14.5 Volatility targeting · M14 Portfolio and going live

Motivation

M02 showed volatility clusters. If expected returns do not rise one-for-one with volatility, holding constant exposure through a high-volatility regime takes more risk for the same return.

Setup

Strategy return $r_t = \mu + \sigma_t\varepsilon_t$ , with $\varepsilon_t$ i.i.d. mean 0, variance 1, independent of $\sigma_t$ ; $\sigma_t$ known at $t-1$ (in practice, an EWMA or GARCH forecast from M13). The volatility-managed strategy holds exposure $k_t = c/\sigma_t$ , possibly capped.

Proposition 14.10Volatility targeting raises the Sharpe ratio when the mean does not scale with volatility

The unmanaged Sharpe ratio is $\mu/\sqrt{\mathbb E\sigma_t^2}$ . The managed Sharpe ratio is

S_{VT} = \frac{\mu\,\mathbb E[1/\sigma_t]}{\sqrt{1 + \mu^2\operatorname{Var}(1/\sigma_t)}}.

If $\mu$ is small relative to volatility (true for daily or weekly returns), $S_{VT} \ge \mu/\sqrt{\mathbb E\sigma_t^2}$ , with strict inequality when $\sigma_t$ is not constant.

Proof

$x_t = k_tr_t = c\mu/\sigma_t + c\varepsilon_t$ . Mean $c\mu\mathbb E[1/\sigma_t]$ . Variance $c^2\mu^2\operatorname{Var}(1/\sigma_t) + c^2$ (the two terms are uncorrelated because $\varepsilon_t$ is independent of $\sigma_t$ with mean zero). Divide. For the comparison, ignore the $\mu^2$ term in the denominator (second-order for small $\mu$ ) and use Jensen twice: $\mathbb E[1/\sigma] \ge 1/\mathbb E\sigma \ge 1/\sqrt{\mathbb E\sigma^2}$ .∎

Example

A strategy with an 8% annual mean that spends half its time at 10% volatility and half at 30% has Sharpe $0.08/\sqrt{0.5(0.01) + 0.5(0.09)} = 0.358$ unmanaged and $0.515$ volatility-managed (annual figures, so the $\mu^2$ term is kept). Moreira & Muir ("Volatility-Managed Portfolios", Journal of Finance 72(4), 2017) document improvements of this kind for several US factor portfolios. Whether it holds for your strategy is an empirical question: test it with the M04 safeguards.

If instead $\mu_t = S\sigma_t$ (constant Sharpe ratio in every regime), the managed strategy has Sharpe exactly $S$ and gains nothing in Sharpe, but it still delivers stable volatility, which makes position-sizing rules (M03) behave as intended.

Practitioner notes

Cap leverage (e.g. $k_t \le 2$ ). Volatility forecasts lag, so the first day of a crash is taken at full size; vol targeting reduces exposure after volatility rises. Gaps (overnight, event days) are not controlled. In Indian equities, exposure reductions in a broad sell-off are made at the same time as everyone else's, so expect worse fills.