I don’t know enough about the etiquette here, but I am having to fight the urge to post a bunch of memes along the lines of “It’s not the bull market, I really am a genius”.
I would strongly advise anyone who’s considering following this to consider doing this with considerably less than their whole portfolio and with much lower expectations than 7x’ing your money.
A lot of people respond to things like this post by assuming that the author was lucky. This is usually correct, at least when applied to random claimants on the interwebs, but we can put some bounds on it: the higher the returns, the more luck required to achieve them, assuming an efficient market. The efficient markets model says that any strategy during this period had expected return of 50%. So, if the post author used a strategy with probability p of achieving 600% returns, and 1-p of losing everything, then efficient markets implies p*(100+600) + (1-p)*0 = (100+50), i.e. p = 0.21 (roughly). This is the highest probability any strategy could have of achieving 600% returns during this period, without exploiting any market inefficiency.
In other words: at most one-in-five people could achieve returns that high without exploiting market inefficiency. It should therefore probably update us nontrivially away from the possibility that the post author just got lucky. (Though depending on your priors, you might still think the post author got lucky.)
Everyone else has already pointed out that you misunderstood what EMH states, so I wont bother adding to their chorus. (Except to say I agree with them).
I will also disagree with:
at most one-in-five people [...] It should therefore probably update us nontrivially away from the possibility that the post author just got lucky.
1 in 5 isn’t especially strong evidence. How many of the other 5 people would you expect to be publishing on the internet saying “You should trade stocks”.
I agree this isn’t a very strong argument. I think theoretically we can probably get a much tighter probability bound than 20% by looking directly at the variance of my strategy, and concluding that given that variance, the probability of getting 600% return by chance (assuming expected return = market return) is <p for some smaller p. But in practice I’m not sure how to compute this variance. Intuitively I can point to the fact that my portfolios did not have very high leverage/beta, nor did I put everything into a single or very few highly volatile stocks or sectors, which are probably the two most common high variance strategies people use. (Part of the reason for me writing this post is that while LW does have a number of people who achieved very high investment returns, they all AFAIK did it by using one of these two methods, which makes it hard to cite them as exemplifying the power of rationality.)
Assuming the above is still not very convincing, I wonder what kind of evidence would be...
Without even checking, I can think of a bunch of assets which 7x’ed since Jan 2020. (BTC/general crypto, TSLA, GME/AMC etc). So yes, I agree this depends on the portfolio you ran.
Personally, I have seen enough people claiming to outperform, but then fail to do so out of sample. (I mean, out of sample for me, not for them) for me to doubt any claim on the internet without a trading record.
Either way, I think it’s very hard to convince me with just ~1.5 years of evidence that you have edge. I think if you showed me ~1k trades with some sensible risk parameters at all times, then I could be convinced. (Or if in another year and a half you have $300mm because you’ve managed to 7x your small HF AUM, I will be convinced).
Without even checking, I can think of a bunch of assets which 7x’ed since Jan 2020. (BTC/general crypto, TSLA, GME/AMC etc).
I figured that’s the first thing someone would think of upon hearing “7x” which is why I mentioned “This was done using a variety of strategies across a large number of individual names” in the OP. Just to further clarify, I have some exposure to crypto but I’m not counting it for this post, I bought some TSLA puts (forgot whether I made a profit overall), and didn’t touch AMC. I had a 0.1% exposure to some GME calls which went to 1% of my portfolio and that’s the only involvement there.
Personally, I have seen enough people claiming to outperform, but then fail to do so out of sample.
Can you please give some examples of such people? I wonder if there are any updates or lessons there for me.
Either way, I think it’s very hard to convince me with just ~1.5 years of evidence that you have edge. I think if you showed me ~1k trades with some sensible risk parameters at all times, then I could be convinced.
I don’t think I’ve done that many trades (depending on how you define a trade, e.g., presumably accumulating a position across different days doesn’t count as separate trades). Maybe in the low hundreds? But why would you need ~1k trades to verify that I was not doing particularly high variance strategies? I guess this is mostly academic though, as it would take a lot of labor to parse my trade logs and understand the underlying market mechanics to figure out what I was doing and how much risk I was taking (e.g., some pair/arbitrage trades were spread across several brokers depending on where I could find borrow). I don’t supposed you’d actually want to do this? (I also have some privacy concerns on my end, but maybe could be persuaded if the “value added” in doing this seems really high.)
Or if in another year and a half you have $300mm because you’ve managed to 7x your small HF AUM, I will be convinced
I’m definitely not expecting such high returns going forward. (“600% return” was meant to be Bayesian evidence to update on, not used to directly set expectations. I thought that went without saying around here...) Obviously there was a significant amount of luck involved, for example as I mentioned the market was particularly inefficient last year. One of the hedge fund managers I follow had returns similar to mine this year and last year, but not in the years before that. I’d guess 20-50% above market returns is a realistic expectation if market conditions stay similar to today’s, and I hope I can still outperform if market conditions go more “out of sample” but I currently have no basis to say by how much.
Also, I’m already starting to feel diminishing returns (is there a more technical term for this in the investing world?) kick in at my AUM level, as I now have to spend multiple days accumulating some positions to the sizes that I want (and they sometimes take off before I finish), or ignore some particularly illiquid instruments that I would have traded in the past.
I figured that’s the first thing someone would think of upon hearing “7x” which is why I mentioned “This was done using a variety of strategies across a large number of individual names” in the OP.
Right, I wasn’t disagreeing with you, just explaining why 7x isn’t strong evidence in my own words.
Can you please give some examples of such people? I wonder if there are any updates or lessons there for me.
Yes, but I don’t think there’s a huge amount of value in doing that. If you spend any time following stock touts on twitter / stock picking forums etc you will see these people quickly.
To be clear, I have no interest in dissuading you from trading. You’ve smashed it—you have confidence in your edge—go wild. I’m more cautioning people from following you thinking this is easy. Financial markets are extremely competitive and hard. It’s easy to mistake luck for skill and I don’t want other people losing money they can’t afford. I generally find posts like this are net-negative EV.
But why would you need ~1k trades to verify that I was not doing particularly high variance strategies?
I wouldn’t with something like that. However, assuming 250 trades each done on 5% of your capital, you’d need to be returning >15% on every single trade to return 7x. My experience say that’s unlikely, but ymmv. If that is the case then yes, you have serious edge and please take my money. (Especially if those are after tax returns!).
I don’t supposed you’d actually want to do this? (I also have some privacy concerns on my end, but maybe could be persuaded if the “value added” in doing this seems really high.)
I’m not sure what the value would be for me doing it? I’m not sure adding my word saying “I think this guy is legit because I saw a track record” would bring much value to either of us. I guess if I worked for a fund which might have interest in hiring people then I guess I might. But at the times when I have been in those roles, if someone turns up and makes implausible claims about returns, I politely show them the door.
I’d guess 20-50% above market returns is a realistic expectation if market conditions stay similar to today’s, and I hope I can still outperform if market conditions go more “out of sample” but I currently have no basis to say by how much.
I wish you the best of luck. I have never achieved anything close to those levels of returns, but would sorely love to do so.
If you spend any time following stock touts on twitter / stock picking forums etc you will see these people quickly.
The people I follow generally don’t advertise their track record? For the hedge fund manager I mentioned, I had to certify that I’m an accredited investor and sign up for his fund letters to get his past returns. For the ones that do, e.g., paid services on SeekingAlpha that advertise past returns, it has not been my experience that they “then fail to do so out of sample” (at least the ones that passed my filter of being worth subscribing to).
I generally find posts like this are net-negative EV.
Personally, I wish I had seen a post like this 10 years ago. My guess is that there’s at least 2 or 3 people on LW who could become good traders if they tried. Even if 10 times that many people try and don’t succeed, that seems overall a win from my perspective, as the social/cultural/signaling and monetary gains from the winners more than offset the losses. In part I want LW to become a bigger cultural force, and clear success stories that can’t be dismissed as “luck” seem very helpful for that.
Especially if those are after tax returns!
Pre-tax.
I have never achieved anything close to those levels of returns, but would sorely love to do so.
Maybe try some of my tips, if you haven’t already? :)
(You’re not wrong, but I wanted to flag: the way I read John’s comment, the word “nontrivially” already admitted this. If he thought it was strong evidence I’d expect him to have used a stronger word. Nothing wrong with adding clarification, but I don’t particularly think you’re disagreeing with him on this point.)
The efficient markets model says that any strategy during this period had expected return of 50%.
Wait, I think this is wrong. It actually says that any beta 1 strategy had the same expected return as the market as a whole. If my portfolio had a beta of 2, for example, either by using leverage or by buying only high beta stocks, then my expected return would be double that of the market.
I wish I could say that I kept my portfolio’s beta at or below 1 at all times, which would make the reasoning easier, but I did sometimes trade derivatives that arguably had high beta. It would be pretty cumbersome to calculate the exact overall beta, but I’d guess that the average over time probably wasn’t more than 1.5, so you could perhaps redo your reasoning using that.
See my answer to Paul below. Same thing applies for the CAPM: it’s an approximation, and the expected value formula is the efficiency condition which correctly accounts for how much margin one should actually expect to be able to get, as well as the effect of margin calls.
(In the expected value formula, beta is wrapped into the discount factor; the CAPM approximation pulls it out.)
As at_the_zoo said, this isn’t quite right. Under EMH there is a frontier of efficient portfolios that trade off risk and return in different ways. E.g. if the market has an expected return of 6% and an expected variance of 2%, then the 3x leveraged policy has expected return 18% and expected variance of 18% (for expected log-return of 9% vs 5% for the unlevered portfolio). And then when you condition on ex post returns it gets even messier.
I think you could get returns this high without being wise and you are basically trusting at_the_zoo that they didn’t implicitly use a bunch of leverage. Though 600% is high enough that it would require a reasonably large amount of leverage and even then a fair amount of luck, even buying UPRO (=3x levered SPY) at market bottom is only 430% returns.
(More generally, I assume “EMH” here is basically captured by the two mutual funds theorem, and “beta” is the correlation of your portfolio with the risky fund.)
The most fundamental market efficiency theorem is V_t = E[e^{-r dt} V_{t+dt}]; that one can be derived directly from expected utility maximization and rational expectations. Mean-variance efficient portfolios require applying an approximation to that formula, and that approximation assumes that the portfolio value doesn’t change too much. So it breaks down when leverage is very high, which is exactly what we should expect.
What that math translates to concretely is we either won’t be able to get leverage that high, or it will come with very tight margin call conditions so that even a small drop in asset prices would trigger the call.
I think the two mutual funds theorem roughly holds as long as asset prices change continuously. I agree that if asset prices can make large jumps (or if you are a large fraction of the market) such that you aren’t able to liquidate positions at intermediate prices, then the statement is only true approximately.
I think leveraged portfolios have a higher EV according to every theoretical account of finance, and they have had much higher average returns in practice during any reasonably long stretch. I’m not sure what your theorem statement is saying exactly but it shouldn’t end up contradicting that.
Maybe the weirdness is that you are assuming all investors have linear utility? (I don’t know what the symbol V_t means, and generally don’t quite understand the theorem statement.) Or maybe in your theorem “E” is an expectation that weights worlds based on the marginal value of $ in those worlds, whereas in other places we are using “probability” to refer to the fraction of worlds in which an event occurs?
NOTE: Don’t believe everything I said in this comment! I elaborate on some of the problems with it in the responses, but I’m leaving this original comment up because I think it’s instructive even though it’s not correct.
There is a theoretical account for why portfolios leveraged beyond a certain point would have poor returns even if prices follow a random process with (almost surely) continuous sample paths: leverage decay. If you could continuously rebalance a leveraged portfolio this would not be an issue, but if you can’t do that then leverage exhibits discontinuous behavior as the frequency of rebalancing goes to infinity.
A simple way to see this is that if the underlying follows Brownian motion dS/S=μdt+σdW and the risk-free return is zero, a portfolio of the underlying leveraged k-fold and rebalanced with a period of T (which has to be small enough for these approximations to be valid) will get a return
r=kS(T)S(0)−k=kexp(ΔlogS)−k
On the other hand, the ideal leveraged portfolio that’s continuously rebalanced would get
ri=(S(T)S(0))k−1=exp(kΔlogS)−1
If we assume the period T is small enough that a second order Taylor approximation is valid, the difference between these two is approximately
E[ri−r]≈k(k−1)2E[(Δlog(S))2]≈k(k−1)2σ2T
In particular, the difference in expected return scales linearly with the period in this regime, which means if we look at returns over the same time interval changing T has no effect on the amount of leverage decay. In particular, we can have a rule of thumb that to find the optimal (from the point of view of maximizing long-term expected return alone) leverage in a market we should maximize an expression of the form
kμ−k(k−1)2σ2
with respect to k, which would have us choose something like k=μ/σ2+1/2. Picking the leverage factor to be any larger than that is not optimal. You can see this effect in practice if you look at how well leveraged ETFs tracking the S&P 500 perform in times of high volatility.
I didn’t follow the math (calculus with stochastic processes is pretty confusing) but something seems obviously wrong here. I think probably your calculation of E[(Δlog(S)2)] is wrong?
Maybe I’m confused, but in addition to common sense and having done the calculation in other ways, the following argument seems pretty solid:
Regardless of k, if you consider a short enough period of time, then with overwhelming probability at all times your total assets will be between 0.999 and 1.001.
So no matter how I choose to rebalance, at all times my total exposure will be between 0.999k and 1.001k.
And if my exposure is between 0.999k and 1.001k, then my expected returns over any time period T are between 0.999kTμ and 1.001kTμ. (Where μ is the expected return of the underlying, maybe that’s different from your μ but it’s definitely just some number.)
So regardless of how I rebalance, doubling k approximately doubles my expected returns.
So clearly for short enough time periods your equation for the optimum can’t be right.
But actually maximizing EV over a long time period is equivalent to maximizing it over each short time period (since final wealth is just linear in your wealth at the end of the initial short period) so the optimum over arbitrary time periods is also to max leverage.
Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
and to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2
where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdz
and the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdz
which is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdz
so now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.
I think the two mutual funds theorem roughly holds as long as asset prices change continuously. I agree that if asset prices can make large jumps (or if you are a large fraction of the market) such that you aren’t able to liquidate positions at intermediate prices, then the statement is only true approximately.
That’s not the key assumption for purposes of this discussion. The key assumption is that you can short arbitrary amounts of either fund, and hold those short positions even if the portfolio value dips close to zero or even negative from time to time.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Maybe the weirdness is that you are assuming all investors have linear utility? (I don’t know what the symbol V_t means, and generally don’t quite understand the theorem statement.) Or maybe in your theorem “E” is an expectation that weights worlds based on the marginal value of $ in those worlds, whereas in other places we are using “probability” to refer to the fraction of worlds in which an event occurs?
Nope, not linear utility, unless we’re using the risk-free rate for r, which is not the case in general. The symbol V_t is a lazy shorthand for the price of the asset, plus the value of any dividends paid up to time t. The weighting by marginal value of $ in different worlds comes from the discount rate r; E is just a plain old expectation.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Imagine someone who is extremely risk-averse. They aren’t willing to invest much in equities, but they are willing to make a margin loan with ~0 risk. They will get lower return than if they had invested in equities, and they’ll have less risk. I don’t see what’s contradictory about this.
I was being a bit lazy earlier—when I said “EV”, I was using that as a shorthand for “expected discounted value”, which in hindsight I probably should have made explicit. The discount factor is crucial, because it’s the discount factor which makes risk aversion a thing: marginal dollars are worth more to me in worlds where I have fewer dollars, therefore my discount factor is smaller in those worlds.
The person making the margin loan does accept a lower expected return in exchange for lower risk, but their expected discounted return should be the same as equities—otherwise they’d invest in equities.
(In practice the Volker rule and similar rules can break this argument: if banks aren’t allowed to hold stock, then in-principle there can be arbitrage opportunities which involve borrowing margin from a bank to buy stock. But that is itself an exploitation of an inefficiency, insofar as there aren’t enough people already doing it to wipe out the excess expected discounted returns.)
Well, think about it, one-in-five is an extremely high probability! We only need 5 people to try what the OP tried, for one of them to be this successful and to write this post, and we won’t hear about most of those who failed.
I don’t know enough about the etiquette here, but I am having to fight the urge to post a bunch of memes along the lines of “It’s not the bull market, I really am a genius”.
I would strongly advise anyone who’s considering following this to consider doing this with considerably less than their whole portfolio and with much lower expectations than 7x’ing your money.
A lot of people respond to things like this post by assuming that the author was lucky. This is usually correct, at least when applied to random claimants on the interwebs, but we can put some bounds on it: the higher the returns, the more luck required to achieve them, assuming an efficient market. The efficient markets model says that any strategy during this period had expected return of 50%. So, if the post author used a strategy with probability p of achieving 600% returns, and 1-p of losing everything, then efficient markets implies p*(100+600) + (1-p)*0 = (100+50), i.e. p = 0.21 (roughly). This is the highest probability any strategy could have of achieving 600% returns during this period, without exploiting any market inefficiency.
In other words: at most one-in-five people could achieve returns that high without exploiting market inefficiency. It should therefore probably update us nontrivially away from the possibility that the post author just got lucky. (Though depending on your priors, you might still think the post author got lucky.)
Everyone else has already pointed out that you misunderstood what EMH states, so I wont bother adding to their chorus. (Except to say I agree with them).
I will also disagree with:
1 in 5 isn’t especially strong evidence. How many of the other 5 people would you expect to be publishing on the internet saying “You should trade stocks”.
I agree this isn’t a very strong argument. I think theoretically we can probably get a much tighter probability bound than 20% by looking directly at the variance of my strategy, and concluding that given that variance, the probability of getting 600% return by chance (assuming expected return = market return) is <p for some smaller p. But in practice I’m not sure how to compute this variance. Intuitively I can point to the fact that my portfolios did not have very high leverage/beta, nor did I put everything into a single or very few highly volatile stocks or sectors, which are probably the two most common high variance strategies people use. (Part of the reason for me writing this post is that while LW does have a number of people who achieved very high investment returns, they all AFAIK did it by using one of these two methods, which makes it hard to cite them as exemplifying the power of rationality.)
Assuming the above is still not very convincing, I wonder what kind of evidence would be...
Without even checking, I can think of a bunch of assets which 7x’ed since Jan 2020. (BTC/general crypto, TSLA, GME/AMC etc). So yes, I agree this depends on the portfolio you ran.
Personally, I have seen enough people claiming to outperform, but then fail to do so out of sample. (I mean, out of sample for me, not for them) for me to doubt any claim on the internet without a trading record.
Either way, I think it’s very hard to convince me with just ~1.5 years of evidence that you have edge. I think if you showed me ~1k trades with some sensible risk parameters at all times, then I could be convinced. (Or if in another year and a half you have $300mm because you’ve managed to 7x your small HF AUM, I will be convinced).
I figured that’s the first thing someone would think of upon hearing “7x” which is why I mentioned “This was done using a variety of strategies across a large number of individual names” in the OP. Just to further clarify, I have some exposure to crypto but I’m not counting it for this post, I bought some TSLA puts (forgot whether I made a profit overall), and didn’t touch AMC. I had a 0.1% exposure to some GME calls which went to 1% of my portfolio and that’s the only involvement there.
Can you please give some examples of such people? I wonder if there are any updates or lessons there for me.
I don’t think I’ve done that many trades (depending on how you define a trade, e.g., presumably accumulating a position across different days doesn’t count as separate trades). Maybe in the low hundreds? But why would you need ~1k trades to verify that I was not doing particularly high variance strategies? I guess this is mostly academic though, as it would take a lot of labor to parse my trade logs and understand the underlying market mechanics to figure out what I was doing and how much risk I was taking (e.g., some pair/arbitrage trades were spread across several brokers depending on where I could find borrow). I don’t supposed you’d actually want to do this? (I also have some privacy concerns on my end, but maybe could be persuaded if the “value added” in doing this seems really high.)
I’m definitely not expecting such high returns going forward. (“600% return” was meant to be Bayesian evidence to update on, not used to directly set expectations. I thought that went without saying around here...) Obviously there was a significant amount of luck involved, for example as I mentioned the market was particularly inefficient last year. One of the hedge fund managers I follow had returns similar to mine this year and last year, but not in the years before that. I’d guess 20-50% above market returns is a realistic expectation if market conditions stay similar to today’s, and I hope I can still outperform if market conditions go more “out of sample” but I currently have no basis to say by how much.
Also, I’m already starting to feel diminishing returns (is there a more technical term for this in the investing world?) kick in at my AUM level, as I now have to spend multiple days accumulating some positions to the sizes that I want (and they sometimes take off before I finish), or ignore some particularly illiquid instruments that I would have traded in the past.
Right, I wasn’t disagreeing with you, just explaining why 7x isn’t strong evidence in my own words.
Yes, but I don’t think there’s a huge amount of value in doing that. If you spend any time following stock touts on twitter / stock picking forums etc you will see these people quickly.
To be clear, I have no interest in dissuading you from trading. You’ve smashed it—you have confidence in your edge—go wild. I’m more cautioning people from following you thinking this is easy. Financial markets are extremely competitive and hard. It’s easy to mistake luck for skill and I don’t want other people losing money they can’t afford. I generally find posts like this are net-negative EV.
I wouldn’t with something like that. However, assuming 250 trades each done on 5% of your capital, you’d need to be returning >15% on every single trade to return 7x. My experience say that’s unlikely, but ymmv. If that is the case then yes, you have serious edge and please take my money. (Especially if those are after tax returns!).
I’m not sure what the value would be for me doing it? I’m not sure adding my word saying “I think this guy is legit because I saw a track record” would bring much value to either of us. I guess if I worked for a fund which might have interest in hiring people then I guess I might. But at the times when I have been in those roles, if someone turns up and makes implausible claims about returns, I politely show them the door.
I wish you the best of luck. I have never achieved anything close to those levels of returns, but would sorely love to do so.
The people I follow generally don’t advertise their track record? For the hedge fund manager I mentioned, I had to certify that I’m an accredited investor and sign up for his fund letters to get his past returns. For the ones that do, e.g., paid services on SeekingAlpha that advertise past returns, it has not been my experience that they “then fail to do so out of sample” (at least the ones that passed my filter of being worth subscribing to).
Personally, I wish I had seen a post like this 10 years ago. My guess is that there’s at least 2 or 3 people on LW who could become good traders if they tried. Even if 10 times that many people try and don’t succeed, that seems overall a win from my perspective, as the social/cultural/signaling and monetary gains from the winners more than offset the losses. In part I want LW to become a bigger cultural force, and clear success stories that can’t be dismissed as “luck” seem very helpful for that.
Pre-tax.
Maybe try some of my tips, if you haven’t already? :)
(You’re not wrong, but I wanted to flag: the way I read John’s comment, the word “nontrivially” already admitted this. If he thought it was strong evidence I’d expect him to have used a stronger word. Nothing wrong with adding clarification, but I don’t particularly think you’re disagreeing with him on this point.)
Wait, I think this is wrong. It actually says that any beta 1 strategy had the same expected return as the market as a whole. If my portfolio had a beta of 2, for example, either by using leverage or by buying only high beta stocks, then my expected return would be double that of the market.
I wish I could say that I kept my portfolio’s beta at or below 1 at all times, which would make the reasoning easier, but I did sometimes trade derivatives that arguably had high beta. It would be pretty cumbersome to calculate the exact overall beta, but I’d guess that the average over time probably wasn’t more than 1.5, so you could perhaps redo your reasoning using that.
See my answer to Paul below. Same thing applies for the CAPM: it’s an approximation, and the expected value formula is the efficiency condition which correctly accounts for how much margin one should actually expect to be able to get, as well as the effect of margin calls.
(In the expected value formula, beta is wrapped into the discount factor; the CAPM approximation pulls it out.)
As at_the_zoo said, this isn’t quite right. Under EMH there is a frontier of efficient portfolios that trade off risk and return in different ways. E.g. if the market has an expected return of 6% and an expected variance of 2%, then the 3x leveraged policy has expected return 18% and expected variance of 18% (for expected log-return of 9% vs 5% for the unlevered portfolio). And then when you condition on ex post returns it gets even messier.
I think you could get returns this high without being wise and you are basically trusting at_the_zoo that they didn’t implicitly use a bunch of leverage. Though 600% is high enough that it would require a reasonably large amount of leverage and even then a fair amount of luck, even buying UPRO (=3x levered SPY) at market bottom is only 430% returns.
(More generally, I assume “EMH” here is basically captured by the two mutual funds theorem, and “beta” is the correlation of your portfolio with the risky fund.)
The most fundamental market efficiency theorem is V_t = E[e^{-r dt} V_{t+dt}]; that one can be derived directly from expected utility maximization and rational expectations. Mean-variance efficient portfolios require applying an approximation to that formula, and that approximation assumes that the portfolio value doesn’t change too much. So it breaks down when leverage is very high, which is exactly what we should expect.
What that math translates to concretely is we either won’t be able to get leverage that high, or it will come with very tight margin call conditions so that even a small drop in asset prices would trigger the call.
I think the two mutual funds theorem roughly holds as long as asset prices change continuously. I agree that if asset prices can make large jumps (or if you are a large fraction of the market) such that you aren’t able to liquidate positions at intermediate prices, then the statement is only true approximately.
I think leveraged portfolios have a higher EV according to every theoretical account of finance, and they have had much higher average returns in practice during any reasonably long stretch. I’m not sure what your theorem statement is saying exactly but it shouldn’t end up contradicting that.
Maybe the weirdness is that you are assuming all investors have linear utility? (I don’t know what the symbol V_t means, and generally don’t quite understand the theorem statement.) Or maybe in your theorem “E” is an expectation that weights worlds based on the marginal value of $ in those worlds, whereas in other places we are using “probability” to refer to the fraction of worlds in which an event occurs?
NOTE: Don’t believe everything I said in this comment! I elaborate on some of the problems with it in the responses, but I’m leaving this original comment up because I think it’s instructive even though it’s not correct.
There is a theoretical account for why portfolios leveraged beyond a certain point would have poor returns even if prices follow a random process with (almost surely) continuous sample paths: leverage decay. If you could continuously rebalance a leveraged portfolio this would not be an issue, but if you can’t do that then leverage exhibits discontinuous behavior as the frequency of rebalancing goes to infinity.
A simple way to see this is that if the underlying follows Brownian motion dS/S=μdt+σdW and the risk-free return is zero, a portfolio of the underlying leveraged k-fold and rebalanced with a period of T (which has to be small enough for these approximations to be valid) will get a return
r=kS(T)S(0)−k=kexp(ΔlogS)−kOn the other hand, the ideal leveraged portfolio that’s continuously rebalanced would get
ri=(S(T)S(0))k−1=exp(kΔlogS)−1If we assume the period T is small enough that a second order Taylor approximation is valid, the difference between these two is approximately
E[ri−r]≈k(k−1)2E[(Δlog(S))2]≈k(k−1)2σ2TIn particular, the difference in expected return scales linearly with the period in this regime, which means if we look at returns over the same time interval changing T has no effect on the amount of leverage decay. In particular, we can have a rule of thumb that to find the optimal (from the point of view of maximizing long-term expected return alone) leverage in a market we should maximize an expression of the form
kμ−k(k−1)2σ2with respect to k, which would have us choose something like k=μ/σ2+1/2. Picking the leverage factor to be any larger than that is not optimal. You can see this effect in practice if you look at how well leveraged ETFs tracking the S&P 500 perform in times of high volatility.
I didn’t follow the math (calculus with stochastic processes is pretty confusing) but something seems obviously wrong here. I think probably your calculation of E[(Δlog(S)2)] is wrong?
Maybe I’m confused, but in addition to common sense and having done the calculation in other ways, the following argument seems pretty solid:
Regardless of k, if you consider a short enough period of time, then with overwhelming probability at all times your total assets will be between 0.999 and 1.001.
So no matter how I choose to rebalance, at all times my total exposure will be between 0.999k and 1.001k.
And if my exposure is between 0.999k and 1.001k, then my expected returns over any time period T are between 0.999kTμ and 1.001kTμ. (Where μ is the expected return of the underlying, maybe that’s different from your μ but it’s definitely just some number.)
So regardless of how I rebalance, doubling k approximately doubles my expected returns.
So clearly for short enough time periods your equation for the optimum can’t be right.
But actually maximizing EV over a long time period is equivalent to maximizing it over each short time period (since final wealth is just linear in your wealth at the end of the initial short period) so the optimum over arbitrary time periods is also to max leverage.
Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
E[R(T)]=E[V(T)]V(0)=T−1∏i=0E[V(i+1)V(i)|Fi]=T−1∏i=0E[kS(i+1)S(i)−k|Fi]=kT(exp(μ)−1)Tand to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdzand the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdzwhich is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdzso now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.
That’s not the key assumption for purposes of this discussion. The key assumption is that you can short arbitrary amounts of either fund, and hold those short positions even if the portfolio value dips close to zero or even negative from time to time.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Nope, not linear utility, unless we’re using the risk-free rate for r, which is not the case in general. The symbol V_t is a lazy shorthand for the price of the asset, plus the value of any dividends paid up to time t. The weighting by marginal value of $ in different worlds comes from the discount rate r; E is just a plain old expectation.
Imagine someone who is extremely risk-averse. They aren’t willing to invest much in equities, but they are willing to make a margin loan with ~0 risk. They will get lower return than if they had invested in equities, and they’ll have less risk. I don’t see what’s contradictory about this.
Indeed there is nothing contradictory about that.
I was being a bit lazy earlier—when I said “EV”, I was using that as a shorthand for “expected discounted value”, which in hindsight I probably should have made explicit. The discount factor is crucial, because it’s the discount factor which makes risk aversion a thing: marginal dollars are worth more to me in worlds where I have fewer dollars, therefore my discount factor is smaller in those worlds.
The person making the margin loan does accept a lower expected return in exchange for lower risk, but their expected discounted return should be the same as equities—otherwise they’d invest in equities.
(In practice the Volker rule and similar rules can break this argument: if banks aren’t allowed to hold stock, then in-principle there can be arbitrage opportunities which involve borrowing margin from a bank to buy stock. But that is itself an exploitation of an inefficiency, insofar as there aren’t enough people already doing it to wipe out the excess expected discounted returns.)
Well, think about it, one-in-five is an extremely high probability! We only need 5 people to try what the OP tried, for one of them to be this successful and to write this post, and we won’t hear about most of those who failed.