Since various folks have commented on the odd behaviour of
leveraged long and short ETF's I have decided to summarize various
past responses I have made and put them in a blog entry for easy
future reference. By the way, if you have any other "math" type
questions don't hesitate to ask, I love this stuff!
Consider an index trading at 100, a 200% long fund tracking the
index also trading at $100 and a 200% inverse fund trading at
$100.
Watch what happens in the following scenario: Over two days the
index goes to 110 (+10%) then back to 100 (-9.09%).
The double long fund would go to $120 (+20%) then $98.18
(-18.18%). Overall the fund is down 1.82%. The double short fund
would go to $80 (-20%) and then to $94.56 (+18.18%). Overall the
fund is down 5.44%
What if instead the index went to 90 (-10%) then back to 100
(+11.11%)?
The double long fund would go to $80 (-20%) then $97.78
(+22.22%). Overall the fund is down 2.28%. The double short fund
would go to $120 (+20%) then to $93.34 (-22.22%). Overall the fund
is down 6.66%.
To summarize:
a) The index is at the same point it started from.
b) Both funds lost money overall.
c) The amount of money they lost is path-dependent.
d) The short fund does worse under each scenario than the
long fund.
This effect becomes more exaggerated the larger the price moves
and the greater the leverage. One area I am not clear on is what
happens intraday. I would expect that arbitrage would result in
intraday compounding as well, further exaggerating these effects
during volatile trading.
So why would anyone want one of these funds? Because in a
trending market they are awesome delivering well over their
advertised 200% returns, and losing less than you would expect:
Consider an index that starts at 100, goes up 1% on day 1 and
down 0.5% on day 2 and so on for a total of 280 days: ending price
on the index = 100 x (1.01)^140 x (.995)^140 = 199.63 (+99.63%). A
double long fund starting at $100 would end up at 100 x (1.02)^140
x (.99)^140 = $391.70 (+291.70%). The double long delivered almost
300% return - 50% more than the advertized 200%!
Consider the same index this time losing 1% on day one and
gaining back 0.5% on day two. After 274 days the index stands at
100 x (.99)^137 x (1.005)^137 = 50 (-50%). The double long fund
starting at $100 would end up at 100 x (.98)^137 x (1.01)^137 =
$24.55 (-75.45%). The double long "only" lost half as much again as
the index rather than double the index' loss.
Combine these effects with correct risk-adjusted position sizing
(i.e. you only take a position half the size that you would have
done betting in the index unleveraged) and you can see you are
better off in trending markets with the leveraged funds.
Here endeth the lesson!
In case you are interested, the number of days I picked above
were calculated as follows:
days to double the index = 2 x log(2) / ((log(1.01)+log(0.995))
= 280
days to halve the index = 2 x log(0.5) /
((log(0.99)+log(1.005)) = 274
It's rare to find explanations as to how these funds work.
I've been surprised by how quick and steep the losses can be for
the inverse leveraged funds (if held too long at the wrong
time). After being bitten hard a few times, I tend to not
hold them for more than a day. Glad to hear that the long leveraged
funds do better over time. I'm holding some now.
Great study on Ultra ETFs.
The perils of ultra ETFs
Posted by coolhat on 9th of Dec 2008 at 08:34 pm
Great study on Ultra ETFs. I was wondering if you took dividends into account?
Why Ultra ETF's act strange
Posted by keithbob on 9th of Dec 2008 at 10:23 pm
From a member of another site, fwiw:
Since various folks have commented on the odd behaviour of leveraged long and short ETF's I have decided to summarize various past responses I have made and put them in a blog entry for easy future reference. By the way, if you have any other "math" type questions don't hesitate to ask, I love this stuff!
Consider an index trading at 100, a 200% long fund tracking the index also trading at $100 and a 200% inverse fund trading at $100.
Watch what happens in the following scenario: Over two days the index goes to 110 (+10%) then back to 100 (-9.09%).
The double long fund would go to $120 (+20%) then $98.18 (-18.18%). Overall the fund is down 1.82%. The double short fund would go to $80 (-20%) and then to $94.56 (+18.18%). Overall the fund is down 5.44%
What if instead the index went to 90 (-10%) then back to 100 (+11.11%)?
The double long fund would go to $80 (-20%) then $97.78 (+22.22%). Overall the fund is down 2.28%. The double short fund would go to $120 (+20%) then to $93.34 (-22.22%). Overall the fund is down 6.66%.
To summarize:
a) The index is at the same point it started from.
b) Both funds lost money overall.
c) The amount of money they lost is path-dependent.
d) The short fund does worse under each scenario than the long fund.
This effect becomes more exaggerated the larger the price moves and the greater the leverage. One area I am not clear on is what happens intraday. I would expect that arbitrage would result in intraday compounding as well, further exaggerating these effects during volatile trading.
So why would anyone want one of these funds? Because in a trending market they are awesome delivering well over their advertised 200% returns, and losing less than you would expect:
Consider an index that starts at 100, goes up 1% on day 1 and down 0.5% on day 2 and so on for a total of 280 days: ending price on the index = 100 x (1.01)^140 x (.995)^140 = 199.63 (+99.63%). A double long fund starting at $100 would end up at 100 x (1.02)^140 x (.99)^140 = $391.70 (+291.70%). The double long delivered almost 300% return - 50% more than the advertized 200%!
Consider the same index this time losing 1% on day one and gaining back 0.5% on day two. After 274 days the index stands at 100 x (.99)^137 x (1.005)^137 = 50 (-50%). The double long fund starting at $100 would end up at 100 x (.98)^137 x (1.01)^137 = $24.55 (-75.45%). The double long "only" lost half as much again as the index rather than double the index' loss.
Combine these effects with correct risk-adjusted position sizing (i.e. you only take a position half the size that you would have done betting in the index unleveraged) and you can see you are better off in trending markets with the leveraged funds.
Here endeth the lesson!
In case you are interested, the number of days I picked above were calculated as follows:
days to double the index = 2 x log(2) / ((log(1.01)+log(0.995)) = 280
days to halve the index = 2 x log(0.5) / ((log(0.99)+log(1.005)) = 274
I've wondered many times- Thanks!
Posted by snowcrow on 9th of Dec 2008 at 10:56 pm
It's rare to find explanations as to how these funds work. I've been surprised by how quick and steep the losses can be for the inverse leveraged funds (if held too long at the wrong time). After being bitten hard a few times, I tend to not hold them for more than a day. Glad to hear that the long leveraged funds do better over time. I'm holding some now.
ultras
Posted by 00paul on 9th of Dec 2008 at 11:25 pm
The time decay is real, like options you pay a premium for the supercharged Beta. Great work by-- unsane
No, I didn't. That would
Posted by unsane on 9th of Dec 2008 at 08:39 pm
No, I didn't. That would affect the short performance I guess. SKF had about 17% in the period and UYG hardly anything.