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Calculating Value-at-Risk

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  • William Fallon

Abstract

The market risk of a portfolio refers to the possibility of financial loss due to the joint movement of systematic economic variables such as interest and exchange rates. Quantifying market risk is important to regulators in assessing solvency and to risk managers in allocating scarce capital. Moreover, market risk is often the central risk faced by financial institutions. The standard method for measuring market risk places a conservative, one-sided confidence interval on portfolio losses for short forecast horizons. This bound on losses is often called capital-at-risk or value-at-risk (VAR), for obvious reasons. Calculating the VAR or any similar risk metric requires a probability distribution of changes in portfolio value. In most risk management models, this distribution is derived by placing assumptions on (1) how the portfolio function is approximated, and (2) how the state variables are modeled. Using this framework, we first review four methods for measuring market risk. We then develop and illustrate two new market risk measurement models that use a second-order approximation to the portfolio function and a multivariate GARCH(l,1) model for the state variables. We show that when changes in the state variables are modeled as conditional or unconditional multivariate normal, first-order approximations to the portfolio function yield a univariate normal for the change in portfolio value while second-order approximations yield a quadratic normal. Using equity return data and a hypothetical portfolio of options, we then evaluate the performance of all six models by examining how accurately each calculates the VAR on an out-of-sample basis. We find that our most general model is superior to all others in predicting the VAR. In additional empirical tests focusing on the error contribution of each of the two model components, we find that the superior performance of our most general model is largely attributable to the use of the second-order approximation, and that the first-order approximations favored by practitioners perform quite poorly. Empirical evidence on the modeling of the state variables is mixed but supports usage of a model which reflects non-linearities in state variable return distributions. This paper was presented at the Financial Institutions Center's October 1996 conference on "

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  • William Fallon, 1996. "Calculating Value-at-Risk," Center for Financial Institutions Working Papers 96-49, Wharton School Center for Financial Institutions, University of Pennsylvania.
    • Handle: RePEc:wop:pennin:96-49
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    1. Lampros Kalyvas & Athanasios Sfetsos, 2006. "Does The Application Of Innovative Internal Models Diminish Regulatory Capital?," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 217-226.
    2. Matthew Pritsker, 1996. "Evaluating Value-at-Risk Methodologies: Accuracy versus Computational Time," Center for Financial Institutions Working Papers 96-48, Wharton School Center for Financial Institutions, University of Pennsylvania.
    3. Matthew Pritsker, 1997. "Evaluating Value at Risk Methodologies: Accuracy versus Computational Time," Journal of Financial Services Research, Springer;Western Finance Association, vol. 12(2), pages 201-242, October.
    4. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2022. "Estimating risks of option books using neural-SDE market models," Papers 2202.07148, arXiv.org.
    5. Charles-Olivier Amédée-Manesme & Fabrice Barthélémy, 2022. "Proper use of the modified Sharpe ratios in performance measurement: rearranging the Cornish Fisher expansion," Annals of Operations Research, Springer, vol. 313(2), pages 691-712, June.
    6. Anthony M. Santomero, 1997. "Commercial Bank Risk Management: An Analysis of the Process," Center for Financial Institutions Working Papers 95-11, Wharton School Center for Financial Institutions, University of Pennsylvania.
    7. Agata Gemzik-Salwach, 2012. "The Use Of A Value At Risk Measure For The Analysis Of Bank Interest Margins," "e-Finanse", University of Information Technology and Management, Institute of Financial Research and Analysis, vol. 8(4), pages 15-29, February.

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