We study a source of bias in value-at-risk estimates that has not previously been recognized. Because value-at-risk estimates are based on past data, a trader will often have a good understanding of the errors in the value-at-risk estimate, and it will be possible for her to choose portfolios for which she knows that the value -at-risk is less than the "true" value at risk. Thus, The trader will be able to take on more market risk than risk limits based on value-at-risk permit. Biases can also arise if she doesn't have a good understanding of the errors, but uses the estimated covariance matrix to achieve certain portfolio objectives. We assess the magnitude of these biases for three different assumptions about the motivations and behavoir of the trader and find that in all cases, value-at-risk estimates are systematically downward biased. In some circumstances the biases can be very large. Our study of the distributions of the biases also suggests a way to adjust the estimates to "correct" the biases.

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