The correlation risk premium is a premium for uncertainty of future correlation of securities among each other or with a benchmark. A rise in correlation reduces diversification benefits. The common adage that in a crash ‘all correlations go to one’ reflects that there is typically not much diversification in large market downturns and systemic crises, except through outright shorts. Correlation risk premia can be estimated based on option prices and their implied correlation across stocks. There is evidence that these estimates are useful predictors for long-term individual stock performance, over and above the predictive power of variance risk premia.

*The post ties in with SRSV’s summary lecture on implicit subsidies, particularly subsidies or premia being paid in equity markets.
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*The below are quotes from the paper. Emphasis and cursive text have been added.*

### Understanding the correlation risk premium

“Correlation plays an important role in determining longer-term risks [*because it reduces*] diversification benefits…__Correlation [ is] a ‘no-place-to-hide’ state variable__, which predicts risks and returns at longer horizons compared to variance.”

“The variance of the aggregate market return is stochastic, and investors are ready to pay a premium to hedge against changes in variance, the variance risk premium, which, to a large extent, serves as compensation for bearing jump risk. Correlations between individual stocks are also time-varying and, __by pricing index options using relatively higher expected variance than for individual options, investors are also willing to pay a correlation risk premium__ to hedge against changes in correlation.”

“The ex-ante variance risk premium [*premium for a forward horizon*] for options with maturity [*of n days*] can be [*estimated*] as the implied variance at the end of day t minus the realized variance from t -n to t…The ex-__ante correlation risk premium [ premium for a forward horizon] is then constructed as the difference between the implied correlation for options with maturity [of n days] observed at the end of day t and the corresponding realized correlation__ from t-n to t.”

“Focusing on the S&P500…the realized variance is, on average, higher than the implied variance for individual stocks. In contrast, the __variance risk premium for the S&P500 itself is always positive, and statistically significant__…variance risk premiums for individual stocks in the S&P500 demonstrate a lot of heterogeneity. That is, while for a majority of the stocks we fail to reject the null hypothesis of an insignificant variance risk premium, there is still a sizeable fraction of stocks for which we can either reject the null of a positive or a negative variance risk premium.”

“Both, aggregate __index variance and average correlation, are co-moving negatively with the market return__, that is, they tend to increase during bear markets, and, hence, should contribute to the equity risk premium….Expected correlation predicts future diversification risks well.”

### How to measure the correlation risk premium

“In a simple model…both the variance and correlation risk premium should contribute to the equity risk premium…We decompose the equity risk premium into three components: (i) the variance risk premium; (ii) the correlation risk premium; and (iii) an orthogonal component. This ‘beta representation’ allows us to derive a theoretically founded forecasting equation for the market excess return.”

“Traditionally, one would simply…regress realized market excess returns on lagged regressors using historical data. The estimated betas could then, together with the variance and correlation risk premium, be used to predict the future market excess return.”

“We propose a new methodology for estimating ‘contemporaneous betas’ by the joint dynamics of market returns and option-implied variables…[*Based on options pricing*] one can obtain, on each day, implied variances and correlations, which are the risk-neutral expected integrated variance and correlation until option maturity. .. Integrated expected variance and integrated expected correlation are highly persistent [*overtime*]…We demonstrate that the __contemporaneous betas…can be estimated from increments of risk-neutral quantities, that is, [options-] implied variance and implied correlation__. This separate ‘estimation equation’ allows us to use up-to-date daily data to estimate the variance and correlation betas, which substantially improves the out-of-sample performance.”

### The predictive power of the correlation risk premium

“Our analysis focuses on three major U.S. indices, and their constituents, namely, the S&P500, the S&P100, and the DJ Industrial Average (DJ30) for a sample period from January 1996 to April 2016…For the option-based variables, we rely on the Surface File from OptionMetrics, selecting for each index and its constituents options with 30, 91, 182, 273, and 365 days to maturity and an (absolute) delta smaller or equal to 0.5.”

“__Expected correlation…has a strong predictive power for future diversification benefits__ for horizons of up to one year, measured by the average future correlation or by the non-diversifiable portfolio risk.”

“Can the variance and correlation risk premiums predict the market excess return out-of-sample?…The variance and correlation risk premium predict the market excess return out-of-sample, with out-of-sample R-squares [*coefficients of determination*] of up to 10% at a quarterly, and up to 8% at an annual horizon… __the correlation risk premium, inferred from major U.S. stock indices, is able to predict market excess returns in-sample and out-of-sample at horizons of up to one year__.”

“Does the correlation risk premium provide non-redundant information, relative to the variance risk premium, in predicting market excess returns? While the predictability by the variance risk premium peaks at the quarterly horizon and declines after that, the predictive power of the correlation risk premium is strongest for long horizons up to 1 year. That is, we provide strong empirical evidence for the existence of two option-implied components in the equity premium that contain non-redundant information, with the __predictability stemming from the variance risk premium being far more short-lived than that of the correlation risk premium__. These predictability results imply highly significant economic benefits for a representative investor, and crucially depend on the use of the novel beta estimation methodology.”