## Bryan Seegmiller and Brian Boyer, Department of Finance

A recent paper by Binsbergen, Brandt, and Koijen (2012)^{i} examined the equity risk-premium on short- versus long-term dividend claims, providing evidence that the large size of the overall equity risk premium was due mostly to the even higher risk-premium earned on short-term dividends. The implication is that the equity risk premium slopes downward across the term structure. These findings are of note because they constitute an anomaly in the literature, as nearly all asset pricing models—such as Campbell and Cochrane (1999)^{ii}—predict an upward-sloping or flat term structure. Some, such as Belo, Collin-Dufresne, and Goldstein (2015)^{iii}, have sought to develop new asset pricing models to reflect this anomaly. Others, such as Boguth, Carlson, Fisher, and Simutin (2012)^{iv}, have disputed the result, showing that the unexpected findings of Binsbergen, et al. could be explained by the presence of small frictions in financial market microstructures.

The goal of this project has been to produce a robust and accurate estimate of the term structure of the equity risk premium across time. Whereas Binsbergen, Brandt, and Koijen (2012) found evidence of a downward-sloping term structure, our research was not able to validate this finding. Using one approach, we find a higher point estimate on the short-term risk premium versus the longterm, which would indeed indicate a downward-sloping term structure. However, the difference between short- and long-term risk premia fails to meet any accepted measure of statistical significance. In another approach, we find that the short-term risk premium appears to be nearly exactly the same as the long term. In both instances we cannot reject the null hypothesis that the term structure of the equity risk premium is flat.

A key formulation in finance and economics shows that the price of any asset at a given time is equal to the discounted sum of all expected future cash flows. We decompose holdings in the S&P 500 equity index into short- and long-term components using a put-call parity formulation that holds under the typical assumption of no arbitrage. There are two equivalent ways to do this, one using puts and calls and the other using futures contracts. Binsbergen, Brandt, and Koijen (2012) use puts and calls, while we try it with both. Here are our estimates on returns to eighteen month short-term holdings (called the “short-term asset) versus returns on the S&P 500 using put and call data from OptionMetric. Standard errors are in parentheses:

Initially this looks like confirmatory evidence for Binsbergen et al, as the mean monthly risk premium on the short-term asset (second colum from left) is larger than the mean monthly risk premium on the S&P 500 (second column from right). However, the t-stat on the difference i(last column) is only about .89 (.0031/.0035). Thus no strong statistical evidence can be found that the true risk-premia are different, and the higher point estimates over the sample period may simply be an artifact of the data.

Next we estimated the price of short-term holdings for 18- and 24-month short-term assets and compare them to the S&P 500, but instead utilize the equivalanet derivation with futures contracts obtained from Bloomberg. This time returns are on annualized basis. Because of noise in the shortterm asset price, we had to take the approach of summing dividends for the life of the short-term asset holding and divide by the price to compute the returns. This method means that we could only get four observations per year, as futures contracts only expire in March, June, September, and December. The estimated annualized excess return on the 18-month short-term asset was 4.78% per year, while the S&P 500 excess return over the same time period was about 4.79%–nearly the exact same! Using a 24-month short-term asset instead, we found a 3.2% annualized short-term risk premium versus a 4.5% risk premium on the S&P 500. This gives evidence of an upward slope in the equity risk premium, directly counter to the findings of Binsbergen et al. None of these differences were even close to statistically significant. Thus our analysis of annualized returns using futures contracts agrees with our monthly return computations using puts and calls on one very important point: we cannot find strong evidence to conclude that the term structure of the equity risk premium is not flat.

This finding is good news for some of the most commonly used asset pricing models, such as Campbell and Cochrane (1999) or Bansal and Yaron (2004), which predict a flat- or slightly upwardsloping term structure of the equity risk premium. However, our results come with an unfortunate caveat. As previously alluded to, Boguth, Carlson, Fisher, and Simutin (2012) argue that estimating returns on short-term holdings using puts and calls or futures contracts is inherently subject to statistical bias. We had hoped to construct a measure of the risk-premium that was more robust to the biases they allude to; however, our tick-by-tick dataset on S&P 500 futures turned out to be insufficient for the question at hand, as it included almost exclusively futures contracts of very short lengths two maturity (generally one to three months). Our analysis required futures contracts of considerably longer length, and thus the data proved inadequate, and our only results on the futures data came from Bloomberg, which only provides one futures price per day, which would seem insufficient to counteract the measurement error.This was unfortunate as we believed this large dataset on S&P 500 futures would provide the means of averaging out the biases in short-term asset price calculations caused by measurement error in the underlying index or futures contract. Thus our findings should be taken with a grain of salt, and certainly more research must be done to provide an estimate of risk-premia on short-term holdings that is sufficiently robust. Still, the critiques of Boguth et al indicate that if bias is present, it should bias the short-term return estimates upward. Since we found no evidence that the short-term risk premium on the S&P 500 is higher than the long-term, this certainly should give pause to the claims of Binsbergen et al, and may provide reason to hold back on the creation of new asset pricing models built to in part to accommodate their findings, such as Belo, Collin-Dufresne, and Goldstein (2015). Our research thus far would say such efforts are nothing if not premature, and without some strong confirmatory results may turn out to be entirely superfluous.

^{i} van Binsbergen, Jules, Michael Brandt, and Koijen, Ralph. June 2012. “On the Timing and Pricing ofDividends.” American Economic Review, Volume 102, Issue 4, pages 1596-1618.

^{ii} Campbell, John Y., and Cochrane, John H. 1999. “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior.” Journal of Political Economy, Volume 107, Issue 2, pages 205-251.

^{iii} Belo, Federico, Collin-Dufresne, Pierre, and Goldstein, Robert S. May 2015. “Dividend Dynamics and the Term Structure of Dividend Strips.” The Journal of Finance, Volume 70, Issue 3, pages 1115-1160.

^{iv} Boguth, Oliver and Carlson, Murray and Fisher, Adlai J. and Simutin, Mikhail. “Leverage and the Limits of Arbitrage Pricing: Implications for Dividend Strips and the Term Structure of Equity Risk Premia” 2012. Available at SSRN: http://ssrn.com/abstract=1931105