The long-run effects of risk

Advanced economies tend to have large but unstable intermediation sectors. In a new paper (also avaliable as a CEPR discussion paper), we study how a higher degree of risk in the financial sector affects long-run macroeconomic outcomes when banks have limited liability in a general equilibrium model. With full deposit insurance, banks expand balance sheets when risk increases, leading to higher investment and output. With no deposit insurance, we observe substantial drops in long-run credit provision, investment, and output. These differences provide a novel argument in favor of deposit insurance. Our welfare analysis finds that increased risk reduces welfare, except when there is full deposit insurance and deadweight costs are small.

Cross-country evidence reveals substantial heterogeneity across countries with respect to bank risk-taking attitudes. More developed countries tend to have a larger, and more unstable, intermediation sector, with variation depending on a number of factors including the nature and extent of regulation (Bezemer et al., 2020, Calomiris and Haber, 2015). In new research, we consider a quantitative equilibrium model in which banks have limited liability which allows us to investigate how exogenous financial sector risk affects risk-taking by banks, and subsequently long run macroeconomic outcomes (Van der Kwaak et al., 2021). By financial risk we mean the volatility or standard deviation of idiosyncratic shocks to the return on banks’ assets (Christiano et al., 2014). Therefore, our paper contrasts with the baseline Real Business Cycle (RBC) or New Keynesian (NK) models where risk does not affect the steady state or balanced growth path.

In our model, intermediate goods producers issue corporate securities that are purchased by banks, which finance these securities through net worth and deposits. Banks have limited liability and freely choose the degree of leverage in the first period by paying out part of their net worth in the form of dividends. They face the above-mentioned shocks which affect the performance of their securities portfolios and therefore the profits they pay to their owners. Because bankers operate under limited liability, they only care about the distribution of financial risk conditional on survival when making their balance sheet decisions (Diamond and Rajan, 2011). Therefore, changes in the distribution of the shock affect credit provision, investment and output. In our model, the government is responsible for the degree of deposit insurance, which amounts to the fraction of deposits that are reimbursed in case of bank default (Clerc et al., 2015). For simplicity, we do not explicitly model other government regulations or capital requirements that the banks have to abide by, as in Gertler and Karadi (2011). However, we assume that the level of financial risk can be influenced by government regulation, but we take the extent of regulation itself and its political origins and motivations as exogenous. We interpret the standard deviation of shocks to bankers’ idiosyncratic risk as a proxy measure of the degree to which banks are regulated, with a large standard deviation representing a regime with little regulation, as the financial sector will be able to invest in riskier projects that have both a higher payoff in the good state of the world, as well as a much lower payoff in the bad state.

Our analysis focuses on long-run outcomes. The combination of limited liability and (partial) deposit insurance introduces moral hazard into our model (Kareken and Wallace, 1978): as a result, an increase in the shock’s standard deviation increases the expected return on corporate securities conditional on the bank surviving the realization of the shock next period, incentivizing bankers to hold more corporate securities (as in Diamond and Rajan, 2011). In Figure 1 we show that for the full deposit insurance case, banks expanding credit in this way leads to an increase of steady state investment and output with respect to no limited liability. Simultaneously, a higher level of financial risk increases the probability of bank default (and therefore the fraction of banks that default ex post), which raises the deadweight costs from default in equilibrium.

Figure 1: Impact on long run outcomes from changes in the level of financial risk shocks (the solid line shows the full deposit insurance case with limited liability and the black dotted the case without limited liability)

Moral hazard can be reduced by lowering the fraction of deposits that are reimbursed in case of bank default: the smaller the fraction the more the probability of default is priced in by banks’ creditors. However, we find that decreasing moral hazard in this way actually increases the probability of bank default in equilibrium because of a nonlinear feedback loop between banks’ funding costs and the probability of default: the larger the fraction of deposits that are not reimbursed, the more creditors increase interest rates to price in the probability of bank default (see Figure 2). This increase in funding costs, however, decreases banks’ (expected) profitability, which then further increases the probability of bank default. This, in turn, further increases banks’ funding costs, which then have amplification effects. As a result, we find that the strength of this feedback loop increases nonlinearly with the fraction of deposits that are not reimbursed by the deposit insurance agency. In equilibrium, the long-run probability of bank default under a regime with no deposit insurance is quadrupled with respect to the case where half of deposits are reimbursed by the government, which in turn is only slightly higher than in the case of full deposit insurance.

Figure 2: Impact on long run outcomes from changes in the level of financial risk shocks with different levels of deposit insurance (in addition to the previous cases, the black dashed line shows the 50% deposit insurance case with limited liability and the black solid line shows the no deposit insurance case with limited liability).

The impact on the real economy from a complete absence of deposit insurance is large: as a result of higher funding costs, credit provision can drop by 80% with respect to the case of no limited liability when moral hazard is entirely eliminated, causing long-run output to drop by more than 25%. This sharply contrasts with the case of full deposit insurance, where credit provision and output increase relative to no limited liability. The negative impact on the macroeconomy from a complete absence of deposit insurance is substantially mitigated when 50% of deposits are reimbursed, although credit provision, investment, and output still decrease with respect to the no limited liability case.

Therefore, our results provide a second argument in favor of deposit insurance, in addition to the Diamond and Dybvig (1983) argument of preventing bank runs (which are absent in our model): deposit insurance eliminates the above-mentioned feedback loop that would otherwise cause banks’ funding costs to increase to such an extent that the probability of bank default increases far above that under full deposit insurance. As such, financial stability increases with more deposit insurance, despite introducing moral hazard along the way.

We also analyze the deregulation of the financial sector that started in the 1980s in several Western economies, which we capture via simultaneously increasing the level of financial risk and the fraction of bank debt that is not covered by deposit insurance (the latter captures the fact that financial institutions started to rely more on non-deposit financing). Both gradually increase over a period of 50 quarters to mimic the progressive deregulation of the financial sector since the 1980s.  We find that simultaneously allowing for a decline in the long-run natural rate of interest (captured by an increase in the subjective discount factor) leads to an expansion in bank credit as has been observed in recent decades (Knott et al 2014). In the absence of a decline in the natural rate we find that under an increase in the level of risk leads to a higher probability of default and higher funding costs for banks which in turn decreases lending to the real economy leading to lower investment and output. These results suggest that the expansion of the financial sector in recent decades is primarily driven by the decline in the natural rate of interest.

Finally, we consider the consequences of an increase in financial risk on welfare, which is defined as the sum of expected discounted utility. In Figure 3, the vertical axis features welfare (defined as consumption equivalent from steady state consumption under no limited liability expressed in percentage). The two horizontal axes show the standard deviation of the idiosyncratic shock (which reflects the level of financial risk) and the deadweight costs from default (Bernanke et al., 1999). From the figure we see that welfare increases with the standard deviation when the deadweight costs are small, as financial intermediaries expand their balance sheet, thereby financing a larger capital stock which expands output. This allows consumption to increase leading to higher welfare. In the paper we additionally consider the cases of partial and no deposit insurance. Consumption is always decreasing with higher risk, which implies that welfare is so as well. For larger and empirically more realistic deadweight costs we see that the beneficial effect of higher risk is more than offset by deadweight costs absorbing a larger fraction of output, leading to lower consumption.

Figure 3: Steady state results for the model version with limited liability and full deposit insurance

Policy implications

We show banks expand their balance sheet when financial risk increases, by taking advantage of limited liability and deposit insurance to shift downside risk to others. Despite the existence of moral hazard this financial sector expansion has long run benefits, increasing investment and output. Eliminating moral hazard by abolishing deposit insurance would force creditors to correctly price in the probability of bank default, and thereby prevent banks from leveraging up. However in our paper we find that doing so in fact deteriorates financial stability (even in the absence of bank runs) because of a feedback loop between the probability of default and banks’ funding costs: depositors demand higher rates under no deposit insurance, which increases the probability of bank default. As a result depositors demand even higher rates, which then increase the probability of default even further. We also show that financial sector deregulation and the decrease in the natural rate of interest since the 1980s has increased the probability of bank default which has made the financial sector more unstable. Even under full deposit insurance we find that welfare decreases with higher financial risk, despite more credit provision and higher investment and output.


Bernanke, B. S., Gertler, M., & Gilchrist, S. (1999). The financial accelerator in a quantitative business cycle framework. Handbook of Macroeconomics, 1, 1341-1393.

Bezemer, D., Samarina, A., Zhang, L., 2020. Does mortgage lending impact business credit?

Evidence from a new disaggregated bank credit data set. Journal of Banking and Finance 113

Calomiris, C.W., Haber, S.H., 2015. Fragile by design: The political origins of banking crises and scarce credit. volume 48. Princeton University Press.

Christiano, L., Motto, R., Rostagno, M., 2014. Risk Shocks. American Economic Review 104, 27-65.

Clerc, L., Derviz, A., Mendicino, C., Moyen, S., Nikolov, K., Stracca, L., Suarez, J., Vardoulakis, A.P., 2015. Capital Regulation in a Macroeconomic Model with Three Layers of Default. International Journal of Central Banking 11, 9-63.

Diamond, D.W., and Rajan, R.G., 2011. Fear of Fire Sales, Illiquidity Seeking, and Credit Freezes. The Quarterly Journal of Economics 126, 557-591.

Gertler, M., Karadi, P., 2011. A model of unconventional monetary policy. Journal of Monetary Economics 58, 17-34.

Kareken, J.H., and Wallace, N., 1978. Deposit Insurance and Bank Regulation: A Partial-Equilibrium Exposition. The Journal of Business 51, 413-438

Knott, S., Richardson, P., Rismanchi, K., Sen, K., 2014. Understanding the fair value of banks’ loans. Bank of England Financial Stability Paper 31

Christiaan van der Kwaak, João Madeira and Nuno Palma, 2021. The long-run effects of risk: an equilibrium approach. CEPR Discussion Paper 15841

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