Masked instability: Within-sector financial risk in the presence of wealth inequality

Choi, Youngna Article Masked instability: Within-sector financial risk in the presence of wealth inequality Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Choi, Youngna (2018) : Masked instability: Within-sector financial risk in the presence of wealth inequality, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 6, Iss. 3, pp. 1-15, https://doi.org/10.3390/risks6030065 This Version is available at: https://hdl.handle.net/10419/195857 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Masked Instability: Within-Sector Financial Risk in the Presence of Wealth Inequality Youngna Choi †ID Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07043, USA; [email protected]; Tel.: +1-973-655-5132 † Current address: 1 Normal Avenue, Montclair, NJ 07043, USA. Received: 4 May 2018; Accepted: 23 June 2018; Published: 27 June 2018   Abstract: We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial instabilities can arise amid seemingly stable economy and positive market data. The theoretical conjecture is verified by historical macroeconomic time series of the United States households among whom a substantial wealth inequality has been officially confirmed. Keywords: wealth inequality; financial stability; agent-based model; dynamical systems JEL Classification: C02; E01; G01 1. Introduction It has been a decade since the outbreak of the 2007–2010 United States Subprime Mortgage Crisis that was followed by the 2009–2014 European Sovereign Debt Crisis. The two financial crises resulted in the Great Recession that suppressed the global economy for several years. The U.S. economy seems to have fully recovered in terms of economic and financial indicators. For example, the labor market is near full employment with the unemployment rate at 3.8% as of June 2018 according to Bureau of Labor Statistics (2018) , and stock indices hit historical highs in the first quarter of the same year (e.g., Bloomberg 2018). However, reports have been made that the Great Recession exacerbated the uneven wealth distribution in the United States (e.g., Kochhar and Cilluffo 2017;Wolff 2014), and, at the same time, wealth inequality has been addressed as an increasing social problem by both organizations and scholars (e.g., Congressional Budget Office 2016;Organization for Economic Cooperation and Development 2016;Saez and Zucman 2016). Concerns about wealth inequality is not a phenomenon restricted to the United States as can be seen from the frenzied worldwide reaction to the publication of Capital in the Twenty-First Century by Piketty (2014). We investigate how wealth inequality can affect correct assessment of financial stability of an economic sector by showing that, when an economic sector is divided into two subsectors by extreme concentration of wealth and lack thereof, it may mask risk exposures within subsectors that are not readily observable from the aggregate. This article is the latest installment of our ongoing research about the role of borrowing capacity in determining the financial instability, and, as such, we are interested in whether the aggregate borrowing capacity could misrepresent the reality in the presence Risks 2018,6, 65; doi:10.3390/risks6030065 www.mdpi.com/journal/risks Risks 2018,6, 65 2 of 15 of extreme wealth inequality. There are many factors that determine the borrowing capacity of an economic agent, but, for households, it is the current level of wealth, income, and debt that are the most important and easily quantifiable ones. According to the Survey of Consumer Finances (2017) (SCF), the top 10% of the United States households possess more than 75% of the total wealth and about half of the total income while the consumer debt, including a dangerously high level of student loans as pointed by many (e.g., Foroohar 2017), reached a new peak in 2017 Q4 (Federal Reserve Bank of New York 2018). This could create an uneven distribution of borrowing capacity within the households, hence different levels of financial instability as well. There are many definitions of financial stability and financial instability (e.g., Alawode and Al Sadek 2008), yet they all have a common theme: financial stability entails resilience and shock-absorbing capacity while financial instability involves impairment of the financial system and its adverse effect on economic activity. The last condition often leads to the definition of systemic risk, which is summarized by Freixas et al. (2015) as an unusual financial shock that causes strong negative shocks to the real economy. We are interested in the financial (in)stability of a single economic agent rather than the system itself, hence we give our own definition: the financial stability of an agent is a state in which the agent absorbs a shock, not necessarily negative, on its wealth to keep steady its outgoing cash flows. Therefore, when an agent goes through financial instability or is financially unstable, a shock on its wealth is transmitted to another agent or even propagates throughout the system. Examples of such shock transmissions are default on a loan payment for a negative shock and over-investment for a positive shock. It should be noted that unlike the post-subprime crisis works by Rajan (2010) and Kumhof and Rancière (2010) that study the link between wealth inequality and financial crisis or those of Atkinson and Morelli (2011) and Bordo and Meissner (2012) that find no evidence of such a linkage, our goal is not to prove or disprove the existence of any nexus between wealth inequality and financial instability leading to a crisis but to show that wealth inequality can mask financial inequalities within an economic sector that appears financially stable. We use the market instability indicator, an early warning system proposed by Choi and Douady (2012), to quantify the level of financial instability of the United States households and nonprofit institutions serving households (HNISH) and its subsectors. HNISH is chosen because its macroeconomic time series are available on Integrated Macroeconomic Accounts for the United States (2018) and the Survey of Consumer Finances (2017) confirms an extreme wealth inequality within the sector, and hence serves our purpose well. The rest of the paper is organized as follows. Section 2explains the background and setup of the work, and Section 3presents the mathematical details of the impact of wealth inequality on financial stability. Section 4is devoted to a case study of the United States HNISH, and Section 5provides the conclusions. 2. Basic Setup Given an economy, consider its domestic part and divide it into four aggregates called agents, households (H), companies (C), financial institutions not restricted to banks (F), and government (G), indexed from 1 to 4 in that order. For each agent i , define its wealth (commonly known as asset) wi as the sum of its equity and debt, and the total wealth of the economy at time t as the vector w(t)=(w1(t) , · · · , w4(t)) , all in chained currency. The market instability indicator of the system is defined to be the spectral radius of the Jacobian matrix B(t) = bij(t)1≤i,j≤4 of a differentiable function f:R4−→ R4 such that f(w(t)) = w(t+ 1 ) . Higher market instability indicator implies more Risks 2018,6, 65 3 of 15 financial instability of the system. Let Fij(t)> 0 be the fund transferred from agent j to i1 and Fii(t) the internal return on the invested asset portion of wi(t), also in chained currency. With these notations, wi(t+1) = wi(t) + 4 ∑ j=1 Fij(t)− 4 ∑ k6=i Fki(t). (1) The elasticity coefficient aij, defined as aij =∂Fij ∂wj for i6=j, (2) measures how susceptible the outgoing cash flow Fij is to a change in wj . Similarly, we define aii as the change rate of internal return on invested asset Fii(t)with respect to wi(t), so for all iand j aij(t) = ∂Fij(t) ∂wj(t). (3) We further assume that changes in one agent’s wealth do not affect the outgoing cash flows of others, hence ∂Fij ∂wk =0 if j6=k. (4) It can be proved that aij differs from the Jacobian component bij only on the diagonal, bii =1+aii − 4 ∑ k6=i aki, and (5) bij =aij for i6=j. (6) By definition bii(t) = ∂wi(t+1) ∂wi(t)(7) and the market instability indicator of agent i (as opposed to the system) is |bii| . In dynamical systems, the threshold that determines the stability type of an equilibrium is 1. More precisely, if x is a fixed point of a dynamical system g:Rn−→ Rn , then x is a stable fixed point if ρ(x)< 1 and unstable fixed point if ρ(x)> 1 where ρ(x) is the spectral radius of the Jacobian matrix Dg(x) . Therefore, agent i is financially stable if |bii(t)|<1 and financially unstable if |bii(t)|>1. However, it is not convenient to use this version of sectoral market instability indicator with real data because wi(t+ 1 ) is a function of wi(t) and other cash flows, so it is not possible to directly calculate the partial derivative of wi(t+ 1 ) with respect to wi(t) . We derive an alternative version of bii(t) using time derivatives of wealth and cash flows. Assume that F (agent 3) is the sole creditor to all other agents and denote by F3i2 the debt payment by agent i to F. For all agents other than F (i.e., i6= 3), the cash flow F3i(t) consists of two components, what it actually pays and what it fails to pay. We call the former αiand latter δi, hence F3i(t) = αi(t) + δi(t). (8) The new loan Fi3(t)and net new loan νi(t)taken during [t,t+1]are, respectively, Fi3(t) = νi(t) + δi(t)(9) 1 See Choi and Douady (2012) or Choi and Douady (2013) for a detailed nature of these cash flows, although this article follows Castellacci and Choi (2014) that uses different methods in selecting Fij. 2A deposit made by ito F is an asset reallocation, hence not part of F3i. Risks 2018,6, 65 4 of 15 and νi(t) = Di(t+1)−Di(t) + αi(t). (10) Now, we express the elasticity coefficient aij(t)in terms of time derivatives, aij(t) = Fij 0(t) wj0(t), (11) where “ 0 " denotes the time derivative. This is derived by applying chain rule and Equation (4) to Fij 0(t), dFij dt =∑ k ∂Fij ∂wk ·dwk dt =∂Fij ∂wj ·dwj dt . (12) With these notations, differentiate w(t+ 1 ) with respect to time. By Equations (6), (4), and (11), we get wi0(t+1) = ∑ j ∂wi(t+1) ∂wj(t)˙dwj(t) dt =bii(t)wi0(t) + ∑ j6=i bij(t)wj0(t) =bii(t)wi0(t) + ∑ j6=i aij(t)wj0(t) =bii(t)wi0(t) + ∑ j6=i Fij 0(t). (13) Hence, wi0(t+1) wi0(t)=bii(t) + ∑ j6=i Fij 0(t) wi0(t) =bii(t) + ∑ j6=i,3 Fij 0(t) wi0(t)+Fi30(t) wi0(t) =bii(t) + ∑ j6=i,3 Fij 0(t) wi0(t)+νi0(t) wi0(t)+δi0(t) wi0(t), which subsequently yields bii(t) = wi0(t+1) wi0(t)−∑ j6=i,3 Fij 0(t) wi0(t)−νi0(t) wi0(t)−δi0(t) wi0(t). (14) Each of the negative terms in Equation (14) represents the ratio of velocities—the velocity of a cash inflow and that of the wealth—which compares the movement of cash inflow and wealth, and subtracting them in aggregate is to get rid wi0(t+ 1 ) of the contribution that are not from wi(t) to get Equation (7). The last term in Equation (14) plays an important role in determining the financial instability level and defined to be the default elasticity of agent iat time t, ∆i(t) = δi0(t) wi0(t). (15) Risks 2018,6, 65 5 of 15 When dealing with discrete time series such as quarterly macroeconomic data, the time derivatives are to be substituted by difference quotient, wi0(t) = wi(t)−wi(t−1) t−(t−1)=wi(t)−wi(t−1)and (16) Fij 0(t) = Fij(t)−Fij(t−1) t−(t−1)=Fij(t)−Fij(t−1). (17) This modification will be used with real data in Section 4. 3. Wealth Inequality and Financial Stability Ten years after the demise of Lehman Brothers that epitomizes the 2007–2010 U.S. subprime mortgage crisis, the United States economy seems to have fully recovered. The employment is nearly full and the current economic expansion, at eight years and ten months as of 1 May 2018, just broke the record to become the second longest in history (Chandra and Golle 2018). Despite this optimistic news, recent Survey of Consumer Finances (2017) shows that the wealth inequality in the U.S. household sector has grown wider since the time of the previous survey, and now the top 10% own more than 75% of the total wealth (Figures 1and 2). Figure 1. For 2010–2013 Survey of Consumer Finances, the income and wealth percentiles are divided at the top 3%, the next 7%, and bottom 90%. The wealth share of the bottom 90% has decreased steadily, but remains around 25% of total wealth. Risks 2018,6, 65 6 of 15 Figure 2. For 2013–2016 SCF, the income and wealth percentiles for the bottom group remain at 90%, but the top 10% are split at the top 1% and next 9%, which suggests more concentration of wealth on the top 1%. Only the wealth share of the top 1% has increased. The wealth share of the next 9% decreased slightly and that of the bottom 90% has dropped below 25%. To model and investigate the impact of this wealth inequality on the financial stability of the households sector H, we divide it into two subsectors, the rich (agent r ) and the poor (agent p ). We further assume that the rich get richer and the poor get poorer, both with acceleration, which mathematically is expressed as wr0(t)> 0, wr00(t)> 0 and wp0(t)< 0, wp00(t)< 0. This assumption is justified by the wealth share changes in the survey. We also assume that the consumption F2r and F2p have reached a full capacity in the sense that the rich do not want to spend more than they do now, and the poor cannot reduce any more the basic cost of living. The assumption on the rich group intuitively makes sense from decreasing marginal propensity to consume (MPC), and is justified by the recent result of Carroll et al. (2017) that the aggregate MPC can differ greatly across wealth and income distribution and wealthy households with high income have low MPCs. By Equation (5), brr =1+arr −a2r−a3r−a4r, and (18) bpp =1+app −a2p−a3p−a4p. (19) Risks 2018,6, 65 7 of 15 The biggest portion of income for the rich tends to be the return on investment, therefore arr is relatively big and positive. The biggest portion of income for the poor is wages and the return on invested asset is a very small part of wp , which implies app ≈ 0. By our assumption, the consumption for both groups are near a limit with little change, hence both a2r and a2p are negligible. When a borrower’s wealth increases, the lenders often offer more favorable perquisites such as lower interest rates and reduced fees, while decreased wealth results in harsher borrowing conditions such as higher interest rates and minimum balance. This argument on the vicious cycle for the poor and virtuous counterpart for the rich is in line with well-established theories on debt dynamics (e.g., Hart and Moore 1998) , asymmetric information (e.g., Akerlof 1970) , and financial accelerator (e.g., Bernanke et al. 1999). Therefore, we can assert that a3i=∂F3i ∂wi <0, i=r,p. (20) Our assumption on decreasing concave wealth function wp(t) can be, without loss of generality, extended such that the poor are highly leveraged with little equity and a small decrease in their wealth can result in a precipitous increase in loan payment obligation and even failure to pay. 3 This implies a3p − 1. As for tax, the rich get various forms of deduction and the poor do not pay much to begin with, so the change in tax with respect to income change would not be significant for either group. 4 Therefore, brr ≈1+arr −a3r1 and bpp ≈1−a3p1. (21) The size of the market instability indicator implies that both subsectors of households are financially unstable, 5 the rich from excess wealth and the poor lack thereof. Highly indebted households are vulnerable to shocks on their wealth, e.g., drop in housing value, the rise of interest rate on loans, and reduced income from layoffs, to name a few. Such events can cause a financial instability contagion along the feedback loop formed by inter-agent cash flows as explained in Castellacci and Choi (2014), and even trigger a financial crisis similar to the 2007–2010 U.S. subprime mortgage crisis. The rich with excessive capital would look for investment opportunities and this can create bubbles outside their economy just like in the emerging market after the 2008 global financial crisis. However, the households as a single sector can appear financially stable despite these dormant seeds of further turmoil because the subsectoral instabilities of opposite causes can ‘offset’ each other. We use Equation (14) to explain this mathematically. With subsectors r and p , Equation (14) can be written as follows: b11(t) = w10(t+1) w10(t)−F12 0(t) w10(t)−F14 0(t) w10(t)−ν10(t) w10(t)−δi0(t) wi0(t) =wr0(t+1) + wp0(t+1) wr0(t) + wp0(t)−Fr20(t) + Fp20(t) wr0(t) + wp0(t)−Fr40(t) + Fp40(t) wr0(t) + wp0(t) −νr0(t) + νp0(t) wr0(t) + wp0(t)−δr0(t) + δp0(t) wr0(t) + wp0(t). (22) We complement the previous assumptions with the following. First, the rich accumulate wealth much faster than the poor such that it mathematically means w10(t)> 0 and w10(t+ 1 )/w10(t)> 1. Second, the total income including wages and government benefit remain flat. In the Survey of Consumer Finances (2017), the income shares of the top 1% and bottom 90% are almost mirror images 3Indeed, this was the mechanism of adjustable rate mortgages that triggered the 2007–2010 subprime mortgage crisis. 4If, as believed by many, the rich end up paying much less tax due to the 2017 tax reform, then a4rwould become negative. 5 Recall the comment after Equation (7) that the threshold to determine the stability type is 1. However, the market instability indicator may stay above 1 all the time, and, in that case, the financial instability should be tracked by spikes of the indicator. See Figures 3–8. Risks 2018,6, 65 8 of 15 of each other, while that of the remaining 9% remains nearly constant, therefore this assumption is reasonable. This means that most of the wealth increase is due to return on investment F11 . The second assumption on income mathematically means F12 0(t)≈ 0, F14 0(t)≈ 0. Finally, no more net new loans are taken because the rich do not need it, and the poor cannot afford it. Then, b11(t)≈w10(t+1) w10(t)−0 w10(t)−0 w10(t)−0 w10(t)−δi0(t) wi0(t)(23) =w10(t+1) w10(t)−δr0(t) + δp0(t) w10(t). (24) The rich are solvent and liquid enough to rarely default on debt payments while the poor are highly leveraged and susceptible to negative shock on wealth, which implies δr0= 0 and δp0> 0. Therefore, b11(t) is by Equation (24) the difference of two positive numbers, and, as a result, |b11| can become far smaller than |brr| and |bpp| measured separately, making the households appear financially stable while its two subsectors are not. This is an extreme case analysis with assumption on constant consumption and tax payment (which are consequences of flat income and benefit assumption used for Equation (23)), but even in real life the two cash flows could stay relatively steady compared with other ones. Therefore, when other assumptions on wealth and debt are satisfied, it is possible for the households of an economy to appear financially stable and safe from economic distress while it has an unstable subsector that can create asset bubbles (and a crisis when they bust) or one that can mass default on debt payment, triggering a solvency crisis, or both. This analysis is not restricted to the United States but can be applied to any country whose households demonstrate intensifying wealth inequality. 4. Case Study: United States Households To verify our conjecture on the impact of wealth inequality on financial stability, we use the historical time series of current and balance sheet accounts of the United States Households and Nonprofit Institutions Serving Households (HNISH, agent 1) from the Integrated Macroeconomic Accounts for the United States (2018), published jointly by the Bureau of Economic Analysis and Board of Governors of the Federal Reserve System. Some modifications and assumption should be made, however. Calculating the net new loan ν1(t) and default elasticity ∆1(t) in Equation (14) requires the actual payment α1(t) and default amount δ1(t) while the liability in the balance sheet account already reflects the default amount and represents only the remaining debt, i.e., the outstanding balance. Besides we are interested in quantifying how susceptible to an external shock on wealth an agent is. In this regard, we use a predetermined maximum debt-to-wealth (D/W) ratio ρ∗ to set the maximum allowable debt. The D/W ratio is bounded above by ρ∗ means that D1(t) should be less than or equal to ρ∗w1(t) , and the debt exceeding this amount is defined as the default amount δ1(t) . In other words, δ1(t) is what HNISH would fail to pay if the D/W ratio ρ∗ were enforced at time t . Similarly, the actual payment α1(t) is assumed to be 0 and the net new loan νi(t) is the increment in debt level since last time, hence δ1(t) = max(D1(t)−ρ∗w1(t), 0), (25) α1(t) = 0, and (26) ν1(t) = D1(t+1)−D1(t). (27) Figure 3shows the market instability indicator |b11| and default elasticity ∆1 of HNISH from 1960 Q3 to 2017 Q4. The maximum D/W ratio ρ∗ is chosen as the three quartiles of the historical Risks 2018,6, 65 15 of 15 http://www.pewresearch.org/fact-tank/2017/11/01/how-wealth-inequality-has-changed-in-theu-s-since-the-great-recession-by-race-ethnicity-and-income/ (accessed on 30 April 2018). Kumhof, Michael, and Romain Rancière. 2010. Inequality, Leverage and Crises. IMF Working Paper 10(268). Available online: https://www.imf.org/external/pubs/ft/wp/2010/wp10268.pdf (accessed on 18 June 2018). Organization for Economic Cooperation and Development. 2016. Income Inequality Update. Available online: https://www.oecd.org/social/OECD2016-Income-Inequality-Update.pdf (accessed on 30 April 2018). Phillips, Matt, and Jim Tankersley. 2018. Investment Boom from Trump’s Tax cUt Has Yet to Appear. The New York Times. Available online: https://www.nytimes.com/2018/04/30/business/the-tax-cut-buybacksbusiness-investment.html (accessed on 4 May 2018). Piketty, Thomas. 2014. Capital in the Twenty-First Century. Cambridge and London: Belknap Press: An Imprint of Harvard University Press. ISBN 978-0-674-43000-6. Rajan, Raghuram. 2010. Fault Lines. Princeton: Princeton University Press. ISBN 978-0-691-14683-6. Saez, Emmanuel, and Gabriel Zucman. 2016. Wealth inequality in the United States since 1913: Evidence from capitalized income tax data. Quarterly Journal of Economics 131: 519–78. doi:10.1093/qje/qjw004. [CrossRef] S&P Dow Jones Indices. 2018. S&P Corelogic Case-Shiller U.S. National Home Price NSA Index. Available online :http://us.spindices.com/indices/real-estate/sp-corelogic-case-shiller-us-national-homeprice-nsa-index (accessed on 30 April 2018). Survey of Consumer Finances. 2017. Changes in the U.S. Family Finances from 2013 to 2016: Evidence from the Survey of Consumer Finances. Federal Reserve Bulletin 103(3). Available online: https://www. federalreserve.gov/publications/files/scf17.pdf (accessed on 30 April 2018). Wolff, Edward N. 2014. Household Wealth Trends in the United States, 1962–2013: What Happened Over the Great Recession? NBER Working Paper (20733). Available online: http://www.nber.org/papers/w20733 (accessed on 30 April 2018). c  2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).