Note that with using leverage, there’s always margin calls, so calculating expected return becomes more complicated.
e.g. if you’re at 5x leverage, a 20% loss in the market will wipe you out, and 20% losses happen pretty frequently in stocks.
I’m not sure what’s the best quick and dirty way to adjust your expected return estimates to take account of this.
Maybe you could take your regular estimate (i.e. asset expected return * leverage ratio), then multiply it by the probability of wipe out in the relevant time period?
e.g. If you think the S&P has a 10% expected return, and you invest at 5x leverage for a year, the overall expected return is very roughly:
(1 - p ) 50% + p 0%
Where p is the probability of wipeout occurring in the period. In this case, p is the probability of a 20% loss occurring at some point in the next year.
In this case, it’s only worth investing with the leverage if p < 0.8, and that’s if you’re fully risk neutral. I’m not sure what p is, but it could easily be ~80%, which would mean using 5x leverage has lower expected returns than unleveraged investing. Which could help to explain why so few people use that much leverage.
Paul might have a better way of thinking about this!
If you buy index ETFs instead of stocks, the chance of a 20% loss is much smaller. (There have been few 20%-per-year losses in stock-market history.)
I wrote a simulation of 2X margin investing. I’ll be writing up a description of it more formally once it’s debugged and tweaked, but here are the current results for the present value of accumulated savings in US$ assuming an investor contributes $30K/year to buy an S&P 500 ETF at 2X margin over 30 years:
Mean regular = 937563
Mean margin = 1060336
0th percentile regular = 194161
0th percentile margin = 185719
10th percentile regular = 524319
10th percentile margin = 563046
25th percentile regular = 527721
25th percentile margin = 566873
50th percentile regular = 876835
50th percentile margin = 965868
75th percentile regular = 1212209
75th percentile margin = 1366938
90th percentile regular = 2042918
90th percentile margin = 2504675
100th percentile regular = 2042918
100th percentile margin = 2504675
The expected returns from leverage are much less than double ordinary returns but are still nontrivial.
Note that since I haven’t fully tested the program, these results may not be correct. :)
EDIT: A draft of the write-up about these resutls now here.
Impressive you only end up with 10% more over 30 years! Not much gain for far more risk. Explains why so few people invest with that much leverage.
Just eyeballing the data, −20% annual returns seem to occur about once every 20 years.
What’s the frequency of peak-trough losses of −20% though? That’s what actually matters for getting wiped out.
(And does your analysis take account of those? You’d need to be using daily data rather than annual to pick them up).
It also seems like ex ante returns should be lower than historical returns, because the last 50yr or so has in the US been unusually good for equities, and there are various reliable indicators that predict lower returns (e.g. Shiller PE).
The full write-up is now available here. Comments are welcome. The numbers have changed somewhat since before, and I trust them more now because they generally agree with theory.
I used 5.6% as the ex ante annual expected return and included black swans in my simulation. The simulation uses daily returns.
Note that with using leverage, there’s always margin calls, so calculating expected return becomes more complicated.
e.g. if you’re at 5x leverage, a 20% loss in the market will wipe you out, and 20% losses happen pretty frequently in stocks.
I’m not sure what’s the best quick and dirty way to adjust your expected return estimates to take account of this.
Maybe you could take your regular estimate (i.e. asset expected return * leverage ratio), then multiply it by the probability of wipe out in the relevant time period?
e.g. If you think the S&P has a 10% expected return, and you invest at 5x leverage for a year, the overall expected return is very roughly:
(1 - p ) 50% + p 0%
Where p is the probability of wipeout occurring in the period. In this case, p is the probability of a 20% loss occurring at some point in the next year.
In this case, it’s only worth investing with the leverage if p < 0.8, and that’s if you’re fully risk neutral. I’m not sure what p is, but it could easily be ~80%, which would mean using 5x leverage has lower expected returns than unleveraged investing. Which could help to explain why so few people use that much leverage.
Paul might have a better way of thinking about this!
If you buy index ETFs instead of stocks, the chance of a 20% loss is much smaller. (There have been few 20%-per-year losses in stock-market history.)
I wrote a simulation of 2X margin investing. I’ll be writing up a description of it more formally once it’s debugged and tweaked, but here are the current results for the present value of accumulated savings in US$ assuming an investor contributes $30K/year to buy an S&P 500 ETF at 2X margin over 30 years:
The expected returns from leverage are much less than double ordinary returns but are still nontrivial.
Note that since I haven’t fully tested the program, these results may not be correct. :)
EDIT: A draft of the write-up about these resutls now here.
Very interesting, thank you.
Impressive you only end up with 10% more over 30 years! Not much gain for far more risk. Explains why so few people invest with that much leverage.
Just eyeballing the data, −20% annual returns seem to occur about once every 20 years.
What’s the frequency of peak-trough losses of −20% though? That’s what actually matters for getting wiped out.
(And does your analysis take account of those? You’d need to be using daily data rather than annual to pick them up).
It also seems like ex ante returns should be lower than historical returns, because the last 50yr or so has in the US been unusually good for equities, and there are various reliable indicators that predict lower returns (e.g. Shiller PE).
Thanks for those caveats. :)
The full write-up is now available here. Comments are welcome. The numbers have changed somewhat since before, and I trust them more now because they generally agree with theory.
I used 5.6% as the ex ante annual expected return and included black swans in my simulation. The simulation uses daily returns.
Over what period are you measuring these drawdowns? I can look it up for you.