Measuring Risk in Multimodal Rolling Return Distributions

Summary

The document defines rolling returns from stock prices using a chosen observation window and shift, then notes that the resulting return distribution may have multiple peaks. It asks how volatility should represent risk when returns are multimodal. The candidate approaches include reporting separate volatility estimates by mode, combining them with probability weights, fitting one distribution across all observations, or using another measure.

No preferred method, analysis, or empirical evidence is supplied. The question highlights that a single dispersion statistic may obscure distinct return regimes, while assigning separate mode-specific risks requires a way to identify modes and estimate their probabilities. The choice also depends on the intended risk task; the document does not specify a loss function, horizon, or decision context that would determine an appropriate measure.

Key ideas

• Rolling returns depend on both the return window and the shift between observations.
• A rolling return distribution can have multiple peaks rather than one dominant mode.
• A single volatility estimate may conceal differences among regimes in a multimodal distribution.
• Mode-specific or probability-weighted estimates require a method for identifying and characterizing the modes.
• The document poses the measurement question but does not recommend a particular risk statistic.

Tags

# Volatility of multimodal distribution of returns


# Volatility of multimodal distribution of returns












Take $x_1, x_2, \ldots, x_T$ to be the price of a stock, indexed by $t=1, 2, \ldots, T$. Define rate of return at time $t>W$ for a window size of $W$ to be $$r_t = \frac{x_t - x_{t-W}}{x_{t-W}}$$ Rolling returns for a shift $\delta$ are thus given by the series $r_{t}, r_{t+\delta}, r_{t+2\delta}, \ldots, r_T$ (for $t > W$). As shown below, the distribution of rolling returns could be multimodal, meaning that there may be more than one peak in the distribution.

What is the appropriate way to describe the volatility of the rate of return when determining risk? Is it:

- a set of 2-tuples, where the first value of each 2-tuple is the index of the mode (e.g., 1, 2, 3, etc.) and the second value is the volatility of the mode,

- the average of the volatilities of each mode, weighted by the probability of being in each mode,

- the volatility of a single unimodal distribution fit over the entire dataset, or

- something else?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.