Does Precautionary Savings Drive the Real Interest Rate? Evidence from the Stock Market 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 details our data and the VARs at the end of the paper, and describes additional features of the theoretical model in the main text. Pflueger: University of British Columbia. E-mail: carolin.pflueger@sauder.ubc.ca Siriwardane: Harvard Business School. E-mail: esiriwardane@hbs.edu. Sunderam: Harvard Business School and NBER. E-mail: asunderam@hbs.edu.
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 + e 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
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 + q X t + e 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
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.4 reports the results of the following regression: ROE Vol t!t+4 = a + b X t + e 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.4 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): q i,t = r i,t+1 e i,t+1 + rq i,t+1 + n it where q 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. r is a log-linearization constant and n i,t is an approximation error, such that q i,t r i,t+1 e i,t+1 + rq 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 # "  q i,t i2low Vol t 1 N H,t #  q i,t i2high 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 et+1 PV S + r pvs t+1 " # " t+1 1 r PV S e PV S t+1 "  r i,t+1 N L,t i2low Vol t 1 N L,t  i2low Vol t e i,t+1 # " 1  r i,t+1 N H,t i2high Vol t 1 N H,t  i2high Vol t e i,t+1 # # (2) 3
In addition, we assume that pvs t follows an AR(1) process, pvs t+1 = a + f pvs t + x t+1. Next, combining the AR-process with Equation (2), plus some rearranging yields: Cov(Real Rate t, pvs t ) (1 rf) 1 [Cov Real Rate t,rt+1 PV S Cov Real Rate t,et+1 PV S + rcov(real Rate t,x t+1 )] Dividing both sides by Cov(Real Rate t, pvs t ) delivers a simple covariance decomposition: 1 = Y r Y e + Y x (3) where Y r (1 rf) 1 Cov Real Rate t,rt+1 PV S /Cov(Real Rate t, 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 f and r. We fit a simple AR(1) for pvs and find that f =.88 for quarterly data. With regards to r, we consider a range of values from.9 to.97. 1 All of the other components needed for the covariance decomposition are estimated from simple covariances in the data. For all of the ranges of r that we consider, Y r is never less than 7% and approaches 1% for larger values of r. Moreover, for all of the ranges of r considered in Vuolteenaho (22), Y 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.5 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 + x t+4 1 Vuolteenaho (22) sets r =.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
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.5, X t is the realized return volatility of the low-minus-high volatility portfolio, s 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 s t (LMH-Vol Portfolio). Column (2) adds the real rate as a regressor. In both cases, s 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 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 s (Low Vol) s (High Vol). The regressions presented in Table A.6 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.5.2 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 5
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 189-21, 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. 2 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.7: Real Rate t = a + b 1 E t [g t+1 ]+b 2 V t [g t+1 ]+e t 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. 2 Hartzmark (216) uses a sample from 1947 to 21, whereas our sample runs from 197 to 216. 6
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 = b + b 1 t + r t (4) 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)). 3 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 +er t (5) where the cyclical component of R t is captured by the regression residuals, denoted here by er t. Importantly, this filtering methodology is relatively agnostic about the underlying trend driving the series. 4 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 (5) 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 (er t ). A visual inspection shows that r t and er 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: er t =.2 +.71 r t, R 2 =.56 (.1) (8.56) Der t =.1 +.99 Dr t, R 2 =.85 (.29) (4.44) 3 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. 4 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. 7
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.8 shows regressions of the form: Y t = a + b PV S t + q X t + x 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, er 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 (217) before including them in the regression. Echoing our results in the main text, the relationship between PV S and er 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.8 adds the raw output gap and inflation to the level regression of er 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 3.65. 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 8
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 + x t+k, (6) 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.9 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. Results with R t Panel B of Table A.9 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.1 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.1, 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. How- 9
ever, 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: 5 D/P = r f + E r m 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.9 demonstrate that the real rate contains no forecasting power for excess market returns. In Table A.11, we show that the real rate both the Hamilton-filtered 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 er 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. 5 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. r f g 1
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 =.62 + 3.44 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 + r)/(1 r), where r is the autocorrelation of the regression residuals. r =.76 means that C 7.3, thereby implying that the OLS t-statistics need to be divided by a factor of p C = 2.71. The parametric correction therefore shrinks the t-statistic on PV S from 11.41 to 4.21, so the point estimate is still statistically significant. For completeness, we repeat the analysis using the Hamilton-filtered real rate er t. In this case, a univariate regression of er t on PV S t gives: er t =.59 + 3.28 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 11.57 should be adjusted to 4.96. 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 =.44 + 2.47 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 er t : er t =.49 + 2.59 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. 11
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 3.28. 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 tha twe 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. C Data Construction In this section we provide details on how we construct our main variables. C.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 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. 6 6 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. 12
C.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 COMPUS- TAT, 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. C.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. C.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 the book value of equity from COMPQ as of March in that same year. We prefer this definition because it uses up-to-date balance sheet information (allowing for reasonable lags in information release). If COMPQ does not have a valid book value, we obtain book equity from COMPA, again assuming a one-quarter information lag for balance sheet information. If COMPA also does not have a valid book value for a firm, we check the book equity values from Davis, Fama, and French (2), which we downloaded from Ken French s website. For the book equity in Davis, Fama, and French (2), we use their assumption that book values are known as of June 3 of the Last_Moody_Year variable. For the purposes of computing book-to-market ratios, we use the trailing 6-month average of market capitalization using CRSP Monthly. For instance, in June of a given year we 13
take the average end-of-month market capitalization from January through June of that year. We prefer this definition because it smoothes out any high-frequency movements in equity valuations. Book-to-market ratios for a given firm then follow naturally. We have also used the Fama and French (1993) definition of book-to-market ratios and obtain very similar results. Fama and French (1993) assume a more conservative lag in terms of when balance sheet is known and also use lagged market capitalization (e.g. in June of a year, use the previous December s market capitalization). Because our story is that the stock market incorporates precautionary savings motives into prices in real time, we like using a more up-to-date measure of value. C.2 Volatility Calculations C.2.1 Volatility Used for Portfolio Sorts At the end of each quarter, we use daily stock data from the previous two months to compute a high-frequency measure of volatility. We exclude firms that do not have at least 2 observations over this time frame. This approach mirrors the construction of variance-sorted portfolios on Ken French s website. 7 We define a firm s total volatility as the standard deviation of ex-dividend returns (variable RETX) in that month. We also compute idiosyncratic volatility for each firm by taking the standard deviation of the residuals from a regression of daily ex-dividend returns within the month on the three Fama and French (1993) factors. In earlier versions of the paper, we used idiosyncratic volatility as our main sorting variable. The results using total return volatility versus idiosyncratic volatility are virtually identical, mainly because idiosyncratic volatility comprises most of total return volatility at the firm level (Herskovic et al. (216)). C.2.2 Realized Quarterly Volatility In some of our tests, we require a measure of realized volatility over each quarter. For each firm and quarter, we use daily returns (including dividends) to compute within-quarter volatility. C.2.3 Macroeconomic Uncertainty Measures We compute what we call quarterly macroeconomic uncertainty measures from two sources: (i) the Fama and French (1993) factors and (ii) aggregate TFP growth. When using financial returns, we compute quarterly volatility using daily data from within each quarter. To compute the quarterly volatility of aggregate TFP growth we follow Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (212). Specifically, we download the quarterly_tfp.xls file from John Fernald s website (downloaded 1/4/216) and fit an asymmetric GARCH(1,1) to the dtfp series. We model the mean of the dtfp series as a constant. Panel A of Table A.1 contains the point estimates of the GARCH model. Panel B of Table A.1 shows the persistence of our financial market volatility measures, computed by fitting an AR(1) process to each series. At a quarterly frequency, the volatility of the Fama 7 Our long-short portfolio effective replicates the one on Ken French s website. If we regress our portfolio on his, the point estimate is.84, the constant in the regression is statistically indistinguishable from zero, and the R 2 is 96%. 14
and French (1993) is not particularly persistent, with AR coefficients ranging between.5-.65. In the data, quarterly shocks to return volatility appear to die out quickly. Panel C of Table A.1 shows the correlation of our macroeconomic volatility measure (TFP volatility) and our financial market measures. The table shows that the financial market measures are themselves highly correlated, but none are particularly correlated with TFP growth volatility. C.3 Market and Cash Flow Betas C.3.1 Market Betas (CAPM) We compute three types of market (CAPM) betas. In all cases, our benchmark index is the CRSP Value-Weighted Index. For the first two measures of CAPM Beta, all of our individual firm data derives from the CRSP Monthly dataset. We deal with delisted returns as in Shumway (1997) by setting missing delisted returns with codes 4-591 to a value of -3%. C.3.2 1. The first CAPM beta we compute is a two-year rolling beta. In a given quarter, we use the previous twenty-four months worth of monthly return data to compute a CAPM beta. In order to have a valid two-year beta, a firm must have at least 8% of its observations over the previous two years. 2. The second CAPM beta we compute is a long-run beta. We first aggregate monthly returns into six-month returns. Then at the end of each quarter we use the previous ten years worth of data to compute betas from our six-month return series (e.g. 2 observations per regression). Once again, firms must have 8% of their observations in order to have a valid long-run beta. 3. The third CAPM beta we compute uses a two-month window. For each firm, we use daily stock data from the previous two months to compute a high-frequency measure of CAPM beta. We exclude firms that do not have at least 2 observations over this time frame. Cash Flow Beta We consider two alternative methods in computing cash flow betas. In both, betas are computed with respect to quarter-on-quarter growth in national income. We obtain the national income series from the FRED database (series A53RC1). Our first cash flow beta series measures cash flow growth at the firm level as quarter-on-quarter EBITDA growth (series oibdpq from COMPUSTAT quarterly). At the end of each quarter, we use the previous twelve quarters to compute a cash flow beta, where a firm must have at least 8% of its observations to be included. To compute our second cash flow beta series, we construct a mimicking portfolio for national income growth. We first regress national income growth on the five Fama-French factors, as well as the ten Fama-French industry portfolios. Our regression also includes a constant. For the industry portfolios, we subtract out the risk-free rate (Fama-French) in order to make all of our projection portfolios zero cost. This means that the coefficients in the regression represent portfolio weights. The mimicking portfolio is then the fitted value from the regression, but ignoring the estimated constant. At the end of each quarter, we use the mimicking portfolio to compute cash flow betas for each firm. We use the previous two years of monthly return data (dividend-adjusted and corrected for 15
delistings as noted above) to compute a beta for each firm. As with our other beta calculations, a firm must have 8% of its observations to have a valid beta at the end of a given quarter. C.4 Double Sorts In the main text we run several robustness checks regarding the relationship between the real rate and PV S t. These include constructing double-sorted versions of PV S t by sorting stocks on volatility and another characteristic Y. We then construct a Y-neutral book-to-market spread by sorting stocks on volatility within each tercile of characteristic Y. This spread measures the difference in valuations of low volatility and high volatility stocks that have similar values of characteristic Y. The sorting variables we use are described in Section A.1 of this appendix. For the dividendadjusted double sort, we separate firms into two buckets based on whether they paid a dividend over the previous two months. We determine dividend yields by looking at the total return and the ex-dividend return in CRSP. Within the non-dividend payers, we compute the low-minus-high volatility book-to-market spread. We do the same in the set of dividend paying stocks. We then average the spreads across the two bins (e.g. the dividend and non-dividend payers) to compute the the dividend-adjusted version of PV S. D Impulse Response Estimation Details Y t = P t = k  i=1 k  i= B i Y t i + D i Y t i + k  i=1 k  i= C i P t i + A y v y,t (7) G i P t i + A p v p,t. (8) We can rewrite the system (7)-(8) in VAR form with only lagged variables on the right-handside and estimate by OLS. Let u p = A p v p be the VAR residuals in the policy block that are orthogonal to the VAR residuals in the non-policy block. To recover the structural shocks, including v MP, Bernanke and Mihov (1998) require a specific model relating the VAR residuals and the structural shocks in the policy block. Taking the real rate as the policy instrument and the PV S t as an indicator of demand for the risk-free bond, we assume that the market for the risk-free bond is described by the following set of equations: u PV S = av MP + v PV S, (9) u rr = fv PV S + v MP. (1) Eq. (9) is the innovation in investors precautionary savings demand for bonds. It states that the demand for low-volatility assets depends on monetary policy shocks v MP and the structural innovation v PV S. Eq. (1) describes central bank behavior. We assume that the Fed observes and responds to precautionary savings shocks. The model described by Eqs. (9) and (1) has four unknown parameters: a,f, and the two structural shock variances,s 2 PV S, and s 2 MP. We estimate the model using a two-step efficient GMM procedure, as in Bernanke and Mihov (1998). The first step is an equation-by-equation OLS estimation of the VAR coefficients. The 16