# Is this a valid implementation for volatility targeting?

Let positions size for the instrument be

$$K_t \frac{V_t \Sigma^{target}}{\tilde{N_t}} \frac{1}{\sigma_t P_t}$$

where $$K_t = \bar{K}_{t-1} \frac{\Sigma^{target}}{\Sigma^{realized}_{t-1}}$$

where

$$V_t$$ is portfolio capital
$$\Sigma^{target}$$ is target annualized volatility
$$\tilde{N}_t$$ is the number of positions to be in
$$\sigma_t$$ is annualized volatility of instrument
$$P_t$$ is price of instrument
$$\bar{K}_{t-1}$$ is the average of the previous $$K_t$$'s for the past M periods and
$$\Sigma^{realized}_{t-1}$$ is the realized annualized volatility of the portfolio.

I found this implementation in a blog (https://hangukquant.substack.com/p/volatility-targeting-the-strategy and https://hangukquant.substack.com/p/volatility-targeting-the-asset-level). I have changed the notation a bit for simplicity.

The math adds up for me but I'm unsure about using a moving average of previous values to calculate $$K_t$$. He seems to reason that it's an empirical approach that bypasses the need for calculating the covariance matrix, but I would think that simply setting $$K_t = \frac{\Sigma^{target}}{\Sigma^{realized}_{t-1}}$$ would take care of that; while multiplying by $$\bar{K}_{t-1}$$ would simply stablize/smooth out the values of $$K_t$$ through time.

I ran some experiments and it appears that not multiplying by the $$\bar{K}_{t-1}$$ term tends to undershoot the vol target by about 40%, while multiplying by the $$\bar{K}_{t-1}$$ term actually gets you pretty close to the vol target.

Is this a valid implementation? Is there a better implementation I should look at? Did this guy invent the use of the moving average for $$K_t$$ in this context or did he take it from somewhere else? The papers he's cited don't use this implementation and I haven't seen it anywhere else either.

• I have found that a challenge in Volatility Targeting is that we don't know what the long run volatility will be in the future. In my implementation I chose a fixed value $K^*$ based on my best judgement in 2018. In the next 3-4 years the vol turned out to be higher than this and as a result my position was always underweight and underperformed the S&P500. At that time I desided to increase my $K^*$. Using a moving average makes sense to me since it would cause some automatic adjustment of K based on the long term development of vol, instead of having to make a human decision. Commented Jul 11 at 18:29
• ...so yes, the concept is valid and was something I (and probably many others) have thought of doing as well. But more generally, while having less equity exposure when volatility is temporarily above normal and more when it is below normal (the essence of vol targeting) makes a lot of sense it is tricky to identify the "normal" level. Any technique (judgemental or automatic) is subject to error. Commented Jul 12 at 12:59