Co-Movement Between Bond and Stock Risk Premia

1. Introduction

I compare the covariance between the bond risk premium as captured by the Cochrane and Piazzesi (2005) factor and the stock risk premium as captured by the logarithm of the price to dividend ratio as used in, say, Shiller (2006). This covariance measure gives a rough view of whether or not the same risk factors are driving the excess returns in each market.

In section 2, I recreate the Cochrane and Piazzesi (2005) factor and verify its properties. In section 3, I compute the log price to dividend ratio from Robert Shiller’s online data. Finally, in section 4, I conclude by computing the covariances between the Cochrane and Piazzesi (2005) factor, the log price to dividend ratio and the excess returns on the S&P 500.

2. Bond Risk Premium

In this section, I recreate the Cochrane and Piazzesi (2005) bond risk factor. The raw data are $1$ through $5$ year zero coupon bond prices which come from the CRSP Fama-Bliss risk free bond prices hosted on WRDS with run from January 1964 to December 2003. From these raw series, I convert the price as $p_t = \ln \left( P_t/100 \right)$ and then compute bond yields, forward rates and returns as described below where the $t$ subscript represents the current date and the $\tau$ argument represents the bond maturity:

(1) $\begin{align*} y_t(\tau) \ &= \ - \dfrac{p_t(\tau)}{\tau} \\ f_t(\tau) \ &= \ p_t(\tau - 1) \ - \ p_t(\tau) \\ r_t(\tau) \ &= \ p_t(\tau - 1) \ - \ p_{t-12}(\tau) \end{align*}$

Next, I run a regression of the average excess returns for the $2$ through $5$ year maturity bonds over the $1$ year spot rate:

(2) $\begin{align*} \overline{r}^x_{t+1} &= \begin{bmatrix} \gamma_0 \\ \gamma_1 \\ \gamma_2 \\ \gamma_3 \\ \gamma_4 \\ \gamma_5 \end{bmatrix}' \begin{pmatrix} 1 \\ y_t(1) \\ f_t(2) \\ f_t(3) \\ f_t(4) \\ f_t(5) \end{pmatrix} + \varepsilon_{t+1} \\ &= \Gamma' F_t + \varepsilon_{t+1} \end{align*}$

Using monthly data from 1964-2003, I find the regression results below which match up closely with those reported in Table 1 of Cochrane and Piazzesi (2005):

$\begin{align*} \begin{array}{l|cc} & \textit{est} & \textit{s.e.} \\ \hline \hline \gamma_0 & -0.03 & 0.0056 \\ \gamma_1 & -2.06 & 0.255 \\ \gamma_2 & 0.59 & 0.523 \\ \gamma_3 & 2.92 & 0.440 \\ \gamma_4 & 0.90 & 0.323 \\ \gamma_5 & -1.95 & 0.267 \\ \hline R^2 & 0.34 \end{array} \end{align*}$

These results say that, for instance, a $1\%$ increase in the $3$ year forward rate predicts an increase in the average excess return over maturities from $2$ to $5$ years by $3\%$ , while a $1\%$ increase in the $5$ year forward rate predicts a $2\%$ decrease in the same measure. The figure below plots the coefficients above along with a $95\%$ confidence interval.

This figure reports the estimates of gamma from the regression depicted above of average excess returns on bonds of maturities 2 through 5 on forward rates at maturities 1 to 5 as well as a constant corresponding to Panel A of Figure 1 in Cochrane and Piazzesi (2005).

I then use this regression estimate to compute the Cochrane and Piazzesi (2005) factor as the predicted value of the average excess returns:

(3) $\begin{align*} \textit{CP}_t &= \Gamma' F_t \end{align*}$

This figure plots the level and first difference of the Cochrane and Piazzesi (2005) bond risk premium measure using monthly observations of annual returns from January 1964-December 2003 as well as the analogous bond risk premium measure implied in Gabaix (2011).

I also compute the analogous measure of the bond risk premium implied by Gabaix (2011):

(4) $\begin{align*} \textit{CP}_t^G &= \frac{-y_t(1) + 2 \cdot f_t(3) - f_t(5)}{4} \end{align*}$

Below I report the summary statistics of the both bond risk premium measures. This table reads that the average excess return on bonds at maturities $2$ – $5$ years is $0.84\%$ per year in this monthly sample from January 1964 to December 2003 with a standard deviation of $2.4\%$ .

$\begin{align*} \begin{array}{l|cc} & \mathbb{E}[ \cdot ] & \mathbb{S}[ \cdot ] \\ \hline \hline \textit{CP}_t & 0.0084 & 0.024 \\ \textit{CP}_t^G & 0.0014 & 0.0027 \end{array} \end{align*}$

3. Stock Risk Premium

In this section, I compute the logarithm of the price to dividend ratio on the S&P 500 as a proxy for the risk premium in the US stock market. I use data from Robert Shiller’s website which reports the real price level, real dividends and real earnings on the S&P 500 on a monthly frequency dating back to the 1800’s. Using this data, I compute the annual cum dividend excess return on the S&P 500 as:

(5) $\begin{align*} r_{t+12}^e &= \frac{P_{t+12} + D_{t+12}}{P_t} - 1 \end{align*}$

This data delivers the summary statistics below for the monthly sample running from January 1964 to December 2003 corresponding to the Cochrane and Piazzesi (2005) interal. The excess returns are annualized while the log price to dividend and log price to earnings ratios, $p_t - d_t$ and $p_t - y_t$ respectively, are on a month by month basis. This table reads that the average excess return on the stock market over the next year was $6.6\%$ per year with a volatility of $16\%$ , while in any given month the average of the logarithm of the price to dividend ratio is $3.496$ with an average month to month increase of $0.0013$ .

$\begin{align*} \begin{array}{l|cc} & \mathbb{E}[ \cdot ] & \mathbb{S}[ \cdot ] \\ \hline \hline r_{t+12}^e & 0.066 & 0.16 \\ p_t - d_t & 3.496 & 0.413 \\ p_t - y_t & 2.759 & 0.423 \\ \Delta (p_t - d_t) & 0.0013 & 0.0359 \\ \Delta (p_t - y_t) & 0.00039 & 0.041 \end{array} \end{align*}$

Below I plot the annualized excess return on the S&P 500 along with the log price to dividend and log price to earnings ratios in both levels and changes.

This figure shows monthly observations of the annual excess returns on the S&P 500 along with the log price to dividend and price to earnings ratio from January 1964 to December 2003 using data from Shiller (2006).

This figure shows monthly observations of the annual excess returns on the S&P 500 along with the first difference of the log price to dividend and price to earnings ratio from February 1964 to December 2003 using data from Shiller (2006).

I regress the excess return on the S&P 500 over the next year on the current log price to dividend ratio using monthly observations over the period from January 1964 to December 2003. I report these regression results below. The standard errors have not been corrected for overlapping samples. The point estimates indicate that a $1\%$ increase in the price to dividend ratio in the current month is associated with a $0.06\%$ or $6\textit{b.p.}$ decrease in the returns on the S&P 500 over the next year.

$\begin{align*} \begin{array}{l|cc} & \textit{est} & \textit{s.e.} \\ \hline \hline \alpha_{p_t - d_t} & 0.24 & 0.049 \\ \beta_{p_t - d_t} & -0.063 & 0.018 \\ \hline R^2 & 0.026 \end{array} \end{align*}$

I also regress the level and the change in the log price to dividend ratio on the level and the change in the annual real interest rate using monthly observations over the period from January 1964 to December 2003 and report the regression coefficients below. The standard errors have not been corrected for overlapping samples. The annual real interest rate each month is the difference between the Fama-Bliss riskless rate over that year and the log change in inflation using the data from the CPI located here.

$\begin{align*} \begin{array}{l|cc} \mathtt{LHS:} \ p_t - d_t & \textit{est} & \textit{s.e.} \\ \hline \hline \alpha & 3.60 & 0.024 \\ \beta & -2.52 & 0.78 \\ \hline R^2 & 0.42 \end{array} \end{align*}$

$\begin{align*} \begin{array}{l|cc} \mathtt{LHS:} \ \Delta_1 (p_t - d_t) & \textit{est} & \textit{s.e.} \\ \hline \hline \alpha_{\Delta_1 r_{t+12}^f} & 0.00065 & 0.0015 \\ \beta_{\Delta_1 r_{t+12}^f} & -1.70 & 0.26 \\ \hline R^2 & 0.034 \end{array} \end{align*}$

$\begin{align*} \begin{array}{l|cc} \mathtt{LHS:} \ \Delta_{12} (p_t - d_t) & \textit{est} & \textit{s.e.} \\ \hline \hline \alpha_{\Delta_{12} r_{t+12}^f} & 0.012 & 0.02 \\ \beta_{\Delta_{12} r_{t+12}^f} & -2.08 & 1.07 \\ \hline R^2 & 0.13 \end{array} \end{align*}$

The raw correlations here are given in the tables below where the $\tilde{r}^f$ denotes the real riskless rate over the next year computed using an $AR(2)$ implied inflation rate.

$\begin{align*} \begin{array}{l||cccc} & p_t - d_t & p_t - y_t & r_{t+12}^f & \tilde{r}_{t+12}^f \\ \hline \hline p_t - d_t & 1 & 0.88 & -0.14 & -0.14 \\ p_t - y_t & & 1 & -0.15 & -0.15 \\ r_{t+12}^f & & & 1 & 0.995 \\ \tilde{r}_{t+12}^f & & & & 1 \end{array} \end{align*}$

$\begin{align*} \begin{array}{l||cccc} & \Delta (p_t - d_t) & \Delta (p_t - y_t) & \Delta r_{t+12}^f & \Delta \tilde{r}_{t+12}^f \\ \hline \hline \Delta (p_t - d_t) & 1 & 0.88 & -0.28 & -0.25 \\ \Delta (p_t - y_t) & & 1 & -0.29 & -0.26 \\ \Delta r_{t+12}^f & & & 1 & 0.820 \\ \Delta \tilde{r}_{t+12}^f & & & & 1 \end{array} \end{align*}$

We can see from the plots below that this negative correlation between the log price to dividend ratio and the real riskless rate comes entirely from a spike during the 1970’s.

Level and month over month changes in the log price to dividend ratio and annual real riskless rate.

4. Co-Movement

Finally, to get a sense of how much the risk premia in the bond and stock markets co-move, I report the correlation between the Cochrane and Piazzesi (2005) bond risk factor, the Gabaix (2011) implied bond risk premium analogue, the log price to dividend and priec to earnings ratios and the excess returns on the S&P 500 in both levels and changes below:

$\begin{align*} \begin{array}{l||cccccc} & \textit{CP}_t & \textit{CP}_t^G & p_t - d_t & p_t - y_t & r_{t+12}^e & r_{t+12}^f \\ \hline \hline \textit{CP}_t & 1 & 0.885 & - 0.155 & - 0.033 & 0.311 & 0.39 \\ \textit{CP}_t^G & & 1 & 0.135 & 0.286 & 0.246 & 0.19 \\ p_t - d_t & & & 1 & 0.908 & -0.169 & -0.14 \\ p_t - y_t & & & & 1 & -0.163 & -0.15 \\ r_{t+12}^e & & & & & 1 & 0.44 \\ r_{t+12}^f & & & & & & 1 \end{array} \end{align*}$

$\begin{align*} \begin{array}{l||ccccc} & \Delta \textit{CP}_t & \Delta \textit{CP}_t^G & \Delta (p_t - d_t) & r_{t+12}^e & r_{t+12}^f \\ \hline \hline \Delta \textit{CP}_t & 1 & 0.930 & 0.018 & -0.029 & - 0.00035 \\ \Delta \textit{CP}_t^G & & 1 & 0.011 & -0.0121 & -0.00053 \\ \Delta (p_t - d_t) & & & 1 & 0.081 & 0.044 \\ r_{t+12}^e & & & & 1 & 0.44 \\ r_{t+12}^f & & & & & 1 \end{array} \end{align*}$