Financial Econometrics and Empirical Market Microstructure

Integrating Stop-Loss Limits into VaR Limits

The idea of accounting for the traders’ P&L over a specific period in the available exposure limit is developed by Beeck et al. (1999) for a single equity position in a discrete time. In their model, an annual risk limit is defined for the trader at the beginning of the year as a yearly VaR of the position. The annual limit is scaled down to a daily VaR limit with a square-root-of-time scalar, and then translated into a position limit under the assumption that stock returns are normally distributed with a zero or a non-zero mean. The authors consider three types of annual VaR limits: (1) a “fixed” limit, when the trader has the same risk budget and position limit every day, (2) a “loss-constraining” limit, when realized losses reduce the available annual VaR limit while profits can increase it back to its initial size, and (3) a “dynamic” limit, which differs from the stop-loss limit in that there is no cap on the recognition of realized profits in the annual VaR and the limit may increase above its size at the beginning of the year (see Table 1).

Based on a simulation study, Beeck et al. (1999) show that the stop-loss limit is the most conservative option, while the dynamic limit yields the highest profit potential at the expense of the largest P&L volatility. The results of the study also indicate that enforcing the risk limits makes the actual confidence level of the yearly VaR much lower than the one presumed in the VaR model.

Lobanov and Kainova (2005) extend the approach of Beeck et al. (1999) to include historical simulation for calculating VaR limits. They also propose an approach to adjusting position limits for model risk based on the results of the regulatory back-testing (Basel Committee on Banking Supervision 2006) and an alternative procedure of live-testing.

The methodology developed by Beeck et al. (1999) can be used for managing a linear position with a single risk factor to ensure a single exposure limit for a given risk limit. Extending this approach to a portfolio with multiple risk factors leads to non-unique solutions for the position limit, i. e. the composition of the portfolio.

Other drawbacks of this approach make its use problematic even for a single­factor position. Strafiberger (2002) observes that an annual VaR implies that positions are fixed for a 1-year horizon, which is unrealistic for proprietary trading. Scaling an annual limit down to the daily VaR using the square-root-of-time rule leads to an underestimation of the daily position limit and to severe underutilization of economic capital.[35] Alternatively, if scaling is done with a square-root-of-time remaining to the year-end, this leads to an uneven distribution of limits over the year.

Finally, the approach by Beeck et al. (1999) appears to be overly conservative, as the annual VaR will almost never be exceeded by losses if trading is halted after

Table 1 Deriving position limits for a single-factor position Limit type Annual risk limit {YL) Daily risk limit {DL) One-day position limit (V) Fixed YL = YLq = const YL DLt = ___ _ = const (for (i = 0) л/Т DL, = YL (for Ц ф 0) fil — ka(Jv і V, = (for ji = 0) DLt V, = (for /і ф 0) Lt. kaot Loss-constraining YL, = ■ t YLa, ^AF,_s+1 >0, 5 = 1 t t ylq + j2 ah_s+1 , J2 д^-*+і < ° 5=1 5=1 YL, DL, = —j= (for /і = 0) VT 77 — kJJ DL, = YL, —------------ a—= (for /і ф 0) jl T — kaO sJT DL, V, = -— (for /і = 0) ka@t V, - (for /і ф 0) Lt ka (Jt Dynamic t YL, = YL0 + J2 AF,_s+1 5 = 1 YL DL, = —ф (for /і = 0) 77 — k„u DL, = YL, —------------- a—= (for /і ф 0) ji T — kao s/T V, = ^- (for /1 = 0) ka@t V, = (for /і ф 0) Lt ka (Jt

Notation: YL Annual VaR limit, DL Daily VaR limit, V Daily position limit, /1 Average expected return, a Average standard deviation of returns, ka Quantile of the standardized normal distribution

the annual risk limit is depleted. In reality, however, the daily position limit is not always fully utilized by the trader, and the trading book is not necessarily closed till the year-end after the cumulative loss has surpassed the allocated annual risk limit.