A. Technologies and information structure

The economy lasts two periods t = 1, 2 and is endowed with 2 N linear production technologies that use consumption goods at t =1 as input and produce consumption goods at t = 2 as output. Technologies are evenly distributed distributed across N markets m = 1,…, N, so that each market is associated with exactly two technologies j = 1, 2. Each technology is uniquely identified by a pair (m, j). In every market m, one technology is low risk and the other excess risk. The return on the excess risk technology in m is a mean preserving spread of the return on the low risk technology in m. The realization of a random type θ_m ∈ {1, 2} decides that technology j = θ_m is the excess risk technology in m (and thus j = 3 − θ_m is the low risk technology in m). Call Θ = (θ₁,…, θ_N) the vector of types and T = {1, 2}^N the set of such vectors. Call r_m,j the return between t =1 and t = 2 of technology (m, j) and assume that

The random variables i_m and e_m are i.i.d. across markets, while S is the systemic shock common to all markets. All shocks are independent of each other. I normalize to unity the expected returns by assuming that E[i^m] = E [e^m] = E[S] = 1 and I denote with ${\sigma }_i^2$, ${\sigma }_e^2$, ${\sigma }_S^2$ the variances of the shocks.

Figure 2 gives a graphical representation of markets and technologies. The setup is open to various interpretations. For instance, markets can represent different products, or different groups of customers segmented by geographical area, wealth or income. For every market, some choices of the production and marketing technologies are more risky than others. For example, a certain production technology may rely too much on inputs whose prices are very volatile, or a certain marketing strategy may target customers whose payment ability is very uncertain. Excess risk is modeled as a mean preserving spread over the low risk technology, in fact

where the excess risk z_m = i_m(e_mS − 1) satisfies

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Economy divided in N = 4 markets. Within each market there are two linear technologies j = 1,2 with random productivity r_m,_j. The realized type is Θ = (2, 1, 2, 1) so technology j = 2 has excess risk in markets 1, 3 and j = 1 has excess risk in markets 2, 4.

Citation: IMF Working Papers 2010, 266; 10.5089/9781455210732.001.A001

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Given a realization of r_m,_3−θm the source of excess risk in z_m is twofold: excess idiosyncratic risk e_m and excess systemic risk¹ S.

The joint distribution of types is summarized by a probability function F (Θ) that satisfies a symmetry condition

for every Θ ∈ T. Condition (3) simply requires that the probability of realization of a type Θ equals the probability of realization of its opposite type $\overline{\Theta }$ defined as $\overline{\Theta }=3-\Theta$. The ad hoc assumption (3) greatly simplifies our calculations but is not very restrictive, since it encompasses a wide range of behaviors for the joint distribution of types. For instance, consider the case of perfect positive correlation among market types θ_m. Here the support of F (·) is the pair of vectors Θ₁ = (1,…, 1) and Θ₂ = (2,…, 2). Assumption (3) then simply requires that $F\left({\Theta }_1\right)=F\left({\Theta }_2\right)=\frac{1}{2}$. To the other extreme, the case of i.i.d. types automatically satisfies (3), since $F(\Theta )=\frac{1}{2^N}$for any Θ ∈ T.

We are ready to describe the information structure of the economy, i.e. the way in which information about the realization of Θ flows to the agents in the economy. For now, we conveniently assume that the information structure is exogenously imposed to the economy, but in Section D I will derive in detail the microeconomic mechanism that allows this structure to emerge as an equilibrium outcome of the model.

First of all, at the beginning of time 1 all agents are informed that the current realization of types satisfies Θ ∈ T_Θ where

In other words, after the types are realized the only initial uncertainty for agents is whether the realized type is the true Θ or its opposite. Notice that, conditional on this information, assumption (3) implies that the two types are equally likely. Next, agents have two ways to further refine their knowledge. The first is to become an active investor, a choice that reduces the investor’s wealth to a fraction² 1/ξ^A of its original value, with ξ_A > 1, and allows the observation of a private signal I^A(Θ) such that

Alternatively, an investor can choose to be passive. Passive investors face investment costs that reduce their wealth to a fraction 1/ξ_P, with 1 < ξ_P < ξ_A, but observe a public signal I^P(Θ),

where I^ρ is a binary random variable, independent on Θ, which equals 1 with probability ρ ∈ [0,1] and zero otherwise. Active investors are able to take fully informed investment decisions, but have to pay a high cost. Instead, passive investors pay a smaller cost but can rely only on the public signal I^P, which is less precise. In fact, I^P is informative if and only if I ^ρ = 1, an event realized with probability ρ, while if I ^ρ = 0 the public signal is uninformative and the types Θ and $\overline{\Theta }$ remain perfectly confounded. The precision of the public information is summarized by the revelation rate ρ. More formally, signal observation refines an agent’s believe about the true realization of the types. In particular, an agent who observes a signal I ∈ {0, Θ} learns that Θ ∈ T_Θ,_I where

Define η ∈ [0,1] the fraction of active investors at time 1. To make public information a positive spillover of private information collection I assume that

for A^E ∈ (0,1). As mentioned above, the initial observability (4), the particular binary choices for the public signal I^P ∈ {0, Θ} and the form of the spillover function (8) are all derived from first principles in Section D.

All agent have to pay a cost to access investment opportunities, but active agents, who have superior information, face higher costs. For this reason, we can identify with ξ > 1 the relative information cost, where

In terms of points b) and c) in Section II, financial innovation appears to be associated with in costs ξ_A and ξ_P, but not with a decrease in ξ. The remaining point a), relating financial innovation to greater diversification opportunities, can be modeled in the following way. Assume that each investor draws a specialization h in market h = 1,…, N. Call p_h the fraction of investors with specialization h. Agents are constrained to invest at most a fraction $1-\overline{\alpha }$ of their portfolio outside their market of specialization, where $\overline{\alpha }\in \left[\frac{1}{N},1\right]$ is the barrier to diversification. The degree of financial development of an economy is summarized by the vector of distortions $D=\left({\xi }_A,{\xi }_P,\overline{\alpha }\right)$. As financial innovation decreases D, investment costs are reduced and diversification possibilities improve. As a word of caution, it is worthy to emphasize that the separate treatment of the barriers $\overline{\alpha }$ and of the investment costs ξ_A, ξ_P is a convenient artifact but should not be taken too literarily. These variables are all rooted in the same types of market imperfections that financial innovation, when beneficial, helps to eradicate. For instance, high barriers $\overline{\alpha }$ may be due the presence of asymmetric information and moral hazard preventing investors to venture in unfamiliar markets, thus forcing them to concentrate their wealth in a few markets; lack of a centralized financial market for a certain class of assets raises liquidity costs and reduces the number of investors who trade in those assets; explicit regulatory requirements can restrict the possibility to trade some types of assets. These same imperfections are also the cause of high investment costs ξ_A, ξ_B, which must be interpreted in a broad sense as to include all “the cost of transmitting information from one party to another” (Merton [1987]). Examples are costly contracting schemes needed, in the framework of a principal-agent problem, to implement a truthful communication and reduce moral hazard (Holmström and Milgrom [1987], Townsend [1979], Benmelech et al. [2009]). Finally, it is important to stress that the focus of the paper is not on developing a theory of endogenous emergence of financial innovation, the distortions D are in fact treated as exogenous to the economy.

B. Preferences

Agents’ preferences are given by an homothetic utility function, which allows a parametric separation between risk aversion and intertemporal elasticity of substitution. Call {c_t,_I} any consumption process for time t = 1, 2 chosen by the agent after observing a signal I ∈ {0, Θ} at the beginning of time 1. Conditional on I, consumption c₁,_I is deterministic, but consumption c₂,_I is stochastic, since it depends on the random return at t = 2 of the agent’s investement. A conditional consumption process yields a value V_I given by

for β ∈ [0,1]. The ex-ante utility U to the agent is the following,

Similarly to Epstein and Zin (1989), the utility function disentangles the relative risk aversion coefficient γ > 0, γ ≠ 1 from the elasticity of intertemporal substitution ψ > 0, ψ ≠ 1. As we will see, this choice is made to allows a better interpretations of the results. Notice also that by setting $\gamma =\frac{1}{\psi }$ the usual CRRA utility is obtained.

C. The problem of the investor

At time t = 1, an investor first chooses whether to be active (τ = A) or passive (τ = P), then observes the corresponding signal I^τ, and finally makes investment decisions. Investment choices are given by a saving rate x and a portfolio share α_m,_j for every technology (m, j) in the economy, subject to the barrier to diversification $\overline{\alpha }$. Therefore, given an exogenous realization of Θ, an investor endowed with initial unit wealth and specialization h solves

where,

Refinements (7) imply that for agents with an informative signal I = Θ uncertainty about the portfolio return ${\hat{r}}^h$ stems only from uncertainty on the realization of the productivity shocks i, e, S. Agents with an uninformative signal I = 0, instead, face further uncertainty about the types realization. The solution for the optimal portfolio shares is quite intuitive³. Call ${\alpha }_{m,j}^{h*}\left(I\right)$ the optimal portfolio shares of an investor specialized in h when she observes a signal I. When I = Θ the investor is informed about the true type and avoids investing in excess risk technologies. Moreover, since the returns of low risk technologies are i.i.d. across markets, the best strategy to minimize the remaining risk is to distribute the total investment as evenly as possible across markets, conditional on the barrier $\overline{\alpha }\ge 1/N$. Therefore,

When I = 0 the signal is uninformative and, from the point of view of the investor, returns on all technologies are identically distributed. Therefore, to minimize risk, the investment is spread as evenly as possible across technologies, which implies that a share $\overline{\alpha }/2$ is assigned to each of the two technologies in market h and the remaining share $1-\overline{\alpha }$ is equally distributed across the other 2(N − 1) technologies,

Substituting for the optimal portfolio shares, we conclude that, for a given Θ, the portfolio return ${\hat{r}}_I^h$ of an investor with information I and specialization h is

Notice that ${\hat{r}}_I^h$ is independent on Θ. Clearly, an informed investor holds a less risky portfolio than that of an uninformed investor, in fact

where

and $E[{\epsilon }^h|r_{\Theta }^h]=0$. For a realization Θ, call R_I the risk-adjusted return to the portfolio of an investor with information I and specialization h,

Risk adjusted returns are independent both on h and on the particular realization Θ. Since ${\hat{r}}_0^h$ is a mean preserving spread of $r_{\Theta }^h$, a simple application of Jensen’s inequality gives

Using the results obtained so far and the homotheticity of the utility function, the optimal saving rates are found by rewriting $V_I(\tau ,h)=\frac{1}{{\xi }_{\tau }}{\hat{V}}_I$ where

For every signal I, the first order conditions of the problem give the optimal saving rate $x_I^{*}$,

Since all agents, informed or uninformed, allocate a fraction $\overline{\alpha }$ of their investment into their own market of specialization and a fraction $\frac{1-\overline{\alpha }}{N-1}$ to each of the other markets, we can derive aggregate market shares ω_m, representing the fraction of aggregate investment devoted by the economy to market m,

For instance, when financial development is at its minimum $\overline{\alpha }=1$, only the fraction p_m of agents with specialization in m invest in m, and thus ω_m = p_m. From a diversification perspective, some markets will receive too little capital and others too much. Instead, when financial development is at its maximum $\overline{\alpha }=1/N$ investors can perfectly diversify their portfolio beyond their original specialization, and thus ω_m = 1/N for all m.

D. The microfundation of the information structure

When collection of information is costly markets cannot be informationally efficient (Grossman and Stiglitz [1980]), otherwise the superior information available to active investors would be costlessly revealed to passive investors by investment choices, which are publicly observable. However, if the precision of the public information is reduced by the presence of an aggregate noise, then costly collection of private information can still be valuable. This idea is formalized by assuming that passive investors observe the investments made by active investors plus a noise. The noise can be interpreted as created by mistakes in investment choices or in its measurement, or by the activity of another class of agents, the liquidity investors, whose portfolio decision is influenced by random factors, such as liquidity needs.

Assume that there is a constant amount $\overline{L}$ of noisy investment in the economy, distributed across markets according to the market shares ω_m. A random variable | with support $[0,\overline{L}]$ determines the total amount of noisy investment which is directed towards excess risk technologies. Call y (m, j) the sum of active and noisy investment made in technology (m,j). We then have,

where as usual η is the fraction of investors active in each market. Each active investor has a net wealth 1/ξ_A and a saving rate $x_{\Theta }^{*}$. The quantity $\frac{{\omega }_m}{{\xi }_A}{x}_{\Theta }^{*}\eta$ is then the overall amount of active investment directed to the low risk technology in market m. Call y = (y(1,1), y(1, 2), …,y(N, 1), y(N, 2)) and assume that $\overline{L}<x_{\Theta }^{*}/{\xi }_A$. Passive investors use the Bayes rule to deduce the realization of the types from the observation of y. Fix a reference market m and, for j = 1, 2, define

The value L_j corresponds to the realization of the noise L consistent with a observation y(m,j) and a realized type θ_m = j in our reference market m. For j = 1, 2 define Θ _j ∈ T_Θ the vector of types consistent with a realization θ_m = j in our reference market m. Appendix B shows that if (3) holds then,

where u(·) is the probability density function of L. First of all, notice that Prob{Θ₁|y} + Prob{Θ₂|y} = 1 and this is consistent with our initial assumption (4) that agents are always informed about $T_{\Theta }=\{\Theta ,\overline{\Theta }\}$. Equation (16) generates the structure (6) for the public signal if, for instance, L is uniformly distributed on $[0,\overline{L}]$, as the following example shows. Assume that θ_m is the true realization of the type in the reference market m, so that $\Theta ={\Theta }_{{\theta }_m}$ is the true realization of the vector of types. If there is a low realization $L<\frac{\eta }{{\xi }_A}{x}_{\Theta }^{*}$ then it is easy to verify that $L_{3-{\theta }_m}>\overline{L}$. Therefore $u(L_{{\theta }_m})=1/\overline{L}$, $u(L_{3-{\theta }_m})=0$. Then $\text{Prob}\left\{{\Theta }_{{\theta }_m}\right|{y}\}=1$ and $\text{Prob}\left\{{\Theta }_{3-{\theta }_m}\right|{y}\}=0$. Suppose instead that the realization of the noise is sufficiently large, $L\ge \frac{\eta }{{\xi }_A}{x}_{\Theta }^{*}$. Then $L_{3-{\theta }_m}\le \overline{L}$, $u(L_{{\theta }_m})=u(L_{3-{\theta }_m})=1/\overline{L}$ and hence ${Prob}\left\{{\Theta }_{{\theta }_m}\right|{y}\}={Prob}\left\{{\Theta }_{3-{\theta }_m}|{y}\right\}=\frac{1}{2}$. In conclusion, if $L<\frac{\mu }{{\xi }_A}{x}_{\Theta }^{*}$ types are revealed to passive agents by the low level of investment in the excess risk technologies, while if the noise is sufficiently large the true type $\Theta ={\Theta }_{{\theta }_m}$ is undistinguishable from $\overline{\Theta }={\Theta }_{3-{\theta }_m}$. Notice that the revelation rate is

The externality function (8) is obtained by setting

E. Equilibrium

In an equilibrium where the revelation rate is strictly positive the fraction of active investors is also strictly positive. In this situation, the equilibrium revelation rate ρ* makes indifferent the marginal investor between being active or passive, that is

where the left hand side is the utility (9) from being active and the right hand side is the expected utility from being passive. Solving we obtain

An internal solution ρ* ∈ (0, 1) is obtained whenever

We have then the following Proposition.

Proof. Substitute (15) into (14) to obtain (18), which is then substituted into (17) to get (19).                                                                                                                                                                                Q.E.D.

The equilibrium revelation rate is ultimately a function of the exogenous vector of distortions D, so we write ρ*(D). Since ρ* equalizes the value of being active and passive the revelation rate turns out to be, not surprisingly, a function of the relative information cost ξ and not of ξ_A and ξ_B separately. We can perform comparative statics exercises to see how financial innovation affects ρ* through reductions in the distortions D. In particular, to mimic the stylized facts suggested in Section II, I consider exogenous reductions in ξ_A, ξ_B and $\overline{\alpha }$ leaving ξ unchanged. I restrict my attention to two interesting cases that help us give a clear interpretation of the forces behind the relation ρ*(D). The first case is obtained by setting i_m = 1, so that the low risk technology is risk free and excess risk is both idiosyncratic e_m and systemic S. In the second, I set e_m =1 so that all technologies have an idiosyncratic component i_m and the excess risk is only systemic S. I will also focus on interior equilibria ρ* > 0.

Proposition 2 show that information collection decreases whenever financial innovation favors mostly passive investors relative to active investors. To explain this point it is useful to analyze the ratios ζ_x = x_Θ/x₀ and ζ_R = R_Θ/Ro representing, respectively, the saving rate and risk adjusted return of an informed investor relative to those of an uninformed investor. In case i) informed investors face no risk at all since R_Θ = 1, while passive investors face idiosyncratic and system risk. Financial development decreases the relative return ζ_R of informed investors by boosting only the return R₀ of uninformed investors, who can better diversify the excess risk they hold. Since active investors are always informed while passive investors are sometimes uninformed, financial development favors passive investors relatively more than active investors. In case ii) we have ζ_R = Ω^–1 where

The excess risk held by uninformed investors is all systemic, hence the relative return ζ_R is unaffected by the degree of diversification. In this case, therefore, the channel that allows D to affect ρ* is not simply the change in ζ_R, but is the behavior of the relative saving rate ζ_x. When the elasticity of intertemporal substitution is smaller than one ζ_x < 1, because the share of individual wealth consumed by informed investors, who face a higher return R_Θ, is bigger than that of the uninformed investors, who save relatively more to compensate for their lower return R₀. Therefore, even though financial innovation doesn’t change ζ_R, it still favors more uninformed investors (and thus passive investors) who are the ones that are more exposed to investment risk and then reap overall greater fruits from the improvement in diversification possibilities. The opposite i true when ψ > 1, but this latter case seems much less relevant for macroeconomics, where both the econometric (Hall [1988]) and the quantitative (Kydland and Prescott [1982]) literature traditionally consider values of ψ smaller than 0.5. For e_m = 1, the elasticity ψ determines the direction (decrease/increase) in which ρ* changes in response to a decrease in D, while the relative risk aversion γ influences the size of such change. In fact, substituting ζ_R = Ω^–1 into (19) we obtain

where g(R_Θ, Ω) is a strictly positive function. Risk adjusted returns increase with diversification⁴, so $\frac{\partial R\ominus }{\partial (1-\overline{\alpha })}>0$ for any value of the parameters. The elasticity ψ determines the sign of ${\rho }_{1-\overline{\alpha }}^{\prime *}$ while, for given R_Θ and Ω, $\frac{\partial R_{\Theta }}{\partial (1-\overline{\alpha })}>0$ influences its absolute value. The higher the risk aversion, the larger is $\frac{\partial R_{\Theta }}{\partial (1-\overline{\alpha })}$, since the greater is the positive diversification effect on R_Θ generated by a small reduction in the distortion $\overline{\alpha }$. For instance as γ approaches zero investors become risk neutral and R_Θ = 1 for any $\overline{\alpha }$, so that $\frac{\partial R_{\Theta }}{\partial (1-\overline{\alpha })}=0$. These different roles for ψ and γ cannot be uncovered in the standard formulation $\gamma =\frac{1}{\psi }$.

Having established that financial innovation tends to decrease information, it is interesting to assess how this affects aggregate welfare and the business cycle. For equilibria with interior ρ*, investors’ welfare is simply given by the welfare of active investors ${\hat{V}}_{\Theta }/{\xi }_A$. Welfare gains from financial innovation come from lower costs ξ_A and from a lower barrier $\overline{\alpha }$ to diversification, which allows R_Θ in (18) to increase. However, these net positive welfare gains can be greatly reduced by a decrease in the amount of information ρ*. The starkest example is provided by case a), where a decrease in $\overline{\alpha }$ does not change ${\hat{V}}_{\Theta }$. It follows that if we keep ξ_A and ξ_B constant, welfare is unaffected by a strict decrease in the barrier $\overline{\alpha }$. But how is it possible that higher diversification does not rise the welfare of passive investors, who are likely to be uninformed and thus to face excess idiosyncratic risk? The answer is that, in this case, the positive effect of a lower $\overline{\alpha }$ is exactly offset by an endogenous decrease in the quality ρ* of public information. The mechanism is that of a typical moral hazard problem, working through the information externality (8). As diversification possibilities increase, investors become “more insured” against excess risk and thus, ceteris paribus, find it less valuable to take the costly action of collecting private information to eliminate the chance of facing such risk. In other words, information collection and diversification are strategic substitutes for investors. The lower collection of private information reduces the quality ρ* of the public signal, and this impacts negatively on the welfare of passive investors, exaclty counterbalancing the positive effect of a low $\overline{\alpha }$.

Lower public information means that more investors (lower η*) invest more often (lower ρ*) in the “wrong technologies”, i.e. in technologies with excess risk. This result is important and suggests that the finding of Proposition 1 regarding the positive welfare value of financial innovation could be quite fragile. Indeed, in Section IV I extend the basic framework to a more dynamic setting in which risk is time varying and where periods when excess risk is mainly idiosyncratic are followed by periods when excess risk becomes systemic. I show that financial development increases the probability of “liquidation crises”, i.e. situations in which investments in excess risk technologies, initially mistakenly undertaken by uninformed investors, are suddenly liquidated when the increase in systemic risk triggers a wave of information collection. If liquidation crises impose aggregate costs to the economy, then financial innovation can reduce welfare.

To present in simple ways the effect of financial innovation on the business cycle, I characterize the equilibrium values of the statistics ${\sigma }_A^2$ and ${\gamma }_{A,S}={Cov}(A,S)/{\sigma }_S^2$ giving, respectively, the volatility of the multifactor productivity A and the regression parameter measuring the sensitivity of the productivity to systemic shocks. The productivity A is defined as

The multifactor productivity is the overall output at time 2 of the production technologies divided by time 1 total investment, and is expressed in (20) as the weighted average of the investors’ portfolio return divided by total investment. Define ∑² the volatility of the excess risk technologies minus the volatility of i and S, and define ${\widehat{\sigma }}_m^2$ the dispersion of the market shares ω_m,

Appendix C shows that when total noise $\overline{L}$ is small⁵ the business cycle statistics are given by

Financial innovation, accompanied by a reduction in ρ*, has ambiguous effects on aggregate volatility ${\sigma }_A^2$. It is a simple exercise⁶ to show that a lower barrier to diversification $\overline{\alpha }$ always reduces the dispersion ${\hat{\sigma }}_w^2$ of market sizes and thus lowers ${\sigma }_A^2$. Dispersion of market shares ${\hat{\sigma }}_w^2$ amplifies the effects of idiosyncratic volatility since it allows idiosyncratic shocks to markets with above average share ω_m to affect aggregate volatility ${\sigma }_A^2$. This amplification operates both on the volatility of shocks i, and to the excess idiosyncratic volatility ∑² generated when, with probability 1 − ρ*, the public signal is uninformative and passive investor allocate half of their portfolio to excess risk technologies. In the limit of perfect diversification $(\overline{\alpha }=1/N)$ and of a large number of markets (N = ∞), ${\hat{\sigma }}_w^2=0$ because idiosyncratic volatility is washed away in the aggregate. In this case only the systemic volatility ${\sigma }_S^2$ remains and is amplified by reductions in ρ*.

Clear-cut results are obtained for the sensitivity γ_{A, S} which always increases in $1-\overline{\alpha }$. In fact, the smaller revelation rate increases the frequency at which aggregate productivity is exposed to the systemic shock which, for low $\overline{L}$, affects approximately half of the aggregate investment portfolio.

Figure 3 depicts, for a particular numerical example, how the probability density function of A and the revelation rate ρ* change with $\overline{\alpha }$. In particular, I consider the case where idiosyncratic shocks are symmetric, while the systematic shock is, most of the time, slightly above it average and rarely much below it. This generates an asymmetric distribution of S characterized by negative skewness, which captures the notion that negative systematic events are rare but very destructive since, for instance, they trigger the disruption of part of the financial market and large scale liquidations. I also use the standard parametrization $\psi =\frac{1}{\gamma }=0.5$ and the relative information cost ξ is set equal 1.005%. As expected, the equilibrium revelation rate decreases as the barrier $\overline{\alpha }$ decreases. Since ${\sigma }_A^2$ decreases, the reduction in ${\hat{\sigma }}_w^2$ turns out to be the dominating effect on aggregate volatility, while the sensitivity γ_A,_S to the asymmetric systematic shocks, representing negative “tail risk”, increases.

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Probability density function of A (left panel) and revelation rate (right panel) for minimum $(\overline{\alpha }=1/N)$ and maximum $(\overline{\alpha }=1)$ barrier to diversification. With equal probability i_m ∈ {1.18, 0.82} and e_m ∈ {1.18, 0.82}; S = 1.1 with probability 0.82 and S = 0.57 otherwise; N = 120; ξ = 1.005; β = 0.97; ψ = 1/γ = 0.5; p_h = 0.18 0.82 ^h-1. Computed volatilities: ${\sigma }_A^2(\overline{\alpha }=1)=6.3\cdot {10}^{-3}$ ,${\sigma }_A^2(\overline{\alpha }=1/N)=5.5\cdot {10}^{-3}$.

Citation: IMF Working Papers 2010, 266; 10.5089/9781455210732.001.A001

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The next section shows how systemic tail risk can be endogenously generated by financial innovation.