Rigorous proof that volatility target strategies actually tend to the target

I'm working on a paper about volatility timing and target strategies, practical implementation included.

While writing down the mathematical description of the model I wanted to include a rigorous proof that given target $$\tau$$, estimated volatility (for example via EWMA) $$\sigma_t$$ and leverage $$\mathcal{L}_t = \frac{\tau}{\hat{\sigma}_t}$$ in period $$t$$, then the volatility of:

$$r_{t,strategy} = \mathcal{L}_{n,t}r_{t, risk} + (1 - \mathcal{L}_{n,t})r_{t, f}$$

actually tends to target $$\tau$$ as $$t$$ increases, on average, where $$r_{t, risk}$$ and $$r_{t, f}$$ are respectively the risky and risk-free returns of the assets and bonds of the underlying portfolio/index. Which is supposedly what happens in practice.

I taught two main approaches:

• Use the fact that we are weighting more a less fluctuating part than a more fluctuating one (risk-free vs risk)
• Define an optimization problem to minimize volatility to the target

However I am a bit stuck on the technicalities and I suspect I might be looking at it wrong, it would be of great help to get some hints :)

• I suppose $\tau$ is a volatility target, right? – FunnyBuzer Mar 22 '20 at 13:07

Suppose that you are riskless asset with return $$r_{ft}$$ and a risky asset with return $$r_t$$ and conditional volatility $$\sigma_t(r_t) := \sqrt{V_t(r_t)}$$. We build a portfolio using weights $$(w_1, w_2) \in \mathbb{R}$$, or as you wrote it $$w_t := w_{1t}$$, $$w_{2t} := 1 - w_t$$. This portfolio will have a time $$t$$ return of $$r_{pt}$$. Its volatility is given by $$\sigma(r_{pt})$$, defined in a similar fashion as above. We also define conditional volatilities and variances in a similar way as $$(\sigma_t(.), \sigma_t^2(.))_{t \geq 0}$$, respectively.

The target volatility for this portfolio is $$\tau$$ and we're looking for portfolio weights. By definition: \begin{align} \sigma_t^2(r_{pt+1}) &= w_t^2 \sigma_t^2(r_{t+1}) + (1-w)^2 \sigma_t^2(r_{ft+1}) \\ \sigma_t^2(r_{pt+1}) &= w_t^2 \sigma_t^2(r_{t+1}) + 0 \\ \sigma_t(r_{pt+1}) &= w_t \sigma_t(r_{t+1}) \\ \rightarrow w_t &= \frac{\sigma_t(r_{pt+1})}{\sigma_t(r_{t+1})} \\ \rightarrow w_t^* &= \frac{\tau}{\sigma_t(r_{t+1})} \end{align} so, if you knew is the conditional volatility over the next period, you could trivially choose the portfolio weights that would ensure you hit your target level of volatility exactly at every single point in time. But your question is rather about what happens if I use an ESTIMATED level of volatility?

Assume an additive error structure such that $$\hat{\sigma}_t(r_t) := \sigma_t(r_t) + \epsilon_t$$. Some of the movements you see in conditional variance is due to sampling variance, i.e. $$V(\hat{\sigma}_t(r_t)) = V(\epsilon_t) \neq 0$$. If you happen to have a consistent estimator of the conditional variance process for the returns on your risky asset, then your convergence results would be $$$$\forall \delta > 0 \; \lim_{T \rightarrow \infty} \text{Pr}( |\hat{w}_t - w_t^*| > \delta) = 0$$$$ trivially because $$\tau$$ is known and the denominator converges (I am assuming that it convergences in the same sense). In essence, it's not really complicated to prove, as long as you can show you do have an approriate (in an asymptotice sense) estimator for conditional variance, it's fine.

Now, the more problematic issue is that you work with a finite sample, hence: \begin{align} \sigma_t^2(r_{pt+1}) &= \tau^2 V_t \left( \frac{r_{t+1}}{\hat{\sigma}_t(r_{t+1})} \right) \\ \sigma_t^2(r_{pt+1}) &= \tau^2 \left[ \sigma_t^2(r_{t+1}) + \sigma_t^2(1/\epsilon_t) + 2 cov_t\left(r_{t+1}, 1/\epsilon_t\right) \right] \\ \sigma_t(r_{pt+1}) &= \tau \sqrt{\left[ \sigma_t^2(r_{t+1}) + \sigma_t^2(1/\epsilon_t) + 2 cov_t\left(r_{t+1}, 1/\epsilon_t\right) \right]} \end{align} and might be quite a bit more volatility than you wanted to have. Just to be clear, I don't assume that $$\epsilon_t$$ is known at time $$t$$, so the above expressions make sense. One thing you could do to alleviate the problem is instead of relying on an argument about the asymptotics of estimators, you choose weights to minimize the distance between the target and the estimated conditional volatility of your portfolio, knowing that you're using an estimate and that it is hence not a perfect measurement.

And, if you wanted to be extremely fancy, you actually have a degree of freedom over how you estimate the conditional volatilities for both the risky asset and the portfolio. In other words, you could taylor your choice of estimates to the need of getting as close as possible to your target level of volatility.

• Thank you a lot that's exactly the help I was looking for! And thinking of the degree of freedom actually comes in handy when explaining the differences between different methods for estimating conditional volatilities, a fancy bonus indeed. – Emanuele Mar 27 '20 at 16:34