Kupiec VaR Backtesting and the Role of Holding Period

Summary

The document raises a question about the Kupiec unconditional coverage test for value at risk (VaR). It describes counting exceptions over a backtest of a given number of days and comparing the observed count with a binomial distribution parameterized by the VaR tail probability. The question is whether that same exception-count distribution applies to VaR estimates with different holding periods, such as one-day and five-day horizons.

No answer or adjustment is supplied. The note highlights that the basic exception-count formulation does not explicitly include a holding-period parameter, but it does not address how overlapping horizon returns, serial dependence, or the construction of multi-day VaR might affect the test’s assumptions. It is therefore a useful prompt for examining test design, rather than a complete method for incorporating horizon. A backtest should make its horizon and exception definition explicit and check whether the observations satisfy the assumptions used by the chosen test.

Key ideas

• The Kupiec unconditional coverage test compares the observed VaR exception count with a binomial model.
• The document asks whether the test changes when VaR is measured over different holding periods.
• It provides no proposed formula or resolution for incorporating the horizon.
• Overlapping returns and dependence may matter when assessing whether exception observations fit the test assumptions.

Tags

# Kupiec Test Backtesting VaR


# Kupiec Test Backtesting VaR












I am currently analyzing the Kupiec test used for backtesting $VaR$. Suppose that I backtest a $VaR$ system for $n$ days (for example 250), with a confidence interval of $1-\alpha$ (for example a $1-\alpha =0.99$, thus $\alpha = 0.01$). According to Kupiec test (and using the $VaR$ definition) we know that the probability of having $x$ exceedances is given by a Binomial distribution with parameters $n$ and $\alpha$.

In this formulation, however, the holding period of the VaR does not appear as a parameter. In other words, if I backtest a 1-day $VaR$ or a 5-day $VaR$ with same $n$ and $\alpha$, the probability of the exceedances is always given by the same binomial distribution.

Is there a way to introduce the VaR holding period as a parameter of the Kupiec test?

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