On the Time Scaling of Value at Risk with Trading

Portfolio risk measures such as Value at Risk is traditionally measured using a buy and hold assumption on the portfolio. In particular the 10-day market risk capital is commonly measured as the 1-day Value at Risk scaled by the square root of 10. While this scaling is convenient to obtain n-day Value at Risk numbers from 1-day Value at Risk it has some deficiencies. This includes the implicit assumption of a normal iid distribution as well as 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 i.e., that of assuming a buy and hold portfolio. Indeed, understanding the impact of an approximating buy and hold assumption is a key concern in validating the institutions Value at Risk model. Using stock data that covers the period from 6th April 2001 to 17th June 2009, including data for the recent crisis period, we compare the Value at Risk profiles for four different stylized daily trading methods in estimating 10-day Value at Risk. 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 Value at Risk. This effect is especially pronounced in case of buy volatility trading strategies where risk is amplified by trading into volatile instruments - yielding significantly higher risk than 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 that measures of risk should take into account traders style and portfolio level trading strategies if risk is to be accurately measured. This means that financial institutions need to validate their current Value at Risk model trading assumptions against the actual trade behavior