This question assumes an investor with a very low risk-aversion coefficient. The risk-aversion coefficient is basically the 2nd-derivative of an investor's utility function as a function of money. An investor that desires to increase their median wealth as rapidly as possible and that has no other income or expenses other than their investments has a utility function which is the log function.

For such a hypothetical investor, what is the leverage ratio that is rebalanced daily that would produce the maximum average median return for the S&P500 index based on past performance? In other words, what leverage ratio if held constant since the inception of the S&P 500 to the present, would provide the maximum median return on an initial investment where all dividends are being reinvested?

How would the answer differ if you only looked at the volatility of the last 20 years or last 10 years? Is the ideal ratio tending in any particular direction over time, that is, has volatility changed over time and is it more or less volatile today?

There are various 2x and 3x funds out there, but they seem to suggest in their prospectuses that they are only for short term holding and thus imply that any such ideal ratio is probably less than 2x. One could potentially approximate whatever this ideal ratio is by combining 1x and 2x and adjusting ones positions across the two to maintain the ideal ratio. Perhaps the ideal ratio is even less than 1x? Perhaps there is even a fund out there that tries to approximate an ideal ratio?

For example, imagine a hypothetical investment where each day you have 50% chance of quadrupling your investment and 50% chance of losing it all. Is this a good investment, absolutely, would you use a 1.0 leverage ratio, definitely not, it is way to volatile. If I recall correctly, you would choose a 0.25 leverage ratio to maximize your median return using this investment, not your mean return. In investing, a mean return is much less important than your median return, you only get once chance at retirement. For example, if you had a 1% chance of making a trillion dollars and a 99% chance of having nothing, the mean is huge, but the median is zero. Based on the volatility of the S&P500, I'm asking what is the ideal ratio for it to maximize your median return. I think it is clear that the answer is not 1.0 as that would be too coincidental.

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    I would like to know more about this too, for the S&P500 is there a mathematical formula towards how big x-leverage you should take. For sure theres an optimal way here depending on how much a certain s&p500 swings. Im new to the site but are you able to send private messages over this site? Maybe WilliamKF found something about this, if so it would be really nice to hear about it! an infinite leverage would obviously not work since if it dropped 0.001% one day you would be infinitely broke.
    – user51708
    Dec 23, 2016 at 15:26
  • @MT1984 I agree, but it seems the folks here here don't really understand my question it being downvoted a lot and with the answer I got and being upvoted so much. Perhaps you see a way to clarify my question so we can can get a real answer that is a function of volatility?
    – WilliamKF
    Dec 24, 2016 at 3:25
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    Why was this question down-voted? Perfectly acceptable question.. Mar 9, 2020 at 6:09
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    @NicSzerman I suspect the six down-voters don't understand what I'm asking. I tried to get them to understand back in 2012 but failed. Perhaps you can suggest an edit to the question that would make it clearer? The only answer, voted up high appears to my sense to be mathematically incorrect. But again, i failed to convey that too and many people think infinite leverage is the answer although to my sense obviously incorrect.
    – WilliamKF
    Mar 9, 2020 at 18:00
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    The most liked answer is partially correct. What they mean with long term and short term doubled returns is right but it doesn't capture the essence of the error caused by the timing of the rebalance (which is that fact that rebalancing is not done immediately but once per day - doing it immediately is practically impossible though). It's also wrong in failing to explain that those who are lending money prefer to have lower volatility than the volatility of the optimal risk, and in exchange others can have higher returns with higher volatility without necessarily assuming greater risks. Mar 11, 2020 at 3:53

1 Answer 1


The reason that UltraLong funds and the like are bad isn't because of the leverage ratio. It's because they're compounded daily, and the product of all the doubled daily returns is not mathematically equivalent to the double the long-term return. I'd consider providing big fancy equations using uppercase pi as the 'product of elements in a sequence' operator and other calculus fanciness, but that would be overkill, I don't think I can do TeX here, and I don't know the relevant TeX anyway.

Anyway. From the economics theory perspective, the ideal leverage ratio is 1X - that is, unlevered, straight investment. Consider: Using leverage costs money. You know that, surely. If someone could borrow money at N% and invest at an expected N+X%, where X > 0, then they would. They would borrow all the money they could and buy all the S&P500 they could. But when they bought all that S&P500, they'd eventually run out of people who were willing to sell it for that cheap. That would mean the excess return would be smaller. Eventually you'd get to a point where the excess return is... zero?

.... well, no, empirically, we can see that it's definitely not zero, and that in the real world that stocks do return more than bonds. Why?

Because stocks are riskier than bonds. The difference in expected return between an index like the S&P500 and a US Treasury bond is due to the relative riskiness of the S&P500, which isn't guaranteed by the US Government to return your principal.

Any money that you make off of leverage comes from assuming some sort of a risk. Now, assuming risk can be a profitable thing to do, but there are also a lot of people out there with higher risk tolerance than you, like insurance companies and billionaires, so the market isn't exactly short of people willing to take risks, and you shouldn't expect the returns of "assuming risk" in the general case to be qualitatively awesome.

Now, it's true that investing in an unlevered fashion is risky also. But that's not an excuse to go leveraged anyway; it's a reason to hold back. In fact, regular stocks are sufficiently risky that most people probably shouldn't be holding a 100% stock portfolio. They should be tempering that risk with bonds, instead, and increasing the size of their bond holdings over time.

The appropriate time to use leverage is when you have information which limits your risk. You have done research, and have reason to believe that you understand the future of an individual stock/index better than the rest of the stock market does. You calculate that the potential for achieving returns with leverage outweighs the risks. Then you dump your money into the leveraged position. (In exchange for this, the market receives information about anticipated future returns of this instrument, because of the price movement which occurs as a result of someone putting his money where his mouth is.)

If you're just looking to dump money into broad market indicies in a leveraged fashion, you're doing it wrong. There is no free money.

(Ed. Which is not to say there's not money. There's lots of money. But if you go looking for the free kind, you won't find it, and may end up with money that you thought was free but was actually quite expensive.)

Edit. Okay, so you don't like my answer. I'm not surprised. I'm giving you a real answer instead of a "make free money" answer. Okay. Here's your "how to make free money" answer.

Assume you are using a constant leverage ratio over the length of time you've invested your money, and you don't get to just jump into and out of the market (that's market-timing, not leverage) so you have to stay invested. You're going to have a scenario which falls into one of these categories:

  • Over the course of your investment, the S&P500 outperforms the interest rate you pay to borrow money. Ideal leverage ratio: Infinite. Borrow all the money you can to buy all the stock you can (as an individual, I assume you're still not going to be dealing with enough money to substantially move markets.)
  • The S&P500 does not outperform your interest rate, but it outperforms your bank account. Ideal leverage ratio: 1X. Invest the cash you have and don't borrow any.
  • The S&P500 does not outperform your bank account, but its performance is flat or declines slightly. The magnitude of the rate of decline is less than your interest rate to borrow money. Ideal leverage ratio: 0. Keep your money in the bank account, earn interest on that.
  • The S&P500 declines, and the decline is larger than the rate you pay to borrow money. Ideal leverage ratio: Negative infinity. Borrow all the stock you can, get money, put money in bank account to earn a little interest on the side.

The S&P500 historically rises over time. The average rate of return probably exceeds the average interest rate. So the ideal leverage ratio is infinite. Of course, this is a stupid answer in real life because you can't pull that off. Your risk tolerance is too low and you will have trouble finding a lender willing to lend you unsecured money, and you'll probably lose all your money in a crash sooner or later. Ultimately it's a stupid answer because you're asking the wrong question. You should probably ask a better question: "when I use leverage to gain additional exposure to risk, am I being properly compensated for assuming that risk?"

  • I don't feel like this has answered the question. I expect there is a mathematical formula for which you can input the average volatility of the S&P500 and come up with a leverage ratio which may be less than one or more than one. I doubt it exactly equals one. Using this ratio you would maximize your return based on historical performance.
    – WilliamKF
    Sep 26, 2012 at 12:56
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    WilliamKF - I have amended my answer to conform to your expectations. In summary, if you want the best return based on historical performance, you need infinite leverage. Unfortunately, that's a stupid answer. I blame your expectations.
    – user296
    Sep 26, 2012 at 17:46
  • To my understanding, infinite leverage would be incorrect since if the market ever went down, even an infinitesimal amount, your investment value would then be zero or negative and no upside, no matter how leveraged, helps a zero investment grow beyond zero.
    – WilliamKF
    Sep 27, 2012 at 13:19
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    "if the market ever went down, even an infinitesimal amount" -- well, if you compound daily like an UltraLong ETF, yes. but that's the wrong way to do it. anyway, I'm done here. this transmission out.
    – user296
    Sep 27, 2012 at 17:00

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