi, the at least one function associated with Z, and the random sample having index msi of ΔZ, and
• wherein if said first time horizon comprises exactly one time step, said previous value of Xm, Yms, and Zmsi is the start value associated with X, Y and Z respectively, for all m from 1 to M, for all s from 1 to S, and for all i from 1 to I;
• and computing said N simulated loss samples by simulating the first sub-portfolio over the N scenarios over said first time horizon;
• and
• for each (Xm,Yms) for all m from 1 to M and for all s from 1 to S of said N scenarios for said first sub-portfolio, calculate a conditional loss distribution FXmYms for said first time horizon to generate said MS conditional loss distributions for the first sub-portfolio;
• generating MS conditional loss distributions for said second sub-portfolio by performing acts comprising:
• identifying MS scenarios for said second sub-portfolio, wherein said MS scenarios for said second sub-portfolio comprise each of the MS scenarios for said first sub-portfolio defined by MS sets of X and Y values (Xm,Yms) for all m from 1 to M, and for all s from 1 to S obtained for the first sub-portfolio;
• and
• analytically deriving a conditional loss distribution FXmYms for each of said MS scenarios for said second sub-portfolio to generate said MS conditional loss distributions for said second sub-portfolio;
• for each m from 1 to M and each s from 1 to S, convoluting the ms-th conditional loss distribution for said first sub-portfolio with the ms-th conditional loss distribution for said second sub-portfolio;
• wherein MS conditional loss distributions for said portfolio are produced;
• producing by the at least one processor an unconditional loss distribution {circumflex over (F)} for said first time horizon as a mixture of the MS conditional loss distributions for said portfolio;
• and
• providing the unconditional loss distribution {circumflex over (F)} for said first time horizon for calculating one or more risk measures from said unconditional loss distribution {circumflex over (F)}, said one or more risk measures for use in evaluating risk associated with said portfolio.