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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.

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2 Answers 2

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it sounds like the binomial distribution should describe the volatility of a return, i.e. $$\sigma=\sqrt{p(1-p)252}\times 0.1\%$$

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    $\begingroup$ Someone flagged it as low quality post. While I am not agreeing, could you tell if this would be more fitting as a comment? $\endgroup$ Jul 27, 2023 at 2:48
  • $\begingroup$ In the context of the question, what is $p$? $\endgroup$ Jul 27, 2023 at 7:01
  • $\begingroup$ Thanks for the answer! I'm not sure this approach is entirely correct in my scenario because each "trial" is not independent. The decision for switching between the asset and cash is not necessarily random. In fact, it could very well be something like "risk-free from Jan to June and Cash between July and December". $\endgroup$ Jul 27, 2023 at 15:26
  • $\begingroup$ @RichardHardy p is the probability of holding risk-free asset as the question was posed initially. in subsequent comments OP suggests the decision to hold assets is not random $\endgroup$
    – Aksakal
    Jul 28, 2023 at 0:45
  • $\begingroup$ @SirFart-A-Lot if you remove all randomness then the question will not belong to this forum $\endgroup$
    – Aksakal
    Jul 28, 2023 at 0:46
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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.

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  • $\begingroup$ Dave, thanks! If it's not too much to ask, I have a follow-up question. Imagine two scenarios: 1) Holding the asset 90% of the time and cash 10% 2) Holding the asset 50% of the time and cash the remaining 50%. Why would [1] have a lower "risk" ( 0.9*[1-0.9] = 0.09 ) than [2] ( 0.5*[1-0.5] = 0.25)? $\endgroup$ Jul 27, 2023 at 17:37
  • $\begingroup$ I think you are using the term "risk" inappropriately, but that's really a question of the financial economics, not the statistics. $\endgroup$
    – Dave
    Jul 27, 2023 at 17:48
  • $\begingroup$ Ok! I think the question still applies if we talk about standard deviation. Why would the standard deviation be lower [1] relative to [2]? Sorry for constantly following up with another question... $\endgroup$ Jul 27, 2023 at 19:03
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    $\begingroup$ There is variability when you mix the two investments. Sometimes you earn $0.1\%$, and sometimes you earn nothing. Hence, you have positive standard deviation. When you are completely in either investment, you always earn the same return, so the variance and standard deviationa are zero. $\endgroup$
    – Dave
    Jul 27, 2023 at 19:13

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