Value at Risk and Self-Similarity

Abstract

The concept of Value at Risk measures the "risk" of a portfolio and is a statement of the following form: With probability q the potential loss will not exceed the Value at Risk figure. It is in widespread use within the banking industry.It is common to derive the Value at Risk figure of d days from the one of one-day by multiplying with √ d. Obviously, this formula is right, if the changes in the value of the portfolio are normally distributed with stationary and independent increments. However, this formula is no longer valid, if arbitrary distributions are assumed. For example, if the distributions of the changes in the value of the portfolio are self-similar with Hurst coefficient H, the Value at Risk figure of one-day has to be multiplied by d H in order to get the Value at Risk figure for d days.This paper investigates to which extent this formula (of multiplying by √ d) can be applied for all financial time series. Moreover, it will be studied how much the risk can be over-or underestimated, if the above formula is used. The scaling law coefficient and the Hurst exponent are calculated for various financial time series for several quantiles.JEL classification: C13, C14, G10, G21.

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