Market risk budgeting with value-at-risk-limits for linear positions

Value-at-Risk limits shall be used for allocating capital within financial institutions. Commonly the management decides on an annual basis about the amount of capital a stock trader can lose with a prespecified probability within the year. Since the holding period for trading business is much shorter (we assume a day), the annual limit is transformed into a daily VaR limit by the ‘square root of time’ rule. In particular, we explore three different Value-at-Risk-Limits. The first limit is called ‘fixed limit’. The trader has the same limit each day of the year. The second limit is called ‘stop loss limit’ because realized losses during the year reduce trader's daily limit. Third, the ‘dynamic limit’ differs from the ‘stop loss limit’ in that realized profits can increase the daily limit. We show in a simulation study for real German stock returns from 1974–1995 that the ‘fixed limit’ outperforms the ‘stop loss limit’. It yields slightly higher profits combined with a smaller standard deviation. Under the ‘dynamic limit’ a trader can earn higher annual profits, but the standard deviation and the probability of losses increase as well. The frequency of violations of the annual VaR limit is much smaller than the expected probability. This shows that the application of the ‘square root of time’ rule leads to an overestimation of risk on an annual basis.