Disentangling the Measurement of Risk and Uncertainty

In this paper, we study whether it is possible to measure Knightian uncertainty and disentangle it from the conventional risk measure in the location-scale family of normal distributions. First, we establish and prove the impossibility theorems that no quantitative uncertainty measure would be mutually agreeable among investors with general risk-and-uncertainty-averse preferences. Then, we show it is possible to partially rank the uncertainty of the location-scale family of normal distributions by the first, second, fourth, and sixth moments. Lastly, we propose two ways to assess the stock uncertainty, the ranges of location and scale of the family of normal distributions computed from the 1st, 2nd, 4th and 6th moments, and a composite uncertainty measure proposed by Izhakian (2020), but with necessary and critical modification. We assess the uncertainty of stock market indices, S&P 500 in the U.S and CSI 300 in China, and show that the uncertainty and risk measures are indeed distinct.