The variance risk premium of an asset is the difference between options-implied and actual expected return variation. It can be viewed as a price for hedging against variation in volatility. However, attitudes towards volatility are asymmetric: large upside moves are fine while large downside moves are scary. A measure of aversion to negative volatility is the downside variance risk premium, the difference between options-implied and actual expected downside variation of returns. It is this downside volatility risk that investors want to protect against and whose hedging price is a valid and apparently robust indicator of future returns. Similarly, the skewness risk premium, the difference between upside and downside variance risk premia, is also a powerful predictor of markets.

*The post ties in with SRSV’s lectures on implicit subsidies (particularly the part on volatility markets) and setback risk (particularly the part on exit risk).
*

*The below are excerpts from the paper. Emphasis and cursive text have been added.*

### The asymmetry of risk premia

“A fundamental relationship in asset pricing posits a positive relation between risk and asset returns…__This study highlights the importance of asymmetry in the assessment of risk__…Investors like good uncertainty as it increases the potential of substantial gains, but dislike bad uncertainty, as it increases the likelihood of severe losses…We define ‘good uncertainty’ and ‘bad uncertainty’ as volatility associated with positive or negative shocks to fundamentals such as dividend and consumption growth.”

“The **variance risk premium** [VRP] is defined as the __difference between risk-neutral [ options-implied] and physical expectations of returns variation__… We view the VRP as the premium that a market participant is willing to pay to hedge against variations in future realized volatilities.”

“The (total) variance risk premium lumps together market participants’ (asymmetric) views about good and bad uncertainties. As a result, __a positive (total) variance risk premium reflects the investors willingness to pay more in order to hedge against changes in bad uncertainty than for exposure to variations in good uncertainty__…a small positive VRP quantity does not necessarily imply a lower level of risk and/or risk aversion. Rather, it is an indication of a smaller difference between what agents are willing to pay for downside variation hedging versus upside variation exposure.”

“__We propose a new decomposition of the VRP in terms of upside and downside variance risk premia__…The asymmetric views of investors on good uncertainty [*point to*] proneness to upward variability versus bad uncertainty and aversion to downward variability.”

“Presence of asymmetry in physical and risk-neutral distributions gives rise to **skewness risk premium**…We construct a measure of skewness risk premium that closely resembles the variance risk premium. The skewness risk premium is defined as __the difference between risk-neutral and objective expectations of the realized skewness__. It can be shown that this measure of the skewness risk premium is the spread between the upside and downside components of the variance risk premium.”

### Calculation

“We need reliable measures for realized and risk-neutral variance and skewness.”

“We build the variance risk [*measures*]…in…three distinct steps: building the upside and downside realized variances, computing their expectations under the physical measure, and then doing the same under the risk-neutral measure.

- We construct the realized variance of returns on any given trading day as [
*the sum of a number of*] intraday squared log-returns…We add the squared overnight log-return and we scale the realized variance series to ensure that the sample average realized variance equals the sample variance of daily log-returns….We decompose equity price changes into positive and negative returns…__[__….By construction, the cumulative realized variance adds up the cumulative realized upside and downside variances.*We*] separate daily positive from negative quadratic variation using intraday data - To get genuine expected values for realized measures that are not contaminated by forward bias or the use of contemporaneous data,
__we perform an out-of-sample forecasting exercise to predict…realized variances, at different horizons [__[*from 1 to 24 months*]…based on the random walk model*assuming that future expected realized variance for a horizon will be equal to past realized variance over the same horizon backward*]. The difference between realized upside and downside variance can be perceived as a measure of (realized) skewness…To build this measure of skewness…we simply subtract downside variance from upside semi-variance. - To build the risk-neutral expectation of realized variances, we follow the methodology of Andersen and Bondarenko (2007)…[
*based on*] prices of European put and call options…[*which produces*] ‘risk-neutral semi-variances’…of returns. Traditional option-based estimates of asymmetry…are noisy. Our proposed risk-neutral skewness measure, in contrast, is well-behaved, easy to build, and easy to interpret.”

### Empirical findings

“We…compute the excess returns by subtracting 3-month treasury bill rates from log-differences in the S&P 500 composite index, sampled at the end of each month. Since our study requires reliable high-frequency data and option-implied volatilities, our sample runs from September 1996 to March 2016.”

“The variance risk premium is expected to be positive…and proportional to the volatility-of-volatility. We confirm [*this*]… Investors tend to hedge against downward movements to avoid losses…Conversely, investors often find upside movements desirable. They are willing to pay for exposure to such movements…__We find downside variance risk premium to be generally positive, reflecting the compensation required by an investor to bear the downside risk__, whereas upside variance risk premium is generally negative, as it is the discount conceded by an investor to secure exposure to such shocks…We… plot the time series of the VRP, its components, and the skewness risk premium in [*the figure below*].”

“__The VRP…is a robust predictor of asset returns at maturities of 3 to 6 months__. Because of its significant predictive power for short-term asset returns, the VRP is often viewed as reflecting investors’ appraisal of changes in near future volatility…Intuitively, the variance risk premium proxies the premium associated with the volatility-of-volatility, which not only reflects how future random returns vary but also assesses fluctuations in the tail thickness of future returns distribution.”

“[*We*] show that [*the VRP’s*] prediction power stems from the downside variance risk premium embedded in this measure…Empirically, __we demonstrate that the downside variance risk premium – the difference between option-implied, risk-neutral expectations of market downside variance and historical, realized downside variances – demonstrates significant prediction power__ that is at least as powerful as the variance risk premium, and often stronger for excess returns.”

“We also show that __the difference between upside and downside variance risk premia – our proposed measure of the skewness risk premium – is both a priced factor in equity markets and a powerful predictor of excess returns__. The skewness risk premium performs well for intermediate prediction steps beyond the reach of short-run predictors such as downside variance risk or variance risk premia and long-term predictors such as price-dividend or price-earning ratios alike. The skewness risk premium constructed from one month’s worth of data predicts excess returns from eight months to a year ahead. The same measure constructed from one quarter’s worth of data predicts monthly excess returns from four months to one year ahead. We show that __our findings demonstrate remarkable robustness to the inclusion of common pricing variables__. Downside variance risk and skewness risk premia have similar or better forecast ability in comparison with common predictors.”