The long-run relationship between market risk and return
Bibliographic record
Abstract
Many finance applications require an annual measure of the market premium for equity. Using a long sample combined with a very parsimonious conditional variance function, we find a positive relationship between market risk and expected excess returns. Unlike traditional exponential-smoothing filters, our specification has a well-defined unconditional variance and allows for mean reverting volatility forecasts. Although total volatility is significantly priced, the smooth long-run component in volatility is more important for capturing the dynamics of the premium. This parameterization produces realistic time-varying market equity premium estimates over the entire 1840-2003 period. For example, our results show that the premium was relatively low in the mid-1990s but has recently increased. Results are robust to univariate specifications that condition on either levels or logs of past realized volatility (RV), as well as to a new bivariate risk-return model of returns and RV for which the conditional variance of excess returns is the conditional expectation of the realized volatility process.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".