To study the effect of Knightian uncertainty – as opposed to statistical estimation error – in the evaluation of value‐at‐risk (VaR) of financial investments. To develop methods for augmenting existing VaR estimates to account for Knightian uncertainty.
The value at risk of a financial investment is assessed as the quantile of an estimated probability distribution of the returns. Estimating a VaR from historical data entails two distinct sorts of uncertainty: probabilistic uncertainty in the estimation of a probability density function (PDF) from historical data, and non‐probabilistic Knightian info‐gaps in the future size and shape of the lower tail of the PDF. A PDF is estimated from historical data, while a VaR is used to predict future risk. Knightian uncertainty arises from the structural changes, surprises, etc., which occur in the future and therefore are not manifested in historical data. This paper concentrates entirely on Knightian uncertainty and does not consider the statistical problem of estimating a PDF. Info‐gap decision theory is used to study the robustness of a VaR to Knightian uncertainty in the distribution.
It is shown that VaRs, based on estimated PDFs, have no robustness to Knightian errors in the PDF. An info‐gap safety factor is derived that multiplies the estimated VaR in order to obtain a revised VaR with specified robustness to Knightian error in the PDF. A robustness premium is defined as a supplement to the incremental VaR for comparing portfolios.
The revised VaR and incremental VaR augment existing tools for evaluating financial risk.
Info‐gap theory, which underlies this paper, is a non‐probabilistic quantification of uncertainty that is very suitable for representing Knightian uncertainty. This enables one to assess the robustness to future surprises, as distinct from existing statistical techniques for assessing estimation error resulting from randomness of historical data.
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