Value at Risk

An overviewValue at Risk1

Value at RiskThe Need for VaR Definition of VaRUses of VaR VaR Methods VaR - Historical Simulation Changes since the Financial Crises of 2008Strengths and WeaknessSummary Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.2

Value at RiskThe Need for VaR Different Asset Classes use their own measuresFixed Income DurationInterest Rates DV01FX Currency positionCommodity Number of contractsEquities Number of sharesUsing these to compare risk in these portfolios is like comparing apples and oranges. An investor or owner needs a simple measure that can be used in a consistent way to compare risk between these portfolios. And with the ability to aggregate risk appropriately Value at Risk is that risk measure Definition of VaRVAR is a measure of losses due to normal market movementsValue at Risk (VAR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence. VAR is the answer to the question How much could we lose over a specified holding period with a defined probability.There are three key elements of VAR - a specified level of loss in value, - a fixed time period over which risk is assessed - a confidence level.Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.4So if a portfolio has a VaR of $20 millionWe need to know the confidence level used to calculateWe need to know the holding period (time horizon)For example there is only a 5% chance that our company's losses will exceed $20M over the next five days;. This is the classic VaR measure. VaR does not provide any information about how bad the losses might be if the VaR level is exceeded. 09/26/10 Last Updated: September 26, 2010

Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.5Suppose that it is determined that a $100 million portfolio could potentially lose $20 million (or more) once every 20 trading days.The VaR of this portfolio equals $20 million with a 95% level of confidence over the coming trading day; 19 out of 20 trading days (95% of the time), losses are less than $20 millionAt the 95% confidence level, VaR represents he border of the 5% left tail of the normal distribution, also known as the fifth percentile or .05 quantile of the normal distribution

Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.6Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.7

This diagram shows that: 95% of the time, the portfolios value remains above $80 million 5% of the time, the portfolios value falls to $80 million or lessThe VaR of this portfolio is therefore $100 million - $80 million = $20 millionOptions, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.8FORMALIZATION AND APPLICATIONSVaR can simply be expressed asOptions, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 200518.9