Portfolio Leverage as a Pessimistic Volatility and VaR Measure
Summary
The document examines whether gross exposure divided by capital is a useful definition of portfolio leverage. It points out that this measure omits cash and can overstate risk when offsetting holdings are highly correlated, then derives a risk-based interpretation under a deliberately pessimistic dependence assumption.
Assuming equal asset volatility and correlations of +1 for positions with the same sign and −1 for positions with opposite signs makes portfolio volatility proportional to the sum of absolute holdings. With mean-zero multivariate normal returns, the document then relates leverage to VaR per unit of capital. This derivation depends on restrictive assumptions about volatility, correlations, and return distributions; it does not establish that gross leverage reflects risk for ordinary portfolios. The stated one-month VaR example illustrates the scale of volatility needed for the proportionality to become equality.
Key ideas
- Gross exposure divided by capital ignores cash holdings.
- Offsetting positions can make gross leverage look large despite low net exposure.
- Under the stated worst-case correlation model, portfolio volatility is proportional to gross exposure.
- The resulting leverage interpretation is proportional to VaR relative to capital.
- The interpretation depends on equal volatility, extreme correlations, and normally distributed returns.
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Full text
# What is portfolio leverage?
# 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:
- Cash holdings are not included in the summation
- 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...
## Answer by user357269 (score 1, accepted)
https://quant.stackexchange.com/a/67779
Expanding on noob2's comment:
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 $\sigma \approx 150\%$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.