# Alpay Kaya, CFA

La vérité existe. On n'invente que le mensonge. —Braque (Truth exists. One only invents falsehood.)

# What Price Leverage? (Excerpt from Leveraged ETFs)

From Sec. 4.3.1 “Effective Log Leverage”
The difference between an exponentially leveraged system (eLETF) and ‘normal’ leveraged fund (LETF) can also be shown by expressing LETFs as eLETFs with modified log leverage.  Define log leverage  $\gamma$ such that

$y_\mathrm{LETF}^{} (\rho, \beta) = y_\lambda^{} (\rho, \gamma)$

the growth factor of a beta-leveraged ETF is expressed as that of a gamma-leveraged eLETF.  The effective log leverage of a leveraged ETF

$\gamma (\rho, \beta) \, \stackrel{\text{\scriptsize def}}{=} \; {\displaystyle \frac{1}{\;\! \rho \;\!}} \;\! \ln \bigl[ \;\! 1 + \beta \left( \exp (\rho) - 1 \right) \;\! \bigr] \qquad (4.15)$

is a function of leverage $\beta$ and index log return $\rho$.  Fig. 4.3 shows the effective log leverage for both long & short LETFs.

Figure 4.3: Effective log leverage. Shown is log leverage against index log return for triple-leveraged long (left axis) and short (right axis) LETFs. Although γ(0, β) is undefined, the limit is β by l’Hôpital’s rule.

For the long LETF, effective log leverage is less (greater) than the target for positive (negative) index return.  The short LETF’s log leverage is less than the target (i.e., greater magnitude) for positive index returns.  For both cases, the magnitude of log leverage increases as the LETF loses value and decreases as the LETF gains value, a consistent bias which leads to negative drift.  The bias is more pronounced for the short LETF, making its graph more vertical.

The definition in Eq. 4.15 allows a $\beta$-LETF to be written as a $\gamma$-eLETF or a $\beta$-eLETF with correction term

$y_\mathrm{LETF}^{} (\rho , \beta ) = \exp (\gamma \rho ) = \exp (\beta \rho ) \exp \bigl[ (\gamma -\beta ) \rho \bigr] \qquad (4.16)$

with the difference $\gamma - \beta$ serving as the log leverage correction and the argument $(\gamma - \beta) \rho$  the log return correction necessary for expressing negative drift within an eLETF framework.