The Economics of the Comovement of Stocks and Bonds

Transcription

1 Chapter Fifteen The Economics of the Comovement of Stocks and Bonds Alexander David University of Calgary Pietro Veronesi University of Chicago 15.1 Introduction The purpose of this chapter is to provide some recently uncovered facts about the dynamics of the covariance of stocks and bonds as well as to provide a brief literature survey that attempts to explain this covariation. Given our taste for explaining stylized facts about asset prices with models of investors learning about the state of fundamentals, we give more than a fair share of space to this line of research. In particular we develop a model based measure of investors uncertainty about fundamentals and explicitly consider its implications for the stock-bond covariance. We then show that beliefs of professional forecasters extracted from survey data collected by the the Federal Reserve Bank of Philadelphia are highly correlated with the model based beliefs, and have almost the same implications for the stock-bond covariance. We also find some supporting evidence for our model s predictions in international data as well. 509

2 510 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds The left panel of Figure 15.1 shows the quarterly realized series of covariance of returns on the S&P 500 (stocks) and 5-year Treasuries (bonds) constructed from daily returns in each quarter. Quite dramatically, the covariance turned from being mostly positive until about 2000 to being mostly negative since. The right panel shows that the realized beta of bonds also experienced this same sign reversal. That is, since 2000, bonds have become an important hedge against stock market fluctuations. But why weren t bonds a hedge historically? What has changed recently? The answers to these questions have obvious first order consequences for asset allocation between two of the largest financial asset classes A Brief Literature Survey We start with the early work on the stock-bond covariance. Important theoretical relations of the stock and real bond covariance are developed in Barksy (1989). In a two period general equilibrium model, he considers the roles of increased risk and reduced productivity growth in determining the stock-bond covariance. Both disturbances unambiguously lower the real riskless rate, but may cause the stock market to respond negatively depending on the elasticity of intertemporal substitution. This model does not have inflation, but provides valuable intuition for at least a part of the stock and nominal bond covariance. Campbell and Ammer (1993) decompose the returns on stocks and bonds into Stock Bond Covariance Bond Beta FIGURE 15.1 Stock-Bond Covariance and Beta of 5-Year Treasury Bonds

3 15.2 A Brief Literature Survey 511 cash flow and discount rate components and argue that stock-bond covariance should be low. The only component that is common to both assets is the real interest rate, which has relatively low variability. They do find a common component to news about future excess returns on bonds and stocks but find that it is unable to produce a large stock-bond covariance since news is not a large component of bond returns. The massive sign flip in the stock-bond covariance in the data has been picked up in more recent work, and efforts have been made to understand this time variation. Baele, Bekaert and Inghelbrecht (2010) use a linear dynamic factor setting to study the ability of economic factors to explain the dynamics of the stock-bond correlation. 1 The set of factors that they use is large, including interest rates, inflation, the output gap, and cash-flow growth, as well as measures of risk aversion, and uncertainty proxies for real growth and inflation. These authors conclude that fundamental factors contribute little to the stock-bond correlation, but instead, argue that liquidity factors play a larger role in explaining its dynamics. Despite having much of the same macroeconomic information as our model, these authors do not explain the stock-bond correlation, mainly due to their linear specification. In contrast, we will see below that in a learning based framework, asset volatilities and covariances are nonlinear functions of fundamentals. Building on the proxy hypothesis of Fama (1981), researchers have generally thought of rising inflation as a signal for deteriorating fundamentals. For example, using a state space framework, Piazzesi and Schneider (2006) assume that investors dislike surprise inflation not only because it lowers the payoff on nominal bonds, but also because its bad news for future consumption growth. This effect is large when investors have recursive utility with preference for early resolution of uncertainty, as in the long-run risks channel of Bansal and Yaron (2004). Hasseltoft (2009) extends this framework by including dividend growth in the specification and studies the implications for the Fed Model and the stock-bond covariance. A high stock-bond covariance is generated by the common variation in bond and stock risk premia, which is induced by the variation in macro-economic volatility. The model is able to generate a mild negative stock-bond correlation in the last decade by using the covariance between fundamentals as state variables. Indeed, the author estimates a positive covariance between inflation and real dividend growth in the last decade, which in turn induces a mild negative correlation between stocks and bonds in his model. Indeed, more recent models are now able to generate both positive and negative stock-bond covariance. Campbell, Sunderam, and Viceira (2013) use a quadratic term structure model with a latent variable that determines the sign of the relation between real and nominal variables in the economy. After fit- 1 Their model is as follows: Let rt = (r e,t, r b,t ) be the vector of stock and bond returns. Then r t = E t 1 (r t) + β t Ft + ɛt, where F t is the vector of economic factors, and ɛ t N(0, Σ t). The betas as well as Σ t are allowed to be regime dependent.

4 512 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds ting their model to macroeconomic, yield, and stock data from mid 1950s to late 2000, they show that their model implies a stock-bond covariance that endogenously switches sign as a function of the latent state variable. Their fitted model also implies that bond risk premia should have switched sign over time, as bonds turned from being inflation bets in the 1980s to deflation hedges after In fact, especially during the financial crisis, nominal bonds were considered a safe haven that provides insurance against particularly severe adverse economic conditions. That is, bad economic news would make the stock market drop but at the same time make nominal bond gain value, as investors dump stocks for bonds in search of safety. Their status as a hedge against severe economic conditions would induce investors to purchase those bonds even at a premium, that is, purchase them with a yield below expected inflation. Campbell, Pflueger, and Viceira (2014) use a New Keynesian framework to understand the time variation in the stock-bond covariance. A key intuition of their model is that the response of monetary policy changes over time in reaction to supply versus demand shocks which leads to a different stock-bond relation. The effects are magnified by changes in risk aversion, particularly in bad times, which make the model s covariance closer to that in the data. Kozak (2013) proposes a stylized model of production featuring two investment technologies, adjustment costs, and agents with time varying risk aversion. The model s equilibrium produces a time varying covariance between returns to the risky technology and long-term claims to safe debt. The main driver of this covariance is the time variation in the level of risk aversion, which induces a negative covariance between stocks and bonds ( flight to safety ) especially during periods of high risk aversion. An interesting observation made by Connolly, Sun, and Stivers (2005) is that the stock-bond covariance becomes more negative in periods of high stock market volatility. This has certainly been true since These authors argue that in periods of high stock market volatility, investors seek the safety of bonds, so that bond returns are high precisely when stock returns are low. We discuss this issue below, since the second half of the last century showed the opposite empirical pattern The Stock-Bond Covariance and Learning About Fundamentals In David and Veronesi (2013), we study an endowment economy in which the drift rates of real earnings growth and inflation follow an unobserved composite regime-switching model. Investors also observe a smoothed version of true consumption growth with some noise, although its drift is unknown but con-

5 15.3 The Stock-Bond Covariance and Learning About Fundamentals 513 stant. 2 Market participants cannot observe the current regime and thus must learn about it by observing fundamentals and other signals. 3 Inflation has a signaling role in predicting real growth as in the proxy hypothesis of Fama (1981). In addition, we allow for very low inflation (close to zero) to also be an adverse signal about real fundamentals, an observation that is influenced by the recent evidence in Japan, the US, and perhaps more recently, in Western Europe. This nonlinear signaling role of inflation is tractably modeled with the specification of the transitions among the composite regimes of real fundamentals and inflation. We extract investors beliefs about fundamentals from data on both fundamentals as well as asset prices. We use asset prices since they undoubtedly contain forward looking information that investors have about fundamentals, while fundamental data by definition are at best current. A key aspect of our methodology is to develop closed-form formulae for stock and Treasury bond prices, as well as their volatilities and covariances as functions of investors beliefs about fundamentals. We then estimate an overidentified Simulated Method of Moments objective function, which attempts to explain not only the stockbond covariance, but also the complete dynamics of fundamentals, stock and Treasury bond (1-year and 5-year) prices, and volatilities of stocks and bonds. 4 In the following discussion, we leave out the estimates of all the parameters and implications for a range of issues in David and Veronesi (2013) and focus mainly on the narratives of the composite regimes and their implications for the stock-bond covariance. We start by looking at the closed form expressions for the volatility of stocks and bonds, and discuss their determinants. The volatility of nominal stock returns is: 5 n σ N i=1 (π t ) = σ Q + σ E + G i π it (ν i ν(π t )) (Σ ) 1. (15.1) P/E (π t ) There are three sources of variation in nominal stock returns: the volatilities of stocks due to inflation Q t, to real earnings E t, and to beliefs n i=1 π itg i. In the latter, π it denotes investors belief to be in the composite regime i in period 2 The smoothing coefficient of consumption (we estimate that true consumption is about 5 times more volatile than aggregate consumption) is estimated by our procedure and reflects the government s aggregation procedure across households as well as errors in its sampling and data collection. Having a more volatile true consumption brings our model s equity premium closer to the data. 3 In our model, investors also suffer from some degree of money illusion, that is, they partly discount future real cash flows using a nominal stochastic discount factor. The main impact on asset prices is that ceteris paribus, it raises the real rate in periods of higher expected inflation. We show that this feature is mainly useful for getting a better fit for the term structure, but has only a small impact on the stock-bond covariance. 4 We must use SMM rather than GMM since we specify a continuous time model, with discrete observations of fundamentals whose likelihood function is not available in closed-form. 5 With some abuse of terminology, we refer to the diffusion of a return process as its volatility. Equations (15.1) and (15.2) show the diffusion vectors of the return processes. Strictly speaking, the volatility of stock returns, for instance, is the scalar given by σ N (π)σ N (π).

6 514 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds t, G i is the P/E ratio for stocks in regime i, ν i is a vector of fundamental drifts of inflation and earnings growth, and ν(π t ) is its conditional expectation. In particular, the last term in (15.1) is a learning-based, time-varying endogenous component that depends on beliefs π it. It is important to note, that unlike the recent literature on modeling uncertainty (see e.g. Bansal and Yaron (2004) or Bloom (2009)), we assume homoskedastic fundamentals (σ Q and σ E are constant), so that any variation in volatility is endogenous in our model and arises from learning. Also, importantly, our model implies that stock valuation has an important impact on volatility and that uncertainty about regimes in itself does not fully explain the impact of learning on volatility. In fact, valuations (G i in Equation 15.1) change the relative weights given to the alternative sources of uncertainty, inflation and earnings, during different periods in our sample. The volatility of nominal zero coupons bond returns is n σ B i=1 (π t, τ) = B i(τ) π it (ν i ν(π t )) (Σ ) 1. (15.2) B(τ) The form of the bond s volatility in (15.2) is similar to the stock s volatility in (15.1), except that the conditional P/E ratio G i in any regime i is replaced by the conditional bond price B i (τ) in that regime. In addition, there is no exogenous fundamental component since the bond payoff is fixed at maturity. Intriguingly, bond return volatility is non-linearly related to the long-term yield, being higher when the long-term yield is both high or low. This result is consistent with agents uncertainty about entering either a hyper-inflation regime or a deflationary regime, each of which has weak real fundamentals. In either case, bond return volatility is high because uncertainty is high, but the level of longterm yields are at their opposite extremes. This insight may also explain the weak evidence in favor of a linear relation between volatility and bond yields (see e.g. Collin-Dufresne and Goldstein (2002)). From these two volatilities, the stock-bond covariance at different maturities is simply ( dp N Cov t (π t ) Pt N (π t), db ) t(π t, τ) = σ N (π t ) σ B (π t, τ) (15.3) B t (π t, τ) INVESTORS BELIEFS ABOUT COMPOSITE REGIMES Figure 15.2 shows the implied beliefs of investors about each composite growth and inflation regime using our methodology. The beliefs are for the period from 1958 to 2013, based on parameters estimated until 2010 in David and Veronesi (2013). Our estimation procedure revealed that six regimes were required to provide a reasonable fit for the time series of fundamentals, prices, and volatilities (12 series overall) and also to match the unconditional Sharpe ratio on stocks of 0.3 (consistent with the equity premium literature). The left panels show the conditional probabilities of the three low growth regimes (recessions), while the right panels show those for the medium and high

7 15.3 The Stock-Bond Covariance and Learning About Fundamentals 515 growth regimes (booms). The top-left panel shows that investors believed the recessions in the second half of the 1900s typically had low growth accompanied by medium inflation. The probability of such recessions increased in the recessions in the 1970s, 1980s and 1990s. The middle-left panel shows that the beliefs about stagflation only spiked once in our sample around , a period that preceded the Volcker regime when interest rates increased dramatically and were consequently followed by two recessions. Finally, the bottom panel shows the beliefs about deflationary (zero inflation) recessions, which spiked in the recessions of the current millennium. Overall, the three panels show the changing nature of recessions in our sample. Low Growth, Medium Inflation Medium Growth, Low Inflation Low Growth, High Inflation Medium Growth, Medium Inflation Low Growth, Zero Inflation High Growth, Low Inflation FIGURE 15.2 Investors Beliefs About Fundamentals Implied From Regime Switching Model of David and Veronesi (2013)

8 516 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds The right panels show the beliefs about the three kinds of booms. The top panel shows the most likely regime of low inflation and medium growth. There were long periods in the 1960s, 1990s, and the 2000s in which investors strongly believed that they were in such a regime. The middle panel shows beliefs about the overheating boom. We call it that since the estimated transition probability (not shown) shows a higher than typical likelihood of falling into stagflation from this regime than other boom regimes. This instability in real earnings growth not only leads to more volatile beliefs, but also lower stock price valuations. Finally, the bottom panel shows the beliefs of the low inflation and high growth regime, which was nontrivial only during the dot com boom and some part of the mid 2000s. These were periods of unusually strong productivity growth in the US. While it is natural to expect high valuations in such periods, these was also an unusual positive correlation between fundamental news and uncertainty. Why was that? As seen, investors belief of this regime was at its highest points around 0.2, so that positive earnings during these periods increased the uncertainty on whether this extremely strong regime was really driving the fundamentals VALUATIONS AND THE FED MODEL 16 Fed Model (Data) 12 Fed Model (Model) Percent Percent S&P 500 Yield (Data) 5-Year Treasury Bond Yield (Data) S&P 500 Yield (Model) 5-Year Treasury Bonds Yield (Model) FIGURE 15.3 Fed Model for Data and For Composite Regime Switching Model of David and Veronesi (2013) Some time around the 1990s, practitioners on Wall Street noticed that the correlation between the earnings yield on the S&P 500 index and 5-year Treasuries was positive. This relative valuation commonality was called the Fed

9 15.3 The Stock-Bond Covariance and Learning About Fundamentals 517 Model, since the Fed s efforts to fight inflation generally raised rates and yields for both stocks and bonds. In the left panel of Figure 15.3, we do see the strong positive correlation in the period until 2000 (correlation coefficient of 0.7). However, quite remarkably, this correlation turned negative after 2000 (correlation coefficient of -0.6) with the two series showing opposite movements especially during and after the two recent recessions. What happened? Why did the relative valuation relationship break down at the turn of the millennium? We think it is in fact the same mechanism that generated a change in covariance between stocks and bonds, that is, the inversion is due to investors beliefs about the composite regimes that we discussed earlier. Until 2000, the recessions were generally followed by periods of increased inflation risk (the overheating booms). In such periods, positive inflation news would lead to not only higher bond yields due to higher expected inflation, but also higher stock yields due to a drop in stock valuations in anticipation of lower future earnings growth. After 2000, the recessions were of the deflationary type, and thus negative inflation news lead to lower bond yields due to lower expected inflation, but higher dividend yield, again due to the drop in stock valuation in anticipation of lower earnings growth. Therefore, the changing signaling role of inflation, which investors picked up on, implied the reversal of the relative valuation relationship, which is the Fed model. The right panel of Figure 15.3 indeed shows that our fitted model yields such inversion in comovement between the dividend yield and the 5-year bond yield EXPLAINING THE TIME VARIATION IN THE STOCK-BOND COVARIANCE Investors beliefs about composite regimes not only explain the dynamics of the relative valuations between stocks and bonds, but also the comovement of their returns, as can be seen in Figure The signs of the historical and model-based covariance almost always agree, and the magnitudes are similar as well. Overall, the model covariance explains 42 percent of the variation in the data covariance. It must be noted that the stock-bond covariance is one of the overidentifying moments used in our estimation. In the following sections, we will provide evidence that the economic logic of our model has additional evidence (from surveys), and the success of our model in not just an artifact of our fitting procedure. Why does the model covariance explain the time varying dynamics in the data? Once again, during the overheating boom periods of the 1970s and 1980s, any CPI reading above market expectations was taken as an indication that the economy could shift to stagflation battering the returns on both stocks and bonds. Hence, stock and bond covariance was positive. Since 2000, the opposite has happened. With investors now experiencing fears of a deflationary recession regime, positive inflation news is good news for stocks (but bad for bonds) as it signals that the bad deflationary regime could be averted, a channel that leads to a negative covariance between stocks and bonds. Thus, the changing

10 518 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds Stock Bond Covariance (Data) Stock Bond Covariance (Model) FIGURE 15.4 Stock-Bond Covariance in the Data and Implied by Regime Switching Model of David and Veronesi (2013) signaling role of inflation explains the change in sign of the covariance between stocks and bonds. Our model not only provides a unified economic framework explaining the relative valuations of stock and bonds, but it also provides several additional testable relationships between stocks and bonds valuations, and their volatilities, which we test in David and Veronesi(2013) Beliefs from Surveys and from the Model Above we discussed investors beliefs about fundamentals extracted from the estimation of a structural model. Another way to measure investors beliefs is from surveys. Here we briefly describe a method for calculating the average beliefs of agents, which we introduced in David and Veronesi (2013), and compare these beliefs with our model based beliefs. We use data from the Survey of Professional Forecasters (SPF), which we obtained from the Federal Reserve Bank of Philadelphia. The survey asks re-

11 15.4 Beliefs from Surveys and from the Model 519 spondents to provide probabilities of economic fundamentals falling in specific ranges. We average these probabilities across forecasters. To compare survey based beliefs with those from our model, we form intervals of fundamentals centered around the means of our estimated regimes. For example, for inflation, we form four intervals and label them as high, medium, low, and zero, similar to those for our model. In addition, we use the average beliefs to form first and second moments of expected fundamentals in the following quarter. For example, average inflation uncertainty is measured as n ( UncInf t = V t [I] = p j,t I j E t [I] ) 2 j=1 (15.4) where I j are midpoints of inflation unit intervals and p j,t is the average forecasters probability that inflation will be in the given interval I j. A similar procedure can be used to compute a proxy for real economic growth uncertainty. The question on the SPF only asks about the probability, p t, of a decline in real GDP, without specifying the exact amount of the decline (or the increase in case of growth). Hence, we rely on the following proxy: 6 UncEarn t = p t (1 p t ) (15.5) It is useful to note the distinction between average uncertainty and the dispersion of beliefs across forecasters. The latter is used by a number of researchers as a proxy for uncertainty, which might explain a large part of its variation, but it can also arise from disagreement among forecasters about information or opinions on the structure of fundamentals. In Figure 15.5 we directly compare the beliefs from the surveys and those from our model fitted to fundamentals, stock, and bonds data. The top and middle panels show the two sets of beliefs about inflation in the following quarter. As can be seen, the two sets of beliefs mostly move together with correlations ranging from 41 percent for zero inflation to 81 percent for high inflation. Moreover, the magnitude of the variations are very similar. Here the model beliefs exceed 70 percent, while the survey beliefs were only in the 30 percent range. The bottom left panel shows that the two sets of beliefs about low real growth in the following quarter are each highly countercyclical, and strongly move together with a correlation of 50 percent. The model s asset prices and volatilities are functions of beliefs, which could also be called factors, like those in the exponential affine models prevalent in the term structure literature. However, the evidence presented in Figure 15.5 casts some doubts on such a reduced-form interpretation. In fact, by construction our model-based beliefs are fully consistent with Bayes laws 6 We note that (15.5) is proportional to the conditional variance of GDP growth in the case of two regimes ( decline and no decline ), as V t[θ] = 2 i=1 p ( it θ i E ) 2 t[θ] = p1t (1 p 1t )(θ 1 θ 2 ) 2.

12 520 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds 100 A. High Inflation 100 B. Medium Inflation corr. = 81% corr. = 57% percent 50 percent C. Low Inflation 100 D. Zero Inflation corr. = 43% percent 50 percent 50 corr. = 77% percent E. Low Growth corr. = 50% F. Legends Model SPF FIGURE 15.5 Marginal Beliefs About Fundamentals: Survey of Professional Forecasters and Implied by Regime Switching Model of David and Veronesi (2013) and observations of fundamental data. Moreover, the strong similarity between model-based beliefs and survey-based beliefs is quite reassuring for our modeling methodology and interpretation because the survey beliefs are generated completely independently from any aspect of our model. Instead, the similarity of the two sets of beliefs suggests that our filtering model and professional forecasters are tracking similar historical information and forecasting fundamentals similarly.

13 15.5 Survey and Model Beliefs and the Stock-Bond Covariance Survey and Model Beliefs and the Stock-Bond Covariance As discussed in Section 15.3, our model implies that the stock and bond volatilities depends on the two fundamental uncertainties, inflation and earnings, which we have modeled. In periods of higher uncertainty, investors have low confidence in their knowledge about fundamentals and revise their beliefs more in response to news, leading to higher volatility. However, depending on valuations, the two uncertainties are weighted differently over time. In Tables 15.1 and 15.2, we test whether SPF-based and model-based uncertainty measures are able to explain the variation in the stock-bond covariance. We provide results for the covariance of stocks with 1-year and 5-year Treasury bonds. In both tables, regression 1 tests if the stock-bond covariance depends on inflation and earnings uncertainty from the two sets of beliefs. For both sets of beliefs, we find that inflation uncertainty has a positive coefficient and is statistically significant, while earnings uncertainty has a negative but insignificant coefficient. The model-based R 2 is almost twice as high as the SPF-based R 2. The negative coefficient on earnings uncertainty implies that the stock-bond covariance declines on average in recessions, when earnings uncertainty has been high. In regression 2, we use expected inflation and earnings rather than their uncertainties. The results are similar to regression 1, although the R 2 are higher for the SPF-based measures. Why did we use expectations? The mechanism suggested by our model is in fact very much related to expected inflation. A regime of low earnings growth occurs either during periods of very high inflation (stagflation regime) or during periods of very low inflation (deflation regime). When expected inflation is high, positive inflation news (inflation realization above expectation) increase expected inflation and also the probability of stagflation, which is bad for the stock market. Hence, we should expect a positive covariance between stocks and bonds when expected inflation is high. In contrast, when expected inflation is very low, positive inflation news raises expected inflation and lowers the probability of deflation, which is good for the stock market. Hence, we should expect a negative relation between stocks and bonds when expected inflation is very low. Overall, we should expect a positive relation between stock-bond covariance and expected inflation, as corroborated by regression 2 in Tables 15.1 and No such direct channel exists for expected earnings, as the covariance between stocks and bonds could be positive or negative when expected earnings is low, depending on whether we are in a high inflation regime or low inflation regime. Indeed, the lack of significance on ExpEarn in regression 2 is consistent with this theoretical prediction. To see if an earnings-based measure enters nonlinearly, in regression 3, we continue to use a measure of inflation expectations but use the belief of a boom for the real measure. The R 2 of both sets of beliefs for both 1-year and 5- year bonds increase, and for 5-year bonds we get statistical significance for the boom probability. The regressions show that stock-bond covariance is higher in booms. In Section 15.3 we discussed that this arises because higher inflation

14 522 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds Table 15.1 Model Beliefs and the Stock Bond Covariance Variable α β 1 β 2 β 3 R 2 Regression 1: Cov = α + β 1 UncInf + β 2 UncEarn + ɛ Stock 1-Year Bond Cov [-.7085] [4.2433] [ ] Stock 5-Year Bond Cov [ ] [ ] [ ] Regression2: Cov = α + β 1 ExpInf + β 2 ExpRCP + ɛ Stock 1-Year Bond Cov [ ] [ ] [ ] Stock 5-Year Bond Cov [ ] [ ] [0.7350] Regression 3: Cov = α + β 1 ExpInf + β 2 (1 RECESS) + ɛ Stock 1-Year Bond Cov [ ] [ ] [ ] Stock 5-Year Bond Cov [ ] [ ] [ ] Regression 4: Cov = α + β 1 PrHigh + β 2 PrLow + ɛ Stock 1-Year Bond Cov [2.0562] [6.0368] [ ] Stock 5-Year Bond Cov [1.4556] [ ] [ ] Regression 5: Cov = α + β 1 UncInf + β 2 UncEarn PrHigh + β 3 UncEarn PrLow + ɛ Stock 1-Year Bond Cov [ ] [ ] [ ] [ ] Stock 5-Year Bond Cov [ ] [ ] [ ] [ ] Regression 6: Cov = α + β 1 StockVol PrHigh + β 3 StockVol PrLow + ɛ(t) Stock 1-Year Bond Cov [1.1137] [4.5101] [ ] Stock 5-Year Bond Cov [0.3568] [ ] [ ] Notes: t-statistics are in brackets. Bold type indicates significance at 1% level. Standard errors are Newey-West adjusted for heteroskedasticity and autocorrelation.

15 15.5 Survey and Model Beliefs and the Stock-Bond Covariance 523 Table 15.2 Survey Beliefs and the Stock Bond Covariance Variable α β 1 β 2 β 3 R 2 Regression 1: Cov = α + β 1 UncInf + β 2 UncEarn + ɛ Stock 1-Year Bond Cov [ ] [3.0856] [ ] Stock 5-Year Bond Cov [ ] [3.4423] [ ] Regression2: Cov = α + β 1 ExpInf + β 2 ExpRCP + ɛ Stock 1-Year Bond Cov [ ] [4.0778] [0.7469] Stock 5-Year Bond Cov [ ] [3.4012] [1.2307] Regression 3: Cov = α + β 1 ExpInf + β 2 (1 RECESS) + ɛ Stock 1-Year Bond Cov [ ] [3.6682] [1.1152] Stock 5-Year Bond Cov [ ] [3.2930] [1.5302] Regression 4: Cov = α + β 1 PrHigh + β 2 PrLow + ɛ Stock 1-Year Bond Cov [1.2167] [3.4995] [ ] Stock 5-Year Bond Cov [2.0485] [1.9383] [ ] Regression 5: Cov = α + β 1 UncInf + β 2 UncEarnPrHigh + β 3 UncEarn PrLow + ɛ Stock 1-Year Bond Cov [ ] [2.7608] [4.1114] [ ] Stock 5-Year Bond Cov [ ] [3.1091] [2.1207] [ ] Regression 6: Cov(t) = α + β 1 StockVol PrHigh + β 3 StockVol PrLow + ɛ Stock 1-Year Bond Cov [1.9400] [2.1170] [ ] Stock 5-Year Bond Cov [1.6078] [1.4604] [ ] Notes: t-statistics are in brackets. Bold type indicates significance at 1% level. Standard errors are Newey-West adjusted for heteroskedasticity and autocorrelation.

16 524 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds news in booms typically hurts both stock and bond returns as it signals future instability of earnings growth rate. The key point in our explanation of the changing sign of the stock-bond covariance is the changing signaling role of inflation and the difference in inflation expectations in different recessions. Therefore, in periods of high inflation, stocks and bonds have positive covariance, and in periods of zero inflation, they have negative covariance. In regression 4, we directly use the beliefs of high and low inflation as explanatory variables. For both sets of beliefs, and for both maturities, we now obtain statistical significance and the right signs. The R 2 of the two sets of beliefs are also quite similar. To resolve the issue about statistical insignificance of earnings uncertainty, in regression 5, we now interact it with low and high inflation probabilities. Our model predicts that during the two subperiods with high and low inflation, we should have positive and negative coefficients, respectively. We indeed find that this is the case, with strong significant coefficients for the interaction terms, and R 2 as high as 36.9 percent (for the SPF measures). We finally return to the issue of the flight-to-safety phenomenon that was discussed in Connolly, Sun, and Stivers (2005). Using stock options, these authors show that in recent data the stock-bond covariance is more negative during periods of higher implied volatility. Our model is consistent with their findings, but it also implies that bonds do not offer the safe haven comfort in periods of high inflation (which has not really been a concern in recent data). So in regression 6, we use stock volatility as an explanatory variable, and interact it with high and low inflation probabilities from the two sets of beliefs. We find that in both specifications, the stock-bond covariance is high when volatility is high and the probability of high inflation is high, and negative when volatility is high but the probability of low inflation is high. The R 2 for these specifications are the highest among all the specifications considered Some International Evidence Most of the literature discussing the time variation in stock-bond covariance, including David and Veronesi (2013), only used data from U.S. In an early paper, Li (2002) estimated the stock-bond correlation across several countries, and documented that it relates both to expected inflation and inflation uncertainty, which is consistent with the predictions of David and Veronesi (2013). The paper, however, proxied for both expected inflation and inflation uncertainty by using the results of predictive regressions of future inflation. In this section, we consider the evidence across four countries, namely, Germany, Japan, Switzerland, and United Kingdom, for which daily stock and bond data are available on a relatively long sample, and importantly, default risk may be considered small on an ex ante basis. Daily bond indices are from DataStream while daily stock indices are Bloomberg. Bond indices are only available at the 10-year horizon. For these countries, however, we do not have available

17 15.6 Some International Evidence 525 survey forecasts as we do for the U.S. Because we are interested in using ex ante forecasts of future inflation and future real GDP growth as explanatory variables, we employ the projections of future inflation (GDP deflator) and future real GDP growth published by the OECD s semi-annual economic outlook publications. Unfortunately, this data set does not allow us to compute measures of uncertainty like for the Survey of Professional Forecasters, but they still allow us to discuss the relation between fears of deflation and stock-bond covariance. Indeed, the model by David and Veronesi (2013) predicts that as investors inflation expectations become too low, agents increase the probability of entering a deflationary regime, which is bad for the stock market. Thus, stock-bond covariance should turn negative in an environment of low expected inflation. The overall sample across these four countries is shorter than the one used for U.S. and spans 1985 to 2014 at the semi-annual frequency. The four panels of Figure 15.6 plot the OECD-based 1-year ahead inflation expectations across the four countries, together with the stock-bond covariance (multiplied by 1000 for a better visual impression). Across the four panels, we see similar dynamics of expected inflation and stock-bond covariance. In particular, Japan s expected inflation is lower than other countries, and in fact Japan experienced negative expected inflation on several occasions starting from the mid 1990s. Indeed, we see also that the stock-bond covariance turned negative in Japan earlier than in other countries, around early 1990s. Moreover, most countries exhibit two negative spikes in stock-bond covariances around 2002 and during the financial crisis of , which resemble similar events observed in the U.S. (see Figure 15.1). The top panel of Table 15.3 reports the results of individual regressions of stock-bond covariance on expected inflation and expected GDP growth. These regressions correspond to Regression 2 reported in previous Tables. Across the four countries, the covariance between stocks and bonds is positively related to expected inflation but not expected GDP growth. This result is similar to what we found in Regression 2 of Tables 15.1 and The R 2 for the cases of Germany, Japan, and U.K. are also relatively large, between 23% and 32%, while the R 2 for Switzerland is just 7%. In the last line of the top panel of Table 15.3 we also report the same regression but for the United States, now using the same sample and data sources as for the other four countries. As we can see, the same result as in Regression 2 of Table 15.2 holds with the alternative data sources and alternative sample. We check the robustness of the regression results in Table 15.3 by adding a lag of the dependent variable as an additional explanatory variable in the regression. Such lags are meant to instrument for additional unobservable factors that may be affecting both the explanatory variables and the dependent variable. The bottom panel of Table 15.3 reports the regression results. Expected inflation remains a significant explanatory variable for the covariance between stocks and bonds for three of the five countries, namely, Germany, United Kingdom, and United States. Instead, ExpInf is no longer significant for Japan and

18 526 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds 0.1 Germany 0.1 Japan Switzerland 0.1 United Kingdom Expected Inflation Stock Bond Covariance ( 1000) FIGURE 15.6 Inflation Expectation and Stock-Bond Covariances Across Countries. Switzerland. 7 While it is known that simple measurement error of the explanatory variables may be responsible for their lack of significance once a lag of the dependent variable is inserted in the regression, these weaker results for Japan and Switzerland caution against over-interpreting the causal effect of expected inflation on the covariance between stocks and bonds for those two countries in the top panel of Table We did not report results with lags of the dependent variables as regressors in Tables 15.1 and 15.2 as almost all explanatory variables remain significant once lags are included. The only exception are the regressions involving PrLow in Table 15.2 and in regression 4 in Table 15.1, whose coefficients turn insignificant. Of course, measurement error and the short sample in which PrLow is different from zero may well explain this result.

19 15.6 Some International Evidence 527 Table 15.3 OECD Expectations and the Stock-Bond Covariance across Countries α β 1 β 2 β 3 R 2 Regression: Cov = α + β 1 ExpInf + β 2 ExpGDP + ɛ Germany [ ] [3.8698] Japan [3.0395] Switzerland [3.6716] United Kingdom [ ] [4.0401] [3.3277] United States [ ] [4.7759] Regression: Cov = α + β 1 ExpInf + β 2 ExpGDP + β 3 Cov( 1) + ɛ Germany [2.1854] [5.1954] Japan [4.0415] Switzerland [2.608] United Kingdom [2.8772] [3.7832] United States [2.3468] [4.5809] Notes: t-statistics are in brackets. Bold type indicates significance at 1% level. Standard errors are Newey-West adjusted for heteroskedasticity and autocorrelation. A related implication of the model of David and Veronesi (2013) concerns the relation between the stock-bond covariance and the long-term bond yield. Indeed, nominal yields equal real yields plus expected inflation plus a risk premium. The real interest rate itself in David and Veronesi (2013) is also positively related to expected inflation because of money illusion, which makes long-term nominal yields positively related to expected inflation both directly and indirectly through its impact on the long-term real rate. 8. It follows that 8 The empirical literature has documented a positive relation between long-term real rates and expected inflation, although the evidence at the short-horizon is mixed. See e.g. Ang, Baekert and Wei (2008), Haubrich, Pennacchi, and Ritchen (2013), and David and Veronesi (2014) for recent contributions.

20 528 CHAPTER 15 The Economics of the Comovement of Stocks and Bonds Table 15.4 Ten-year Yields and the Stock-Bond Covariance across Countries α β 1 β 2 t(α) t(β 1 ) t(β 2 ) R 2 Regression: Cov = α + β 1 10-year Yield(-1)) + ɛ Germany Japan Switzerland United Kingdom United States Regression: Cov = α + β 1 10-year Yield(-1) + β 2 Cov( 1) + ɛ Germany Japan Switzerland United Kingdom United States Notes: Standard errors are Newey-West adjusted for heteroskedasticity and autocorrelation. if the time variation in the covariance between stocks and bonds is related to hyperinflation fears (for positive covariance) and deflation fears (for negative covariance), we should find a positive relation between long-term yields and the covariance between stocks and bonds. David and Veronesi (2013) report empirical evidence to this effect for the U.S. on the long sample 1960 to The top panel of Table 15.4 shows this to be the case also for international data. In particular, in this table we regress the stock-bond covariance for each individual country onto their 10-year bond yield, whose data are obtained from DataStream. Because yields themselves are endogenous quantities, we lag the 10-year yield in these regressions to partly address issues of reverse causality. The sample is still 1985 to 2014, but the frequency of observation is now quarterly as in David and Veronesi (2013) since for this empirical exercise we are not limited by the semi-annual frequency of OECD economic projections. The results support the prediction from David and Veronesi (2013) that the level of yields and the covariance between stocks and bonds should be positively related. Indeed, the bottom panel of Table 15.4 shows that adding a lag of the dependent variable to the explanatory variables as an instrument for unobservable factors does not change the significance of the long-term yield as an explanatory variable. Therefore, so long as the main variation of the long-term yield over the

21 15.7 Summary 529 sample is due to time variation in expected inflation, the prediction of David and Veronesi (2013) is supported in the international data Summary The covariance of stocks and bonds switched signs from being predominantly positive in the second half of the last century to being predominantly negative in the current century. The main intuition that we offer is that investors fear of high inflation in the 1980s switched to fear of deflation in recent years, both of which signal weakness in the real growth of fundamentals. Thus, bonds offer a safe haven only in periods of low inflation. During high inflation periods, they are hurt by the same fundamental factors as stocks, and have positive conditional betas. Our explanation is line with other recent work on the dynamics of the stock-bond covariance that also feature a time varying relation between inflation and the real economy. Acknowledgements We thank Ben Charoenwong and Paulo Mateus for excellent research assistance. REFERENCES Ang, A., Bekaert, G. and Wei, M. (2008), The Term Structure of Real Rates and Expected Inflation, The Journal of Finance, vol. 6, pp 797-Ű849. Barksy, R. B., (1989), Why Don t the Prices of Stocks and Bonds Move Together?, American Economic Review, vol. 79, pp Bekaert, G., Engstrom, E., and Grenadier, S. (2010), Stock and Bond Returns with Moody Investors, Journal of Empirical Finance, vol. 17, pp Baele, L., Bekaert, G., and Inghelbrecht, K. (2010), The Determinants of Stock and Bond Return Comovements, Review of Financial Studies, vol. 23, pp Bansal, R. and Yaron, A., (2004), Risks for the Long Term: A Potential Resolution of Asset Pricing Asset Pricing Puzzles, Journal of Finance, vol: 59, pp Bloom, N., (2009), The Impact of Uncertainty Shocks, Econometrica, 77, Campbell, J. Y., and Ammer, J. (1993), What Moves the Stock and Bond Markets? A Variance Decomposition for Long-Term Asset Returns, Journal of Finance, vol. 48, pp The positive relation between the covariance between stocks and bonds and long-term yields may also be due to time varying risk premia. Indeed, in a CAPM-like world the lower covariance between stocks and bonds may in fact imply lower risk premia and therefore lower long-term yield (see also Campbell, Sunderam, and Viceira (2013)).

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