Multi-factor approximation : An analysis and comparison ofMichael Pykhtin's paper “Multifactor adjustment”

Sammanfattning: The need to account for potential losses in rare events is of utmost importance for corporations operating in the financial sector. Common measurements for potential losses are Value at Risk and Expected Shortfall. These are measures of which the computation typically requires immense Monte Carlo simulations. Another measurement is the Advanced Internal Ratings-Based model that estimates the capital requirement but solely accounts for a single risk factor. As an alternative to the commonly used time-consuming credit risk methods and measurements, Michael Pykhtin presents methods to approximate the Value at Risk and Expected Shortfall in his paper Multi-factor adjustment from 2004. The thesis’ main focus is an elucidation and investigation of the approximation methods that Pykhtin presents. Pykhtin’s approximations are thereafter implemented along with the Monte Carlo methods that is used as a benchmark. A recreation of the results Pykhtin presents is completed with satisfactory, strongly matching results, which is a confident verification that the methods have been implemented in correspondence with the article. The methods are also applied on a small and large synthetic Nordea data set to test the methods on alternative data. Due to the size complexity of the large data set, it cannot be computed in its original form. Thus, a clustering algorithm is used to eliminate this limitation while still keeping characteristics of the original data set. Executing the methods on the synthetic Nordea data sets, the Value at Risk and Expected Shortfall results have a larger discrepancy between approximated and Monte Carlo simulated results. The noted differences are probably due to increased borrower exposures, and portfolio structures not being compatible with Pykhtin’s approximation. The purpose of clustering the small data set is to test the effect on the accuracy and understand the clustering algorithm’s impact before implementing it on the large data set. Clustering the small data set caused deviant results compared to the original small data set, which is expected. The clustered large data set’s approximation results had a lower discrepancy to the benchmark Monte Carlo simulated results in comparison to the small data. The increased portfolio size creates a granularity decreasing the outcome’s variance for both the MC methods, and the approximation methods, hence the low discrepancy. Overall, Pykhtin’s approximations’ accuracy and execution time are relatively good for the experiments. It is however very challenging for the approximate methods to handle large portfolios, considering the issues that the portfolio run into at just a couple of thousand borrowers. Lastly, a comparison between the Advanced Internal Ratings-Based model, and modified Value at Risks and Expected Shortfalls are made. Calculating the capital requirement for the Advanced Internal Ratings-Based model, the absence of complex concentration risk consideration is clearly illustrated by the significantly lower results compared to either of the other methods. In addition, an increasing difference can be identified between the capital requirements obtained from Pykhtin’s approximation and the Monte Carlo method. This emphasizes the importance of utilizing complex methods to fully grasp the inherent portfolio risks.