A Measure of Risk Appetite for the Macroeconomy

Transcription

1 A Measure of Risk Appetite for the Macroeconomy Online Appendix Carolin Pflueger Emil Siriwardane Adi Sunderam Abstract In this appendix, we present additional results demonstrating: (i) that the link between the real rate and PV S is not subsumed by other stock characteristics; (ii) that neither PV S nor the real rate forecast future accounting ROE for the volatility-sorted portfolio; (iii) 9% of the covariation between the real rate and PV S t stems from the fact that the real rate forecasts future returns on the vol-sorted portfolio; (iv) both PV S and the real rate do not forecast future macroeconomic or financial market volatility; and (v) the real rate and PV S do not line up with alternative ways of measuring contemporaneous risk. In addition, we show that our main conclusions are unchanged if we use the raw real rate series or alternative methods of detrending it. We also show that our results are robust to alternative standard error corrections and generalized least square (GLS) estimation techniques. The appendix also contains (vi) additional details on the VARs and (vii) describes the stylized theoretical model. Pflueger: University of British Columbia. carolin.pflueger@sauder.ubc.ca Siriwardane: Harvard Business School. esiriwardane@hbs.edu. Sunderam: Harvard Business School and NBER. asunderam@hbs.edu.

2 A Additional Results A.1 The Real Rate and Other Valuation Spreads We now explore alternative explanations for the empirical relationship between the real rate and stock portfolios sorted on volatility. Specifically, we examine the possibility that volatility is simply correlated with another characteristic that is more important for explaining the real rate. We sort stocks along a variety of dimensions and form book-to-market spreads based on the sorting variable. For instance, when examining size as a characteristic, we sort stocks in quintiles based on their market capitalization, then compute the difference between the book-to-market ratio of the smallest and the largest stocks. We then run the following regression relating the real rate to the spread in book-to-market based on each sort: Real Rate t = a + b Y t + ε t (1) where Y t is the book-to-market spread based on sorting on characteristic Y. The results are displayed in Table A.2. In row (1), we relate the real rate to the spread in book-to-market sorting stocks based on the expected duration of their cash flows. If high volatility stocks simply have higher duration cash flows than low duration cash flows, then their valuations should fall more when real rates rise. This is one sense in which low volatility stocks may be more bond-like than high volatility stocks (e.g., Baker and Wurgler (212)). In this case, a mechanical duration effect could explain the relationship between the real rate and PV S. To examine this possibility, we follow Weber (216) and construct the expected duration of cash flows for each firm in our data. We then sort stocks based on this duration measure and calculate the spread in bookto-market between high and low duration stocks. As row (1) shows, the relationship between this duration spread and the real rate is negative. However, it is not consistently statistically significant across specifications and is in general much smaller in magnitude than PV S. Row (2) displays the same exercise when looking at the relative valuations of low-leverage versus high-leverage stocks. We define leverage as the book value of long-term debt divided by the market value of equity. It seems natural to think that high-leverage firms have high volatility, and since these firms effectively are short bonds, their equity may suffer disproportionately from a decrease in the real rate. The positive coefficient in row (2) indicates that this intuition bears out in the data. When the real rate falls, the book-to-market spread between low-and-high leverage firms also falls. In other words, high-leverage firms become cheaper when the real rate falls. In rows (3)-(5), we sort stocks based on various measures of traditional market beta. For example, row (3) indicates that the book-to-market spread based on a monthly CAPM beta is correlated with the real rate. In this case, we compute beta using rolling 5-year windows. Row (4) sort stocks based on CAPM betas that we compute using long-horizon returns. Specifically, long-run CAPM betas are computed using semi-annual returns over a rolling ten year window. The reference index is the CRSP Value-Weighted index and we require 8% of observations in order to compute a long-run CAPM beta. The idea here is that long-horizon returns are largely driven by cash flow news rather than discount rate news. Thus, long horizon CAPM betas can be viewed as a measure of aggregate cash flow beta. Row (4) indicates a positive relationship between the bookto-market spread based on long-run CAPM beta in levels, but the relationship is not particularly strong in a statistical sense when moving to first-differences. Row (5) uses a measure of CAPM beta that is computed using daily data over rolling 6-day windows. This construction mimics how 1

3 we compute volatility (and hence PVS). There is again a positive relationship between 2-month beta and the real rate, but not one that is robust across specifications. In row (6), we sort stocks on the estimated beta of their cash flows with respect to aggregate cash flows. Specifically, cash flow betas are computed via rolling twelve quarter regressions of quarter-on-quarter EBITDA growth on quarter-on-quarter national income growth. EBITDA is defined as the cumulative sum of operating income before depreciation. We require a minimum of 8% of observations in a window to compute a cash flow beta. If high volatility stocks have higher cash flow betas than low volatility stocks, then their valuations should fall more when aggregate growth expectations are low. In this case, our results using PV S could be explained by changes in aggregate growth expectations rather than change in the precautionary savings motive. Row (6) shows that the book-to-market based on cash flow betas is not significantly correlated with the real rate. Keep in mind that the preceding regressions are all univariate. The relevant question for us is whether PV S is just picking up on the information carried in these various book-to-market spreads. Three pieces of evidence strongly suggest that PV S carries independent information about the real rate. First, in Table 3 of the main text, we run bivariate horse races of PV S against each of these alternative sorting variables. None of these alternative sorting variables drive out PV S from the regression. This is true when running the horse races in levels, first differences, and across different subsamples. Table A.3 of this appendix extends these horse races to include all of our variants for computing CAPM Beta. Second, in Table 3 of the main text, we also create double-sorted versions of PV S. For example, we sort stocks into terciles based on their CAPM beta, and then within each CAPM beta tercile, we compute the difference in book-to-market ratios of low and high volatility stocks. Finally, we average the spread between low- and high-volatility stocks across the CAPM beta terciles. This procedure delivers us a version of PV S that is immunized to CAPM beta but differentially exposed to volatility. As Table 3 of the main text shows, these double-sorted versions of PV S perform just as well as the standard PV S. Third, in row (9) of Table A.2 we run a kitchen-sink regression of the following form: Real Rate t = a + b PV S PV S + θ X t + ε t where X t is a large set of control variables. It contains all of the valuation spreads discussed above, plus Shiller s CAPE, expected GDP growth from the Survey of Professional Forecasters, the aggregate book-to-market ratio, the within-quarter volatility of the three Fama and French (1993) factors, the realized volatility of portfolios sorted on volatility (portfolio level and within-portfolio average), the average idiosyncratic volatility factor of Herskovic et al. (216), and the volatility of TFP growth. Row (9) of the table reports the estimated b PV S, its associated t-statistic, and the adjusted R 2 from the regression. The simple takeaway from the kitchen-sink regression is that none of the control variables drive out the explanatory power of PV S for the real rate. The coefficient on PV S remains statistically significant in both the levels and first-differenced specifications, and the point estimate compares favorably to those found in the main text. If anything, including the other control variables increases the economic relationship between PV S and the real rate. These results suggest that the relative valuation of high and low volatility stocks contains unique information about the real rate. 2

4 A.2 Forecasting ROEs Analogous to Campbell and Shiller (1988), Cohen et al. (23) present a present-value identity showing variation in a firm s book-to-market ratio can be decomposed into variation in discount rates, expected cash flows (ROE), or future valuation ratios. We use this logic to understand why PV S moves around in the first place. In the main text, we show that PV S (and the real rate) strongly forecast future returns on the portfolio of low-minus-high volatility stocks. Based on this forecasting regression, we deduce that PV S variation is driven by discount rate variation in the relative pricing of low and high volatility stocks. Because of the real rate correlates strongly with PV S and also predicts future returns to high volatility stocks, we also surmise that real rate variation is driven by this same discount rate variation. We formalize this logic in Section A.3. To further support this argument, we use PV S and the real rate to forecasts the future accounting return on equity (ROE) of stock portfolios formed based on volatility. We compute the annual ROE of this long-short portfolio using the accounting measures in Cohen et al. (23). Table A.6 reports the results of the following regression: ROE Vol t t+4 = a + b X t + ε t t+4 where ROE Vol t t+4 is the ROE of the low-minus-high volatility portfolio from time t to t + 4. X t is either PV S or the real rate. Table A.6 makes clear that neither PV S nor the real rate forecast future cash flows (ROE) of the high-minus-low volatility portfolio. A.3 Covariance Decomposition Vuolteenaho (22) derives the following relation tying a firm i s log book-to-market ratio to its future log return and log accounting return (ROE): θ i,t = r i,t+1 e i,t+1 + ρθ i,t+1 + ν it where θ i is the log book-to-market of firm i, r i,t+1 is its log stock return, and e i,t+1 is the log ROE. ρ is a log-linearization constant and ν i,t is an approximation error, such that θ i,t r i,t+1 e i,t+1 + ρθ i,t+1. To map this expression to the current setting, we define the log version of PV S t, denoted by pvs t, as follows: pvs t [ 1 N L,t ] [ θ i,t i Low Vol t 1 N H,t ] θ i,t i High Vol t where, for example, N L,t is the number of firms in the low vol portfolio at time t. The Vuolteenaho (22) decomposition then implies that: pvs t rt+1 PV S epv t+1 S + ρ pvs t+1 [ ] [ t+1 1 r PV S e PV S t+1 [ r i,t+1 N L,t i Low Vol t 1 N L,t i Low Vol t e i,t+1 ] [ 1 r i,t+1 N H,t i High Vol t 1 N H,t i High Vol t e i,t+1 ] ] (2) 3

5 In addition, we assume that pvs t follows an AR(1) process, pvs t+1 = a + φ pvs t + ξ t+1. Next, combining the AR-process with Equation (2), plus some rearranging yields: ( ) ( ) Cov(Real Rate t, pvs t ) (1 ρφ) 1 [Cov Real Rate t,rt+1 PV S Cov Real Rate t,et+1 PV S + ρcov(real Rate t,ξ t+1 )] Dividing both sides by Cov(Real Rate t, pvs t ) delivers a simple covariance decomposition: 1 = Ψ r Ψ e + Ψ ξ (3) where Ψ r (1 ρφ) 1 Cov ( Real Rate t,r PV S t+1 ) /Cov(Real Ratet, pvs t ), and so forth. Equation (3) states that covariation between today s real rate and pvs t can arise for three reasons: (i) today s real rate forecasts future returns to the volatility-sorted portfolio, r PV S ; (ii) today s real rate forecasts future cash flows on the same portfolio, e PV S ; or (iii) today s real rate forecasts future innovations in tomorrow s pvs. To operationalize the decomposition, we need to first estimate φ and ρ. We fit a simple AR(1) for pvs and find that φ =.88 for quarterly data. With regards to ρ, we consider a range of values from.9 to All of the other components needed for the covariance decomposition are estimated from simple covariances in the data. For all of the ranges of ρ that we consider, Ψ r is never less than 7% and approaches 1% for larger values of ρ. Moreover, for all of the ranges of ρ considered in Vuolteenaho (22), Ψ r is never below 9%. This is rather unsurprising given that the real rate does not forecast future ROE for the low-minus-high volatility portfolio. We therefore conclude that a large majority of the covariation (around 9%) between PV S t and the real rate comes from the real rate forecasts future returns on the volatility-sorted portfolio. Put differently, PV S t and the real rate correlate because discount rate shocks to high-volatility stocks coincide with shocks to the real rate. This simple fact is a large reason we interpret PV S t as measuring the stock market s precautionary savings motive. A.4 Forecasting Returns with the Quantity of Risk Our main finding is that variation in PV S explains variation in the real rate. Moreover, PV S variation is mostly driven by changes in the expected return to holding low-versus-high volatility stocks. In turn, expected returns can vary because the quantity of risk is changing or the pricing of that risk is changing. In the main text, we argue that the quantity of risk channel is unlikely. We show this by running simple regressions of the real rate on the quantity of risk, with no robust relationship emerging. Our conclusion is that the real rate lines up with PV S because of investors attitudes towards holding high volatility stocks changes over time, which impacts both PV S and the real rate. As another way to reinforce this point, Table A.9 analyzes the ability of the real rate and various measures of the quantity of risk to forecast future returns of volatility-sorted portfolios: R t t+4 = a + b 1 Real Rate t + b 2 X t + ξ t+4 1 Vuolteenaho (22) sets ρ =.967 for annual data. We use a range of values to get a sense of how sensitive our decomposition is to the approximation constant. 4

6 where R t t+4 is the annual return on a portfolio that is long low-volatility stocks and short highvolatility stocks. X t is a measure of the quantity of risk. The purpose of the regression is just to check that the quantity of risk does not reliably forecast future returns on the low-minus-high volatility portfolio, thereby implying that expected returns and hence PV S and the real rate are not moving around because risk is changing. In Columns 1 and 2 of Table A.9, X t is the realized return volatility of the low-minus-high volatility portfolio, σ t (LMH-Vol Portfolio). For a given quarter, we compute the realized volatility of the long-short portfolio using daily data. Column (1) runs a univariate forecasting regressing using only σ t (LMH-Vol Portfolio). Column (2) adds the real rate as a regressor. In both cases, σ t (LMH-Vol Portfolio) has no forecasting power for future returns on the portfolio and does not drive out the forecasting ability of the real rate. In Column (3), X t is TFP Growth volatility. Columns (4)-(6) instead use the realized return volatility of the three Fama and French (1993) factors. In all cases, the forecasting ability of the real rate is more or less unchanged in all of these specifications. In Column (4), the volatility of the market does possess some forecasting power over and above the real rate, while leaving the magnitude and statistical significance of the real rate unchanged. High current aggregate market volatility predicts low returns to the low-minus-high volatility portfolio. This finding suggests that market volatility may have some relation with the compensation that investors require for holding volatile stocks, consistent with the precautionary savings interpretation of the risk premium on volatile stocks. However, aggregate market volatility clearly does not drive out PV S t nor does it explain an economically meaningful share of variation in PV S t. For comparison, R-squared in column (4) is only.16, compared to an R-squared of.26 from forecasting annual returns with PV S t in Table 5 in the main paper. Moreover, a univariate regression of PV S on aggregate market volatility has an R 2 of only 1%, indicating that PV S mostly captures variation different from variation in the quantity of risk. A.5 The Real Rate and Alternative Measures of Risk A.5.1 Forecasting Volatility Next, we examine the possibility that PV S t is related to expectations of future volatility instead of contemporaneous or lagged volatility. In Table A.8, we try to forecast volatility directly using either PV S t or the real rate. Formally, we run: Vol t+1 = a + b X t + ε t+1, (4) where X t is either PV S t (Panel A) or the real rate (Panel B). Each column examines a different volatility measure, as specified by the column header. For instance, column (1) examines the spread in average realized return volatilities between our low-volatility portfolio and our high-volatility portfolio, while column (2) examines the common factor in idiosyncratic volatility variable of Herskovic et al. (216). In columns (2) and (3) of Panel A, PV S t has marginally significant forecasting power for aggregate market volatility and the common factor in idiosyncratic volatility, as one might expect if an increase in volatility leads to stronger risk appetite. But again, the R 2 s are small. In Panel B, we find that the real rate does not forecast any of the volatility measures. While the real rate has marginally significant forecasting power for the volatility of the SMB portfolio and the common factor in idiosyncratic volatility, the signs of these coefficients are the opposite of what 5

7 one would expect if a lower rate reflects concerns about higher volatility. 2 It is also possible that movements in PV S t are driven by investor forecasts of future risk that is not ultimately realized. We have explored the relationship between PV S t and survey-based measures of uncertainty and volatility. We have not found strong relationships, though that could be due to the short timespan for which these measures are available. A.5.2 Replicating Main Results Using Average Stock Volatility Our preferred measure of risk in the low-minus-high volatility portfolio is the volatility of the portfolio itself. An alternative way to measure the risk in the portfolio is to look at the average volatility of high-volatility stocks relative to the average volatility of low-volatility stocks. We define this measure as σ (Low Vol) σ (High Vol). The regressions presented in Table A.1 study whether the real rate contemporaneously correlates with this and other measures of the quantity of risk. Echoing our analysis in the main text, there does not appear to be a quantity of risk measure that reliably explains real rate variation across the various level and first-difference regression specifications that we explore. Most importantly, none of these risk measures drive out the explanatory power of PV S for the real rate, which is what we would expect if PV S is simply capturing movements in the amount of risk in the economy. A.5.3 Broader Measures of Uncertainty An important precursor to our study is Hartzmark (216), who studies the relationship between the real interest rate and a measure of macroeconomic uncertainty. Macroeconomic uncertainty is defined as the variance forecast that comes from fitting an ARMA(1,1)-GARCH(1,1) model to several different macroeconomic series (e.g. real GDP growth). Using annual data from , Hartzmark (216) finds that there is a negative relationship between the real interest rate and macroeconomic uncertainty. This observation stands in contrast to our finding that the real interest rate has little correlation to macroeconomic volatility during our sample. To better understand the differences between the two studies, we now reproduce the analysis from Hartzmark (216) for our sample. 3 Our analysis throughout the paper is conducted at a quarterly frequency, so we focus our comparison on Table 5 of Hartzmark (216), which also uses quarterly data. Table 5 of Hartzmark (216) indicates that the strongest explanatory power for the real rate comes when measuring macroeconomic volatility using industrial production growth. Consequently, we stack our results against the same measure of macroeconomic uncertainty (series IPB51SQ from FRED). Denote E t [g t+1 ] as the one period ahead predicted mean equation from fitting ARMA(1,1)- GARCH(1,1) to the log-growth rate of industrial production. V t [g t+1 ] is the associated one-period variance forecast that comes out of the same model. Both expected growth and variance are known at time t from the perspective of the ARMA-GARCH model. Using these forecasts, we run the following regression in Table A.11: Real Rate t = a + b 1 E t [g t+1 ] + b 2 V t [g t+1 ] + ε t 2 In Table A.9, we also show that the quantity of risk also has no ability to forecast excess returns on the long-short portfolio of volatility sorted stocks, which is again consistent with our results being driven by time variation in investor aversion to volatility. 3 Hartzmark (216) uses a sample from 1947 to 21, whereas our sample runs from 197 to

8 Column (1) of the table confirms the general intuition of Hartzmark (216), who finds that higher expected growth is positively correlated with the current real rate. In addition, higher future growth volatility (V t [g t+1 ]) leads to a lower real rate today. Nevertheless, the point estimate on both quantities is imprecisely measured and statistically insignificant. In Column (2), we include PV S in the regression, thereby indicating that neither expected growth nor volatility drive out PV S. Importantly, the 4% R 2 in Column (1) is very low compared to the 45% R 2 in Column (2). This fact highlights the weak explanatory power of E t [g t+1 ] and V t [g t+1 ] for real rate variation, especially in comparison to PV S. B Robustness The purpose of this section is to conduct a several of robustness tests to ensure that our statistical inference regarding the relationship between the real rate and PV S is not driven by specific choices in defining our main variables. We begin by discussing alternative methods of filtering the real rate (e.g. using a deterministic versus stochastic trend). We then show that our results are largely unchanged with these alternative filters or if we simply study the raw real rate. We conclude the section by exploring several ways of adjusting the standard errors in our main regression tests to account for persistent variables. The main takeaway of the section is that there is a robust relationship - both in economic and statistical terms - between the real rate and PV S t. For the remainder of this appendix, we use R t to denote the raw real rate. B.1 Filtering the Raw Real Rate The top panel of Figure A.1 plots the raw real rate R t from 197Q2 to 216Q2. The downward trend in R t has received recent attention from many macroeconomists who argue that it reflects a form of economic secular stagnation (e.g. Summers (215)). In this paper, we do not focus on the longer-run trend in R t, but rather the large cyclical variation around this trend. Our goal is to better understand the determinants of higher frequency (e.g quarterly) movements in the real rate. To achieve this goal, we need to empirically extract the cyclical component of the real rate. In the main text, we use a simple linear deterministic trend to do so: R t = β + β 1 t + r t (5) Here, the detrended real rate r t is just the sequence of residuals from the regression. We chose this approach because it is simple and transparent. Still, it is fair to wonder whether a deterministic (downward) linear trend is a plausible model of the economy s real interest rate. No economic theory would predict the real rate to tend towards negative infinity over the next fifty years. A natural alternative that we explore now is to allow for a stochastic drift in the real interest rate. In short, real rates look extremely similar whether we remove a linear or stochastic trend, consistent with the finding that it is extremely difficult to distinguish between deterministic and stochastic trends in finite samples (Campbell and Perron (1991)). 4 4 We think of the stochastic or non-stochastic drift as a simple way of controlling for long-run output growth. For example, Holston et al. (216) embed this type of thinking in their statistical model of the natural rate of interest. They model the natural rate of interest as the sum of two random walks, one of which also drives the stochastic drift of potential output growth. 7

9 Specifically, we follow Hamilton (217) to extract the cyclical component of R t in the presence of a potentially stochastic drift. For quarterly data, Hamilton (217) recommends the following regression to achieve the filter: R t = k + k 1 R t 8 + k 2 R t 9 + k 3 R t 1 + k 4 R t 11 + r t (6) where the cyclical component of R t is captured by the regression residuals, denoted here by r t. Importantly, this filtering methodology is relatively agnostic about the underlying trend driving the series. 5 This is particularly useful in our context because, again, we are not interested in understanding longer-run trends in R t. Hamilton (217) also provides an extensive argument for why regression (6) is superior to the more standard Hodrick-Prescott filter. The bottom panel of Figure A.1 plots the linearly detrended real rate (r t ) and what we call the Hamilton-filtered real rate ( r t ). A visual inspection shows that r t and r t are quite similar. That is, linearly detrending and using the Hamilton-filter appear to give similar estimates for the cyclical component of the real rate. A regression of one on the other, run in both levels and first-differences, confirms this intuition: r t = r t, R 2 =.56 (.1) (8.56) r t = r t, R 2 =.85 (.29) (4.44) where Newey-West t-statistics with five lags are listed below point estimates. Both specifications indicate that the linearly detrended real rate is fairly close to the Hamilton-filtered rate. The constant in both regressions is near zero, the point estimate on r t is near one, and the R-squared s are pretty large. As a result, we focus on the simpler, linearly detrended real rate in the main text and repeat our core analysis on the Hamilton-filtered rate now. To be certain that detrending (in any fashion) is not driving our conclusions, we also show our results using the raw real rate R t. B.2 Results Using r t and the Raw Real Rate B.2.1 The Real Rate and PVS Table A.12 shows regressions of the form: Y t = a + b PV S t + θ X t + ξ t where X t is a vector of control variables and Y t is either the Hamilton-filtered rate r t or the raw rate R t. Results with r t Columns (1)-(6) run the regression for the Hamilton-filtered real rate, r t. The control variables that we use are the aggregate book-to-market ratio, the output gap, and the inflation rate. For consistency, we extract the cyclical components of these variables using Hamilton 5 In fact, Hamilton (217) argues that it is still a useful method for extracting the cyclical component of a series that has a deterministic time trend. 8

10 (217) before including them in the regression. Echoing our results in the main text, the relationship between PV S and r t is robust across level and first-difference specifications, and is not altered much by the addition of our control variables. Column 2 adds the aggregate book-to-market ratio as a control to the regression, which has very little effect on the point estimate on PV S. Column 3 of Table A.12 adds the raw output gap and inflation to the level regression of r t on PV S. Again, we include these variables to check whether PV S is just picking up on Taylor (1993) rule variables. The Hamilton-filtered rate does load positively and significantly on the output gap, which is what we would expect if the central bank follows some version of a Taylor (1993) rule. The important thing though is that the inclusion of these variables does not impact the point estimate or statistical significance of PV S in the regression. The results using the Hamilton-filtered rate also compared favorably to those using the simple linear detrending. For instance, in Column (3) when we regress r t on PV S and the full set of controls, the point estimate on PV S is The same regression using the linearly detrended rate r t gives a point estimate of 3.28 on PV S. Results with R t Columns (7)-(9) repeat the analysis for the raw real rate R t. Importantly, in this case, we do also not do any filtering to the control variables these regressions only use raw variables. Column (7) runs a univariate regression of the raw real rate on PV S. The regression coefficient of 3.87 is once again economically and statistically indistinguishable from the point estimate of we get when using the detrended real rate (see Table 2 in the main text). The R-squared is again impressively high at.38. Columns (8) and (9) add the aggregate book-to-market and the output gap and inflation as control variables. While the aggregate book-to-market enters significantly, the R-squared in columns (7) and (8) is almost the same, indicating that the explanatory power of PV S t for the real rate is much stronger than that of the aggregate book-to-market. Most important, none of our conclusions regarding the relationship between the real rate and PV S are impacted. Interestingly, when using the Hamilton-filtered rate or the raw rate, the coefficient on the aggregate book-to-market ratio is statistically significant for the levels regression. We explore why this occurs below in Section B.2.4. B.2.2 Returns on Volatility-Sorted Portfolios and the Raw Real Rate As discussed in Section 3 of the main text, one natural explanation for the observed correlation between the real rate and PV S is that the compensation investors demand for bearing uncertainty, and thus their demand for precautionary savings, varies over time. We argue that the data supports this view by showing that both PV S and the real rate forecast the return on a portfolio that is long low-volatility stocks and short high-volatility stocks. We repeat that analysis now by running: R t t+k = a + b x t + ξ t+k, (7) where R t t+k is the return of the low-minus-high volatility portfolio from time t to t + k. x t is either the Hamilton-filtered rate r t or the raw rate R t. Results with r t Panel A of Table A.13 displays the forecasting results using r t. Columns (1) and (2) of the table shos the results for k = 1 and k = 4, respectively. For both horizons, the coefficient on r t is statistically significant at a 5% confidence level, mirroring our findings with the detrended rate in Table 5 of the main text. The R 2 in the regressions are also pretty similar to those found when using the detrended rate, as are the magnitude of the point estimates. 9

11 Results with R t Panel B of Table A.13 shows the results using the raw real rate. For both horizons, the coefficient on the raw real rate is statistically significant, as we found with the detrended rate in Table 5 of the main text and the economic magnitudes are similar to Table 5 in the main text. This leads us to conclude that detrending does not drive our forecasting regression results. Columns (3) and (4) of both panels A and B show that the real rate, regardless of which version we use, does not forecast excess returns on the aggregate stock market. In other words, real rate variation is not driven by the same factors that drive risk premiums on the aggregate stock market, just as we found in the main text for our baseline detrended real rate. B.2.3 The Quantity of Risk and the Raw Real Rate Table A.14 repeats our analysis in Section 3.4 of the main text. There, we show that the detrended real rate is not correlated with observable quantities of volatility, which supports our view that the relationship between the real rate and returns on the long-short portfolio sorted on volatility is likely driven by changing aversion to volatility. As is clear from Table A.14, the Hamilton-filtered real rate and the raw rate also possesses no reliable relationship with quantities of macroeconomic or aggregate financial market risk, nor do our quantity of risk measures impact the relationship between the real rate and PV S. B.2.4 The Aggregate Stock Market and the Real Rate In Section B.2.1 we document a positive and statistically significant relationship between the aggregate book-to-market ratio and the raw real rate, as well as the Hamilton-filtered real rate. However, the positive relationship between the real rate and the aggregate BM is highly dependent on specification and the method of detrending. For example, the relationship is nonexistent when we difference the data. Moreover, the relationship in levels disappears when linearly detrending the real rate and the aggregate BM ratio. In addition, there is ample empirical evidence that variation in the aggregate value of the stock market is largely disconnected from real rate variation (e.g. Campbell and Ammer (1993)). In sum, we do not view the evidence in Section B.2.1 to reveal a robust link between the real rate and the aggregate BM ratio. Even if there is a relationship between the real rate and the aggregate value of the stock market in our sample, it is probably mechanical. Standard Gordon growth model logic suggests that the aggregate dividend-yield is driven by the risk-free rate r f, the market risk premium E [ r m r f ], and the growth rate of aggregate dividends g: 6 D/P = r f + E [ r m r f ] g The simple formula immediately illustrates the mechanical relationship between the risk-free rate and the dividend-yield. Of course, D/P and r f may also correlate if the risk-free rate is also related to the market risk premium or aggregate dividend growth. However, we know this is not the case. Table 5 in the main text and Table A.13 demonstrate that the real rate contains no forecasting power for excess market returns. In Table A.15, we show that the real rate both the Hamilton-filtered 6 A similar argument holds for the aggregate book-to-market ratio, but the dividend-price ratio is easier for the purposes of this illustration. As an empirical matter, the two are 98% (6%) correlated in levels (first-differences) for our sample. 1

12 and raw series has no forecasting power for aggregate real earnings growth or aggregate real dividend growth. In conclusion, the link between the aggregate BM ratio and the real rate that appeared in Section B.2.1 appears to be either mechanical, or more likely, nonexistent. B.3 Inference The AR(1) coefficients of the Hamilton-filtered rate r t, the linearly detrended rate r t, and PV S t are.81,.85, and.88, respectively. While the persistence of PV S t may appear high, it is useful to keep in mind that it is much less persistent than the aggregate valuation ratios, where persistent regressor biases have found the most attention in asset pricing (Stambaugh (1999)). While PV S t has a quarterly AR(1) coefficient of.88, corresponding to a half-life of 1.4 years, the aggregate book-to-market has an AR(1) coefficient of.99, corresponding to a much longer half-life of 17.2 years. This simple comparison already suggests that inference problems from persistent regressors are likely to be much less severe in our setting than for aggregate valuation ratios. We ues a battery of approaches to formally establish that our results are not driven by serially correlated regressors. First, we run all our main results in differences, as shown throughout the main text and the appendix. In this section, we explore several ways of adjusting standard errors, GLS, and a bootstrap simulation exercise. B.3.1 Standard Error Corrections Our baseline univariate regression of the linearly detrended real rate (r t ) on PV S yields the following estimates: r t = PV S t (5.2) (11.41) [2.64] [5.36] where the parenthesis below the point estimates contain OLS t-statistics and the square brackets contain Newey-West t-statistics with five lags. The first thing to note from this simple regression is that Newey-West correction still indicates the point estimate on PV S is statistically significant. The second thing to note is that the nonparametric Newey-West correction shrinks the OLS t-statistic by a factor of nearly two. This owes in part to the fact that the regression residuals have a first-order autocorrelation of.76. We address this persistence directly by using a standard parametric correction based on on the estimated residual autocorrelation. Specifically, we multiply the standard errors in the regression by a factor of C = (1 + ρ)/(1 ρ), where ρ is the autocorrelation of the regression residuals. ρ =.76 means that C 7.3, thereby implying that the OLS t-statistics need to be divided by a factor of C = The parametric correction therefore shrinks the t-statistic on PV S from to 4.21, so the point estimate is still statistically significant. For completeness, we repeat the analysis using the Hamilton-filtered real rate r t. In this case, a univariate regression of r t on PV S t gives: r t = PV S t (5.11) (11.57) [2.73] [6.53] The first-order autocorrelation of the residuals for this specification is.69, implying that the OLS t-statistic of should be adjusted to

13 The broader takeaway here is that no matter how we adjust our standard errors, we are still able to comfortably reject the null that the point estimate on PV S is equal to zero. B.3.2 Generalized Least Squares (GLS) For statistically efficiency and to account for the role of outliers, we also estimate the relationship between the linearly detrended real rate and PV S using generalized least squares. This is just a Prais-Winsten regression, which amounts to quasi-differencing the data before running the regression. GLS gives the following estimates: r t = PV S t (1.32) (6.15) where the GLS t-statistics are listed below point estimates. We also estimate the same system using the Hamilton-filtered real rate r t : r t = PV S t (1.9) (6.35) Regardless of the detrending method, the relationship between the real rate and PV S remains economically and statistically significant when using GLS. Moreover, if we run the regression using data up until the financial crisis (pre-29), we get fairly similar point estimates on PV S across simple OLS and GLS estimation methods. For example, when using the Hamilton-filtered real rate, OLS gives a point estimate on PV S of 3.44 and GLS gives a point estimate of B.3.3 Simulation Evidence Finally, one might be concerned that our results are biased in a Granger-Newbold sense. We use simulations to show that the standard error and R 2 from our baseline regression are not just a result of regressor persistence. Specifically, we fit an AR(1)-GARCH(1,1) model to r t and PV S and simulate independent processes mimicking the persistence properties of r t and PV S t and with identical sample length as in the data. In the simulated data, where by construction r t and PV S t are unrelated, we regress r t on PV S t, retaining the Newey-West corrected t-statistic (five lags) for PV S and the R 2 in the simulated regression. Figure A.2 presents histograms of the simulated t-statistics and R 2 from this exercise for 1, independent simulations. The plot also shows the actual t- statistic on PV S and the R 2 that we estimate in the data. The p-values listed in the plot are just the proportion of simulations where the t-statistic (or R 2 ) exceed the actual t-statistic we estimate in the data. For both the t-statistic and R 2, less than.5% of simulations can match the regression of the real rate on PV S that we estimate using actual data. Combined with the other analysis in the paper, this tells us that under the null of no relation between PV S t and r t it would be highly unlikely to observe the t-statistics and R-squareds that we see in the data. This simulation hence again adds to our evidence that the relation between PV S t and r t is highly statistically significant. 12

14 C Evidence from Other Asset Classes and Risk Appetite Measures C.1 Data from other asset classes PVS, our new measure of risk appetite or precautionary savings, derives from the relative valuation of low and high-volatility stocks. Our basic argument is that the pricing of these stocks is informative about investor risk preferences when investors are not perfectly diversified. Given our assertion that PVS measures risk preferences that matter for the whole economy as reflected by the explanatory power for the real rate it is natural to wonder whether PVS relates to the relative pricing of low and high-volatility securities in other asset classes. To make some preliminary progress on this question, we use the test assets from He et al. (217), henceforth HKM. HKM provide portfolios that are designed to represent the cross-section of securities in a wide variety of asset classes: equities, U.S. government bonds, U.S. corporate bonds, options, commodities, and foreign exchange (FX). We refer the reader to HKM for more detail on each of these portfolios. Within each asset class, we form a portfolio that is long the low volatility portfolio in that asset class, and short the high-volatility portfolio. For each portfolio in each asset class, we compute the volatility at each quarter using the trailing 5-year history of monthly portfolio returns, requiring a minimum of four years of data. We are constrained to use monthly data because HKM do not have daily asset class data. For example, suppose we want to form the low-minus-high volatility portfolio for U.S. corporate bonds in quarter t. 7 We then compute the volatility of each of the 1 HKM corporate bond portfolios over the previous 5 years. We then go long the portfolio with the lowest trailing volatility and short the portfolio with the highest volatility. We hold this long-short portfolio for one quarter, and then repeat the process. Denote the returns to this long-short strategy as LMHVt Corp, where the superscript denotes the asset class we are studying and the subscript denotes time of the return. In the main paper, we forecast this return using PVS: LMHV Corp t+1 = a + b Corp PV S t + ε t+1 We also run this forecasting regression using the real rate instead of PVS. The idea behind the exercise is simple: if the risk appetite that matters for the real rate is truly reflected in the relative valuation of low and high volatility assets, then PVS (and the real rate) should correlate with the risk premium on this type of trade in all asset classes. When running this analysis for U.S. equities, we do not use the test assets from HKM, since we form our own portfolio of equities using the entire cross-section of CRSP firms. 8 We do indeed find that both PVS and the real rate forecast the low-minus-high volatility trade in U.S. corporate bonds, options, and CDS. In addition, PVS and the real rate both correlate negatively with the term spread. Because longer maturity Treasuries tend to be more volatile than shorter maturity Treasuries, the term spread seems like a reasonable measure of expected returns on the high-minus-low volatility trade in Treasuries; thus, PVS and the real rate appear to line up in a consistent manner with Treasuries as well. Broadly speaking, this evidence gives us some comfort that the intuition of our equity-based measure shows up in other, large asset classes. We prefer to measure risk appetite from the stock market, as opposed to 7 This corresponds to US_bond11-US_bond2 in HKM s data. 8 If we use HKM s equity assets, which are just the 25 Fama-French portfolios sorted on size and value, we get very similar forecasting results. 13

15 other asset classes, because equity markets are relatively more liquid and have a longer time series of available data. C.2 Risk appetite from the aggregate stock market A natural way to measure risk appetite from the aggregate stock market is based on the expected return of the market in excess of the risk-free rate. Intuitively, when risk appetite is low, expected excess returns need to be high in equilibrium to induce investors to hold the aggregate stock market. To operationalize this intuition empirically, we therefore need a good measure of expected excess market returns. We obtain a statistically optimal measure of expected excess returns following the methodology developed in Kelly and Pruitt (213). Specifically, we use the three-pass regression filter (3PRF) in Kelly and Pruitt (213). In particular, we use the entire sample to estimate the 3PRF and assume two latent factors. In our experimentation with the procedure, using two factors balances the desire to have a good in-sample predictor of market returns against overfitting. We denote our estimate of expected excess market returns using the full sample as E t [Mkt-Rf t,t+4 ]. The t + 4 in this notation indicates that we are predicting one-year head annual returns. The E t [ ] means that the forecast is formed using variable values at time t. 9 The variables that we use as the predictors in the Kelly and Pruitt (213) procedure are five BM ratios from sorting on each of the following variables: size, BM ratios, cash-flow duration (Weber (216)), leverage, cash-flow beta with respect to aggregate cash flows, leverage, beta with respect to the aggregate market (using a 5-year window and a 1-year window), and total volatility. See Section D for additional details on how we construct these BM ratios. In addition, we include the aggregate BM ratio, aggregate dividend yield, and CAY from Lettau and Ludvigson (24). This gives us a total of 43 predictors that we feed into the 3PRF to forecast annual excess market returns. The R-squared in the forecasting regression is 14.2%. As a point of comparison, we are able to nearly double the forecasting power (in-sample) of CAY alone, which gives a forecasting R- squared of 7.5%. The sample size for this analysis is 18, and is lower than our main sample (N = 185) because the duration sorted portfolios that we include as predictor variables have a shorter sample. As we show in the paper, E t [Mkt-Rf t,t+4 ] is basically uncorrelated with the real rate. This is not particularly surprising given that the aggregate book-to-market ratio is itself basically uncorrelated with the real rate. D Data Construction In this section we provide details on how we construct our main variables. D.1 Building Valuation Ratios Our valuation ratios (book-to-market) derive from the CRSP-COMPUSTAT merged databases. We augment CRSP-COMPUSTAT with the book value data used in Davis, Fama, and French (2). We provide additional details of our variable construction below, but at a high level our procedure is as follows: for a given firm f on date t, we look for a valid value of book equity in COMPUSTAT 9 We have tried a truly out-of-sample version of the 3PRF and obtain similar conclusions regarding the correlation with the real rate. 14

16 Quarterly, then COMPUSTAT Annual, and finally the book values contained in Davis, Fama, and French (2). We assume balance sheet information is known with a one-quarter lag. Finally, we combine the aforementioned book value with the trailing 6-month average of equity market capitalization to form a book-to-market ratio for firm f. Our results are not sensitive to these variable definition choices. 1 D.1.1 COMPUSTAT Quarterly From COMPUSTAT Quarterly (COMPQ). Specifically, we obtain information on all firms (INDFMT = INDL) with a standardized data format (DATAFMT = STD) that report financial information at a consolidated level (CONSOL = C). In order to avoid the well-known survival bias in COMPU- STAT, we only include firms once they have at least 2 years of data. We define book common equity (BE) according to the standard Fama and French (1993) definition. Specifically, BE is the COMPUSTAT book value of shareholder equity, plus balance-sheet deferred taxes and investment tax credit, minus the book value of preferred stock. We use the par value of preferred stock in COMPQ to estimate the value of preferred stock. D.1.2 COMPUSTAT Annual When using COMPUSTAT Annual (COMPA) for balance sheet information, we obtain information on all firms (INDFMT = INDL) with a standardized data format (DATAFMT = STD) that report financial information at a consolidated level (CONSOL = C). In order to avoid the wellknown survival bias in COMPUSTAT, we only include firms once they have at least 2 years of data. For firms that change fiscal year within a calendar year, we take the last reported date when extracting financial data. This leaves us with one set of observations for each firm (gvkey) in each year. We define book common equity (BE) according to the standard Fama and French (1993) definition. Specifically, BE is the COMPUSTAT book value of shareholder equity, plus balance-sheet deferred taxes and investment tax credit, minus the book value of preferred stock. Following Fama and French (1993), we use the redemption, liquidation, or par value (in that order) to estimate the value of preferred stock. D.1.3 Defining Valuation Ratios For all of our reported results, we build book-to-market ratios at end of each quarter t as follows: The book equity comes from COMPQ, and we assume this data is known with a 3-month lag. This means we add three months to the DATADATE field in COMPQ to define the KNOWNDATE. Then at the end of each quarter, we take the book equity on the last available KNOWNDATE. For instance, this means that in June of a given year, we are using 1 Specifically, in unreported results, we have experimented with several permutations in terms of: (i) balance sheet timing conventions (including the usual Fama and French (1993) setup); and (ii) equity market capitalization (e.g. the latest market value, the median market capitalization in the latest month, etc.). All of our main conclusions in the paper are robust to these alternatives. 15