Historical Simulation

1)Introduction

These years have been characterized by significant instabilities in financial market worldwide. This has led to increasingly more concern about risk management. In financial mathematics and financial risk management, Value at Risk (VaR) is a widely used risk measure of the risk of loss on a specific portfolio of financial assets. There are three main methods of calculating VaR: Delta-Normal VaR, Historical VaR, and Monte Carlo VaR. This report focuses on the historical VaR (Historical Simulation Model), which unlike the other two methods, is a non-parametric approach that uses an Independent and Identically Distributed Random Variable.

Historical Simulation is the procedure for predicting value at risk by “simulating” or constructing the cumulative distribution function of asset returns over time. It does not require any statistical assumption beyond stationary of the distribution of returns or, in particular, their volatility. Put simply, in this method, the distribution of profits and losses is constructed by taking the current portfolio, and subjecting it to the actual changes in the key factors experienced during each of the last α periods. Once the hypothetical mark-to-market profit or loss for each of the last α periods have been calculated, the distribution of profits and losses and the value-at-risk can then be determined.[pic 1]

Suppose we observe data from day 1 to day t, and      is the return of portfolio on day t, then we get a series of return                             . The value of risk with coverage rate “p” is calculated as the (100*p)% of the sequence of past portfolio returns[pic 2]

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Historical VaR is the most popular approach for calculating value at risk (VaR) and expected shortfall (ES) for market risk, according to a survey conducted by Perignon and Smith (2006). The results showed that 73% of banks among 60 US, Canadian and large international banks over 1996-2005 have reported that their VaR methodology used was historical simulation.