When panic makes you blind:
a chaotic route to systemic risk

The system is composed by $N$ financial institutions (banks) creating portfolios by investing in some of the $M$ available risky assets. In the following, we adopt the convention of labeling risky investments as $i,j,k,\ldots$ and banks as $a,b,c,\ldots$. All financial institutions are assumed to be equivalent, that is they have the same initial equity capital and the same capital requirements, they solve the same portfolio problem and they use the same scheme in forming risk expectations. This is of course a simplifying assumption in order to achieve analytical tractability. However, [Corsi et al., 2016] show how heterogeneity does not affect crucially the results of the model.

We assume that banks update their expectations of risk at time intervals of unitary length and accordingly they make new decisions about leverage and portfolio diversification (see Fig. 1). In the time interval $(t,t+1]$, ($t\in\mathbb{Z}$), banks rebalance their portfolio to target the leverage, but without changing the risk expectations. The rebalancing takes place in $n$ time subintervals within $(t,t+1]$, i.e. at $\{t+\frac{1}{n},t+\frac{2}{n},t+\frac{3}{n},...,t+\frac{n}{n}\equiv t+1\}$. $\frac{1}{n}$ is the timescale associated with the portfolio rebalancing (the red axis in Figure 1) whereas the unit value is the timescale associated with the portfolio decisions (the black axis in Figure 1). Clearly, $n\in\mathbb{N}\setminus\{0\}$. As clarified below, [Corsi et al., 2016] considered the case of removing the endogenous process of formation of risk expectations, i.e. portfolio decisions are taken without consideration on the price dynamics. This is equivalent to optimize the portfolio only once with exogenous risk expectations and the process of portfolio rebalancing follows without changing the value of the optimal leverage.

Here we generalize the model of [Corsi et al., 2016] by specifying how banks form risk expectations in Eq. 2.6 to take portfolio decisions. Several empirical and experimental studies, for example [Hommes, 2009, Bao et al., 2013], have shown that financial agents use statistical models of past observations of the price to forecast the future, the so-called backward-looking expectations. Here we introduce a similar expectation scheme which exploits past observed prices in forecasting future risks.

By summarizing the building blocks of our model, the dynamics of the financial system is described by a slow-fast random dynamical system at discrete time which is specified by the following equations,

where $\widehat{\Sigma}_{d,t}$ and $\widehat{\Sigma}_{u,t}$ are the maximum likelihood estimators of the (diversifiable) variance and covariance of returns aggregated at the time scale of portfolio decisions and can be obtained as explained in Appendix A, see Eq. A.19.

Eq. 2.17 describes the dynamics of banks’ portfolio decisions, i.e. the slow component of the dynamics described by the slow variables $\lambda_{t}$, $\Sigma_{d,t}^{\omega}$, and $\Sigma_{u,t}^{\omega}$. Eq. 2.18 describes the price evolution of the risky investments, i.e. the fast component of the dynamics described by the fast variables $r_{i,s}$.

The model must satisfy a stationarity condition, namely covariance stationarity of the autoregressive process, which is $\lambda<\gamma+1$⁸⁸8The VAR(1) process in Eq. 2.18 is covariance stationary when the largest eigenvalue of the matrix $\bm{\Phi}$ is smaller than one, that is equivalent to assume $\lambda_{t}<\gamma+1$ as shown in [Corsi et al., 2016].. It is also $\lambda\;\geqslant\;1$ and $0\;\leqslant\;m\;\leqslant\;M$, by construction.

Summarizing the main steps of the dynamics: (i) portfolio decisions at time $t-1$ determine the value of leverage depending on risk expectations, see Eqs. 2.17, and (ii) affect the price dynamics between $t-1$ and $t$ because of the autoregressive coefficient $\bm{\Phi}(\lambda_{t-1},\Sigma^{\omega}_{d,t-1},\Sigma_{u,t-1}^{\omega})$, of the return process, see Eqs. 2.18; (iii) at time $t$, banks estimate asset risks by using past observations of prices, $\widehat{\Sigma}_{d,t}(\{\bm{r}_{t-1+k/n}\}_{k=1,2,...,n})$ and $\widehat{\Sigma}_{u,t}(\{\bm{r}_{t-1+k/n}\}_{k=1,2,...,n})$. Then, (iv) banks form new risk expectations $\Sigma^{\omega}_{d,t}$ and $\Sigma^{\omega}_{u,t}$, and (v) make new portfolio decisions at time $t$.

We analyze the model in the asymptotic limit via a dynamical systems approach by investigating the map in Eq. 4.2. Table 1 shows the values of the parameters used to illustrate the main properties of the model.

A comprehensive insight of the skeleton dynamics is obtained by looking at the bifurcation diagram in the top panel of Figure 2. The bifurcation diagram of a map shows the long run dynamics after a transient time has passed as a function of a parameter. For our model we can use as the bifurcation parameter either the memory parameter $\omega$ or the VaR parameter $\alpha$, or the diversification cost $c$. Later, we will show the model dynamics depending on $\alpha$ and $c$ to have more insights regarding policy implications.

The top panel of figure 2 shows the bifurcation diagram as a function of $\omega$. Decreasing the memory parameter, a period-doubling cascade to chaos occurs. In other words the deterministic skeleton of the financial system shows cycles of increasing complexity with more and more periods until chaos occurs¹⁰¹⁰10In this analytical approach to the study of dynamics of the financial system, the domain for leverage and risk expectations is $\{\lambda,\Sigma^{\omega}_{d},\Sigma_{u}^{\omega}\}\in[1,\gamma+1)\times\mathbb{R}^{+}\times\mathbb{R}^{+}$. In particular, bounds for leverage are equivalent to the conditions of stationarity for the process of Eq. 2.18.. The signature of the chaotic behaviour is the positive Lyapunov exponent associated with it. The bottom panel of Figure 2 shows the estimated Lyapunov exponent $\Lambda$ as a function of $\omega$ and positive values of $\Lambda$, observed for small $\omega$, signal the presence of deterministic chaos.

Note that the unit of $\omega$ in Figure 2 is the inverse of the time scale of the portfolio decisions, see Eq. 2.16. By using as unit the time scale of portfolio rebalancing, the time scale becomes $\tilde{\omega}=\omega^{\frac{1}{n}}$. For example, if banks rebalance their portfolios at daily frequency but the portfolio decisions are taken once a month, a memory parameter $\tilde{\omega}=0.97$ at the daily time scale corresponds to an effective memory of $\tau\approx 33$ days. The corresponding value of the memory parameter at the monthly time scale is $\omega\approx 0.4$. Thus small values of $\omega$, which in the bifurcation diagram corresponds to cycles or chaos, are associated with values of $\tilde{\omega}$ close to $1$.

One can understand the behaviour in Figure 2 by considering that, when memory is large enough, banks learn from past history and, at least in the long-run, they make exact forecasts of future risks. Thus large memory stabilizes the financial system and portfolio decisions about leverage and diversification reach a fixed point equilibrium. When $\omega$ decreases below a given threshold, risk expectations with smaller memory break the fixed point equilibrium by inducing cycles of period two. Indeed, a shorter memory has the potential of creating panic-induced fall in portfolio holdings reflecting sudden decrease in leverage and diversification. When the adopted financial leverage and the overlap between banks’ portfolios are high, the price impact of the portfolio rebalancing increases significantly the observed risks. Small memory in risk expectations tends to overestimate future risks and as a consequence banks reduce suddenly their positions. Hence, leverage and diversification suddenly decrease. The opposite situation occurs when leverage and diversification are small due to previously overestimated risk. The 2-period cycle is characterized by a mismatch between the expectation and the realization of portfolio volatility and leverage cycles reflect the mismatch between the banks’ perceived risk and the ‘true’ risk.

When the memory decreases further, our model predicts cycles of larger periods and, eventually, a chaotic dynamics. Notice that the amplitude of both periodic and chaotic leverage cycles increases by decreasing $\omega$. When $\omega>\omega^{*}$, $\lambda$ oscillate within $[1,\gamma+1)$ and the covariance stationarity of the return is guaranteed, while when $\omega<\omega^{*}$ the process becomes non-stationary. Hence, $\omega^{*}$ corresponds to a transition from stationarity to non-stationarity for the model dynamics. In the left panel of Figure 3, we show the contour map of $\omega^{*}$ as a function of the square root of the idiosyncratic variance $\sqrt{\Sigma_{\epsilon}}$ and the total number of investments $M$. The larger is $\sqrt{\Sigma_{\epsilon}}$, the smaller is the value of $\omega^{*}$: this means that larger exogenous risk reduces the value of the financial leverage because of the Value at Risk constraint and, as a consequence, the feedback effects of leverage targeting. More surprisingly, the value of $\omega^{*}$ increases with $M$, that is introducing financial innovations and new instruments requires investors with larger memory in forming risk expectations to keep the system in stationary equilibrium. Indeed, a larger number $M$ of available asset investments leads to an underestimation of risk because of larger portfolio diversification which is followed by an increase of the financial leverage. Then, the combined effects of the overlap of banks’ portfolios with the price impact of the leverage targeting make the financial system more unstable.

The breaking of fixed point equilibrium occurs by a period-doubling bifurcation or flip bifurcation (see for example [Crawford, 1991]). In the case of $\omega$ as bifurcation parameter, we refer to the value of the memory parameter for which the period-doubling bifurcation occurs as $\omega_{2}$. The type of bifurcation can be analyzed by studying the Jacobian associated with the map of Eq. 4.2. The map describes a three-dimensional dynamical system but one eigenvalue of the Jacobian is identically null which suggests that the two equations associated with the process of expectation formation in Eq. 4.2 are not independent at the first order in the expansion. The positive eigenvalue describes the eigenspace associated with the independent component of these equations. The dynamics in this eigenspace is characterized by a fixed point since the corresponding eigenvalue is inside the unit circle for all $\omega$. The third eigenvalue is instead negative. The right panel of Figure 3 shows the non vanishing eigenvalues of the Jacobian as a function of $\omega$. When the negative eigenvalue hits the critical value $-1$ for $\omega=\omega_{2}$, a period-doubling bifurcation occurs. After that, the attractor for the dynamics is not the fixed point anymore and a new solution appears describing the 2-period cycles. The new solution is the fixed point for the map of Eq. 4.2 iterated twice. In a self-similar way, the period-doubling bifurcation repeats again but for the iterated map giving rise to the period-doubling cascade to chaos. The phenomenon is very general, was noticed for the first time by studying the logistic map [May, 1976] and was described in [Feigenbaum, 1978].

We also study the stability properties of the financial system by analyzing the value of $\omega_{2}$ as a function of the other model parameters. In the top left panel of Figure 4, we show the relation between the value $\omega_{2}$ and the liquidity parameter $\gamma$. The larger is the market liquidity of risky investments, the less important are the effects related to the expectation feedbacks. Hence, the larger is the market liquidity, the more stable is the dynamics of the financial system. Intuitively, when the price impact of the portfolio rebalancing to target the financial leverage does not affect significantly the investment prices, then volatility and portfolios do not fluctuate much. Liquid markets are more stable because endogenous impact of market players’ strategy affects less the risk [Danielsson et al., 2012a, Danielsson et al., 2012b]. Finally notice that, since the transition values, such as $\omega_{2}$, depend on liquidity, an abrupt change in liquidity can drive the market from a stable to an unstable regime. In the top right panel of Figure 4, we show the relation between $\alpha$ and $\omega_{2}$. The decreasing behaviour indicates that when Value at Risk is less stringent, the financial market becomes dynamically stable if the memory used by agents to build expectations is longer.

In the bottom panels of Figure 4, we show the fixed point analysis of the analytical mapping in Eq. 4.2 in the neighbourhood of the first bifurcation point. We compare the case of adaptive expectations of risk described in Eq. 2.15 with naive expectations, i.e. the case of zero memory ($\omega=0$ in Eq. 2.15). The bottom left panel shows the stable points of the map (black dots) for the variable $\lambda$ when memory is different from zero, i.e. the fixed point of the map in Eq. 4.2 or the periodic orbit that is the fixed point for the map iterated twice. When the bifurcation parameter $\alpha$ decreases below the threshold determining the period-doubling bifurcation, the new solution for the map iterated twice appears and the fixed point of the map becomes unstable (red dots). This behaviour holds for any $\omega$ different from zero. The bottom right panel of Figure 4 shows the case of naive expectations of risk. At the bifurcation point, no solutions for the map iterated twice exist. However, the fixed point is not attractive anymore. In this case, the dynamics of the system is cyclical but characterized by amplitude increasing in time¹¹¹¹11Clearly, the map of Eq. 4.2 makes sense as long as the variables are in the domain.. Hence, when financial agents behave naively in risk forecasting, the breaking of fixed point equilibrium corresponds to a transition from stationary to non stationary dynamics for the financial system.

From the point of view of financial policy, our results can be summarized as follows.

• •

  Decreasing the parameter of the VaR, thus allowing a less stringent capital constraint, makes financial system more unstable. For example, in the described financial system characterized by an average investment volatility of $3\%$ in the holding period of the portfolio, diversification costs about $10\%$ of the equity value and a memory in risk expectations equal to $\omega=0.1$, relaxing the probability of loss $P_{VaR}$ from $5\%$ to $6\%$ ($\alpha=1.64\rightarrow 1.5$) breaks the fixed point equilibrium inducing leverage cycles. Furthermore, allowing probability of loss associated with Value at Risk of about $7-8\%$ leads the financial system towards chaotic evolution, see the left panel of Figure 5.

• •

  Decreasing the cost of diversification tends to increase the dynamical instability of the financial system. Let us notice in the right panel of Figure 5 that decreasing diversification cost, e.g. from $10\%$ to $5\%$ of equity value, breaks systemic equilibrium. A similar effect is obtained by introducing financial innovations and new instruments which increase the number of available asset investments $M$, inducing an underestimation of risk because of larger diversification, see Figure 3.

• •

  Imposing to financial institutions operating in the market to adopt larger memory in their expectation schemes for risk forecasting has always stabilizing effects on market stability. Not surprisingly, increasing $\omega$ means to take into account more information about the system, converging asymptotically to the fixed point equilibrium. However, larger memory means slower reaction to risk news. What could seems meaningless from a micro-prudential point of view, as it is if adopted only by one or few financial institutions, becomes convenient in terms of dynamical stability when all institutions behave similarly. Thus, our analysis suggests that an appropriate financial policy is necessary to promote a systemic behaviour which prefers smooth adjustments driven by more information to overreactions driven by news.

The dynamics of the reduced model in the limit $n\rightarrow\infty$ is governed by the following one dimensional map for $\lambda_{t}$,

where $\lambda\in[1,\gamma+1)$ and $\Sigma_{\epsilon}=\lim_{n\rightarrow\infty}n\sigma_{\epsilon}^{2}$ is the variance of the idiosyncratic noise at the slow time scale of portfolio decisions, see Eq. 2.19. In Appendix A, we show how to compute analytically the variance of the aggregated return $\sum_{q=1}^{n}r_{t-1+q/n}$ in the limit $n\rightarrow\infty$ to obtain Eq. 4.3. The map $f$ in explicit form provides valuable analytical insights into the dynamical behavior of the system. The fixed point of the map, i.e. $\lambda^{*}=f(\lambda^{*};\omega,\alpha,\gamma,\Sigma_{\epsilon})$, corresponds to the asymptotic equilibrium when $f^{\prime}(\lambda^{*};\omega,\alpha,\gamma,\Sigma_{\epsilon})$ lies inside the unit circle. When $f^{\prime}(\lambda^{*};\omega,\alpha,\gamma,\Sigma_{\epsilon})$ hits the value $-1$, the period-doubling bifurcation occurs. The left plot of Figure 6 shows the shape of the map, under the linear transformation $x\equiv\frac{\lambda-1}{\gamma}$, as a function of $\omega$. Notice that the value of the fixed point does not depend on $\omega$ and the maximum of the map is a decreasing function of $\omega$. Furthermore, the value $\omega^{*}$ at which the transition to non stationarity occurs corresponds to the value of $\omega$ at which the maximum of $f$ is equal to $\gamma+1$ (or the unit value when we rescale the invariant interval of the map as in Figure 6).

It is interesting to notice that the complex behavior of the period-doubling cascade to chaos may occur when the exogenous variance of the idiosyncratic noise is below a threshold value. In the right panel of Figure 6 we show $\omega_{2}$ and $\omega^{*}$ as a function of $\sqrt{\Sigma_{\epsilon}}$. When we approach the threshold (black dotted line), $\omega_{2}$ and $\omega^{*}$ tend to coincide and both values become arbitrarily low. Above the threshold, no breaking of the fixed point equilibrium occurs. The curves describing $\omega_{2}$, $\omega^{*}$, and the threshold partition the parameter space in regions characterized by specific dynamical properties, see Fig. 6.

For large but finite $n$, the estimator $\hat{\bm{\Sigma}}_{t}$ of the covariance matrix fluctuates around the covariance matrix $\widetilde{\bm{\Sigma}}$. These fluctuations in risk estimation affect portfolio decisions about leverage and diversification and, as a consequence, also the dynamics of the financial system. Here, we want investigate how much fluctuations in risk estimations affect the deterministic skeleton of financial system. In the following, we present an analytical argument to answer this question. Then, in Subsection 5.2 we justify it by comparing the analytical results with Monte Carlo simulations of the reduced version of the model.

The asset return dynamics is governed by the VAR process of Eq. 2.18. By estimating the autoregressive process between $t-1$ and $t$ when $n$ is finite, financial institutions can obtain the exogenous components of the VAR (1) process, i.e. the residuals $\{\epsilon_{i,s},f_{s}\}_{s=t-1+k/n,\>k=1,2,...,n}^{i=1,...,M}$¹²¹²12From a mathematical point of view, the process 2.18 is equivalent to the process for the endogenous components of returns, see Equation 2.11. As shown in [Corsi et al., 2016], it can be decomposed in $M+1$ independent AR(1) process. The residuals of Eq. 2.18 are simply linear combinations of the residuals associated with the $M+1$ AR(1) processes.. The estimator of the idiosyncratic and systematic risks at the time scale of portfolio rebalancing is the realized variance of the residuals,

Since by assumption residuals are i.i.d. and normal, the quantities $(n-1)\hat{\sigma}_{\epsilon}^{2}/\sigma_{\epsilon}^{2}$ and $(n-1)\hat{\sigma}_{f}^{2}/\sigma_{f}^{2}$ follow a chi-squared distribution $\chi^{2}_{n-1}$ with $n-1$ degrees of freedom, and the $90\%$ confidence interval of $\hat{\sigma}_{\epsilon,f}^{2}$ is

We use this confidence interval as a measure of the fluctuations in risk estimation. Let us notice that it goes to zero when $n$ goes to infinity, as numerically confirmed in the simulation of the model. Fluctuations in the estimation of the covariance matrix in Eq. 4.1 are obtained according to formulas in Appendix A, see Eqs. A.5, A.6, A.16¹³¹³13From the point of view of VAR(1) process estimation, model variables $\lambda_{t-1},m_{t-1}$ in formulas in Appendix A correspond to parameters of the VAR(1) process, $\Phi_{t-1}$ in Equation 2.18. Similarly to the variance of residuals $\hat{\sigma}_{\epsilon}^{2}$ and $\hat{\sigma}_{f}^{2}$, in the ideal process of estimating $\bm{\Phi}_{t-1}$, we obtain in turn the estimator $\widehat{\bm{\Phi}}_{t-1}$ within confidence intervals. For simplicity, we make mean field approximation by assuming a picked distribution on the real values..

Portfolio decisions are directly affected by fluctuations in risk estimation. Through the analytical mapping of Eq. 4.2 and by using $\Sigma_{\epsilon}\equiv n\sigma_{\epsilon}^{2}$ and $\Sigma_{f}\equiv n\sigma_{f}^{2}$, we obtain the range in which leverage varies depending on $\delta\hat{\sigma}_{\epsilon,f}^{2}$. That is, the $90\%$ confidence interval of leverage, $\delta\lambda$, with respect to the skeleton dynamics when $n$ is finite¹⁴¹⁴14$\delta\lambda$ is obtained by differentiating Eq. 4.2 with respect to $\lambda_{t}$ and $\sigma_{\epsilon,f}^{2}$..

Figure 7 shows the deterministic skeleton and the fluctuations around it when $n$ is finite. When $n$ is large, the dynamical evolution of the system is very close to its skeleton. By decreasing the value for $n$, fluctuations become more and more important. Empirically we observe that $\delta\lambda$ is inversely proportional to $n$. However, at least for the 2-period cycles, properties of the dynamical model are conserved when $n$ is of order $\mathcal{O}(10^{3})$ or larger.

Notice that the approach presented here can not be applied to the chaotic dynamics. In dynamical systems theory, when chaos occurs, it is not possible to describe the system through the linearization around an equilibrium because there is not anymore a generalized equilibrium point (such as a stable fixed point or a periodic orbit), see [Eckmann and Ruelle, 1985]. Hence, this analytical argument based on the perturbation of a generalized equilibrium can not be applied when the deterministic skeleton is chaotic.

In this Subsection we study numerically the model for finite $n$. For tractability, we focus on the reduced version of the original model by considering one financial institution investing in one risky asset. In the reduced model, we lose the aspects related to the diversification of the portfolio. In turn, the comparison of numerical simulations with the theoretical results of the previous Subsection is easily obtained, as it will be clear below. Finally, this approach via the reduced model focuses more clearly on the role of $n$ in systemic stability of the financial system. In the numerical simulations, we assume that the bank correctly perceives the autoregressive dynamics of the asset price and the variance $\hat{\sigma}_{t}^{2}$ is obtained by estimating the AR(1) process on observed returns, see Eq. 2.19. We consider for simplicity the case of large memory in risk expectations ($\omega=0.4$) in such a way that the deterministic skeleton is a fixed point equilibrium. The case of 2-period cycles is equivalent from the point of view of the analytical approximations.

In Figure 8 we show the fluctuations of the financial leverage $\delta\lambda$ as a consequence of the fluctuations in the estimator of risk. We compare the analytical approximation (black dots) with Monte Carlo simulations (blue dots). The analytical approximation is obtained via Eq. 5.2 with $M=1$ risky investment. Numerically, we obtain $\hat{\delta\lambda}$ in the following way. We simulate the reduced version of the slow-fast random dynamical system. Then, we collect the data after the initial transient is passed. Hence we obtain the empirical probability distribution $\hat{F}_{\lambda}$ associated with a simulated path $\{\hat{\lambda}_{t}\}_{t=0,...,T}$. Coherently with $\delta\lambda$ obtained via the analytical approximation, we determine $\hat{\delta\lambda}$ as the $90\%$ confidence interval

The theoretical prediction and the numerical results agree when $n\;\geqslant\;10^{4}$.