Risky Assets, Variance Drag, and the Role of Risk Premia
Summary
The document examines the claim that any asset with positive volatility must have a negative expected return over a sufficiently long horizon. The argument applies a relation between arithmetic and geometric averages, reasoning that variance drag eventually dominates as the forecast horizon grows. The responses distinguish this effect from the compensation investors may receive for bearing risk.
In a setting with no risk premium, volatility can reduce expected compounded growth, consistent with the claim’s intuition. In actual financial markets, however, expected returns on risky assets may include a premium large enough to offset that drag. One answer describes the drag as related to half the variance under a normal-return assumption, but offers only a rough comparison and does not derive the horizon formula or establish a universal premium. A second response frames risk as deviation around a zero expected return. The discussion therefore challenges the blanket conclusion without proving that every risky asset has positive long-run growth; outcomes depend on assumptions about return distributions and risk compensation.
Key ideas
- Volatility can reduce expected compounded growth through variance drag.
- Without a risk premium, the argument for negative expected growth can hold.
- Risk premia may offset variance drag in financial markets.
- The discussion does not prove that every risky asset has positive long-run returns.
- The variance-drag approximation depends on assumptions about the return distribution.
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Full text
# Do all risky assets have negative expected return over long enough time horizons? # Do all risky assets have negative expected return over long enough time horizons? I stumbled across a site that claims > "given a long enough forecast horizon H, all assets with positive volatility have an unbiased expected return that is negative". They base this on the formula: > Expected log return over Horizon periods = (1- Horizon/Sample)*(Sample Arithmetic Avg) + Horizon/Sample *(Sample Geometric Average) (which I found comes from this paper) and the logic that since Arithmetic>Geometric, for large enough Horizon, this will become negative. I can't find a flaw in their logic, but this seems to run counter to our intuitions about the stock market. It implies that over a long enough time horizon, the market portfolio has a negative expected return, which also implies that it will approach zero. Do all risky assets have negative expected returns as the time horizon approaches infinity? If not, what's the flaw in the argument? If so, how do you reconcile that with our experience and intuitions about risky assets? ## Answer by demully (score 2) https://quant.stackexchange.com/a/68700 Short answer [the link/URL doesn't work]. In a world without risk premia, this logic would be correct. Variance drag would cause all volatile risk assets to have negative expcted returns. Which is precisely why most financial markets apply a discount rate to risky assets that more-than compensates for these Kelly Betting problems. Even if we took the argument at face value, betting/investing in fractional-size would still be profitable. Let alone start to play games like "find me any 20 year period where investing in the S&P500 or MSCI World has been un-profitable after dividends". Put simply, the vol dynamics are right. But the risk-reward of risk assets tends to have an average that blows this argument out of the water ;-) Variance drag costs half sigma sqaured (assuming a normal distribution). If sigma "costs" c.25% of sigma (ie Sharpe Ratio of 0.25% is the price of risk-taking). then this becomes irrelevant rather quickly ;-) best, DEM ## Answer by user7056 (score 0) https://quant.stackexchange.com/a/68531 The risk measures the deviation from the zero expected return (no free lunch). If the asset produces a profit of +p or a loss of -p, it is equally risky.
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