# The Acceleration of Least Squares Monte Carlo in Risk Management

Lu Xiong*

Department of Mathematical Sciences, Middle Tennessee State University, Faculty of Actuarial Science, Faculty of Computational Science, USA

Received Date: January 29, 2021;  Published Date: February 09, 2021

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#### The Application of LSMC in Risk Management

Solvency Capital Requirement (SCR) of Solvency II requires the computation of the economic capital, the minimum capital giving the insurance company a 99.5% survival probability over a oneyear horizon via a full probability distribution forecast [9,10].

The SCR at level α=99.5% can be computed as    #### Distributed Regression for LSMC Speedup

When it comes to multi-factor risks modeling approximation, the multi-dimensional polynomial would be extremely complicated. This would make the regression slow or not possible to finish within reasonable time.

To over the computational complexity of multi-risk factor LSMC, we propose distributed regression for LSMC. The idea of distributed regression is fairly simple: instead of running the regression on one computer, we distribute the regression task to multiple computers (usually using cloud computers), then average the regressed coefficients to get the final regression equation. In this way the computing time can be significantly reduced. We can mathematically prove this simple idea can actually obtain the optimal regression results .

There are several advantages of distributed regression: First, the computing time for the traditional least square regression is O(n3), where n is number of observations in data. While for distributed regression, it’s O(n3/m2), where m is the number of distributed computers. If we distributed the regression task to 10 computers, we could reduce to computing time to 1% of the original regression, 50 computers to 0.04%. Second, distributed regression can protect the data privacy, because very little or no communication is required when computing from distributed computers. Therefore, almost no data exchanged happened between different data platforms. If we have policy data stored in different platforms and we don’t want to share the data across, we can use distributed regression to obtain the regression coefficients from each platform then average the coefficients to get the total regression equation.

We propose the following distribute regression algorithm for LSMC: There are several advantages using distributed regression to accelerate the LSMC. 1) The current parallel algorithms for LSMC require the parallel computing of the big matrix inverse, while using distributed regression we only need compute the small matrix inversion for each chunk of data. 2) When comes to multi-risk modeling, the amount of the outer scenarios would be huge that no single computer can handle it. For a N risk-factor problem, it will require 10000N outer scenarios if we simulate 10,000 outer scenarios for each risk-factor. If we use distributed regression, each computer only needs processes a smaller chunk of data assigned. 3) This divide-and-conquer type distributed learning method can also be applied to speed up other algorithms like clustering, treebased method, deep learning etc. 4) Easy to be scaled on distributed framework like Map-reduce, or Spark .

None.

#### Conflict of Interest

No conflict of interest.

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