Risk Budgeting with Value at Risk Limits

Abstract

Our analysis focuses on the risk budgeting process for banks using value at risk limits. In this context, we investigate three major practical problems: a) differences in time horizons between the bank’s total risk budget and the trading divisions’ activities; b) adjustment for accumulated profit and losses to risk budgets, and c) incorporation of correlations between assets into the risk budgeting process. To analyze these practical problems, we use Monte Carlo simulation. Thereby, it can be shown that differences in time horizons among risk budgets and trading units can be adjusted by the square root of time rule. Three types of limits are proposed for the adjustment of accumulated profit and losses: the fixed, stop loss and dynamic limits. While the two latter restrict the maximum loss to the ex ante specified limit and show a symmetric profit and loss distribution, the dynamic limit’s distribution is skewed to the right. We further illustrate that the average usage of total risk capital is only 31.45 % for a trading division with thirty independently deciding traders. This shortfall is due to diversification effects. This setting is compared with a benchmark model in which total risk capital is always used at the full capacity of 100 %. The comparison shows that the average profit in the former model is only 33.13 % of the generated profit in the benchmark model. The results may have interesting organizational implications on the banking sector.