Term Structure(s) of Equity Risk Premia

Transcription

1 Term Structure(s) of Equity Risk Premia Leandro Gomes Gávea Investimentos Ruy Ribeiro PUC-Rio First draft: 4th April 2016 Current draft: 10th April 2017 Abstract Using dividend and variance swap data simultaneously, we provide a series of novel facts on the two term structures of equity risk premia: holding period returns and maturity. These two dimensions are distinct and jointly important for a better understanting of risk premia and its time variation. We also uncover new information on the level and variation of the term structure of the (physical) expectation of dividend growth. Moreover, we find that liquidity is priced in the cross-section of variance and dividend swaps and affect many conclusions in related literature. Among many facts, we show that: risk premium in increasing and concave on maturity; short-maturity and short-horizon risk premium is lower than market risk premium; risk premium is also increasing and concave on investment horizon, but increases much faster than maturity; independently of maturity, most of the risk premium is associated with short-horizon risk premium; dividends are highly predictable once we account for time variation in term structure of holding period returns; and liquidity is relevant for short-term holding period returns and to explain the high returns of dividend contracts. Using both dimensions jointly highlights additional challenges to asset pricing models, but, unlike previous literature, we show that all these facts are consistent, for instance, with a long-run risk model with disaster risks.

2 1 Introduction Since Mehra and Prescott [1985], finance researchers have been trying to understand the reason for the supposedly high level of equity risk premium. In parallel, additional empirical evidence suggested that the equity premium was not constant over time and actually predictable by price ratios and other variables. Many models have followed to explain both features of the equity premium, including some that tried to explain both jointly such as Campbell and Cochrane [1999] s habit formation model, Bansal and Yaron [2004] s long run risk model and the rare disaster risk models by Barro [2006], Wachter [2013] and Gabaix [2012]. More recently, new data has allowed us to learn more about the term structure of the equity premium. van Binsbergen et al. [2012] calculated market s prices for dividend strips using option prices, i.e. the price today of each of future s dividends, and found that the equity premium is concentrated on the short term, which is inconsistent with many of the above proposed solutions. Moreover, Dew-Becker et al. [2017] suggested that the equity premium is related to short-term realized variance risks. In their paper, the market only hedges shocks to realized equity market s variance, while hedging against long term expected variance shocks are free, which also contradicts most of prevailing finance models and some new macroeconomic models, where shocks to uncertainty are important drivers of the business cycle. Martin [2017] addresses the term structure of risk premium from another angle, showing that a new variance swap price gives us direct information on the market risk premium and show that this measure indeed predicts future returns at different horizons. In this paper, we provide additional facts on the term structure of risk premium and a model that captures the main features of the data: a high time-varying risk premium with increasing and concave average term structure(s). We argue that there are two different term structures of risk premium: horizon and maturity. Risk premium is increasing and concave on both horizon and maturity, but it increases faster on the horizon dimension. These term structures are closely related, even though they appear to be quite distinct at a first glance. Their relation can be used to uncover the term structure of (physical) expectation of dividend growth. We exploit jointly the information on equity returns, dividend swaps and variance swaps, taking advantage of short- and long-maturity contracts. Return and dividend growth predictability becomes considerably stronger once we account for the time-variation in the term structure of equity premium and dividend growth. We show that an illiquidity factor is priced in the cross-section of dividend and variance contracts and that it explains why dividend strips appear to have high returns. 1

3 We propose a model that captures the main facts discussed here. We integrate disaster risk into a long run risk model combining model features previously discussed in Barro and Jin [2016] and Drechsler and Yaron [2011]. Interestingly, we also show that the long run risk model with jumps proposed by Drechsler and Yaron [2011] with an alternative calibration can quantitatively match most of the patterns in the term structure of equity risk, dividend risk and variance risk premium. Changing the calibration is sufficient to change the conclusion in Dew-Becker et al. [2017] on the failures of this model. A new calibration shows that it is consistent with higher relative importance for short term variance risk versus longermaturities. Also, some of results are improved when we account for market liquidity risk, which is a risk not present in all models in this particular literature. This is needed, for example, to show that Drechsler and Yaron [2011] s model is also capable of matching the correlation matrix of dividends and variance returns, which adds even more robustness to models ability to explain asset prices. Drechsler and Yaron [2011] extends the reach of Bansal and Yaron [2004] long run risk models, which basically rely on Epstein-Zin utility functions and exogenously consumption and dividend growth dynamics that are subject to small but very persistent Gaussian shocks that generate the time-varying risk premia while maintaining low risk-free rates. This kind of model was already used to price other assets, like US Treasury bonds on Piazzesi and Schneider [2007], exchange rate volatility on Colacito and Croce [2011], and the Variance Premium, i.e. the difference between one month priced and realized volatility as on Drechsler and Yaron [2011]. The remainder of this paper is organized as follows. Section 2 discusses the importance of understanding separately the different dimensions of term structure of equity premium: horizon and maturity. We show the the average equity premium in both dimensions is increasing and concave and how these term structures vary over time. We also describe how we obtain this information from variance and dividend swaps. Section 3 shows the relation between the two term structures and how we uncover the term structure of the physical expectation of future dividend growth. Section 4 revisits return and dividend growth predictability regressions through new lens, showing that both are more predictable once we account for information on the three term structures: expected dividend growth, maturity and investment horizon. In section 5, we test the importance of market liquidity risk and how it affects our previous results. Section 6 presents models that are consistent with the empirical evidence. Section 7 concludes. 2

4 2 Term Structure of the Equity Premium In this section, we present measures of equity risk premia for different holding periods and maturities constructed using variance and dividend swap data. We show that risk premium is increasing and concave in both dimensions and increases much faster in the horizon dimension. In order to understand the pricing of different equity portfolios, we need to understand better how investor value time and risk in multiple dimensions. Interestingly, we have financial instruments nowadays that can help us understand better horizon and maturity effects in equities. Dividend swaps can tell us about maturity effects, while variance swaps can be informative of horizon effects. Independently of the maturity of a certain claim, investors may require different returns for different holding periods. Independently of the holding period, investors may require different returns for claims paying in different maturities. Additionally, the price of each claim depends on its expected growth. There are connections among all these dimensions and we start with horizon and maturity in this section. In the next section, we introduce expected dividend growth. We first show how dividend swaps and variance swaps can help us in the analysis of term structure of risk premium. Then, we show how they are related to each other. The distinction between horizon and maturity is far more clear in the case of fixed income securities. Moreover, it is also a lot more common to analyze the term structure of prices in fixed income. We commonly use interest rates for zero-coupon bonds with different maturities to discuss term structure issues as these bonds have just one cash flow at maturity. Even very long bonds will have a required rate of return when we hold them for a shorter period such as one year. This rate will only be the same across bonds with different maturities if we assume that expectation hypothesis holds. Otherwise, it is safe to assume that a bond with longer maturity will require a higher rate of return and we would call this difference a term premium. If expectation hypothesis holds, then expected holding period return is equal to the expected rate of return of the bond with maturity equivalent to the investment horizon. If expectation hypothesis does not hold, this relation between holding period and maturity does not hold anymore, but we would still expect that changes in required rates in one dimension could affect the other. 2.1 Dividend Swaps and Maturity Dividend swaps can help us create intruments similar to zero-coupon bonds but in the equity space: dividend strips. Dividend swap prices are not market expectations of future dividends 3

5 as they reflect in part a risk premium associated with the maturity of the dividend payment. We can use dividend swap data to construct dividend strips which are the current value of the dividend to be paid in a particular year t + j. The price of a stock in time t, P t, is the expected sum of all future dividends discounted by each correspondent stochastic factor, M t+i. Dividend strip corresponds to each element of this sum. Hence, the dividend strip price, S i, represents the risk-adjusted present value of the expected dividend payment to be in period t+i. Dividend swaps prices, DS t+j, will be equivalent to dividend strips multiplied by the gross risk free rate from period t to period t + j. We describe additional details in the Appendix, but the basic relations are the following: [ ] P t = E t M t+i D t+i = S t+i = R f,t:t+i DS t+i (1) i=1 i=1 i=1 Here, we use a proprietary database from an important derivatives player for US S&P500 Index Dividend Swaps for annual maturities of 1 to 9 years. Our data ranges from January 2003 to June Figure 1: Dividend Growth Price: Time series evolution of the dividend swap price relative to dividends in the year before for 4 different horizons from January 2003 to June This measure corresponds to the priced expected dividend growth. In our sample, we saw significant variation in the term structure of dividend prices. Figure 1 plots the time series of dividend swaps. With the exception to the Great Financial 4

6 Table 1: Term Structure of Equity Premium (Maturity) Rm-rf 1y 2y 3y 4y 5y 6y 7y 8y 9y Div Swaps Mean Excess Return 6.9% 4.4% 5.9% 6.9% 7.4% 8.1% 9.1% 9.6% 9.8% 10.3% Annualized Sharpe Ratio Div Strips Mean Excess Return 6.9% 5.0% 7.0% 8.7% 9.9% 10.9% 12.2% 12.9% 13.4% 13.9% Annualized Sharpe Ratio Cum Div Strips Mean Excess Return 6.9% 5.0% 6.0% 6.9% 7.6% 8.2% 8.9% 9.4% 9.7% 13.9% Annualized Sharpe Ratio This table presents the annualized mean returns and annualized Sharpe ratios of dividend swaps, dividend strips and cumulative dividend strips from January 2003 to June Crisis, there was always an implied dividend growth priced in the market. In this figure, we divide the swap price by the past one-year dividend. Hence, this measure corresponds to the dividend growth priced in the market. We are able to calculate the return, risk premium and other return measures for an investment in a dividend of a particular maturity. We compute these returns following a constant maturity approach where we reinvest into a particular contract on a monthly or yearly basis. By doing so, we are able to measure the term structure of equity returns relative to the maturity of the payments. In this section, we analyze the patterns generated by contracts, but, in a latter section, we return with a discussion of other determinants of level and slope in this curve. Later we propose an alternative explanation for the shape of this curve. We consider the possibility that the overall curve in higher than it would be and potentially with a different slope if market for dividend contracts were more liquid. Later, we show that the return of all contracts, including the intermediate ones, becomes lower than the return of S&P 500 index, but the curve remains increasing and concave. Table 1 shows the main result of the section, which is the fact that, in our dataset, the equity premium is increasing and concave with maturity. While the annualized excess return of S&P 500 is of 6.9 %, the annualized mean monthly return of the 1-year dividend strip is only of 5%. Nonetheless, all other dividend strips of longer maturities average more than 7% during this time span. Similar to van Binsbergen et al. [2012], we can compute the returns of the residual dividend strip, i.e. those beyond 9 years. This average return would be lower than the respective equity index. In 1, we also consider the cumulative dividend strip, which corresponds to the price of all dividends up to year i. In all cases, we find that premium is increasing with the term of the contract and becomes higher than the overall market risk premium only after years 2 to 4 depending on the case. This result is inconsistent with the evidence in van Binsbergen et al. [2012] and partially consistent with results in van Binsbergen and Koijen [2017] but not with their interpretation of these results. We should emphasize that the time span of this sample is different and results are sensitive due to relatively small sample. 5

7 [Mention SR of Market and contracts above] Alternatively, van Binsbergen et al. [2012] uses options data to infer the implied short term, D ST, and long term dividends, D LT, prices. Formally, the objects they are aiming to construct are the following, where i corresponds to month units: [ 24 ] S ST = E t M t+i D t+i i=1 [ ] S LT = E t M t+i D t+i i=25 (2) In other words, their dataset allows them to form two blocks of dividend strips. The short term, which implied by options prices, is the sum of the next 2 years of monthly dividends, while the long-term is the residual prices needed to make (1) hold. They find that from January 1996 to October 2009, a substantial amount of the S&P s equity premium is on the short term, as this dividend strip generated an annualized 10% return, against only 3.2% from the index. Therefore, they argue, long-term returns would have had to be lower to make the identity (1) hold. 2.2 Variance Swaps and Horizon In most finance models, some kind of variance is usually the determinant of the level of risk premium. As Martin [2017] points out, market risk premium should be related to a measure of priced variance and he derives a simple relation using properties of the stochastic discount factor. Here we relate the term structure of variance risk prices to the term structure of the equity premium, following very closely the approach in Martin [2017]. In this context, we are discussing the term structure along different investment horizons and not the term structure of maturities. In his derivations, the asset in question remains the same, and therefore its maturity is also stable. To be clear, we analyze the market risk premium, but change the holding period of the investment. It is not the same we did with dividend swaps as there we kept the holding period constant and only changed the maturity of the dividend contracts. Representing the equity risk premium lower bound as a equality, we would infer that market risk premium would vary over time t and over horizon j depending on the changes to the risk free rate between time t and t+j and priced variance for returns between t and t+j: E t [R t:t+j R f,t:t+j ] = R f,t:t+j V ar Q t [R t:t+j ] (3) 6

8 Here we use variance swap data. One disavantage is that we depart from the setup in Martin [2017] as his derivations are based on a simple variance swap that he proposes and computes directly from option data. However, the same paper argues that diffences are most often very small to be economically important. Our advantage is that we have access to multiple contracts, including very long maturities. Usual maturities are 1, 2, 3, 6, 12 months as well as annual versions from 1 to 12 years maturities. We use a proprietary database on variance swaps with maturities of 1 month, and 1 to 10 years (same provider of the dividend swap data). Our time series ranges from January 2001 to May Whenever we need to compute returns, we use the same spline interpolation used for dividend swaps. The main difference of our database to Dew-Becker et al. [2017] is that they have shorter and fewer maturities after the 1-year contract. For long-term variance swaps contracts, for example, their database covers only , likely a small window in a period where we faced an extreme financial crisis, as we can see in Figure 2. Consequently, our database is more suitable to analyze medium- and long-term holding periods. Nonetheless, our results resemble Dew-Becker et al. [2017] s average term structure of variance prices as we also observe an upward slope. Realized variance shocks may lead to a downward-sloping curve from time to time. Figure 3 show that the shape of the curve conditional on the slope of the first two contracts. An inverted curve was very frequent during the during the epicenter of Global Financial Crisis as seen in Figure 2. Figure 2: Variance Swap Price: Time series evolution of variance swap prices for 4 different horizons in variance points from January 2003 to June

9 Figure 3: Conditional Variance Swap Curves: Variance swap curves conditional on the slope between the 1-month and 1-year contract prices. Sample is divided in two parts corresponding to positive and negative slopes. Values are in variance points from January 2003 to June Figure 4: Equity Risk Premium: Time series of a measure of equity risk premium following the methodology in Martin [2017] for 4 different horizons. Using simplifying assumptions on Martin [2017] s approach, we find the the holdingperiod term stucture of equity risk premium is on average increasing and concave on the investment horizon. In fact, the curve is positively sloped XX% of the times. Figure 4 8

10 Table 2: Term Structure of Equity Premium (Holding Period) 1m 1y 2y 3y 4y 5y 6y 7y Average 4.97% 6.08% 6.35% 6.62% 6.84% 7.07% 7.29% 7.52% Median 3.05% 5.40% 5.62% 5.83% 6.17% 6.37% 6.55% 6.78% Max 38.76% 20.28% 17.83% 17.27% 16.45% 15.70% 15.18% 15.06% Minimum 1.13% 1.91% 2.23% 2.39% 2.50% 2.57% 2.54% 2.47% Notes: This table presents descriptive statistics (average, median, maximum and minimum) for the implied holding-periodterm structure of risk premium using variance swaps prices and the term structure of treasury rates from January 2003 to June 2014, following the calculation proposed by Martin [2017]. shows the time series evolution of the term structure, while Table 2 shows basic statistics for all holding periods. This table shows the maximum for each investment horizon, which is usually associated with an inverted curve. We compute a time series for equity risk premium for holding periods from 1 month to 10 years. As a proxy for the risk free rate for all the holding periods, we use the term structure of zero-coupon government bonds. 2.3 Liquidity Risk and Swap Contracts In this section, we show that some of the previous results are partially explained by liquidity exposure of the swap contracts. We find that liquidity factors affect the performance of these constracts. Additionally, we find supporting evidence that liquidity are priced in the cross-section of dividend and variance swaps. We considered two measures based on marketwide liquidity. First, we use the liquidity factor proposed by Pastor and Stambaugh [2003]. Second, we use the change in the TED spread and also the lagged value of the TED spread to capture market illiquidity. For the sake of completeness, we also include Fama-French factors in the analysis to evaluate whether other risks are relevant. We include the market return even though this term may generate confounding effects. Table 3 shows that dividend strip returns are sensitive to liquidity measures with the expected sign and that short-horizon variance swaps are also strongly affected. In all regressions, we add the market returns measured with the CRSP market portfolio. In Model 1, we also include the Fama-French size and value factors, while we add lagged TED spread to Model 2. Note that dividend strip betas with respect to the change in TED spread are all positive and of similar magnitude. Therefore, in this dimension the effect of illiquidity is similar on the average returns to all dividend contracts. At the same time, there is significantly more dispersion in betas for variance swaps, where the front end more sensitive to liquidity measures indicating that market liquidity is very important for short horizon investing. We 9

11 also find a direct relation between market returns and liquidity factors in this sample but not reported here. We also tested a cross-sectional model not reported here where we find that the TED spread is priced in the cross-section of these assets and that their expected returns of dividend strips net of liquidity risk are in fact lower than market returns net of liquidity risk for all contracts. 10

12 Table 3: Time series regressions of dividend strip and variance swap returns on liquidity variables Excess Dividend Strips Returns Variance Swap Returns 1y 2y 3y 4y 5y 6y 7y 8y 9y 1m 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y Model 1 (including Fama-French Factors) CONST DIFFTED MKT SMB HML TRLIQ Model 2 (including lagged TED spread) 11 CONST DIFFTED MKT TRLIQ TED(-1) Notes: This table presents time series regressions of dividend strip and variance swap returns on both lagged level (TED(-1)) and change in TED spread (DIFFTED) and also on Pastor and Stambaugh [2003] s traded liquidity factor (TRLIQ). Model 1 includes Fama-French factors, while Model 2 incorporates the lagged effect of TED spread. T-statistics are reported below coefficient estimates.

13 Intuitively, lower market liquidity can generate greater realized variance by construction, which would consequently influence the returns of 1-month variance swaps. Taking the great financial crisis as an example, there was a relevant lack of liquidity on the market that mechanically could have increased the realized variance or vice versa. Therefore, we conclude that the 1-month variance swap return is related to two potential risks: pure realized variance and aggregated-market liquidity. A substantial amount of the 1-month variance risk is related to the aggregated market liquidity. When we subtract the aggregated market liquidity effect of the 1-month variance swaps and other instruments, correlations between returns of these "new" instruments become much smaller. On the same line, the new Sharpe Ratio of liquidity-free 1-month variance swap is lower and closer to zero but still negative as expected. Additionally, we later consider whether the alleged failure of some of the models to price some long-term assets may be due to illiquidity. We are not able to analyze the direct liquidity of the underlying assets as we do not have access to volumes traded and bid-ask spread that could help us measure it directly. This does not mean that other returns are not influenced by their own illiquidity. Nevertheless, these results indicate that the relevance of market aggregate liquidity is concentrated on the short-end of the variance price curve but important for all dividend contracts. 3 Horizon, Maturity and Dividend Growth In this section, we show that the relation between maturity and investment horizon is not as simple as normally assumed. We find that the pricing of dividend swaps of any maturity is dependent mostly on the expected holding period return for an investment horizon that is normally shorter than maturity. 3.1 Relating Horizon and Maturity The pricing of all dividends strips with maturities of up to 9 year are basically determined by an investment horizon of one year. Using an investment horizon equivalent to the maturity does not add much value in most cases. Our analysis shows that the risk premium on dividend strip prices is mostly determined by an exposure to a short-horizon risk premium. Table 4 shows the regression of dividend strip prices on the ERP for different holding periods. Panel A considers a specification where a dividend strip with maturity t+j depends on all ERP with horizon shorter than and equal 12

14 to t + j. We considered other alternatives such as the case where only ERP t+1 matters (Panel B) or only ERP t+j matters (Panel C). The case where both may matter is consistent with the above. Putting together all these results, we find that ERP t+1 appears to be the most important term in all cases. In theory, we would expect a negative and statistically significant coefficient on the approprite measure of risk premium. We find it to be negative on the shorter horizons relative to maturity and it is always significant for ERP t+1. The importance of this term is decreasing in the maturity of the dividend contract, but the fraction of the variation in dividend strip prices due to risk premium is more than 50% of the total variance. Panel B shows that it is sufficient to keep the ERP t+1 term. We should emphasize that focusing on the horizon equal to maturity as in Panel C leads to much weaker results and a quick loss in goodness of fit. We believe that part of this contribution to variance is explained by a possible positive covariance between expected dividend growth and expected return news, but the effect is nevertheless economically significant. 3.2 Implied Maturity Risk Premium and Implied Dividend Growth Using these regressions, we can divide the pricing of dividend strips into two components: implied expected dividend growth and risk premium. We perform this decomposition using dividend swap prices as they are close in theory to measures of expected future values as there is no time discounting. We find the implied expected dividend growth as the residual of the projection of the priced dividend growth on all ERP terms with horizon shorter than the maturity of the respective dividend swap. Figure 5 shows the time series evolution of the implied growth. Table X shows some basic statistics, while Figure 6 shows the comparison of priced and expected dividend growth. 13

15 Table 4: Price of Dividend Growth on Zero-Coupon Variance Swaps D1Y D2Y D3Y D4Y D5Y D6Y D7Y Panel A (all terms) CONST ERP1y ERP2y ERP3y ERP4y ERP5y ERP6y ERP7y R2 57% 65% 66% 70% 75% 74% 76% Adjusted R % 64.13% 65.07% 68.48% 73.61% 73.01% 74.52% Panel B (only one-year ERP) CONST ERP1y R % 53.38% 49.27% 44.64% 41.69% 38.31% 34.08% Adjusted R % 53.00% 48.85% 44.19% 41.21% 37.81% 33.54% Panel C (only diagonal terms) CONST ERPiy R % 33.47% 18.47% 5.92% 1.46% 0.15% 0.02% Adjusted R % 32.92% 17.80% 5.15% 0.65% -0.66% -0.80% Notes: Here we regress the implied price of dividend growth relative to the 12-months trailing realized dividends using dividend swaps prices on the zero-coupon variance swaps prices 14

16 Figure 5: Expected Dividend Growth under P-Measure: estimate of the expected dividend growth under the physical measure based on the residual of the projection of dividend swap prices on ERP measures for horizons shorter than the respective period. Figure 6: Priced vs Expected Dividend Growth: a comparison of the priced and expected dividend growth based on model for 1- and 3-year horizons. 15

17 4 Predictability In this section, we confirm that we have more information about future returns and future dividends once we separate the two effects using information on dividend and variance swaps. Before we proceed with this analysis, it is important to realize that this particular sample differs substantially from results that were originally obtained with return and dividend predictability regression. Standard textbook such as Cochrane [2001] and Campbell et al. [1997] would show that returns were predictable while future dividend would be unpredictable. These results have, however, changed in the past decades as dividend seem more predictable. We find that returns have become less predictable while dividends have been more predictable more recently. One possibility is that expected dividend growth has become more variable and/or less correlated with expected returns. The online appendix shows dividend and return predictability regressions using price-dividend ratio as predictor variable. We run these regression for both S&P 500 and CRSP market portfolios and consider 1 to 3-year holding periods. Each panel does that for a different horizon: , and We confirm a significant change in the predictability patterns. Here we find stronger return predictability once we also consider ERP as shown in Table 5. 16

18 Table 5: Return Predictability with Equity Risk Premium and Dividend Yield months 24 months 36 months a b c d e a b c d e a b c d e Intercept log(dy) ERP1m ERPiy R Notes: This table presents 12-, 24- and 36-month predictability of market returns using different combinations of the follwing variables: log dividend yield, 1-month equity risk premium, and the maturity-equivalent i-year equity risk premium based on Martin [2017]. The first line for each regressor is the estimated coefficient, while the second shows the Newey-West corrected t-statistics.

19 Table 6: Dividends Predictability by Physical-Expectation of Dividend Growth months 24 months 36 months a b c d e f a b c d e f a b c d e f Intercept log(dy) ERP 1m ERPi RESi R Here we present the 12-months predictability of dividends using the implied physical-expectation of dividend growth of different maturities relative to the trailing 12-months realized dividend. The physiscal-expectation of dividend growth is the residual of the price of dividend growth regressed on the equivalent maturity equity risk premium implied by the variance swaps. The first line for each regressor is the estimated coefficient, while the second shows the Newey-West corrected standard deviation

20 5 A Model with Long-Run Risk and Jumps In this version of the paper, we consider a long run risk model with jumps. This model can explain many features in the data. In the next version, we treat the current model as a special case and introduce a more general model with disaster risk. 5.1 Framework A common feature on this nascent literature of term-structure of dividends and variance prices is to test if standard finance models can provide the same kind of stylized facts seen on data. For instance, Dew-Becker et al. [2017] show that Drechsler and Yaron [2011] and Wachter [2013] s models fail to generate Sharpe ratios close to zero for Variance Swaps of longer maturities, while Gabaix [2012] s model of rare disaster can match this stylized fact, even though failing to quantitatively match other moments. On the other hand, van Binsbergen et al. [2012] show that Campbell and Cochrane [1999], Bansal and Yaron [2004] and Gabaix [2012] generate too much equity premium on the medium and long term, which would be at odds with their data, while not with ours. Thereafter, both these papers are very skeptical on the ability of standard financial models to provide adequate answers to the term structure of risk premium. Our results, however, are clearly more positive. Here we focus on Drechsler and Yaron [2011] model, which is an extension of Bansal and Yaron [2004] long-run risk model, where agents have preference for early resolution of uncertainty and thereafter are willing to pay to protect themselves against news shocks to the state variables which are governed by a persistent long-term risk. The new feature of this model is to make this long-term risk vulnerable not only to Gaussian, but also, jump shocks. They demonstrate, then, that the time-variation on economic uncertainty generate positive variance premium that can predict excess stock market returns. Also, this model is also able to quantitatively match mean, volatility, skewness and kurtosis of consumption growth and stock market returns, as well as mean and standard deviation of variance premium. We show that if we properly calibrate DY model to match the term structure of variance swaps, instead of focusing on the 1-month variance premium solely as in the original paper, we are able to match additional results. While former results are maintained, now we can also match variance and dividend swaps returns. Therefore, not only we have an upward sloping equity premium up to 9 years, but, contraty to what DGLR found, we can also quantitatively match the Sharpe ratios for variance and dividend swaps and variance risk prices. 19

21 5.2 The Model First, the model starts with a maximization problem of a representative agent with Epstein- Zin utility, V, subject to a budget restriction of an endowment W. He can either consume this endowment or reinvest a portion for a total return of R c,t+1 max V t = [(1 δ)c 1 γ θ t + δ ( E t [V 1 γ t+1 ] ) ] 1 θ 1 γ θ subject to W t+1 = (W t C t )R c,t+1 The dynamics of the economy is as follow: both consumption ( c) and dividend ( d)) growth are exogenously modeled with a common long-run risk variable (x) and subject to non-independent Gaussian Shocks. Long-run risk term is a very persistent variable subject to both Gaussian and jump shocks, generating large and infrequent negative pulses and more often small positive effects. c t+1 = µ c + x t + z c,t+1 x t+1 = ρ x x t + z x,t+1 + J x,t+1 d t+1 = µ d + φx t + z d,t+1 The other main component of the model is the σ 2, with a long-run mean σ 2, which is also a persistent process that determinate the intensity and probability of jumps and Gaussian shocks on the long run risk variable. σ 2 t+1 = ρ σ σ 2 t + z σ,t+1 σ 2 t+1 = ρ σ σ 2 t + (1 ρ σ ) σ 2 t + z σ,t+1 + J σ,t+1 The Gaussian shocks, z, are calibrated so that there are correlations only between consumption and dividend growth processes, but their intensity is time-varying and depend on the σ 2. z t+1 = (z c,t+1, z x,t+1, z σ,t+1, z σ,t+1, z d,t+1 ) N(0, h + Hσ 2 t ) Specifically, the jump shocks, for each j, are compound-poisson processes in the form 20

22 below: J i,t+1 = N t+1 i ξ j i j=1 i = σ, x where ξ j i are conditionally independents. N j t+1 is the Poisson counting process with intensity λ t = l σ σt 2 for both jumps. Therefore, we emphasize that σt 2 is the variable that determinate the expected number and also the intensity of all shocks, both jump or Gaussian. It has a major role in generating risk premium. Calibration strategy follows Drechsler and Yaron [2011] very closely. The key points are: Risk aversion coefficient: γ = 9.5 Inter-temporal elasticity of substitution: ψ = 2 ξ x exp(1/v x ) + v x, to achieve small and frequent positive but large and infrequent negative jumps ξ σ exp(1/vσ) l σ = 0.75 so that jump shocks arrive on average 0.75 times per years, and negative jump long run jumps once in 4 years Originally, Drechsler and Yaron [2011] calibrated the model to match 1-month Variance Swap (l σ = 0.8) Here, I use the parameter to match long-term Variance Swap prices What is the difference of our calibration and the original, which was also used on Dew- Becker et al. [2017]? While Drechsler and Yaron [2011] s original results still hold, as seen in the next session, if we calibrate the average number of shocks to match the variance premium instead of the long-term price of variance, we end up overshooting these prices, which is what Dew-Becker et al. [2017] found. However, because our intention is to match the long-term variance swaps, we calibrated the average number of shocks accordingly. Surprisingly, all other results remain within confidence intervals. The results are based on 1000 simulations of 79 years of monthly observations. However, each variable is calculated using the equivalent sample size on the data. 5.3 Results First, it is worth emphasizing that despite being a slightly different calibration, the original results are maintained as seen in Table 7. 21

23 Second, contrary to Dew-Becker et al. [2017], we find that the model can properly match both variance swaps prices and their Sharpe ratios as seen in Figures 8 and 9. Therefore, variance risk is concentrated on the short term of the curve, just like it happens with data. Moreover, Sharpe ratios for medium- to long-term maturities are close to 0, and positive results appear within the confidence interval, indicating that even though these risks are relevant and agents would like to hold hedges against them, states of nature where infrequent large jumps occur on the simulated economy are enough to generate in-sample positive returns. Figure 7: Selected original results Notes: Here we show the on the first column moments of consumption and dividend growth, as well as market-related variables: mean return, standard deviation and kurtosis for the market portfolio. Also, moments related to the variance-related variable such as expected price of variance, mean and standard deviation of variance premium. Results from the model are based on 1000 simulations of 79 years. We present the confidence intervals of 5-95%. Figure 8: Term-Structure of Variance Prices Notes: Here we present the results for the term-structure of annualized zero-coupons variance risks. Results from the model are based on 1000 simulations of 78 years. Confidence intervals correspond to 5-95% intervals. Also, even though confidence intervals are considerably large, actual dividend swaps returns and Sharpe ratios are within the expected range, as seen in Figures 10 and

24 Figure 9: Variance Zero-Coupons Sharpe Ratios Notes: Here we present the results for the term structure of annualized zero-coupons variance Sharpe Ratios. Results from the model are based on 1000 simulations of 78 years. Confidence intervals correspond to 5-95% intervals. Figure 10: Dividend-Swaps Mean Returns Notes: Here we present the results for the term structure of annualized mean returns of Dividend Swaps. Confidence intervals do not appear on the plot for aesthetic reasons as they are too wide. Results from the model are based on 1000 simulations of 78 years. What about the results that appear to be sensitive to the liquidity risk on data? Because the model does feature liquidity risk, it is reasonable to expect that the model may fail in this respect. Indeed, the cross-correlations matrix is only matched, when we correct for aggregated market liquidity in the data. The bottom line is that DY model framework does a very decent job explaining not only the standard moments of the data, but also shows robustness to explain the term structure of variance and dividends prices, albeit not being able to match non-corrected liquidity aspects, as we would expect. 23

25 Figure 11: Dividend-Swaps Sharpe Ratios Notes: Here we present the results for the term structure of annualized Sharpe Ratios of Dividend Swaps. Results from the model are based on 1000 simulations of 78 years. Confidence intervals correspond to 5-95% intervals. 6 Conclusion This paper show that the equity premium is on the medium term, contrary to previous research. We also find evidence that contradicts the interpretation that there free hedges against expected variance shocks. Short-term variance risk premium is more important to the whole term structure of equity premium. We identify term-structure variables that forecast both future dividend growth and future market returns. Both the level and the slope of the variance curve are good predictors for market and dividend returns, while a clean version of the dividend yield forecasts dividend growth. Finally, we show that a different calibration of Drechsler and Yaron [2011] s long run risk model with jumps can account for most of the relevant stylized facts, such as mean returns, Sharpe ratios and return correlations of market and of the term structure of dividend and variance returns. We find that overall market liquidity is relevant for the returns of shortdated variance-related assets, but not for long-dated variance nor dividends (long- or shortdated). 24

26 7 Appendix 7.1 Dividend Swap In order to identify similar terms, we use dividend swaps. Dividend swaps (DS) are financial derivatives where we exchange realized dividends RD of a specific period, e.g. year 2020, for a strike price DS. Therefore, the payoff of implementing a strategy of a DS is simply: P ayoff n t = t+n j=t+n 1 RD j DS n t In our case, we use different data that allow us to separate the market price into dividend strips prices of annual maturities and the residual part. Here, we use a proprietary database from an important derivatives player for US S&P500 Index Dividend Swaps for annual maturities of 1 to 9 years. Our data ranges from January 2003 to June Therefore, we identify: [ 12 ] [ 24 ] S 12 = E t M t+i D t+i, S 24 = E t M t+i D t+i, (4) i=1 i=13 [ 36 ] [ 108 ] [ ] S 36 = E t M t+i D t+i,..., S 108 = E t M t+i D t+i, S res = E t M t+i D t+i i=25 i=97 Our strategy to calculate dividend strip prices has some advantages over BBK due to the use of dividend swaps. Specifically, we have more granularity as we have more periods. Additionally, our definition of "long-term" is close to what most investors would define as actual long-term. Another advantage is observe directly the market price of a future dividend. However, being a forward contract, what we get is not exactly a dividend strip. In order to calculate the today s price of a dividend strip, we proceed by simply discounting the swap price using the US Treasury yield of equivalent maturity. Because we have data for yearly maturities, i.e. between 12 and 24 or 48 and 60 months ahead, in order to calculate monthly returns, we follow the literature and interpolate using splines our dividend swaps to create other contracts not in the database. In other words, we have the prices for dividends between months 12 and 24 as well as for dividends between 13 i=108 25

27 and 25 and thereafter. Finally, the monthly return for holding an n Dividend Swap is: Return n t+1 = DSn 1 t+1 DS n t DS n t 7.2 Variance Swap Before we proceed, we introduce the type of data we use and our sample. Variance Swap is a financial instrument where we exchange the realized variance RV over n periods for a strike price V S n with the payoff defined as: t+n P ayofft n = RV j V St n j=t Under regular conditions, the fair price of a maturity n Variance Swap is by the following equation, where Q indicates the risk neutral probability measure. V S n t = E Q t n RV t+j j=1 Following Dew-Becker et al. [2017], using a non-arbitrage argument, we construct the instrument for expected variance n periods ahead as Z n t = E Q t [RV t+n ] = V S n t V S n 1 t Also Zt 0 = RV t, the realized variance and Zt 1 = V St 1, simply the price of the 1-month variance. The return of betting on the price of expected variance in t for n periods ahead, which we expect to be negative on average, is then simply Return n t+1 = Zn 1 t+1 Z n t Z n t 26

28 7.3 Common Risks in Variance and Dividends We now focus on the relation between the term structures of variance and dividends prices. We test this relation using returns on variance swaps and dividend swaps of all maturities. To do so, we also considered the zero-coupon versions of variance swap returns. On the dividend side, we also used returns on dividend strips and retuns on cumulative dividend strips. In order to have a common source of risk, we first need to find a common source of risk. As we can see in Table 2, the cross-correlations of dividend and variance swaps monthly returns are mostly negative. This may indicate that there may be priced variance risk, both on short and long-term dividends. Table 8 shows that the magnitude of the 1-month variance swap return cross-correlation with all dividend swap zero-coupon returns is significantly higher than other maturities, close to -40% against others near to 0. The 1-month variance swap may be a more relevant source of risk for the term-structure of the equity premium, whatever the maturities of dividend swaps we consider. We also found similar results when using dividend strip returns and cumulative dividend strip returns. In all cases, we find a stronger correlation in the first column (1-month) and close to the diagonal (same year). Table 9 shows that variance swap returns explain a significant part of dividend swap return variation for all contracts from 1- to 9-year maturity. In these regressions, we use both 1-month variance swap returns and n-year variance swap returns where n corresponds to the same year of the dividend swap. We expected that both coefficients would be positive, but we find that only for contracts with maturity longer than 5 years. In all cases, only the 1-month variance swap is significant. Note that the intercept is economically and statistically insignificant. Hence, these two factors describe well average returns of dividend swaps. 27