To VaR, or Not to VaR, That is the Question

We consider economic obstacles that limit the reliability and accuracy of value-at-risk (VaR). Investors who manage large market transactions should take into account the impact of the randomness of large trade volumes on predictions of price probability and VaR assessments. We introduce market-based probabilities of price and return that depend on the randomness of market trade values and volumes. Contrary to them, the conventional frequency-based price probability describes the case of constant trade volumes. We derive the dependence of market-based price volatility on the volatilities and correlation of trade values and volumes. In the coming years, that will limit the accuracy of price probability predictions to Gaussian approximations, and even the forecasts of market-based price volatility will be inaccurate and highly uncertain.

💡 Research Summary

The paper “To VaR, or Not to VaR, That is the Question” investigates a fundamental source of error in conventional Value‑at‑Risk (VaR) calculations: the randomness of trade volumes and trade values that accompany large market transactions. Traditional VaR models assume that price changes follow a stationary distribution estimated from historical price movements, implicitly treating trade volume as constant. In reality, institutional investors, hedge funds, and sovereign wealth funds often execute orders whose size can dramatically shift market depth, causing price impact that is not captured by a simple frequency‑based probability model.

Key Contributions

1. Two Probability Frameworks – The authors distinguish between (a) a frequency‑based probability derived from the empirical distribution of past price changes under the constant‑volume assumption, and (b) a market‑based probability that explicitly models trade volume (V_t) and trade value (Q_t) as stochastic processes. By treating price change (\Delta P_t) as a function of the product (Q_t V_t), they embed liquidity‑driven price impact directly into the probabilistic description.

2. Derivation of Market‑Based Price Volatility – Assuming that (Q_t) and (V_t) are jointly normally distributed with means (\mu_Q, \mu_V), standard deviations (\sigma_Q, \sigma_V), and correlation (\rho_{QV}), the paper derives a closed‑form expression for the variance of price changes:
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