Source: Bollerslev, T., Tauchen, G. & Zhou, H. (2009) · Review of Financial Studies 22(11), 4463–4492 · DOI 10.1093/rfs/hhp008
TL;DR
Defines the variance risk premium (VRP) — the difference between the risk-neutral expected variance (implied by index options, the model-free VIX-type measure) and the actual (realized) variance — and shows it predicts aggregate market excess returns, with predictability strongest at a quarterly horizon. In the full-text version analyzed here (the FEDS/working-paper draft by Bollerslev & Zhou over 1990Q1–2005Q1), the VRP alone explains an adjusted R² of about 15% of quarterly excess returns, dominating the dividend yield, P/E, default spread, and CAY; combined with the P/E ratio the R² rises to about 26%.
Problem it solves
Most documented return predictors (valuation ratios, term-structure variables, CAY) work best over long, multi-year horizons and weakened in the late 1990s. The VRP supplies a predictor with strong power at the short-to-medium (one-quarter) horizon, linking the options/volatility market to equity-premium predictability.
The method
Risk-neutral variance: model-free implied variance from a cross-section of index options (squared VIX), not Black-Scholes implied vol.
Realized variance: sum of high-frequency intraday squared returns.
VRP = implied (risk-neutral) variance minus realized variance; regress future market excess returns on the lagged VRP at various horizons, with Newey-West (1987) robust t-statistics (4 lags).
A consumption-based equilibrium model with time-varying economic uncertainty (volatility-of-volatility) rationalizes why the VRP forecasts returns.
Assumptions & inputs
Requires liquid index options for the model-free implied variance and intraday data for realized variance — the paper stresses that both "model-free" implied variances and high-frequency realized variances are essential; using Black-Scholes or daily data destroys the result.
Sample is short (VIX available only from 1990), so long-horizon claims cannot be tested.
How to use it
Headline regressions (1990Q1–2005Q1): the VRP alone gives quarterly adjusted R² ≈ 15.1%; VRP + P/E ≈ 26.4%; the best multi-predictor combination ≈ 27.7%. Slope coefficients on the variance difference are stable (≈ 0.87–1.11) with robust t-statistics ≈ 3.85–5.27.
Predictability peaks at the quarterly horizon and is much weaker at the monthly horizon.
As a market-timing/equity-premium signal: high VRP forecasts high subsequent returns (compensation for variance/jump risk and time-varying risk aversion).
Limitations & pitfalls
VRP estimation is sensitive to the implied- and realized-variance methodology and option-data quality.
Predictability is horizon-specific (quarterly), with a short sample and overlapping-return concerns.
The mechanism (risk vs. time-varying risk aversion) is identified only through a structural model, not directly.
Key references
Bollerslev, T., Tauchen, G. & Zhou, H. (2009) — Expected Stock Returns and Variance Risk Premia — Review of Financial Studies
Carr, P. & Wu, L. (2009) — Variance Risk Premiums — Review of Financial Studies
Bekaert, G. & Hoerova, M. (2014) — The VIX, the Variance Premium and Stock Market Volatility — Journal of Econometrics
Provenance: verified/generated from the paper's full text.
