## Abstract

Using weekly stock-bond correlations estimated with highfrequency data, the authors find that a lower (more negative) stock-bond correlation forecasts falling 10-year interest rates over the coming weeks. It also forecasts falling oneyear interest rates over the next year. The reverse is true when the stock-bond correlation is higher (more positive). Therefore investors, in particular those with long-term, bond-like liabilities, should take greater duration risk when the recent stock-bond correlations are lower. The authors propose two possible explanations of such predictive power: (1) the markets and/or policymakers’ underreaction to the changing economic conditions the stock-bond correlation implies; and (2) the markets’ initial underreaction to the long-term bonds’ safe-haven status.

**TOPICS:** In markets, VAR and use of alternative risk measures of trading risk

Using weekly stock–bond correlations estimated with high-frequency (HF) data, we find that a lower (more negative) stock–bond correlation (SB-Correl, henceforth) forecasts falling U.S. 10-year Treasury yield over the coming weeks and falling one-year bond yield over the next year. For brevity, we refer to the SB-Correl’s predictive power as the correlation effect. Investors should take greater duration risk when the recent SB-Correl is lower. We obtain our empirical results using a simple decile portfolio based on a ranking of the weekly HF SB-Correl on its historical values.

Our findings contribute in three ways: 1) we document the HF SB-Correl’s strong predictive power over Treasury bond (henceforth, bond) returns; 2) we document an inverse relationship between the 10-year bond’s time-varying equity betas and returns; 3) we provide possible explanations for the correlation effect, such as policymakers and/or markets’ underreactions to changing economic conditions, and markets’ initial underreaction to the long-term bond’s changing status as a safe haven.

**DATA AND METHODOLOGY**

To study SB-Correls’ short-term and cyclical dynamics, we reduce information latency by using HF futures price data. From January 11, 1985, to September 6, 2013, we obtain one correlation estimate per week with no overlapping window, totaling 1,496 weekly observations. We obtain the realized weekly correlations using S&P 500 Index futures and 10-year Treasury bond futures price data with 10-minute frequency from tickdata.com. A frequency of 10 minutes is still a low enough frequency to avoid microstructure noise. To ensure good futures liquidity, we use electronic futures contracts when they are liquid enough (Exhibit 1).

As Exhibit 2 shows, both HF and low-frequency (LF) SB-Correls exhibit a secular downward trend. HF SB-Correls are also highly persistent in the short and medium term.

We also obtain three-month, one-year, two-year, five-year, and ten-year constant-maturity Treasury yields from the FRED database. We estimate forward rates and zero coupon bond log returns by linearly interpolating the zero-coupon yield curves. We obtain equity log returns using the S&P 500 Total Return Index (SPTR)^{1} and three-month Treasury yields.

Let *n* = 20 weeks for illustrative purpose. At the end of each week (*t*), we rank the last 20 weeks’ HF SB-Correls (*HF*_*Correl _{T}
*,

*T*=

*t*–19,

*t*–18,…,

*t*) and de- termine the decile of

*HF SB*_

*Correl*, or

_{t}*Decile20*, accordingly. For convenience, we also refer to week (

_{t}*t*) as week 0 and week (

*t*+ 1) as week 1. A simple HF SB-Correl-sorted portfolio allocates greater weight to bonds on week 1 when the decile is lower on week 0: 1

To show a more representative result, we use different sampling windows and use the averages of their deciles. As HF SB-Correls are highly persistent, we may also use the lagged deciles’ equally weighted moving averages. A portfolio is hence defined by its underlying asset, the object used for ranking, the sampling windows, and corresponding lag terms. For example:

[ Asset = *Bond10Yr*; RankObj = *HF_Correl*; n = 20,40; Lag = 1, 4] means
2

For the resulting time series of returns, we calculate averages, standard deviations, and Sharpe ratios. To test for the statistical significance of the Sharpe ratios and decile differences, we use a bootstrap to estimate the *t*-values. For a given strategy, suppose we have a sample of *T* weekly observations of SB-Correl, the associated contemporary return, and the date. To estimate the *P*-value, we draw *N* = 10,000 samples of *T* observations (with replacement) from the empirical distribution. For each boostrap sample, we sort the observations sequentially according to their new date. We then use the defined SB-Correl-sorted decile strategy, re-establish a new portfolio, and calculate the average return for each decile. The Sharpe ratio’s *P*-value is
3

And the *t*-values are given by *t* = *N*
^{-1} (1 - *p*), where *N* is the *cdf* of the normal distribution with zero mean and unit variance.

We also use a simple regression to disentangle the correlation effect from other factors. We construct the factor portfolios in a similar manner to the SB-Correl-sorted portfolio and define curve spread as the difference between ten-year and one-year interest rates. In Equation (2), instead of letting RankObj=*HF_Correl,* we let RankObj=*Yield*, RankObj= –*Curve_Spread* and RankObj= –*VIX* respectively. We use the resulting time series of returns *R _{Yield_Momentum}
*,

*R*, and

_{Curve_Spread}*R*as the factors. By regressing the returns of the HF SB-Correl-sorted portfolio to the underlying bond returns—the equity returns as well as these factors—we control for possible systematic exposures: 4

_{VIX}where *R* is the HF SB-Correl-sorted portfolio’s returns; α is the adjusted alpha; and β_{Yield_Momentum}, β_{Curve_Spread}, and β_{VIX} are the estimated factor exposures.^{2}

Following Ilmanen [2003], Campbell, Pflueger, and Viciera [2013] and Johnson et al. [2013], we consider a potential structural break in January 1997. We also consider another potential structural break in December 2008, when the fed funds rate is essentially zero.

**EMPIRICAL RESULTS**

Exhibit 3 shows the correlation effect over the 10-year bond from 1997 to 2013. We choose a geometric series—[20, 40, 80, 160] weeks—as the sampling window.^{3} We use no lag terms. The 10-year bond underperforms exceptionally in the bottom deciles. D10 underperformed D1 by 32% on week one and by 13% per week on average from week 2 to week 8 (returns are annualized). But such decile dispersion is absent from week 9 to week 52. The simple strategy’s excess return and volatility are 3.9% and 5.9%, respectively. The Sharpe ratio is 0.66 and statistically significant.

The outperformance of the long-term (10-year) bond in D1 is related to a flight-to-safety phenomenon. Gulko [2002] found that, during stock market crashes, the correlation between U.S. stock and bond returns switches sign from positive to negative. Connolly, Stivers, and Sun [2005] ascribed the negative SB-Correls since 1997 to flight to safety, where increased stock market uncertainty induces investors to flee stocks in favor of the long-term bond and realized SB-Correls drop sharply.

On the flip side of flight to safety, we ascribe the short-lived underperformance of the bond in D10 to a desire to restore normality, where investors bid down the price of the safe long-term bond in times of increased SB-Correls, inducing corresponding negative returns for the long-term bond.

For each decile in Exhibit 3, we estimate the equity market beta by regressing the excess bond returns on week 1 to their contemporary equity returns. For D1 and D10, the stock betas are -0.14 and -0.05, while the excess bond returns are 19% and –13%, respectively. A plot of the results reported in Exhibit 3 would show that there is an inverse relationship between the 10-year bond’s equity betas and returns. Hence, the 10-year bond has higher return when it is less risky, if risk is measured by its time-varying equity betas.

As Dopfel [2003] warned, investors find a low SB-Correl beneficial in an asset-only context but detrimental when there is a bond-like liability, as a lower SB-Correl increases surplus risk. Our empirical findings suggest that a lower SB-Correl (D1) is beneficial for an asset-only investor. Not only does a lower SB-Correl indicate lower risk for a stock–bond portfolio, but it also forecasts higher bond returns in the short run. On the other hand, a lower SB-Correl hurts equity investors with a long-term bond-like liability. Not only does a lower SB-Correl increase surplus risk, but it also forecasts increased liability. Therefore, investors should take greater duration risk when the recent SB-Correls are lower, either with cash bond or with derivatives.

What about bonds with shorter maturities? We study the different persistence and magnitude of the correlation effect over the entire forward curve from 1985 to 2013. There are three sub-periods, divided by the two potential structural breaks (January 1997 and December 2008). D10 and D1 are less significant before 1997, especially at the long end of the forward curve. Before December 2008, D10 and D1 persistently forecast rising and falling interest rates, respectively, at the short end of the forward curve for about 52 weeks. The results are available online.

Therefore, in addition to the drive to restore normality, our strategy is also against the yield curve carry. As shown in Exhibit 3 on the curve spread (10Yr-1Yr), because of the highly persistent short-term interest rate movements over the following 52 weeks, in D1 the curve spread (ten year to one year) steepens from 1.04% to 1.65%, on average, in 52 weeks, while in D10 it flattens from 1.80% to 1.19%.

Compared with previous research, our findings have a few distinct features. First, instead of the transmission from stock market uncertainty to SB-Correls, we document the transmission from SB-Correls to the bond returns. Second, we document a pronounced “restore-to-normality” phenomenon in parallel to the “flight-to-safety” phenomenon. Third, we document an inverse relationship between the 10-year bond’s time-varying equity betas and returns. And finally, we document that a low SB-Correl forecasts a more steepened yield curve over the next year and our strategy is against yield curve carry (10Yr-1Yr).

**CONTROLLING FOR OTHER EFFECTS**

How does the correlation effect relate to other? Could it be that the correlation effect simply captures the yield momentum, curve spread, and stock market uncertainties?

To answer the question, we employ the same decile-based strategy to construct portfolios around a few factors: yield, curve spread, VIX. Similarly to Equation (2), the HF SB-Correl-sorted strategy and a factor portfolio are defined as

[Asset=*Bond10Yr*; RankObj=*SB_Correl* or *Factor*; n=20,40,80,160; Lag=1,2, 4,8]

From the sub-sample from January 1997 to September 2013, after correcting for these factors, the alpha for the HF SB-Correl-sorted strategy remains significant and the weekly alpha is 0.066%: 5

This translates to an annualized alpha of 3.4%. The yield momentum and VIX factor loadings are positive. The equity risk factor loading is also negative. The curve spread factor loading is negative, because our strategy is against yield-curve carry. These factor loadings’ signs are robust and consistent when using other plausible sampling windows and lag terms. The factor loadings of the curve spread and equity returns, which are valuable, are negative. In practice, the HF SB-Correl-sorted strategy can be used as an overlay for a conventional equity and bond portfolio.

For the 10-year bond, we find that the correlation effect is robust by using different sample window and quantiles. HF SB-Correls also have consistently greater predictive power than do the low-frequency SB-Correls. The exhibit is available online.

Given this analysis, we conclude that for the 10-year bond, the correlation effect holds up well after controlling for other classic factors.

For the one-year bond, much of the HF SB-Correl-sorted strategy’s excess returns are explained by the yield momentum, curve spread, and stock market uncertainties factors. The adjusted alpha diminishes. In addition, for the one-year bond, the HF SB-Correls’ predictive power is similar to that of low-frequency SB-Correls. This is not surprising, as the predictive power of HF SB-Correls over the short-term interest rate is moderate but long lasting, and using the most recent data is not an advantage. The test exhibits are available online.

**POSSIBLE EXPLANATIONS**

We first provide a possible explanation for the 1-year bond results by assuming negligible bond risk premia (BRP). We then provide a possible explanation for the 10-year bond results.

We assume that bond and stock prices are driven by two economic drivers: expected growth and inflation. SB-Correls are driven by the following:

a. Sensitivities to economic drivers. Higher expected inflation drives both bond and stock prices down; higher expected economic growth drives bond price down, but drives stock price up.

b. The relative volatility of the expected inflation and growth. SB-Correl is low when the uncertainty of growth dominates the inflation uncertainty. As shown in Exhibit 4, although growth is volatile for both the pre-1997 and post-1997 periods, the level-dependent inflation volatility has subsided since the 1990s. After 1997, SB-Correls are low, as growth uncertainties drive stock and bond in the opposite directions while the impact of inflation uncertainties is relatively muted.

c. Growth-inflation correlation. SB-Correl is low when growth and inflation are positively correlated. As Exhibit 4 shows, in the 1970s and 1980s, supply shocks moved inflation and growth in different directions, making bond returns pro-cyclical. But lately demand shocks make bond returns countercyclical.

How is that related to the predictive power of the SB-Correl over the one-year bond? A lower SB-Correl implies that growth uncertainties have a greater effect than do inflation uncertainties. This is valid for the entire period. Therefore, a lower SB-Correl implies that the central bank may ease the policy rate, because resisting a recession becomes the paramount policy priority, over resisting higher inflation. More importantly, after 1997, a lower SB-Correl implies positive growth–inflation correlation and possible deflationary recession risk. Therefore, a lower SB-Correl implies that the central bank may ease the policy rate as both the growth and inflation outlook require lower interest rates, especially since the 1990s.

In practice, a central bank moves short rates in small steps, so bankers can observe the consequences of their actions and assess the sequential incremental rate changes. The bond markets may forecast that further cuts are likely to follow the latest policy rate cut. In theory, this expectation will be built immediately into the short-term term structure. However, this term structure may not immediately reflect the economic conditions the SB-Correl implies; there will probably be a lag. This is either because the policy rate reacts to the implied economic conditions with a lag, or because the markets react to the future policy rates with a lag. Such underreaction may be more persistent since 1997, because of the positive growth–inflation correlation, and hence easier policy rate directional decision, and because of the increased persistence of monetary policy since 1997 (Campbell, Pflueger, and Viciera [2013]).

This is a possible explanation for the correlation effect on the one-year bond. How about the ten-year bond? Unlike the one-year bond, the BRP is pivotal for the ten-year bond. Following Ilmanen [2011, chapter 9] and in the spirit of Campbell, Sunderam, and Viciera [2013], we let *BRP* (safe haven) be an important component of the overall BRP. Such a safe-haven premium refers to long-term bonds’ equity and/or recession-hedging ability. If a lower SB-Correl reflects a greater ability for long-term bonds to hedge equity and recession risk, then it is possible that markets initially underreact to the bonds’ changing safe-haven premia, such that long-term bonds’ prices lag behind the SB-Correls.

Traditional efficient-markets thinking suggests that asset prices should completely and instantaneously reflect movements in underlying fundamentals. But such thinking need not conform to reality. We use the 10-year bond’s performance in 2013 as an example. As of April 19, 2013, the 10-year interest rate was 1.73%. Consider a 60/40 stock-bond portfolio as the market portfolio, and let the stock/bond weight, correlation, and volatility be

Then the marginal contribution to portfolio risk is

If the portfolio is the tangency portfolio whose Sharpe ratio is maximized, then

Let Exp Return of Equity = 5%, then Exp Return of Bond = -0.83%. Given the same volatility and same equity expected return, Exhibit 5 shows the mapping from SB-Correls to the bond’s expected returns.

As of June 21, 2013, the 10-year interest rate was 2.52%, and the realized HF SB-Correl increased to 0.12. For simplicity, let ρ = 0, and let other inputs be the same. Then according to Exhibit 5, Exp Return of Bond = 0.83%, which is about 1.66% higher than when ρ = -0.6 In this case, the bond yield has to be 1.66% higher instantaneously (without considering roll-down), which translates to an immediate and substantial bond price depreciation. But empirically, the yield only increases by (2.52% – 1.73% = 0.79%) from April 19, 2013, to June 21, 2013, much less than 1.66%. Subsequently, the 10-year interest rate climbs to 2.73% two weeks later and to 2.94% 11 weeks later, while the realized HF SB-Correls remain close to zero. It is possible that the markets learn about bonds’ evolving riskiness and safe-haven status and incorporate that information into bond prices and expected returns with a lag.

This is a hypothetical example for illustrative purpose only, because the real correlations, volatilities, and expected equity returns are unknown.

Why is the high-frequency SB-Correl a stronger and more persistent predictor of long-term bonds’ price after 1997? We provide two additional arguments.

First, quantitatively, the 10-year bond’s expected returns are more sensitive to SB-Correls when SB-Correls are already low. As Exhibit 5 shows, the bond’s expected return increases by 1.85% when ρ increases from -0.8 to -0.2, but it increases by only 1% when ρ increases from 0.2 to 0.8. The average HF SB-Correl is 0.35 before 1997 and is –0.30 after 1997—a substantial difference. On the other hand, the standard deviation of HF SB-Correl is similar. Therefore, after 1997, the changing SB-Correl is a greater driver of the bond’s theoretical expected returns. If the markets’ initial underreaction is roughly proportional to the changing bond’s theoretical expected returns, then after 1997, the changing SB-Correl is also a greater driver of the bond’s subsequent returns.

Second and qualitatively, after 1997, a bond’s safe-haven status matters more to the investors. We follow Kim and Wright [2005], who explicitly incorporate investors’ expectations.^{4} BRP has a secular downward trend, and it dips further into the negative territory in 2011 and 2012. This suggests that investors expect that the cumulative returns for holding cash are similar or may be even higher than that of holding a 10-year bond to maturity. According to Campbell, Sunderam, and Viciera [2013], in recent years and with BRP at such a depressed level, instead of an inflationary bet, investors regard the long-term bond as a deflationary (safe-haven) hedge. Hence, we think that long-term bonds’ prices are more sensitive to the change of *BRP* (safe haven) after 1997.

**CONCLUSIONS**

Asset correlations signal useful information from an interesting perspective about macroeconomic conditions. The connection between the bond–stock correlation and the state of the macro-economy should be of special interest to both investors and policymakers.

Using high-frequency data, we have shown that a lower stock–bond correlation forecasts the best of both return and risk for the 10-year bond. Not only does the 10-year bond have higher returns in the following weeks, but it is also less risky, if risk is measured by a bond’s equity market beta. Lower stock–bond correlation is thus beneficial to an asset-only investor, especially if the investor has leveraged bond positions. But lower stock–bond correlation is detrimental to an investor with long-term bond-like liability, especially if the investor invests aggressively in equities and leverage is not allowed. The reverse is true when the stock–bond correlation is higher.

In accordance with the predictive power of high-frequency stock–bond correlations, we develop simple decile portfolios for the 10-year and 1-year bonds, respectively. For the 10-year bond, the decile portfolio is characterized with significant alpha and negative loading to both the curve spread factor and the equity risk factor. It is thus a good overlay for a traditional stock and bond portfolio. For the 1-year bond, the decile portfolio has a high Sharpe ratio, but the excess returns are primarily attributed to the yield momentum, curve spread, and equity market uncertainties factors. For the 1-year bond, a possible explanation for the correlation effect is the markets or policy markers’ underreaction to the changing economic conditions implied by the stock-bond correlations. For the 10-year bond, a possible explanation of the correlation effect is the markets’ initial underreaction to the 10-year bond’s safe-haven status, which the stock–bond correlations imply.

## ENDNOTES

↵

^{1}From 1985 to 1987, SPTR is not available and we use the S&P 500 Index as the proxy.↵

^{2}We estimate the*t*-value of the alpha and betas using the closed-form formula for OLS instead of simulation.↵

^{3}We find that the empirical results would be much weaker if we use only past-week stock–bond correlations or other shorter windows. This is because short windows involve too much noise. The exhibit is available online.↵

^{4}Data available at http://www.federalreserve.gov/pubs/feds/2005/200533/200533abs.html Compared with the classic C-P BRP by Cochrane and Piazzesi [2005], a survey-anchored BRP may be less prone to possible unrealistic BRP estimates.

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