# 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?

• The strange thing is that the Jacquier, Kane and Marcus article you linked never mentions negative returns AFAIK. Although it does give the formula you also mention. Oct 24 at 18:32
• @noob2 Yeah, I thought the same thing! I would have expected them to mention it as a corrollary, but they only mention that it can go below the geometric average. Oct 24 at 18:39
• inserted working link Nov 10 at 7:52

This is a mathematical fact but imo there's not much to square with intuition, without appropriate fractional sizing (kelly) volatility is guaranteed to eat away returns over very long time horizons but people are not investing over time horizons where this effect will materialise (though volatility drag is still a thing.)

• So, if I just buy some SPY and pass that down to my descendants, eventually it won't be worth anything? My intuition at least is that it would get more valuable over time, even very long time horizons. Nov 8 at 19:35
• @KalevMaricq, no, it may well be worth something, but your geometric average return is likely to be less than one because down 50% up 50% is 75% not 100 (volatility drag).
– TKLF
Nov 9 at 17:54
• Ok, I guess we have different intuitions about it. To me it seems more likely to have a positive average return, but if you don't have that intuition then you're right that there's nothing to square. Nov 12 at 17:59

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

• I think the issue here is that the claim is that this holds true in a world with risk premia too. That (1-k)*(Arithmetic Average return)+k*(Geometric Average return) will be negative for large enough k, no matter how high the average returns are due to risk premia, because arithmetic>geometric when variance>0. Nov 12 at 18:03
• Now the article is available (and read), I get the point. And they're right that using arithmetic is pos-biased while using geometric is neg-biased (for estimates of arithmetic returns). However, the precise highlighted claim is patently false. Imagine an asset that has a 50% chance of a zero 0 and a 50% chance of +20%. That has a positive variance... but how then can the expected payoff be less than zero? ;-) Nov 13 at 21:18
• ps the bit I might have missed in passing (because I'm not sure I accept their sampling methodology, ie your second quote) is that a geometric measure is an unbiased estimator of future geometric returns (in which case the arithmetic is just e^that - 1). Nov 13 at 21:22
• I like your example. If it isn't true then the question is: where is the mistake in the logic? As for the geometric average being an unbiased estimator of geometric returns, the paper seems to say that it is biased, unless the sample period = the horizon. Do you disagree with that? Nov 14 at 1:01

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.