The Probability of Rare Disasters: Estimation and Implications

Transcription

1 The Probability of Rare Disasters: Estimation and Implications Emil Siriwardane 1 1 Harvard Business School Harvard Macro Seminar: 9/21/2015 1/32

2 Introduction 1/32

3 Rare Disasters œ Recent growth in rare disasters (RD) paradigm e.g. Rietz (1988), Barro (2006) œ Main Idea: rare macroeconomic disasters can explain asset prices Equity premium puzzle Can also account for business cycles in production economies œ Why should we care? Unified framework between macroeconomics and asset pricing Capital costs: do some firms do worse in disasters? Is it priced? Welfare: benefits of financial asset trade in the presence of disasters 1/32

4 Empirical Work on Rare Disasters œ International macro data paints a positive picture œ With reasonable risk-aversion, the average frequency and severity of disasters matches unconditional stock market returns œ This is true even if: Disasters unfold over a number of years The economy recovers after disasters œ Most empirical work has assumed a constant probability of a disaster 2/32

5 Time-Varying vs. Constant Disaster Probability œ Time-series variation in the probability of a disaster is critical œ Theoretically: Explains many asset pricing patterns (excess vol, term structure, FX rates, etc.) Necessary to generate realistic business cycles œ Empirically: With constant probability, S&P 500 options imply a disaster probability that disagrees with macro data... But with time-varying disasters, macro data implies realistic option prices 3/32

6 This Paper œ Is there evidence that the probability of a disaster is time-varying? œ Can we estimate this variation? œ How do asset markets and the macroeconomy respond to changes in the probability of a disaster? Is it consistent with theory? œ This paper sheds light on these questions 4/32

7 The Probability of a Disaster 35 Model-Implied Probability of Disaster (%) Russian/LTCM Default Dot Com Bubbles Bursts U.S. Plans to Invade Iraq U.S. Financial Crisis Euro Crisis, Gaza Strip Euro Crisis (Greece/Spain) /32

8 High Level Logic of My Approach œ Start in the options markets. Why? RD hypothesis has unique predictions for options œ Core assumption of all rare disaster models: Disasters happen at same time for every firm in the economy Firms just differ in how exposed they are to the event œ Using options, I form a non-parametric estimate of jump-risk: D it p it {z} {z} i prob. of firm i jumping determined by jump size of firm i œ RD hypothesis says that p it = p t should be same across firms 6/32

9 Summary of Findings Three core facts Time series variation in firm-level jump risk is driven by single factor. bp t Extracted factor. Clearly not constant 2. An increase in bp t forecasts declines in short-run economic activity: Unemployment rises Industrial production, manufacturing, and total capacity utilization fall 3. Disaster risk is priced in cross-section of U.S. equities Disaster risky stocks earn 7.6% in risk-adjusted returns Plus a calibration to sharpen the mapping between the model and the data S&P 500 falls when bp t increases - matches calibrated model 2. Reasonable cross-sectional loss rates of stocks in a disaster Model: disaster risky stocks must lose 57%. Data: 51% Findings are consistent with a standard time-varying rare disasters model 7/32

10 A Model of Rare Disasters: Quick Overview 7/32

11 Outline œ Out-of-the-box endowment economy with variable disasters: Continuous time version of Barro (2006); Gabaix (2012) œ Purpose of going through the model: Overview of RD hypothesis Intuition for how option prices comove under null of the model œ Roadmap: 1. Economy primitives 2. Equilibrium P-D ratios 3. Risk-neutral returns 4. Option prices + disaster likelihood 8/32

12 Consumption and Dividend Dynamics Aggregate Consumption dc t = g C dt + æ C dw C,t + (B t 1)dJ t C t {z } {z } Normal-times Risk Disaster Risk œ J t is macroeconomic jump process œ Disaster occurs over next dt with probability p t dt œ B t > 0 is a stochastic disaster recovery rate Dividend Growth dd it = g id dt + dn D it D + (F it 1)dJ t it {z } {z } Normal-times Risk Disaster Risk œ dnit D is a mean-zero martingale and is independent of disasters œ Stocks with higher F it do better in disasters 9/32

13 Preferences and State Variable Dynamics œ Preferences = Epstein-Zin-Weil risk-aversion. IES = (1 1/ )/(1 ) œ State variable dynamics: One route is to model p t, B t, F it separately Instead, I model them jointly... œ Define resilience: H it º p t E t B t (F it +( 1)B t ) Stocks with high resilience do better in crises Resilience is a linearity-generating (LG) process º AR(1) Mean-reverts to H i at speed H,i œ Gabaix (2012) or Farhi and Gabaix (2015) for other examples 10 / 32

14 Equilibrium œ Price-dividend ratios are given by: P it = D it ± i eh it ± i + H,i! ± = Ω +g C / ± i = ± g id H i œ eh it = H it H i, so measures deviations from mean Stocks with high resilience have high valuation ratios Mean-reversion in resilience generates mean-reversion in P-D ratio œ A useful quantity for empirical work = so-called risk-neutral returns 11 / 32

15 Risk-Neutral Probability Measure A Primer œ Risk-neutral probability is proportional to Arrow-Debreu state price œ Discrete time analogue: Suppose there are s = 1,...S possible states of the world Each state has a probability º(s) Marginal utility in each state is m(s) œ State prices (AD) and risk-neutral probabilities (º ) AD(s) = º(s) m(s) º (s) = AD(s)/ P s AD(s) = (1 +r f ) AD(s) œ I m interested in how º (s = disaster) moves around 12 / 32

16 Equilibrium Returns Under the Risk-Neutral Measure In the model dp it P it = r ft dt + æ it dw it p te t B t (F it 1) dt +(F it 1)dJ t œ J t is the RN macroeconomic jump (disaster) process œ Under the RN-measure, disasters no longer occur with a probability p t œ p t p te t B t is the RN probability of a disaster: When E t [B t ] < 1 and > 0, then p t > p t œ Intuition: risk-averse agent prices assets as if disaster is more likely 13 / 32

17 I. Intuition: Using Options to Learn About Jumps œ For simplicity, shut off time-varying disasters and assume F i ø 1 œ RN returns take a simple form: dp it P it = (r f B (F i 1))dt + æ i dw it +(F i 1)dJ t œ In this case, the value of a short-dated put option is given by: Put(P it,k P ) º p Put BS (F i P it,k P ) {z } + (1 p ) Put BS (P it,k P ) {z } value if there is a disaster value if there is no disaster œ Put BS (X,K) is a Black-Scholes put with initial price X, strike K, and volatility æ i 14 / 32

18 II. Intuition: Using Options to Learn About Jumps Put(P it,k P ) º p Put BS (F i P it,k P ) {z } + (1 p ) Put BS (P it,k P ) {z } value if there is a disaster value if there is no disaster œ Since F i ø 1, a call option with strike K C > P it has no value in disaster: Call(P it,k C ) º (1 p ) Call BS (P it,k C ) œ Next, form an option portfolio called a risk-reversal: RR = Put(P it,k P ) m Call(P it,k C ) m and K C chosen to exactly offset the non-disaster portion of the put Thus, RR = p Put BS (F i P it,k P ) œ Takeaway: every firm s risk-reversal contains information on p 15 / 32

19 A More General Way to Estimate Jumps œ Can estimate jumps for broader class of risk-neutral return processes œ Carr and Wu (2009) and Du and Kapadia (2013): Use a portfolio of options (across strikes) to isolate the jump Resembles the construction of the VIX. Basically, a more general version of a risk-reversal œ For a given date and firm, I call this measure D it : D it = i p t œ i a simple function of the recovery, F it : i = 2 E 1 +ln(f i ) +ln(f i ) 2 /2 F i 16 / 32

20 A Comment on this Jump Risk Estimator D it = i p t œ Testable hypothesis: the model suggests that a panel of D it should obey a single factor structure œ The single factor should capture time-series variation in p t œ The option pricing theory behind D it is actually non-parametric: Doesn t impose that p t is the same across firms i.e. if each firm just has an idiosyncratic jump process, then D it = i p it œ Useful because we can check whether option-implied tails are consistent with the model, without imposing anything on the data 17 / 32

21 The Factor Structure of Option-Implied Tails 17 / 32

22 The Data œ I form a jump risk measure, D it, for each firm and each date Includes the entire universe of U.S. traded options in OptionMetrics January 1996-April 2015 œ D it is a 30-day measure of expected jumps for a given firm. œ 6,762,860 firm-day pairs œ Keep in mind that this measure does not depend on the RD model! No prior structure is assumed on the panel of D it 18 / 32

23 Factor Structure of Jumps (Principal Component Analysis) 50 Percent of Variation Captured by Factor Factor 19 / 32

24 A Single Factor Drives Variation in Option-Implied Jumps œ Panel of D it has clear one-factor structure Very robust to subsamples, inclusion of firms, etc. œ Implies that jump intensities are indeed the same across firms, i.e. p t is common across firms œ This is the core of the rare disasters paradigm œ Can also map the extracted factor, bp t, to the model: bp t = p t = E t B t pt Bottom Line: Time-series variation in factor = time-series variation in p t 20 / 32

25 The Probability of a Disaster Assume constant disaster severity + use Barro (2006) calibration to recover p t 35 Data: p = 3.1%... Barro and Ursua (2008)/Nakamura et al. (2013): p º 3.2% Model-Implied Probability of Disaster (%) Russian/LTCM Default Dot Com Bubbles Bursts U.S. Plans to Invade Iraq U.S. Financial Crisis Euro Crisis, Gaza Strip Euro Crisis (Greece/Spain) / 32

26 Supporting Evidence: p t and Future Economic Activity 21 / 32

27 Motivation: The Probability of Disaster in GE œ Gourio (2012) + Kilic and Wachter (2015) build production economies with disasters œ Example of intuition: 1. Capital stock is reduced by disasters (think wars) 2. Capital is riskier when disasters are imminent 3. So firms invest less when disasters become more likely œ Motivated by these models, I use forecasting regressions as a way to validate the interpretation of p t : 22 / 32

28 Forecasting Regressions œ Basic setup: Y t+1 = c + X 4 i=1 iy t+1 i + 0 X t + bp t +u t+1 œ Y t is one of a few macro-variables œ X t are competing forecasting variables: Real federal funds rate Term structure of Treasury yields (10Y-3M) Aggregate stock market returns œ Monthly horizon from April bp t is standardized. 23 / 32

29 When the Probability of Disaster ", Economic Activity # Unemployment and TCU in % points. Y t+1 = c + X 4 i=1 iy t+1 i + 0 X t + bp t +u t+1 Dependent Variable at t +1 Coefficient on bp t t-stat Adj. R 2 Unemployment % growth in industrial production Total Capacity Utilization % growth in manuf. inventories CFNAI œ Increases in bp t lead to: 6.4bp rise in unemployment, º 100K people (Kilic and Wachter (2015)) 24 / 32

30 When the Probability of Disaster ", Economic Activity # Unemployment and TCU in % points. Y t+1 = c + X 4 i=1 iy t+1 i + 0 X t + bp t +u t+1 Dependent Variable at t +1 Coefficient on bp t t-stat Adj. R 2 Unemployment % growth in industrial production Total Capacity Utilization % growth in manuf. inventories CFNAI œ Increases in p t lead to: 6.4bp rise in unemployment, º 100K (Kilic and Wachter (2015)) Decreased production and investment (Gourio (2012)) 24 / 32

31 Discussion and Robustness œ Macroeconomy almost always responds more to bp t than: Aggregate stock market returns Real federal funds rate Slope of the term structure œ Unemployment most sensitive to changes in bp t œ Results not driven entirely by the crisis: Most go through if I exclude Dec 2007-June 2009 œ For forecasting, bp t generally outperforms the VIX bp t has unique information on the macroeconomy Results in line with bp t measuring variation in disaster probability 25 / 32

32 Supporting Evidence: Disaster Risk and Cross-Section of Equity Returns 25 / 32

33 The Cross-Section of Equity Returns œ In the model, expected excess returns given by: µ it r ft = p t E t B t (1 F it ) œ Firms that recover a lot in disasters (high F it ) have low premiums œ bp t also useful for determining which stocks are disaster risky: r it = a i + Ø Disaster i bp t +error it Disaster-Ø s are increasing in E t [F it ]. œ Punchline: stocks with high disaster-ø s should earn low returns Disaster-Ø > 0, means stock rises when bp t rises. Hedges disaster. 26 / 32

34 A Simple Test œ Sort stocks into portfolios based on disaster-ø s œ Trading strategy done in real time, so fully implementable œ Includes entire universe of CRSP stocks œ Monthly frequency 27 / 32

35 Disaster Risk is Priced in Cross-Section of U.S. Equities Sample Period: February 1996-December All returns annualized. Portfolio Excess Return 4-Factor Æ CAPM-Ø Low Disaster Risk ** ** 0.78 High Disaster Risk * 0.88 High-Low ** For Æ s, ** = 5% significance and * = 10% significance. All CAPM-Ø s significant. Excess returns monotonically increase from low to high disaster risk 28 / 32

36 Disaster Risk is Priced in Cross-Section of U.S. Equities Sample Period: February 1996-December All returns annualized. Portfolio Excess Return 4-Factor Æ CAPM-Ø Low Disaster Risk ** ** 0.78 High Disaster Risk * 0.88 High-Low ** For Æ s, ** = 5% significance and * = 10% significance. All CAPM-Ø s significant. Same pattern after adjusting for exposure to Fama-French factors 28 / 32

37 Disaster Risk is Priced in Cross-Section of U.S. Equities Sample Period: February 1996-December All returns annualized. Portfolio Excess Return 4-Factor Æ CAPM-Ø Low Disaster Risk ** ** 0.78 High Disaster Risk * 0.88 High-Low ** For Æ s, ** = 5% significance and * = 10% significance. All CAPM-Ø s significant. Reject joint hypothesis that all Æ s= 0. p-value = 1.28% 28 / 32

38 Disaster Risk is Priced in Cross-Section of U.S. Equities Sample Period: February 1996-December All returns annualized. Portfolio Excess Return 4-Factor Æ CAPM-Ø Low Disaster Risk ** ** 0.78 High Disaster Risk * 0.88 High-Low ** For Æ s, ** = 5% significance and * = 10% significance. All CAPM-Ø s significant. CAPM-Ø doesn t really capture disaster risk (Gabaix (2012)) 28 / 32

39 Disaster Risk is Priced in Cross-Section of U.S. Equities Sample Period: February 1996-December All returns annualized. Portfolio Excess Return 4-Factor Æ CAPM-Ø Low Disaster Risk ** ** 0.78 High Disaster Risk * 0.88 High-Low ** For Æ s, ** = 5% significance and * = 10% significance. All CAPM-Ø s significant. Takeaway: Disaster risk is priced. Premium for bearing disaster risk º 7.6% 28 / 32

40 Two Calibration Exercises 28 / 32

41 What is the Disaster-Ø of the Aggregate Stock Market? Calibration: Disaster severity, B = F = 0.6. Risk-aversion, = 4. IES = 2 œ Simple regression of S&P 500 returns on bp t : Disaster-Ø = 6.4 t-stat = -7.9 œ The fact that disaster-ø < 0 implies an IES> 1 Barro (2009) Nakamura, Steinsson, Barro, and Ursua (2013) œ In calibrated model, disaster-ø º 11. So in the same ballpark. Can also roughly match range of observed disaster-ø s in cross-section 29 / 32

42 How Much Should Disaster Risky Stocks Fall in the Crisis? Calibration: Disaster Likelihood and Severity, p = and B = 0.6. Preferences unchanged Portfolio Realized Recovery ( F b i ) Model-Implied Recovery (F i ) Model suggests disaster risky stocks should fall slightly more during crisis. They also fall less than low disaster risk stocks. Government distortions? 30 / 32

43 How Much Should Disaster Risky Stocks Fall in the Crisis? Calibration: Disaster Likelihood and Severity, p = and B = 0.6. Preferences unchanged Portfolio Realized Recovery ( F b i ) Model-Implied Recovery (F i ) Model overshoots low disaster risk stocks. Additional factors needed? 30 / 32

44 Conclusion 30 / 32

45 In Summary œ This paper contains a set of facts about options/equities. 1. A panel of option-implied jump measures is driven by single factor, bp t 2. bp t forecasts declines in future economic activity 3. Stocks that fall when bp t rises are risky, so require a premium œ Price of disaster risk is about 7.6% annually œ These facts are consistent with the rare disasters model: Two calibration exercises to sharpen the mapping œ Interpret bp t as the (risk-neutral) probability of a disaster 31 / 32

46 Next Steps œ What are disaster risky stocks? œ What drives movements in bp t? Good place to start: who is pricing the risk? Link to intermediary-based asset pricing? œ Estimate a structural model Non-parametric analysis suggests this is a good idea Identification via entire cross-section of options (firms + strikes) 32 / 32

47 Thank You! 32 / 32

48 Robert J. Barro. Rare disasters and asset markets in the twentieth century. The Quarterly Journal of Economics, 121(3): , doi: /qjec Robert J. Barro. Rare disasters, asset prices, and welfare costs. American Economic Review, 99(1):243 64, Peter Carr and Liuren Wu. Variance risk premiums. Review of Financial Studies, 22(3): , Jian Du and Nikunj Kapadia. The tail in the volatility index Emmanuel Farhi and Xavier Gabaix. Rare disasters and exchange rates. Quarterly Journal of Economics, Xavier Gabaix. Variable rare disasters: An exactly solved framework for ten puzzles in macro-finance. The Quarterly Journal of Economics, 127(2): , Francois Gourio. Disaster risk and business cycles. American Economic Review, 102(6): , September Mete Kilic and Jessica A. Wachter. Risk, unemployment, and the stock market: A rare-event-based explanation of labor market. Working Paper, Emi Nakamura, Jon Steinsson, Robert Barro, and Jose Ursua. Crises and recoveries in an empirical model of consumption disasters. American Economic Journal: Macroeconomics, 5(3):35 74, doi: /mac URL Thomas A. Rietz. The equity risk premium: A solution. Journal of Monetary Economics, 22(1): , July / 32