# Is there a better expected return on investment than with ETFs accepting higher variance? If yes, what is the maximum?

My understanding (a bit oversimplified, ignoring inflation etc.) is:

I do nothing with my money: Variance is 0, expected return is 0.

I put my money in ETFs: Variance is positive, expected return maybe 6%.

Now my question is: If I am willing to increase the variance further, what is the maximum expected return?

One idea would be to use leverage by taking a credit (say with 4% interest rates) and use it to buy ETFs.

From other questions I learned that there are leveraged products for ETFs, however there (of course) also losses are leveraged, so it is not directly clear what the expected return is.

TLDR What is the maximum expected return of investment if the variance can be arbitrary? Is beating ETFs even (statistically / historically proven) possible accepting a higher variance? What is the expected return of leveraged ETFs (considering that also losses are leveraged)?

It matters a lot how you define expected return.

Note that if you start with say \$1,000 and repeatedly flip a fair coin to either gain or lose 1%, then the expectation (arithmetic mean) value of your future wealth is always \$1,000, but you are ultimately more likely to lose than gain overall. That is, the distribution of long-term returns is skewed, because returns compound multiplicatively rather than additively. Your median wealth after say 10,000 flips corresponds to gaining on 5,000 and losing on 5,000, i.e. \$1,000 x 1.01^5,000 x 0.99^5,000 = \$607, a 39% loss. You have a 50-50 chance of ending up on either side of \$607. This is also the geometric mean.

Thus, the "expected return" than is most meaningful is the expectation of log returns, which compound additively and thus have a much more intuitive long-term result. The expected log return for each coin-flip above is 0.5 x ln(1.01) + 0.5 x ln(0.99) = -0.005%. This can also be understood as the expected (arithmetic mean) utility, if utility is measured by the log of wealth, which conveniently captures the diminishing value of each additional dollar (a dollar provides more utility to a thousandaire than to a millionaire).

The idea that you can continue to increase expected return indefinitely by increasing risk is only valid for the arithmetic expectation, which as we've seen isn't very meaningful. For example, buying stocks with leverage increases the arithmetic expectation proportionally, but at the cost of an increasingly likely bust with large losses that you can't recover from. The expected log return captures what matters most to an investor, and turns out to be essentially the arithmetic return minus a penalty proportional to the variance. As you increase leverage, for example, the variance penalty (quadratic in leverage) soon grows faster than the arithmetic return (linear in leverage), so the expected log return declines.

Thus, variance is not just an emotional risk that you can "steel yourself" to and that will average out in the long run, but rather it has a direct mathematical effect on the most likely long-run outcomes.

According to efficient market theory, the highest possible expected log return is generated by the "market portfolio" consisting of all investable assets at market-cap weights, corresponding to a broad unleveraged index fund. Any deviation from this portfolio to seek a higher expected return would be pointless because the variance penalty outweighs any benefit.

• Thanks! Assuming I have 10.000\$ that I want to invest and assuming I can afford to lose all that money, would a leveraged ETF (by a factor of 2) be a valid choice that gives (roughly) 12% expected return (in the sense of arithmetic mean)? What specifically would be the downside? In particular, how would the median behave compared to the median of the normal ETF? Is it that simple that if the normal ETF rises by 5% I make 1.000\$ instead of 500\$ with the leveraged ETF compared to the normal ETF? And if it loses 50% I lose the entire 10.000\$? What happens if it loses more than 50%? Commented Feb 13, 2022 at 21:25