Value at Risk Models

Definitions

  • Value-at-risk (VaR) is a battle-tested measure of risk first developed by J.P. Morgan.

  • VaR measures the maximum loss an investment or portfolio is likely to incur, over a specified period, with a certain level of confidence. It is expressed as a monetary value and a corresponding confidence level (e.g., 95% VaR).

Concrete

  • VaR is an especially valuable measure of risk in the context of leveraged firms and insurance-like applications as these types of mechanisms are highly liable to insolvency in the case of black-swan events.

  • Concrete is unlikely to see a 95% lower extremal outcome over a prolonged period of time; however, VaR at the 99% and 95% confidence levels are valuable cases to monitor as they are most likely to lead to insolvency-potential scenarios.

Model

  • Formally, given a t-day time interval, confidence level α : 0 < α < 1, and portfolio X, the t-day VaRx at confidence level α returns a price value such that the probability of a loss greater than VaRx is at most 1 − α.

  • VaR example

  • Because VaR is a function of confidence level, it is non-convex and discontinuous for discrete distributions (most time series). As a result, VaR may penalize diversification: given two financial positions and their anticipated (random) future returns X and Y it might be that VaR(X + Y )< VaR(X) + VaR(Y).

  • Furthermore, VaR measures potential losses up to a certain probability but does not provide any insight into the magnitude of losses beyond this point (commonly known as the “VaR breach”).

  • For example, a 14-day VaR at confidence interval 5% may be only $X, but the average loss size beyond the VaR breach point could be much larger - VaR gives no insight into the magnitude of tail risk. This is addressed by conditional VaR (CVaR).

  • CVaR can be understood as “filling in the gap” left by VaR. That is, while VaR predicts the minimum loss beyond a certain confidence interval, it does not provide insight into the magnitude of all potential losses beyond the minimum loss incurred at confidence level α.

  • Simply put, CVaR adds up all the values beyond the probability of the corresponding VaR. Imagine adding all the values in the red distribution diagram above.

  • Formally, the CVaR of X with confidence level 0 < α < 1 is the mean of the generalized α-tail distribution:

Further reading

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