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 loose 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 quot;square root of timequot; rule. In particular, we explore three different Value-at-Risk-Limits. The first limit is called quot;fixed limitquot; The trader has the same limit each day the year. The second limit is called quot;stop loss limitquot; because realized losses during the year reduce trader's daily limit. Third, the quot;dynamic limitquot; differs from the quot;stop loss limitquot; 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 quot;fixed limitquot; outperforms the quot;stop loss limitquot;. It yields slightly higher profits combined with a smaller standard deviation. Under the quot;dynamic limitquot; 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 quot;square root of timequot; rule leads to an overestimation of risk on an annual basis