FEDS Notes
April 10, 2014
Some Implications of Knightian Uncertainty for Finance and Regulation
Doriana Ruffino
Introduction
With the recession of 2008, "uncertainty" became a buzzword.^{1} Since then, economists have largely shaped how policymakers, politicians, and the general public think about uncertainty, through, among other means, models that explicitly account for uncertainty.^{2} For example, the robust control methods of Hansen and Sargent (2010) address the concern of model misspecification under uncertainty. Gilboa and Schmeidler (1989) and Klibanoff, Marinacci, and Mukerji (2005) model uncertainty by assigning not one but multiple probabilities to a single uncertain event. Although some may argue that these frameworks only approximate actual decisionmaking under uncertainty, they represent a major improvement over standard risk models. And although models with more reasonable sources of uncertainty are complex and, thus, somewhat difficult to work with, they can yield important insights for supporting policy discussions and guiding regulation. In Paul Samuelson's words, "Good questions outrank easy answers" (1986, p. 561).
The majority of finance models with uncertainty have appeared in the asset pricing literature, particularly in papers on optimal portfolio selection. This literature makes two main contributions. The first is to show that investors generally demand a premium to hold assets with uncertain returns; that is, investors dislike uncertainty--even beyond their dislike of risk.^{3} The second is to explain observationally puzzling investor behaviors, including the low participation in financial markets and the tendency to hold "familiar" assets.^{4} However, decision-making models under uncertainty have, to date, rarely been applied to banking.
The Great Recession has exposed some important challenges that banks, as well as other systemically important financial institutions, face in stressful market conditions when the accuracy of information they rely upon is called into question. Caballero and Krishnamurthy (2008), for example, argue that the complexity and lack of history of some credit products caused excessive uncertainty, culminating in a freezing up of credit markets. The market freezing, say Easley and O'Hara (2010a), stemmed from uncertainty about fair asset prices. Because bid and ask prices did not reflect investors' pessimistic beliefs during the crisis, no trading occurred at the quoted prices. This conjecture elicits two questions: What are fair asset prices under uncertainty, and how can market freezing be prevented by containing uncertainty? The answer to the first question entails modeling investors' portfolio preferences under uncertainty.^{5} As for the second question, Easley and O'Hara (2010b) show how microregulating security exchanges can create a need for liquidity and increase participation.
The papers just mentioned are topical and add to the ongoing development of approaches used to identify and manage threats to financial stability. However, post-crisis regulatory reforms are virtually devoid of concerns about uncertainty. For example, requirements for higher bank capital ratios are based on a risk model that ignores uncertainty. In particular, capital charges are based on a unique loss-distribution function. While such an assumption may be plausible for normal times, it might not be during a crisis, when standard asset-valuation measures fail. Thus, rules aimed at ensuring adequate capital buffers in stressful circumstances could prove inadequate to the extent they are calibrated to a paradigm that abstracts from reality by assuming banks and investors can perfectly foresee all states of the world.
Application to Bank Regulation
The following example illustrates some possible effects of uncertainty on bank risk-taking.^{6} The market is formed of n ambiguous assets with expected rate of return r_{i} and a riskless asset with a known return r_{f} . As in Maccheroni, Marinacci, and Ruffino (2013), an asset is ambiguous if the variance of its expected return under the prior probability μ is positive--that is, σ^{2}_{μ}(E(r_{i})) > 0, i = 1, ..., n. Higher values of this variance correspond to poorer information on a return's possible probability models. We denote by r the vector of n ambiguous returns and by 1 the n-dimensional unit vector. Also, we set
E_{P}[r - r_{f} 1] = [E_{P}(r_{1} - r_{f} ), ..., E_{P}(r_{n} - r_{f} )]^{T}, | (1) |
Σ_{P}[r] = σ_{P}(r_{i} , r_{j}), i, j = 1 ..., n, | |
Σ_{μ}[E[r]] = σ_{μ}(E(r_{i}) , E(r_{j})), i, j = 1 ..., n, |
where Σ_{P}[r] and Σ_{μ}[E[r]] are the variance-covariance matrixes of returns and expected returns under the reference probability P and the prior probability μ, respectively, and E_{P}[r - r_{f} 1] is the vector of expected excess returns under P. Provided that the information in equation 1 is available, the solution to the bank's optimal portfolio-selection problem is
ŵ= [λΣ_{P}[r] + θΣ_{μ}[E[r]]]^{-1} E_{P}[r - r_{f} 1], | (2) |
in which λ and θ capture attitudes toward risk and ambiguity, respectively.
The optimal solution shows that only if there is no ambiguity (Σ_{μ}[E[r]] = 0) or if the bank is neutral to ambiguity (θ = 0) does the Markowitz-Tobin risk model accurately predict the bank's optimal portfolio composition. However, since banks are generally averse to ambiguity, and return correlations go to 1 as ambiguity rises, θΣ_{μ}[E[r]] is positive and the risk model is incorrect. More important, the robust portfolio weights in equation 2 are smaller than the risk weights. So if the bank ignores ambiguity while allocating its assets, it will end up taking on too much risk in its portfolio. And because risk-weighted assets also determine how much capital the bank holds, capital ratios resulting from a suboptimal asset allocation are at best opaque measures of the bank's safety. Thus, in light of the fact that ambiguity is nearly absent in banks' asset-allocation decisions, and that the ultimate goal of regulatory policy is to create a safer financial system, tighter capital reforms are essential; failure to introduce them, or enforce them, encourages banks to behave recklessly.
The Broader Context
While the development of financial-sector models with uncertainty has been limited to date, models of "real" economic activity under uncertainty have received greater attention. Specifically, recent research investigates how uncertainty can generate business cycle fluctuations. Gilchrist, Sim, and Zakrajsek (2010) find that as uncertainty increases, so does it increase the default risk premium of a corporation's bondholders, leading to a higher cost of capital and lower investment. And if uncertainty is so extreme as to impede corporate boards' decisions, investment can be forgone entirely (Garlappi, Giammarino, and Lazrak, 2013). Uncertainty about government policy has real effects too. Fernandez-Villaverde, Guerron-Quintana, and Kuester (2012) show how fiscal uncertainty about the timing and form of budgetary adjustments reduces investment, aggregate output, consumption, and hours worked. Julio and Yook (2012) document that high political uncertainty causes lower investment in election years, while Pastor and Veronesi (2012) measure movements in stock prices after a policy change is announced. Baker, Bloom, and Davis (2013) suggest that the slow recovery from the Great Recession is associated with higher policy uncertainty during the period 2007 to 2009. Together, these papers convincingly prove that expectations over future policy changes affect economic agents' current decisions--an argument made famous by Lucas (1976) and Kydland and Prescott (1977). How to effectively mitigate the adverse effects of uncertainty on agents' decisions, however, remains to be fully worked out.
Some clarity may come from better understanding the "essential" complexity of economic structures through a microfoundation of the macroeconomy--or, within the regulatory framework, through the integration of microprudential and macroprudential regulation.^{7} This is no small feat. At the micro-level, devising precise policy responses to a crisis (for example, setting parameters that banks must obey) is desirable but poses a danger. Because it is impossible to fully anticipate the effects of regulatory measures taken under uncertainty, the ex post evaluation of the effects may be difficult and their interpretation misleading. At the macro-level, modeling the complexity of markets requires sensible, fact-based assumptions, even at the cost of tractability. Developing models that tackle these challenges comprehensively seems worthwhile and I believe it can be achieved through novel approaches and more interdisciplinary discussions and research.
References
Baker, S., Bloom, N., and Davis, S. (2013). "Measuring Economic Policy Uncertainty." Working Paper No. 13-02. Chicago: University of Chicago, Booth School of Business.
Boyle, P., Garlappi, L., Uppal, R., and Wang, T. (2012). "Keynes Meets Markowitz: The Tradeoff between Familiarity and Diversification." Management Science, Vol. 58, pp. 253-272.
Caballero, R., and Krishnamurthy, A. (2008). "Knightian Uncertainty and Its Implications for the TARP." Financial Times Economists' Forum, November 24.
Easley, D., and O'Hara, M. (2009). "Ambiguity and Nonparticipation: The Role of Regulation." Review of Financial Studies, Vol. 29, pp. 1817-1843.
Easley, D., and O'Hara, M. (2010a). "Liquidity and Valuation in an Uncertain World." Journal of Financial Economics, Vol. 97, pp. 1-11.
Easley, D., and O'Hara, M. (2010b). "Microstructure and Ambiguity." Journal of Finance, Vol. 65, pp. 1817-1846.
Epstein, L., and Miao, J. (2003). "A Two-Person Dynamic Equilibrium under Ambiguity." Journal of Economic Dynamics and Control, Vol. 27, pp. 1253-1288.
Fernandez-Villaverde, J., Guerron-Quintana, P., and Kuester, K. (2012). "Fiscal Volatility Shocks and Economic Activity." NBER Working Paper 17317 .
Garlappi, L., Giammarino, R., and Lazrak, A. (2013). "Ambiguity in Corporate Finance: Real Investment Dynamics." Working Paper.
Gilboa, I., and Schmeidler, D. (1989). "Maxmin Expected Utility with Non-unique Prior." Journal of Mathematical Economics, Vol. 18, pp. 141-153.
Gilchrist, S., Sim, J., and Zakrajsek, E. (2010). "Uncertainty, Financial Frictions, and Investment Dynamics." Working Paper.
Gollier, C. (2011). "Portfolio Choices and Asset Prices: The Comparative Statics of Ambiguity Aversion." Review of Economic Studies, Vol. 78, pp. 1329-1344.
Hansen, L., and Sargent, T. (2010). "Fragile Beliefs and the Price of Uncertainty."Quantitative Economics, Vol. 1, pp. 129-162.
Hayek, F. (1974). "The Pretence of Knowledge." Prize Lecture, The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.
Illeditsch, P. (2011). "Ambiguous Information, Portfolio Inertia, and Excess Volatility." Journal of Finance, Vol. 66, pp. 2213-2247.
Izhakian, Y., and Benninga, S. (2011). "The Uncertainty Premium in an Ambiguous Economy." Quarterly Journal of Finance, Vol. 1, pp. 323-354.
Julio, B., and Yook, Y. (2012). "Political Uncertainty and Corporate Investment Cycles." Journal of Finance, Vol. 67, pp. 45-83.
Klibanoff, P., Marinacci, M., and Mukerji, S. (2005). "A Smooth Model of Decision Making under Ambiguity." Econometrica, Vol. 73, pp. 1849-1892.
Knight, F. (1921). Risk, Uncertainty, and Profit. Boston, Mass.: Hart, Schaffner & Marx/Houghton Mifflin.
Kydland F., and Prescott, E. (1977). "Rules Rather than Discretion: The Inconsistency of Optimal Plans." Journal of Political Economy, Vol. 85, pp. 473-492.
Lucas, R. (1976). "Econometric Policy Evaluation: A Critique," in Brunner, K., and Meltzer, A., eds., The Phillips Curve and Labor Markets. Carnegie-Rochester Conference Series on Public Policy, Vol. 1. Amsterdam: North-Holland, pp. 19-46.
Maccheroni, F., Marinacci, M., and Ruffino, D. (2013). "Alpha as Ambiguity: Robust Mean-Variance Portfolio Analysis." Econometrica, Vol. 81, pp. 1075-1113.
Pastor, L., and Veronesi, P. (2012). "Uncertainty about Government Policy and Stock Prices." Journal of Finance, Vol. 64, pp. 1219-1264.
Ruffino, D. (2014). "A Robust Capital Asset Pricing Model." Finance and Economics Discussion Series 2014-01. Washington: Board of Governors of the Federal Reserve System.
Samuelson, P. (1986). "Gold and Common Stocks," in Crowley, K., ed., The Collected Scientific Papers of Paul Samuelson, Vol. 5. Cambridge, Mass.: MIT Press, p. 561.
1. Google searches for the phrase "economic uncertainty," for example, rose to an all-time high in February 2009. The second-highest level was reached in May 2010, as the euro-zone crisis intensified. These findings derive from a chart of the frequency of searches for the phrase "economic uncertainty" generated by the website Google Trends, at http://google.com/trends (accessed November, 2013). Return to text
2. Henceforth I use the term "uncertainty" (or "ambiguity") in the sense of Knight (1921), who first defined risk and uncertainty independently. Knight wrote that "risk is randomness in which events have measurable probabilities". Uncertainty, however, describes events with unknown or objectively unmeasurable probabilities. Return to text
3. The required premium--called the ambiguity premium--is quantitatively significant over and above the classic risk premium, and it increases with aversion toward uncertainty. Maccheroni, Marinacci, and Ruffino (2013); Gollier (2011); and Izhakian and Benninga (2011) lay the foundations for the study of the ambiguity premium. Return to text
4. See, for example, Easley and O'Hara (2009); Boyle, Garlappi, Uppal, and Wang (2012); Illeditsch (2011); and Epstein and Miao (2003). Return to text
5. Ruffino (2014) and Easley and O'Hara (2010a), among others, find that uncertainty increases required asset returns in equilibrium. Return to text
6. A general version of this example is developed in Ruffino (2014). Throughout the analysis I opt for the term "ambiguity" instead of "uncertainty". Return to text
7. On essential complexity, see Hayek (1974): "The social sciences, like much of biology but unlike most fields of the physical sciences, have to deal with structures of essential complexity, i.e. with structures whose characteristic properties can be exhibited only by models made up of relatively large numbers of variables." Return to text
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