# What is portfolio leverage?

The definition of leverage is: $$L = \frac{\sum_i |H_i|}{C}$$ where $$C$$ is the amount of capital, $$H_i$$ is the size of holdings in asset $$i$$.

This strikes me as a weird definition for several reasons:

1. Cash holdings are not included in the summation
2. If asset $$1$$ and $$2$$ are 99% correlated and $$H_1+H_2 = 0$$ then in theory you're leveraged 2x, but in reality you barely have a position. This is realistic for futures or ADRs.

It feels like leverage is some kind of riskiness measure. If $$L$$ exceeds a pre-agreed amount, say $$5$$, the portfolio is deemed too risky and you're asked to reduce your positions.

I would interested to know if it possible to derive this formulation of leverage from something more theoretically motivated, something with some regard for the covariance of asset returns...

• What you define here is Gross Leverage, there is also Net Leverage where you don't take the absolute values (and therefore in your example Net Leverage is zero). Both measures are useful. See for ex this paper www0.gsb.columbia.edu/faculty/aang/papers/HFleverage.pdf Sep 5, 2021 at 16:24
• Your prime broker is worried about the worst case, and the worst case is all your longs go down and all your shorts go up. The Gross Leverage is of some interest to them for this reason. Sep 5, 2021 at 16:57

Suppose that each asset has the same volatility $$\sigma$$ and their correlation is $$\mathcal{C}_{ij} = \begin{cases} 1 & \text{ if } \mathrm{sgn}(h_i) = \mathrm{sgn}(h_j) \\ -1 & \text{ otherwise} \end{cases}$$ or more concisely $$\mathcal{C} = \mathrm{sgn}(h) \mathrm{sgn}(h)'$$
Then the portfolio volatility is \begin{align} \sigma_h &= \sigma \sqrt{h' \mathcal{C} h} \\ & = \sigma \sqrt{h' \mathrm{sgn}(h) \mathrm{sgn}(h)' h} \\ &= \sigma ||h||_1 \end{align}
And assuming mean zero multivariate normal returns, the value at risk is: $$\mathrm{VaR}_\alpha = \sigma_h \sqrt{\Delta T} \Phi^{-1}(1-\alpha)$$
So leverage is proportional to the portfolio VaR (under this pessimistic model) divided by capital: $$L \propto \frac{\mathrm{VaR}_\alpha}{C}$$
Assuming it's a one month, 1% VaR, there is equality at $$\sigma \approx 150\%$$.