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\begin{document}
\title{Rebalancing and leverage}
\author[ ]{Investment Math}
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Rebalancing does not need to have a contrarian flavour. For this reason,
the claim that rebalancing has benefits is empty. It needs further
qualification. I illustrate this using a leveraged portfolio.
As usual, I take a world with two assets, cash bearing zero interest
rate and a risky asset with price \(p\). At inception,
investment is initiated with one dollar. The initial price is
\(p_{0}=1\). The initial amount is leveraged to reach an exposure
to the risky asset of \(\pi>1\). The position in cash is thus
\(1-\pi \lt 0\). Assuming the portfolio is continuously rebalanced
and the price trajectory is smooth, its value as a function of the price
is given by: \[V(p)=p^{\pi},\] and the number of shares held as a
function of the price is: \[n(p)=\pi p^{\pi-1}.\] This is an increasing
function of the price so that the investment policy consists in buying
more shares as they go up, and selling shares as they go down.
Rebalancing to a leveraged exposure is a momentum policy, as opposed to
a contrarian policy.
I show the leveraged trading policy and the leverage value function
versus the corresponding buy-and-hold quantities in the graphs below.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/leverag/leverag}
\caption{{Value functions: the rebalanced leveraged portfolio (red) versus the
buy-and-hold portfolio (blue)%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/tradleverag/tradleverag}
\caption{{Trading rules: the rebalanced leveraged portfolio (blue) versus the
buy-and-hold portfolio (red)%
}}
\end{center}
\end{figure}
In the next two graphs, I show the corresponding configuration for
unleveraged portfolios (\(\pi \lt 1\))\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/dleverag/dleverag}
\caption{{Value functions: the rebalanced unleveraged portfolio (red) versus the
buy-and-hold portfolio (blue)%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/dtradleverag/dtradleverag}
\caption{{Trading rules: the rebalanced unleveraged portfolio (blue) versus the
buy-and-hold portfolio (red)%
}}
\end{center}
\end{figure}
The difference between the two cases is clear:
\begin{itemize}
\tightlist
\item
Momentum case: leveraged portfolios lead to increasing trading rules
and convex value functions
\item
Contrarian case: unleveraged portfolios lead to decreasing trading
rules and concave value functions
\end{itemize}
What about short positions, i.e.~\(\pi \lt 0\) ?
\selectlanguage{english}
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