On the time scaling of value-at-risk with trading

Portfolio risk measures such as value-at-risk (VaR) are traditionally measured using a buy-and-hold assumption on the portfolio. In particular, ten-day market-risk capital is commonly measured as the one-day VaR scaled by the square root of ten. While this scaling is convenient for obtaining n-day VaR numbers from one-day VaR, it has some deficiencies. This includes the implicit assumption of a normal independent and identical distribution and the implicit assumption of a buy-and-hold portfolio with no management intervention. In this paper we examine the potential effect of the second implicit assumption, ie, the assumption of a buy-and-hold portfolio. Indeed, understanding the impact of an approximating buy-and-hold assumption is a key concern in validating an institution's VaR model. Using stock data that covers the period from April 6, 2001 to June 17, 2009, including data from the recent financial crisis period, we compare the VaR profiles of four different stylized daily trading methods for estimating ten-day VaR. The trading methods are the convex, concave and volatility-based trading methods. In our analysis we find that the trading strategy may have a substantial impact on the accuracy of the square-root-of-time rule in scaling VaR. This effect is especially pronounced in the case of buy-volatility trading strategies, where risk is amplified by trading into volatile instruments, yielding significantly higher risk than would be the case under a buy-and-hold assumption or a square-root-of-time rule. On the other hand, risk reduction versus buy-and-hold for strategies that trade into low-volatility instruments may be small. Our findings strongly support the argument that measures of risk should take traders' styles and portfolio-level trading strategies into account if risk is to be accurately measured. This means that financial institutions need to validate their current VaR model trading assumptions against actual trade behavior.