Both losses are quadratic in expected volatility (for small leverage), and expected volatility is linear in the number of days for a GBM.
Let me know if you have a reference to understand this better. :)
It doesn’t seem intuitive to me because with a leveraged ETF, you lose from volatility alone. In contrast, if you bought stocks on margin and those stocks went up (with whatever degree of volatility you want so long as it didn’t trigger a margin call), then you wouldn’t lose from volatility because you’re not buying and selling. Maybe you’re suggesting that the expected losses if you do have to make a margin call are big enough to make margin buying comparable to leveraged ETFs?
Another reason I’m confused is that almost everyone advises against long-term investing in leveraged ETFs, while it’s common to encourage long-term margin investing. So there must be a difference between them.
It doesn’t seem intuitive to me because with a leveraged ETF, you lose from volatility alone. In contrast, if you bought stocks on margin and those stocks went up (with whatever degree of volatility you want so long as it didn’t trigger a margin call)
You can exactly simulate a leveraged position using a leveraged ETF by increasing your position whenever the market goes down and decreasing it whenever the market goes up. If you think that price changes between periods are a martingale, then it’s pretty obvious that changing your investment ilike this won’t increase your net profits.
This strategy is only a good idea if the market is more likely to go up after it goes down. In practice, this has been true on some time scales, while the reverse is true on others. In theory, this shouldn’t predictably happen, because a savvy investor can make money. But whether or not you think this is true, there are more direct ways to earn money if you think that you can predict which way the market will move.
Note that one reason that this strategy looks good on paper is that it’s a thinly-veiled version of the martingale, taking place on a log scale. If you keep doubling down every time you lose, then you will either win a bit or you will lose everything. I.e., you can keep borrowing more to avoid a margin call, but if you do that it just makes an eventual margin call worse.
It’s pretty easy to do the math for a GBM and log utility. I don’t know a reference. An easy way to think about it is to consider a leveraged ETF that rebalances every day vs. every other day. Both have losses that depend on volatility. Which one experiences more volatility on average? The answer is that they are basically the same, because the volatility over two days is double the volatility over one day: half of the time the moves are in opposite directions and the volatility is 0, and half of the time they are in the same direction and the reallized volatility is 4x. Similarly if we go up to every 4 days, or every 8 days. Your leveraged positioan is just rebalancing every few years.
it’s common to encourage long-term margin investing
I haven’t encountered much of this (at least not at 2x leverage), most people caution against the risk of a margin call during a downturn.
It’s going to take me a while to grok all of your arguments. :)
You can exactly simulate a leveraged position using a leveraged ETF by increasing your position whenever the market goes down and decreasing it whenever the market goes up.
This is a key point that I don’t understand. It seems like the two are different, as the following example shows.
Leveraged ETF:
Buy a 2:1 leveraged ETF for $50. They borrow $50. They buy $100 in stocks. During the day, stocks increase 10%, so that the value is $110. To restore a 2:1 leverage ratio, the fund borrows another $10 and uses it to buy stocks, resulting in $120 of stocks and $60 of debt.
Regular margin investing:
Starting with $50, borrow another $50 to buy $100 in stocks. Stocks increase 10% to $110. The ratio of assets to debt is now 2.2:1.
If you had sold off some of the ETF to get assets down to $110, the leverage ratio would have remained 2, not 2.2. So how can buying/selling the ETF replicate regular leverage?
You care about your leverage debt ratio, not the leverage : debt ratio of a fund in which you have invested some of your money.
If you sell $5 of the leveraged ETF in your scenario, then you have exactly the same position as the margin investor, just it’s $55 in a 2x leveraged ETF instead of $110 of the equity. You have $5 in your pocket and $55 of debt from the ETF, rather than $0 in your pocket and $50 of debt. Of course, if you liquidated the whole position, you would just end up with $60 either way. If there is any difference, I don’t see it.
You have $5 in your pocket and $55 of debt from the ETF,
If you sell $5 of the ETF, it seems like the sold $5 would get rid of half equity and half debt, leaving $5 in your pocket and $115 in the ETF, of which $57.5 is debt.
Various articles seem to suggest a difference in performance characteristics between daily rebalancing vs. buy-and-hold. For example, Figure 3 on p. 10 of this paper.
After that day, the ETF is worth $60. If you sell $5 of it, you hold $55 of 2x exposure and $5 of cash, which is equivalent to what you would have if you had done no trading; on net you are long $110 of stocks and short $50 cash.
I think the mistake in your reasoning is saying that you ‘get rid of half equity and half debt’. You actually reduce your equity by 2x, your debt by 1x, and then receive 1x cash also.
Note that the paper you link to says several things that basically echo Paul:
“The above suggests that a leveraged ETF with a positive expected raw return but
negative expected growth, increasingly resembles a lottery ticket over time. As time
passes, the chances of the lottery player ending up with zero approach certainty, but the
payoff if he wins continues to increase to ensure that the lottery itself has favorable odds.”
“We find that
the expected raw return of the levered ETF is the highest simply because the investor
borrows at 2% to invest at 7.5%; however, the distribution of return outcomes is arguably
unattractive. The investor in a levered ETF achieves negative expected growth and also
has the lowest median portfolio return.”
As Paul pointed out, this is close to the opposite of the martingale strategy; in the martingale strategy you eliminate your losses with near-certainty (where how near depends on how much you can afford to keep doubling up) but your losses should you incur them get larger and larger, here you eliminate your gains with near-certainty but the gains get larger and larger.
Something that is explicitly ignored in that paper is the presence of fees and transaction costs. Daily leveraged ETFs do more trading than, say, one that rebalanced monthly would, because standard deviations scale with the square root of time. In the non-theoretical world, there are costs to this which should be considered, and they get bigger as the ETF(s) in question get bigger.
Finally, I would note that there is a surprising amount of nonsense written about ETFs online. I could point to much clearer-cut examples of incorrect or highly misleading statements.
After that day, the ETF is worth $60. If you sell $5 of it, you hold $55 of 2x exposure and $5 of cash, which is equivalent to what you would have if you had done no trading; on net you are long $110 of stocks and short $50 cash.
Thanks for this! Now I understand.
BTW, if you have time:
What’s your opinion on non-rebalancing leverage done manually?
What if you pay for the leverage interest using new income in a similar way as people pay off home loans? (This doc encourages such an approach.)
Let me know if you have a reference to understand this better. :)
It doesn’t seem intuitive to me because with a leveraged ETF, you lose from volatility alone. In contrast, if you bought stocks on margin and those stocks went up (with whatever degree of volatility you want so long as it didn’t trigger a margin call), then you wouldn’t lose from volatility because you’re not buying and selling. Maybe you’re suggesting that the expected losses if you do have to make a margin call are big enough to make margin buying comparable to leveraged ETFs?
Another reason I’m confused is that almost everyone advises against long-term investing in leveraged ETFs, while it’s common to encourage long-term margin investing. So there must be a difference between them.
You can exactly simulate a leveraged position using a leveraged ETF by increasing your position whenever the market goes down and decreasing it whenever the market goes up. If you think that price changes between periods are a martingale, then it’s pretty obvious that changing your investment ilike this won’t increase your net profits.
This strategy is only a good idea if the market is more likely to go up after it goes down. In practice, this has been true on some time scales, while the reverse is true on others. In theory, this shouldn’t predictably happen, because a savvy investor can make money. But whether or not you think this is true, there are more direct ways to earn money if you think that you can predict which way the market will move.
Note that one reason that this strategy looks good on paper is that it’s a thinly-veiled version of the martingale, taking place on a log scale. If you keep doubling down every time you lose, then you will either win a bit or you will lose everything. I.e., you can keep borrowing more to avoid a margin call, but if you do that it just makes an eventual margin call worse.
It’s pretty easy to do the math for a GBM and log utility. I don’t know a reference. An easy way to think about it is to consider a leveraged ETF that rebalances every day vs. every other day. Both have losses that depend on volatility. Which one experiences more volatility on average? The answer is that they are basically the same, because the volatility over two days is double the volatility over one day: half of the time the moves are in opposite directions and the volatility is 0, and half of the time they are in the same direction and the reallized volatility is 4x. Similarly if we go up to every 4 days, or every 8 days. Your leveraged positioan is just rebalancing every few years.
I haven’t encountered much of this (at least not at 2x leverage), most people caution against the risk of a margin call during a downturn.
It’s going to take me a while to grok all of your arguments. :)
This is a key point that I don’t understand. It seems like the two are different, as the following example shows.
Leveraged ETF: Buy a 2:1 leveraged ETF for $50. They borrow $50. They buy $100 in stocks. During the day, stocks increase 10%, so that the value is $110. To restore a 2:1 leverage ratio, the fund borrows another $10 and uses it to buy stocks, resulting in $120 of stocks and $60 of debt.
Regular margin investing: Starting with $50, borrow another $50 to buy $100 in stocks. Stocks increase 10% to $110. The ratio of assets to debt is now 2.2:1.
If you had sold off some of the ETF to get assets down to $110, the leverage ratio would have remained 2, not 2.2. So how can buying/selling the ETF replicate regular leverage?
You care about your leverage debt ratio, not the leverage : debt ratio of a fund in which you have invested some of your money.
If you sell $5 of the leveraged ETF in your scenario, then you have exactly the same position as the margin investor, just it’s $55 in a 2x leveraged ETF instead of $110 of the equity. You have $5 in your pocket and $55 of debt from the ETF, rather than $0 in your pocket and $50 of debt. Of course, if you liquidated the whole position, you would just end up with $60 either way. If there is any difference, I don’t see it.
Thanks for elaborating. :)
If you sell $5 of the ETF, it seems like the sold $5 would get rid of half equity and half debt, leaving $5 in your pocket and $115 in the ETF, of which $57.5 is debt.
Various articles seem to suggest a difference in performance characteristics between daily rebalancing vs. buy-and-hold. For example, Figure 3 on p. 10 of this paper.
After that day, the ETF is worth $60. If you sell $5 of it, you hold $55 of 2x exposure and $5 of cash, which is equivalent to what you would have if you had done no trading; on net you are long $110 of stocks and short $50 cash.
I think the mistake in your reasoning is saying that you ‘get rid of half equity and half debt’. You actually reduce your equity by 2x, your debt by 1x, and then receive 1x cash also.
Note that the paper you link to says several things that basically echo Paul:
“The above suggests that a leveraged ETF with a positive expected raw return but negative expected growth, increasingly resembles a lottery ticket over time. As time passes, the chances of the lottery player ending up with zero approach certainty, but the payoff if he wins continues to increase to ensure that the lottery itself has favorable odds.”
“We find that the expected raw return of the levered ETF is the highest simply because the investor borrows at 2% to invest at 7.5%; however, the distribution of return outcomes is arguably unattractive. The investor in a levered ETF achieves negative expected growth and also has the lowest median portfolio return.”
As Paul pointed out, this is close to the opposite of the martingale strategy; in the martingale strategy you eliminate your losses with near-certainty (where how near depends on how much you can afford to keep doubling up) but your losses should you incur them get larger and larger, here you eliminate your gains with near-certainty but the gains get larger and larger.
Something that is explicitly ignored in that paper is the presence of fees and transaction costs. Daily leveraged ETFs do more trading than, say, one that rebalanced monthly would, because standard deviations scale with the square root of time. In the non-theoretical world, there are costs to this which should be considered, and they get bigger as the ETF(s) in question get bigger.
Finally, I would note that there is a surprising amount of nonsense written about ETFs online. I could point to much clearer-cut examples of incorrect or highly misleading statements.
Thanks for this! Now I understand.
BTW, if you have time:
What’s your opinion on non-rebalancing leverage done manually?
What if you pay for the leverage interest using new income in a similar way as people pay off home loans? (This doc encourages such an approach.)