\documentclass[12pt]{article}
\usepackage{extsizes}
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\usepackage{graphicx}
\usepackage{amsmath, amsfonts}
\newcommand{\sign}[1]{\text{sgn} \left(#1 \right)}
\newcommand{\diag}[1]{\text{diag} \left(#1 \right)}
\newcommand{\transpose}{'}%{^\mathsf{T}}
\newcommand{\var}{\sigma^2}
\title{Crowding}
%\author{G.A.Paleologo}
\date{}
\begin{document}
\maketitle
\tableofcontents
\section{Why Care?}
``Crowding'' has become a popular term nowadays. It wasn't always so. The event that focused attention on crowding and its dangers is the ``Quant Crisis'' of 2007. Khandani and Lo\footnote{A. Khandani and A. W. Lo. ``What Happened To The Quants In August 2007?''. \emph{Journal of Investment Management} \textbf{5}(4) (2007): 29-78.\\
A. Khandani and A. W. Lo. ``What Happened To The Quants In August 2007?: Evidence from Factors and Transactions Data''. \emph{Journal of Financial Markets} \textbf{14} (2011):1-46.} and Pedersen\footnote{M. K. Brunnermeier  and L. H. Pedersen. ``Market Liquidity and Funding Liquidity''. \emph{Review of Financial Studies} \textbf{22} (2009): 2201-2238.\\
L. H. Pedersen, ``When everyone runs for the exit''. \emph{The International Journal of Central Banking} \textbf{5}: 177- 199.}
were among the first to attempt a descriptive analysis of this phenomenon. Since then, crowding has been mentioned in currency trades\footnote{M. Pojarliev and R. M. Levich. ``Detecting Crowded Trades in Currency Funds''. \emph{Financial Analyst Journal} \textbf{67} (1): 26-39 (2010).} and in CDOs\footnote{US Office of Financial Research, ``2012 Annual Report''.}. The latter contains a simple definition of crowding:


\begin{displayquote}
\textbf{Crowded Trade}: \emph{A trade in which the market participants have large and similar positions, creating the risk that there will be insufficient liquidity should market participants seek to unwind their
positions simultaneously.}
\end{displayquote}
Since 2010 the scope and importance of crowding has grown. On the one side, the LTCM crisis has been analyzed and convincingly attributed to crowding by Chincarini\footnote{L. Chincarini. ``The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal'' (2012). Bloomberg Press.}. In addition to this backdating exercise, the industry has been handed by the goddess of fortune a fresh batch of out-of-sample crowding events (see Table \ref{table:events}). At the end of February 2016, several large long-short equity fundamental strategies experienced large drawdowns, first in the energy sector and then market-wide.
\begin{table}[]
\centering
\begin{tabular}{ll}
\hline
 \textbf{Event}  & \textbf{Period} \\ \hline\hline
 LTCM Collapse\footnote{L. Chincarini (2012)} &   August-September 2008    \\
 \hline
 Quant Crisis\footnote{Khandani and Lo (2009)} & August 8-11, 2007 \\ \hline
  Carry Trade Collapse\footnote{Polyarev and Levich (2010)}           &    September 2008   \\ \hline
  Chinese Drawdown\footnote{J. Bian, Z. He, K. Shue, H. Zhou. ``Leverage-Induced fire sales and stock market crashes''. Working paper (2018).} & May-July 2015\\ \hline
Energy E \& P Short Squeeze &  February-March 2016      \\ \hline
Market Correction                         &    October-November 2018
\end{tabular}
  \caption{An incomplete series of unfortunate crowding events.}
  \label{table:events}
\end{table}

In October and November 2018 another prolonged and pervasive drawdown occurred in the long-short equity hedge fund sector. In both cases, the losses in some of the best-performing long-short strategies in the business ranged between 3\% and 5\% of Gross Market Value. These losses are large enough to damage permanently the track record of a hedge fund. Crowding is not the cost of doing business. It is an existential risk and needs to be understood and managed. With regards to the former, we have just slouched past the start line; with regards with the latter, the starter's gun has not fired yet.
\begin{figure}
  \includegraphics[width=\linewidth]{fig_PassiveInvesting.pdf}
  \caption{Left: AUM invested in active, passive investing, and  Passive Investing as percentage of the total Right: relative size of passive mutual funds and ETFs.}
  \label{fig:activepassive}
\end{figure}

What \emph{we know} are a few striking trends that may or may not relate to the incidence of crowding events.
\begin{description}
\item[Retail Investors have gone extinct.] the US equity market has steadily moved toward institutional investing: direct asset ownership was 47.9\% in 1980, 21.5\% in 2007\footnote{K. R. French. ``The Cost of Active Investing''. \emph{Journal of Finance}, \textbf{63} (4): 1537-1573.}. It is less than 10\% today.
\item[Hedge funds assets are not growing.]Hedge funds AUM grew at at an annual clip rate of 23\% in the period 1990 to Q2 2008: \$39Bn to \$1.9Tn\footnote{Data from Hedge FundResearch Database.}, but have effectively stopped growing since. The most recent AUM available is \$3.1Tn; an annualized growth rate of 5\%, while the SP500 grew at \%8.6.
\item[Hedge funds AUM has become concentrated.]
\item[Crowding has not significantly increased.] A common measure of crowding is position overlap.
\item[Equity hedge funds flow has grown.]We do not have hard data to support this conclusion, but we can use some back-of-the-envelope. Current average trading turnover in long-short equity funds (measured as traded value/Gross Market Value) is approximately 15-20x/year. It was certainly smaller five years ago. Meanwhile, AUM for the largest funds has increased compared to 2003. Citadel has grown 250\%; Millennium and Balyasny about 200\%. I assume the following GMVs for the long-short businesses \emph{alone}: \$130Bn for Citadel; \$100Bn for Millennium; \$50Bn for Balyasny; \$50Bn for Point72. This total \$230Bn corresponds to an annual traded GMV of \$3.5Tn today, and half as much five years ago. In the US, annual trading volume is\$55Tn and has not grown significantly in the past five years. Long/Short equity participation \emph{of these four firms alone} in the market is 6.3\% today and was half as much five years ago. However, only about 25\% is fundamentally driven\footnote{Goldman Sachs communication (2013)}. Therefore, four firms are behind one in five fundamental trades today. If we consider the top 20 long/short hedge funds, it is reasonable to assume that they trade with each other in the majority of cases.
\end{description}
There have been efforts to link deleveraging effects to AUM levels and flow in passive investing, to changes in position overlap among firms, and amount of firm investment overlap within a single position. A large number of crowding descriptors has sprung to life\footnote{See, e.g., M. K. Bayraktar, S. Doole, A. Kassam, S. Radchenko. ``Lost in the crowd? Identifying and Measuring Crowded Strategies and Trades''. \emph{Barra Research Insight} (2015). \\J. Jussa \emph{et al.}. ``Strategy Crowding''. \emph{Deutsche Bank Signal Processing} (May 2016)}. I hope I am not being unfair to the authors if I summarize these efforts as inconclusive shots in the dark (by their own standards) that conflate exploratory and confirmatory analysis and do not control for multiple comparisons.

Here a few robust, if stylized, facts about crowding:
\begin{enumerate}
 \item A ``crowded'' portfolio, long assets held by hedge funds, and short names not held by hedge funds, has a positive Sharpe Ratio and small annualized returns, but with sudden drawdowns.
 \item Analogously, a ``crowding'' factor has little explanatory power of the cross-section of stock returns; and yet it is responsible for sizable losses in a concentrated periods, as if its volatility jumped and/or drift changed.
 \item ``Crowding'' is precipitated by an external event. For example, in the case of the LTCM collapse, it was the joint shutdown of the Salomon Brothers Fixed Income prop trading desk and the Russian Default. In 2016, sudden expectations of a mid-year recession.
 \item Crowding drawdowns can happen in the whole market, within factor themes, and within a specific industry as well. They also happen in different asset classes: equities, fixed income, currency crowding events have been documented.
 \item Crowding drawdowns occur contemporaneously with drawdowns in momentum, although the horizon of the momentum may vary.
 \end{enumerate}
 Many questions about crowding do not have an answer. Perhaps,

\begin{itemize}
	\item \textbf{How does crowding happen?} We know that assets that are shared among firms are the most vulnerable in the event of a drawdown. Otherwise, crowded stocks seem to perform well. Why is crowding a bad thing, and only in short periods of time?
	\item \textbf{What determines crowding?} Is it due to \emph{flow} into passive investing? Or rather is it due to relative AUM \emph{size} allocated to passive investing? Or something else?
\end{itemize}
And, of course, the Mother of All Questions:
\begin{itemize}
	\item \uppercase{\bf Can we manage crowding risk?}
\end{itemize}
\section{What's New Here}

In the next sections I describe a \emph{dynamic} model of crowding events. This means that the model allows for trading to occur over many periods, as it is realistic to assume, and not over one, two or three periods, \footnote{Brunnermeier and Pedersen present a three-period model. Stein's model is two-period. See J. C. Stein. ``Sophisticated Investors and Market Efficiency''. \emph{Journal of Finance}, \textbf{64}(4): 1517-1548 (2009).} The model also accommodates for the existence of a population of traders, characterized by different risk tolerances and valuations. In this respect, the approach is closer in spirit to agent-based models by R. Bookstaber\footnote{R. Bookstaber, M. Paddrik and B. Tivnan. ``An Agent-based Model for Financial Vulnerability''. \emph{Office of Financial Research Working Papers}. (2014)} but unlike the latter is can be treated analytically and not only through simulations. The model relies on only two assumptions:
\begin{description}
	\item[Investor behavior:] Investors are mean-variance maximizers, with a given risk tolerance and private valuation of the asset universe.
	\item[Risk tolerance:] When they are profitable, investors increase their risk tolerance. When they lose money, they decrease it.
\end{description}
The first assumption essentially states that, everything else equal, an investor allocates more money when her private valuation differs more from its public price, and that, everything else equal, she holds a bigger portfolio if her risk tolerance is bigger. To a large extent, the model and the results are more general than mean-variance optimization. We are not interested in maximum generality here, and MVO is a good first approximation. The second assumption is empirically true: investors grow their portfolio when it is profitable, and reduce it otherwise. We assume a linear response to loss and gains. As for the first assumption, it is possible to greatly generalize the results, but a linear reduction is empirically a good enough approximation to real-life behavior.

The major take-away messages from this model are:
\begin{enumerate}
	\item The model predicts the onset of crowding and then possible drawdowns through a sequence of four events:
	\begin{enumerate}
	\item a \emph{leveraging} phase, during which investors increase their risk tolerance
	\item  an \emph{equilibrium} phase, characterized by moderate changes in prices and risk tolerances;
	\item a \emph{catalyst} event that is sufficiently intense to take the system out of equilibrium;
	\item the \emph{deleveraging} event proper, during which all investors reduce their risk tolerances, and prices move away from fundamentals.
	\end{enumerate}
	\item The model predicts that drawdowns only occur when there is a shock that affects the risk aversion \emph{change rate} of the investors. In other words: we don't need to assume that a large investor liquidates, but rather that the participation rate in the market in the course of liquidation be high for a short period of time. Analogously, we don't need to assume a large but slow market drawdown. A short market shock can trigger a deleveraging crisis. What prompts deleveraging is an event of high \emph{intensity}, not necessarily of large \emph{size}.
	\item The model suggests that when trading ideas change rapidly, the leveraging phase does not occur, and the environment is less prone to risk of deleveraging. Conversely, when trading ideas do not change, i.e., when investors hold the same assets for a long period of time, leveraging can occur and the equilibrium will be reached, after which deleveraging will eventually happen.
	\item The model can't predict the time of deleveraging, since this follows a random shock. However, it identifies a measure of \emph{crowding vulnerability}. The greater the average cross-sectional correlation among portfolio holdings, the greater the risk of a deleveraging event.
	\item Finally, the model predicts that dispersion in beliefs reduces the occurrence of crowding:
\end{enumerate}
\newpage
\section{Model}
\subsection{Notation}
We have $m$ investors trading $n$ assets. The investors trade over multiple periods $0, \delta t, 2\delta t,\ldots$.

Notation:
\begin{itemize}
	\item $e_m, e_n$, vectors of ones of size $m, n$ respectively.
	\item $\text{diag}(x)$ is a diagonal matrix with the vector $x$ on the main diagonal.
	\item for two vectors $x, y$ having the same size, $xy$ (and $x/y$) is a vector obtained by element-wise multiplication (division).
	\item For a matrix $A$, $A_{i \cdot }$ is the $i$-th row vector, $A_{\cdot j}$ the $j$-th column vector, and $[A]_{ij}$ is the $(i,j)$ element; an analogous notation is used for vectors. For certain special matrices and vectors (e.g., $\Theta$) we use the lowercase symbol shorthand and write $\theta_{ij}$ instead of $[\Theta]_{ij}$.
	\item For a matrix $A$, define the operator $\text{vec}(A)$ that returns a vector composed by the stacked columns of $A$.
	\item For any function $f:R\rightarrow R$, denote $\delta f(t):= f(t)-f(t-\delta t)$.
	\item $\delta_{ij}$ is Kroneker's delta.
\end{itemize}

Data and parameters:
\begin{itemize}
	\item $\Theta\in R^{n\times m}$ the matrix of valuations, constant over time. Investor $j$ has a subjective valuation for  the terminal value of asset $i$ is $\theta_{ij} >0$.
	\item $s\in R^n$ is the supply of assets, constant over time.
	\item $p\in R^n$ is the vector of asset prices, time-varying.
	\item $\rho \in R^m$ is the vector of investors' risk tolerances, time-varying.
	\item $\Sigma\in R^{n\times n}$ is the asset covariance matrix. For simplicity we assume $\Sigma$ is diagonal and constant, with the $i$th diagonal element equal to $\var_i$.
\end{itemize}


\subsection{Laws}
\begin{description}
\item[Mean-Variance Investors.] In each period investor solve a mean-variance problem.
  $\Theta\in R^{n\times m}$ is a matrix such that $\theta_{ij}$ is the valuation of asset $i$ by investor $j$ . Let $p\in R^n$. The optimization problem for each investor is that of selecting a investment $x$ where $q_{ij}$ is the dollar amount invested in security $i$. The investor optimization problem is
	\begin{align*}
		\max_q\; & (\theta_{.j}/p -e_n)' q -\frac{1}{2\rho_j} q'\Sigma q
	\end{align*}
	The solution is
	\begin{align*}
		q_{ij}(\rho, p) = & \frac{\rho_j}{\var_i}  (\theta_{i j}/p_i - 1)
	\end{align*}

	\item[Dynamic Risk Tolerance.] Investors increase or decrease their risk tolerance based on their dollar gains or losses  in the previous period.
	\begin{align}\label{eq:risktolerancedynamics}
		 \rho_j(t+\delta t)-\rho_j(t) =  \gamma_j q_{\cdot j}'(t)r(t)
	\end{align}
	where $r(t) := p(t) / p(t-\delta t) - 1$ is the vector of asset returns and $\gamma_i$ is a positive constant parameter which captures the sensitivity of the investor to losses.
\end{description}
I should emphasize that this is a descriptive model, not a normative one. I choose mean-variance optimization as a behavioral reference not because it is normatively defensible (it is not) but because it is a) followed approximately by many investors; b) intuitive: the investor holds more of a security if she expects a higher risk-adjusted return; and the portfolio is multiplied by a factor that corresponds to its overall size. Many of the results below hold under more general investing models, but the goal here is to provide insights rather than the most general analysis. Similarly, I believe the linear risk tolerance updating behavior to be empirically true.

In each period  we  obtain the price vector from the market-clearing condition
\begin{align*}
s_i p_i &= \sum_{j=1}^m  q_{ij}(\rho, p) \qquad i=1,\ldots, n
\end{align*}
from which we can determine $p (\rho)$. It is
\begin{align}
	p_i(\rho)%=& \frac{-e_m\transpose \rho +\sqrt{(e_m\transpose \rho)^2 +4s_i\var_i[\Theta\rho]_i} }{2s_i\var_i}\\
	=& \frac{e_m\transpose\rho}{2s_i\var_i} \left({ \sqrt{1 +\frac{4s_i\var_i[\Theta\rho]_i}{ (e_m\transpose\rho)^2}} } - 1\right) \label{eq:equilibriumprice}
%	\\
%	\simeq &    \frac{ [\Theta\rho]_i}{ (e_m\transpose\rho)} -  \frac{s_i\var_i}{ (e_m\transpose\rho)^3} [\Theta\rho]_i^2
\end{align}
%where the last approximation holds in the limit $e_m\transpose \rho \rightarrow\infty$.
%where $\rho_{\text{tot}}=\sum_i\rho_i$ and $\bar\theta(t):= \sum_i\rho_i(t)\theta_i /\rho_{\text{tot}}(t)$.
\section{Derisking Dynamics: General Equations}
Given Equations (\ref{eq:risktolerancedynamics}) and (\ref{eq:equilibriumprice}), we identify the behavior of rick tolerances over time.

From Eq. (\ref{eq:equilibriumprice}) we obtain
\begin{align*}
	r_i(t) & =\frac{p_i(\rho(t+\delta t))-p_i(\rho(t))}{p_i(\rho(t))}\\
	&\simeq \frac{1}{p_i} \sum_{j=1}^m\frac{\partial p_i}{\partial\rho_j}(\rho_j(t)-\rho_j(t-\delta t))
\end{align*}
We replace returns in Eq.(\ref{eq:risktolerancedynamics}) to get
\begin{align}
\delta\rho_j(t+\delta t) =   &  \sum_{i=1}^n \sum_{k=1}^m \gamma_j q_{ij} \frac{1}{p_i}\frac{\partial p_i}{\partial\rho_k}\delta\rho_k(t) \notag \\
	  =   & \sum_{k=1}^m  \left( \sum_{i=1}^n\gamma_j q_{ij} \frac{1}{p_i}\frac{\partial p_i}{\partial\rho_k} \right) \delta\rho_k(t) \label{eq:risktolerancedynamics2}
\end{align}
We define
\begin{align*}
A_{jk}(\rho):= &  \sum_{i=1}^n\gamma_j q_{ij} \frac{1}{p_i}\frac{\partial p_i}{\partial\rho_k}
\end{align*}
and introduce the new variable
$$\xi(t):= \delta\rho(t)$$
Rewrite Eq. (\ref{eq:risktolerancedynamics2}) as
\begin{align}
\xi(t+\delta t) =   &  A \xi(t) \label{eq:risktolerancedynamics3}
\end{align}
 From the definition of $\delta\rho$
\begin{align*}
	\rho(t+\delta t)-\rho(t)=\delta\rho(t+\delta t)=A\xi(t)
\end{align*}
The dynamic equations are
\begin{align}\label{eq:generalsystem}
	\begin{cases}
		\xi(t+ \delta t)- \xi(t)= (A-I) \xi(t)\\
		\rho(t+\delta t)- \rho(t)=A \xi(t)
	\end{cases}
\end{align}
In the continuous limit, this becomes a system in $2m$ variables:
\begin{align}\label{eq:generalsystemcontinuous}
\boxed{
	\begin{cases}
		d \xi=(A-I) \xi(t) dt\\
		d\rho= A\xi(t) dt
	\end{cases}
	}
\end{align}
The variables $(\xi,\rho)$ are not observed directly; all we can observe are prices, which are function of $\rho$ through Eq. (\ref{eq:equilibriumprice}). We are not concerned with parameter identification, and focus instead in the behavior or the system over time.
In the following subsections, we consider interesting special instances of the problem, with increasing degrees of generality.
%
\section{Extension: dynamics with changes in valuation}
%
In the previous sections  we assumed that valuations don't change. What would happen if the valuations changed? Would we observe the same behavior? The answer

%\begin{align*}
%\frac{\partial \log p_i}{\partial\theta_{\ell j}}= \left( 1 +\frac{4s_i\var_i[\Theta\rho]_i}{ (e_m\transpose\rho)^2} \right)^{-1/2}\frac{4s_i\var_i\rho_j}{ (e_m\transpose\rho)^2} \delta_{\ell i}
%\end{align*}
\begin{align*}
	r_i(t)\simeq &  \sum_{j=1}^m\frac{1}{p_i}\frac{\partial p_i}{\partial\rho_j}(\rho_j(t)-\rho_j(t-\delta t)) +\\
	& \sum_{j } \frac{\partial \log p_i}{\partial\theta_{i j}} \delta \theta_{i j}(t)
\end{align*}

\begin{align*}
\delta\rho_j(t+\delta t)  =   & \sum_{k=1}^m   A_{jk}  \delta\rho_k(t) + \sum_{i\ell}
 \gamma_{ j} q_{ij}  \frac{\partial \log p_i}{\partial\theta_{i \ell}} \delta \theta_{i \ell}(t)
\end{align*}
Define matrices $B^j(\rho)\in R^{n\times n}$.
\begin{align*}
[B^j(\rho)]_{i\ell}:= &  \gamma_j q_{ij}  \frac{\partial\log p_i}{\partial\theta_{\ell j}}
\end{align*}
After some algebraic manipulations, we obtain the modified equations
\begin{align*}
\boxed{
	\begin{cases}
		d \xi_j=[(A-I) \xi(t)]_j dt + \text{vec}(B^j)\transpose \text{vec}(d\Theta) \qquad j=1,\ldots, m\\
		d\rho= A\xi(t)  dt
	\end{cases}
	}
\end{align*}

\section{Derisking Dynamics: Special Cases}
\subsection{Uniform Beliefs, Uniform Risk Tolerances}
The case of identical investors with homogeneous beliefs is instructive because it is representative of a crowded scenario with a population of investors of similar size.
In this scenario, the columns of $\Theta$ are identical, so we write $\Theta=\theta e\transpose_m$. The element $\theta_i>0$ is the shared valuation of asset $i$. As a result, the price is function of the sum of the risk tolerances:
%
%
\begin{align}
	p_i(\rho)= & \frac{e_m\transpose\rho}{2s_i\var_i} \left({ \sqrt{1 +\frac{4s_i\var_i\theta_i}{(e_m\transpose\rho)}} } - 1\right) \notag\\
	=& g_i(e_m\transpose\rho) \label{eq:priceaggregate}
\end{align}
where $g_i(x)= x/(2s_i\var_i)(\sqrt{1 +4s_i\var_i\theta_i/x } - 1)$.
Furthermore the investors are identical in beliefs and initial risk attitudes. Can then write $\gamma=\bar\gamma e_m$ for some $\bar\gamma>0$. Multiply both equations (\ref{eq:generalsystem}) by $e_m\transpose$:
\begin{align*}
	\begin{cases}
		e\transpose_m\dot \xi=e\transpose_m(A-I) \xi(t)\\
		e\transpose_m\dot\rho= e\transpose_mA\xi(t)
	\end{cases}
\end{align*}
The equations simplify considerably:
\begin{align*}
[e_m\transpose A]_k=&\sum_{j=1}^m A_{jk}\\
 = &  \sum_{j=1}^m\sum_{i=1}^n\bar\gamma \frac{\rho_j}{\var_i}  \left(\frac{\theta_{i}}{g_i(e_m\transpose\rho)} - 1\right) \frac{1}{g_i(e_m\transpose\rho)}\frac{ dg_i}{dx }(e_m\transpose\rho) \\
 =&\bar\gamma e\transpose_m\rho \sum_i \left(\frac{\theta_{i}}{g_i(e_m\transpose\rho)} - 1\right) \frac{1}{\sigma_i^2 g_i(e_m\transpose\rho)}\frac{ dg_i}{dx}(e_m\transpose\rho) \\
 =&\bar\gamma e\transpose_m \rho F(e_m\transpose\rho)
\end{align*}
where
$$F(x):=\sum_i \left(\frac{\theta_{i}}{g_i(x)} - 1\right) \frac{1}{\sigma_i^2}\frac{ d}{dx}\log g_i(x).$$
%
and therefore $e_m\transpose A\xi=\bar\gamma e\transpose_m \rho F(e_m\transpose\rho)(e\transpose_m \xi)$. We now define
\begin{align*}
x:=  e_m\transpose\rho\\
y:= e_m\transpose\xi
\end{align*}
And replace those variables in the original equations:
\begin{align}
\boxed{
	\begin{cases}
		\dot y =  m\bar\gamma   [ F(x)x-1] y\\
		\dot x= \bar\gamma F(x)xy  \label{eq:homogeneousbeliefs}
	\end{cases}
}
\end{align}
\begin{figure}
\centering
  \includegraphics[width=0.7\textwidth]{phaseportrait1.pdf}
  \caption{Phase Portrait for System (\ref{eq:homogeneousbeliefs}).}
  \label{fig:activepassive}
\end{figure}
Next, we want to characterize the qualitative behavior of the solution of system (\ref{eq:homogeneousbeliefs}). To do this, we first need to understand the behavior of $xF(x)$.
How does the equation looks like? First, notice that $g_i(x)$ has the following properties:
\begin{align*}
	g_i\ge & 0\\
	\frac{dg_i}{dx}> & 0\\
	\frac{d^2g_i}{dx^2} < & 0\\
	g_i(0)= & 0\\
	g_i(x)\rightarrow & \theta_i \;\text{as}\; x\rightarrow \infty
\end{align*}
next, we characterize $xF(x)$. It is  $xF(x)=\sum_{i=1}^n f_i(x)$, with
%
$$f_i(x):= \frac{x(\theta_i-g_i)}{\var_i g_i} \frac{d}{dx}\log g_i$$
%
We omit the proof of the following properties, which can be verified by calculus:
\begin{align*}
	\frac{d f_i}{dx} <&0\\
	f_i(x) \rightarrow &\infty \quad\text{as $x\rightarrow 0$} \\
	 f_i(x) \rightarrow & 0 \quad\text{as $x\rightarrow\infty$}
\end{align*}
Therefore, $xF(x)$ also is a decreasing function of $x$ with the same properties as $f_i$. Let $x^\star$ the unique solution of the equation $xF(x)=1$. The function $xF(x)-1$ is positive for $x< x^\star$ and negative otherwise.
\subsection{Uniform Beliefs}
The case of investors with homogeneous beliefs is also instructive because it allows to see the impact of investors with different risk sensitivities $\gamma_i$. In this scenario, only the columns of $\Theta$ are identical. We repeat the same steps as the previous section. Only the $\xi$ can be aggregated into $y:=e\transpose_m \xi$. We obtain a simpler system in $m+1$ variables:
\begin{align}
\boxed{
	\begin{cases}
		\dot y =  (\gamma\transpose\rho)[ F(e\transpose_m \rho)-1] y \label{eq:uniformbeliefs} \\
		\dot\rho_j= \gamma_j F(e_m\transpose\rho)y \rho_j  \notag
	\end{cases}
}
\end{align}
%
\subsection{Heterogeneous Beliefs}
Consider the case of several investors with heterogeneous beliefs. The price function can be approximated in the ``small investor'' regime and the ``large investor'' regime as
\begin{align*}
	p_i(\rho)%=& \frac{-e_m\transpose \rho +\sqrt{(e_m\transpose \rho)^2 +4s_i\var_i[\Theta\rho]_i} }{2s_i\var_i}\\
	=& \frac{e_m\transpose\rho}{2s_i\var_i} \left({ \sqrt{1 +4s_i\var_i[\Theta\rho]_i/ (e_m\transpose\rho)^2} } - 1\right)\\
	\simeq &
\begin{cases}
	\dfrac{\sqrt{  [\Theta\rho]_i  }
}{\sqrt{s_i}\sigma_i  }  &||\rho||\ll 1\\
$ $\\
	\dfrac{  [\Theta\rho]_i}{ e_m\transpose\rho} & ||\rho||\gg 1
\end{cases}
\end{align*}
Furthermore the partial derivative is given by
\begin{align*}
	\frac{\partial}{\partial\rho_k}\log p_i=1+ \left({ \sqrt{1 +4s_i\var_i[\Theta\rho]_i/ (e_m\transpose\rho)^2} } - 1\right)^{-1}  \frac{4s_i\var_i (\theta_{ik} - 2e_m\transpose\rho [\Theta\rho]_i)}{ (e_m\transpose\rho)^4}
\end{align*}
The small and large investor approximation for the derivative is
\begin{align*}
	\frac{\partial}{\partial\rho_k}\log p_i=
\begin{cases}
	\dfrac{1}{2}\dfrac{\theta_{ik}}{[\Theta\rho]_i}
	 &||\rho||\ll 1\\
$ $\\
	\dfrac{\theta_{ik}}{ [\Theta\rho]_i}-\dfrac{1}{e_m\transpose\rho} & ||\rho||\gg 1
\end{cases}
	\end{align*}
The equilibria of the system are the solution of
\begin{align*}
	\begin{cases}
		A(\rho)\xi=\xi\\
		 A(\rho)\xi=0
	\end{cases}
\end{align*}
corresponding to $\xi=0$. Consider the linearized system at any point $(\tilde\rho, 0)$, which is (unsurprisingly)
\begin{align*}
\left(
\begin{array}{c}
\dot \xi\\
\dot\rho
\end{array}
\right) =
\left(
\begin{array}{ll}
A(\rho_0)-I & 0 \\
A(\rho_0) & 0
\end{array}
\right)
\left(
\begin{array}{c}
\xi\\
\rho
\end{array}
\right)
\end{align*}
Let $v$ an eigenvector of the matrix with eigenvalue $\lambda$. Split
$$v=\left(
\begin{array}{c}
v_1\\
v_2
\end{array}
\right)
$$

where $v_i\in R^m$.  By substitution, it is immediate to see that the two vectors must satisfy $Av_1=(\lambda+1) v_1$ and $v_2=\lambda^{-1}(\lambda+1) v_1$.

Define $\bar\theta_i:= [\Theta\tilde\rho]_i/ e_m\transpose\tilde\rho$ and $E_{ij}:= \theta_{ij}-\bar\theta_i$, and let $E\in R^{n\times m}$ be the corresponding matrix.
\subsection{Large Investor Limit}
In the ``large investor'' limit, the matrix is
\begin{align*}
A_{jk}:= &  \sum_{i=1}^n\gamma_j q_{ij} \frac{1}{p_i}\frac{\partial p_i}{\partial\rho_k} \\
=& \frac{\gamma_j}{ e_m\transpose\tilde\rho}  \sum_{i=1}^n\ (\theta_{ij} -\bar\theta_i)  (\theta_{ik} -\bar\theta_i) \\
=& \frac{\gamma_j}{ e_m\transpose\tilde\rho}  \sum_{i=1}^n\ E_{ij}E_{ik} \\
=&\frac{\gamma_j}{ e_m\transpose\tilde\rho} [E\transpose E]_{jk}\\
\Rightarrow A= & \frac{1}{ e_m\transpose\tilde\rho} \diag{\gamma} E\transpose E
\end{align*}
The element $[E\transpose E]_{jk}$ can be interpreted as a covariance between the valuations of investor $j$ and $k$.

From this formula, we can immediately see that, if we inflate the errors $E$ by a factor $\kappa>1$, then $A$ becomes $\kappa^2(e_m\transpose\rho)^{-1} \diag{\gamma} E\transpose E$. The eigenvalues correspondingly are inflated by a factor $\kappa$.

If instead we inflate $\rho$ by a factor $\kappa>1$, the eigenvalues are deflated by a factor $\kappa$. Finally, assume that the valuations of the managers are correlated across assets.
\subsection{Small Investor Limit}
In the limit $||{\rho}||\rightarrow 0$ the matrix $A$ is
\begin{align*}
A = \dfrac{1}{2}\diag{\gamma}\Theta\transpose \diag{\sqrt{s_i}\sigma_i/([\Theta\rho]_i )^{3/2} }\Theta
\end{align*}
Hence $A-I$


%--- StreamPlot[{y, .5*(.2/(x +.3)-1)y},{x,-2,5},{y,-5,5}]

%
%
%\section{Analytic Solution for the Case of Independent Assets}
%\begin{description}
%	\item[Belief Propagation:] Investors $i=2,\ldots, m+1$ are ``active'', in that they communicate, update their valuations based on the outcome of this communication and trade accordingly. During each period, they observe the consensus value and update their priors using the formula
%	\begin{align}
%		 \theta_i(t+1)=&\alpha \theta_i(t)+(1-\alpha)\bar\theta(t)
%%		 \theta_i(t+1)=&\alpha_1 \theta_i(t)+ \alpha_2 r(t) + (1-\alpha_1-\alpha_2)\bar\theta(t)
%	\end{align}
%	where
%	\begin{align*}
%\bar \theta(t) :=&\sum_i\rho_i(t)\theta_i(t)/\sum_j \rho_j(t)\\
%		r(t) :=& p(t) / p(t-1) - 1
%	\end{align*}
%\end{description}
%We solve the problem in the special case of a single asset, investors with the same belief propagation parameters and risk tolerances that are relatively close, and an uninformed investor that simply switches from a high valuation state to a low valuation state for the asset. This is not just a toy model, as it applies also to the case of multiple uncorrelated securities, e.g., to the analysis of idiosyncratic returns.
%
%Let $H$ a matrix with $(H)_{ij}=\delta_{ij}\rho_i$. We assume that $\alpha_i=\alpha_j$ for all investors. Let
%\begin{align}
%A=	\left(
%\begin{array}{cc}
%1 & 0 \\
%0 & \alpha I +(1-\alpha) ee\transpose/m
%\end{array}
%\right)
%\end{align}
%We also assume there is a single stock with volatility $\sigma$. For a fixed vector of risk tolerances $\rho$, the equations become
%\begin{align}
%	\theta(t) = &   A^t
%	 \theta(0)\\
%%	\rho({t+1})= & \rho(t) \circ F(R(t)) \\
%	x^\star(t)  = & \sigma^{-2} H(t) (\theta(t)-p(t)) \\
%	p(t) = & \frac{1}{e\transpose\rho(t)} \left(\rho\transpose(t)\theta(t)- \var s
%	\right)
%\end{align}
%The eigenvalues and eigenvectors of $A$ can be fully characterized. Let $e_{-1}$ be an $(m+1)$-dimensional vector where all elements are equal to $1/\sqrt m$, except for the first element, which is equal to zero. Let $f_1,\ldots f_{m-1}$ be an orthonormal basis for the subspace orthogonal to $e_1, e_{-1}$; i.e., $f_i\transposef_j=\delta_{ij}$, $e_1 f_i=0$,  $e_{-1} f_i=0$. The following pairs are eigenvectors/eigenvalues of $A$: $(e_{1},1)$, $(e_{-1},1)$, $(f_i, \alpha)$, $i=1,\ldots, m-1$. In the following, we denote $\theta_{1}=(\theta_1, 0, \ldots,0)$, $\theta_{-1}=(0,\theta_2,\ldots,\theta_{m+1})$, $\bar\theta_{-1} =m^{-1}\sum_2^{m+1} \theta_i$.
%The spectrum of $A$ implies that $\lim_{t\rightarrow \infty} \theta_i(t) =\bar\theta_{-1}(0)$. This can be interpreted as convergence of belief propagation to the empirical average of active investors. The proof is below.
%\begin{align}
%	\theta(t) = &   A^t\theta(0)  \\
%	= &   (e_1e_1\transpose + e_{-1}e_{-1}\transpose + \alpha^t\sum_i f_i f_i\transpose) \theta(0) \\
%	= & e_1\theta_1(0) + \sqrt m e_{-1} \bar \theta_{-1}(0)  + \alpha^t(\sum_i   f_i f_i\transpose)\theta_{-1}(0)  \\
%	= &  v + \alpha^t w
%\end{align}
%where
%\begin{align}
%	w :=& (\sum_i   f_i f_i\transpose)\theta_{-1}(0)\\
%	v:=&(\theta_1(0), \bar\theta_{-1}(0), \ldots, \bar\theta_{-1}(0))\transpose
%\end{align}
%%where $\kappa:= ({e\transpose\rho})^{-1}\rho\transpose\sum_i f_i f_i\transpose \theta_{-1}(0)$
%The valuation stays constant for the first investor, while converges at a geometric rate to the average valuation for the remaining investors. $w$ is the projection of active valuations on the subspace orthogonal to $e_{-1}$. The element $w_i$ can be interpreted as the deviation of $\theta_i$ from $\bar\theta_{-1}$. Note that $e'w=0$.
%
%Finally, note that $\sum_i   f_i f_i'$ is the orthogonal projection onto the subspace orthogonal to $(e_1, e_{-1})$. This suggests that we can express it more simply. The projection on the subspace spanned by $(e_1, e_{-1})$ is give by $BB'$, where $B := (e_1 | e_{-1})$. But then the projection to the orthogonal subspace is $I-BB'$. Hence $\sum_i   f_i f_i'= I- BB'$. From this we can write an explicit formula for $w$:
%\begin{align}
%	w := & \theta_{-1}(0) -  BB'\theta_{-1}(0) \\
%	  = & \theta_{-1}(0) - B
%	  \left(
%	  \begin{array}{c}
%	  	0 \\
%	  	\sqrt m \bar \theta_{-1}(0)
%	  \end{array}
%	  \right)\\
%	  & = \theta_{-1}(0) - \bar \theta_{-1}(0)
%	\end{align}
%So $w$ is the excess valuation vector for the active investors.
%For the time being, assume that the risk propensity is constant and that $\theta_0(0)$ is also constant. We can write down explicitly the price evolution via spectral decomposition of $A$, and then verify retroactively that the conditions for deleveraging conditions do not occur.
%\begin{align}
%	p(t) = & \frac{1}{\rho'e} \left(\rho' A^t\theta(0) - \var s
%	\right) \\
%	= & \frac{1}{\rho'e} (\rho' (v+\alpha^t w) - \var s )\\
%	= & \frac{1}{\rho'e}(\rho' v-\var s +\rho'w \alpha^t)
% \end{align}
% We isolate the contribution to price of the uninformed investor. Let $\gamma:=\sum_{i=2}^{m+1}\rho_i$.
%\begin{align}
%	p(t) = & \frac{1}{\gamma +\rho_1}(\rho_1\theta_1(0) + \gamma \bar\theta_{-1}(0) -\var s +\rho'w \alpha^t)
%\end{align}
%Notice that $\rho'w$ does not depend on $\rho_1$ or $\theta_1(0)$.  Hence, the price dynamics in the absence of deleveraging depends only on a few aggregate statistics:
%\begin{itemize}
%	\item the aggregate risk propensity $\gamma$ of active investors,
%	\item the risk propensity $\rho_1$ of the passive investor,
%	\item the consensus valuation $\bar\theta_{-1}$ of the active investors,
%	\item the valuation $\theta_1$ of the passive investor,
%	\item the total dollar volatility $\var s$ of the asset,
%	\item the covariance $\rho'w$ between risk tolerances of active investors and their initial excess valuations.
%\end{itemize}
%The price will either increase (decrease) monotonically or decrease monotonically if $\rho'w$ is negative (positive). The term is negative when investors with the lowest risk tolerance have higher valuation for the asset. Because the most bullish investors (regarding valuations) are the most risk averse, their comparatively lower demands are unable to bring up the price. Over time, the valuations converge, and the risk-seeking investors gradually increase their value.
%
% In the absence of changes in risk tolerances, the asset return is
% \begin{align}
% 	r(t) & =   \frac{\rho'w \alpha^t(\alpha-1)}{\rho' v-\var s +{\rho'w}\alpha^t }
% \end{align}
%This will be positive (negative) depending on whether $\rho'w$ is positive (negative).
%
%and the position for $i=1,\ldots,m+1$ is
% \begin{align}
%	 x^\star_i(t)  = & \frac{\rho_i}{\sigma^{2}} \left( \left(v_i - \frac{\rho' v}{\rho'e}\right) +\frac{\var s}{\rho'e}+\left(w_i-\frac{\rho'w}{\rho'e}\right)\alpha^t \right)
% \end{align}
%For $i>1$, the term $v_i -{\rho' v}/{\rho'e} +{\var s}/{\rho'e}$ is independent of $i$. The position is then proportional to the risk tolerance and to the excess starting valuation $w_i$. Less risk averse investors and more bullish (compared to consensus) are the ones with the largest positions; but over time, bullishness becomes irrelevant, whereas risk propensity still determines relative size.
%
%Deleveraging  will occur whenever a change in risk tolerance of the uninformed investor will be high enough to trigger the event in the informed investors. To illustrate a deleveraging event in detail we focus on a specific scenario. Consider first the case $\rho'w>0$, so that returns are positive before the event. Also, assume that
% \begin{align}
% \theta_1(0) < \bar\theta_{-1}(0) + \rho_1^{-1}(\var s - \rho'w \alpha^t)
%\end{align}
%In other words, the uninformed investor does not have a much higher valuation than the consensus of the informed investors. Under these conditions, an \emph{increase} in risk tolerance of the uninformed investor will result in a price decrease. The intuition is that the uninformed investor becomes bigger and ``crowds out'' the informed ones, who have a higher valuation for the asset.
%
%
%Let us denote the uninformed investor's tolerance change by $\delta\rho_1 >0$ and $\sum_i\rho_{Li}:=\gamma -\delta\gamma$. The new price becomes
% \begin{align}
%	p(t) = & \frac{1}{\gamma + \rho_1 -\delta\gamma+\delta\rho_1}((\rho_1+\delta\rho_1)\theta_1(0) + (\gamma-\delta\gamma) \bar\theta_{-1}(0) -\var s +\rho'w \alpha^t)
%\end{align}
%It is easy to show that the new asymptotic price is lower than the original one. The evolution of the asset price is shown below.
%

  \end{document}