Prediction of Volatility and Value at Risk with Copulas for Portfolios of Commodities

Abstract

Value at Risk (VaR) is a popular measurement for valuing the risk
exposure. Correct estimates of VaR are essential in order to properly
be able to monitor the risk.
This thesis examines a copula approach for estimating VaR for
portfolios of commodities. The predictions are made from a semi-
parametric model with Monte Carlo methods. The underlying model
is constructed by choosing the best fit from different (E)GARCH-
models for margins and some of the most common Archimedean and
Elliptical copulas for the dependence.
None of the copulas in the scope gave a good fit to the data for the
dependence. However, the copula with the best fit was the t-copula,
which later was compared with the normal copula, the variance-covariance
method and the method with historical observations. The compari-
son was done with Kupiec’s test for correct number of VaR breaks and
Christoffersen’s test for independent breaks.
The results showed that none of the models in scope performed
well, although the copula approach followed the data better. Backtest-
ing suggested that the copula models overestimated the risk, resulting
in too few VaR-breaks that also were clustered.
The conclusion was that other copulas would be needed to appro-
priately model the dependence, or that more sophisticated modeling
methods in general should be used.

Popular Abstract

Knowing the risk one is exposed to is of great interest, but very difficult to
know. This is as true in everyday life as for investments. The situation gets
more complicated when multiple risks are present, due to the dependence
between them can be very complex. Using new statistical tools could be a
method to improve estimations.
To know what risk one is exposed to is crucial, especially when it comes to investments.Calculating the risk for a single asset can be done by estimating the variance of possible outcomes. If there is a wider variety of possible outcomes, the risk and variance is higher than for assets with concentrated outcomes.
In order to calculate the risk for a portfolio, two things need to be known: the risk for the separate assets and the dependence between them. The dependence describes how different assets are connected, in other words: if the assets tend to increase or decrease in price at the same time.
First of all, the risk for the separate assets are modeled with ARMA and GARCH
models which are fundamental models for time series. The models are trying to capture information from the history of the stock’s movements in order to be able to predict future movements. Secondly, the dependence has in this research been modeled with copulas (from Latin ”to connect”) - a mathematical tool that glues together the risks for the separate assets. The usage of copulas allows modeling of the risk for the assets separately, before
the dependence alone is estimated with the copula. Copulas themselves describe the
dependence and there are a wide range of them.
In the end, the method with copulas did not show better estimations than traditional methods for the commodity portfolio, but they did follow the test data much better.
Testing of more data sets and other copulas could lead to better results. The quest of being able to accurately take calculated risks will continue.

• author: Mörée, Felix
• supervisor: Nader Tajvidi
• organization: Mathematical Statistics
• course: FMS820 20162
• year: 2016
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Commodities, Copula, GARCH, VaR
• language: English
• id: 8957070
• date added to LUP: 2018-08-28 08:15:19
• date last changed: 2018-08-28 08:15:19