Measuring Portfolio Volatility when Risk-Off

External question — Cross Validated Stack Exchange Author: Sir Fart-A-Lot Original post: https://stats.stackexchange.com/questions/622402 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I'm trying to calculate the annualized volatility of a single-asset trading strategy that is either 100% long or 100% cash. Imagine a risk-free asset with a guaranteed and fixed daily return of 0.1%. For some unreasonable reason, we decide to hold this asset 50% of the time, and just hold cash for the remaining 50% of the period. By definition, both the risk-free asset and cash have a 0% volatility of returns. But in the case in which we switch between two assets, we have a non-zero volatility, because we have a bimodal distribution of returns. I guess this is technically correct in a statistical sense, but not in a financial sense (as a risk metric). Is there a way to account for this "unexpected" behavior? Now, let's replace the asset with one that has non-zero volatility. In order to avoid the abovementioned behavior, I thought about calculating the standard deviation only with the returns of the asset and then scaling down to the percentage of days we're holding it during the year. Something like the following: $$\sigma(Asset)_{Anual}=\sigma(Asset)_{Daily}\cdot\sqrt{252\cdot Exposure_{Asset}}$$ I'm sorry in advance if this question is too basic.

External answer — Cross Validated Stack Exchange Author: Aksakal Original post: https://stats.stackexchange.com/a/622403 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. it sounds like the binomial distribution should describe the volatility of a return, i.e. $$\sigma=\sqrt{p(1-p)252}\times 0.1\%$$

External answer — Cross Validated Stack Exchange Author: Dave Original post: https://stats.stackexchange.com/a/622471 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. You get zero volatility when the returns are constant. When you bounce between no-yield cash and that yes-yield investment, your returns are not constant, so there will be some variability in the returns, hence volatility. (Note that $0.1\%$ daily return is $28\%$ annually, which is quite a bit higher than the historical performance of Warren Buffet’s Berkshire Hathaway stock. That is, $0.1\%$ daily yield is quite large!) You have $k$ -many days holding no-yield cash, and you have $252-k$ days holding the asset that has a $0.1\%$ return. Thus, you have a distribution of returns that has $k$ points with values of zero and $252-k$ points with values of $0.001$ . You know how to calculate volatility from a data set of returns. Who cares that so many values are equal? This turns out to be the calculation given in another answer: $0.001\sqrt{252p(1-p)}$ , for $p = \frac{252-k}{252}$ , assuming $252$ trading days in the year. That is, $p$ is the proportion of days where you hold the assets that earns the guaranteed $0.1\%$ yield. Whether or not this makes financial sense is a separate matter. Volatility is supposed to give some sense of uncertainty. For this particular setup, there does not appear to be any uncertainty, as you know what the returns will be and when they will occur, so when you call it volatility, that seems misleading in the financial context. As for why there is variance (or standrd deviation) despite the apparent certainty, there is variability when you mix the two investments: sometimes you earn $0.1\%$ , and sometimes you earn nothing. Hence, you have a positive standard deviation of the returns. When you are completely in either investment, you always earn the same return, so the variance and standard deviationa are zero.