Part 3 of the Inefficient Markets sequence.

[I am not your financial advisor! Examples are illustrations of principles for educational purposes only, to help you make your own financial decisions with due diligence. Read part 1 first.]

Suppose I offer you a deal: You will flip a fair coin. If it's heads, I will pay you a dollar. If it's tails, you will pay me a dollar.

Does this game strike you as unfair? Does it seem dangerous? Your expected value is zero. If we iterate this game, you'll experience some gains and some losses, but they will tend to even out over time. So if anything, it's pointless. Entertaining maybe, but dangerous? No, not really.

Now suppose I let you bet any number of dollars X. If you win, you will pay me X dollars. If I win, I will pay you X dollars.

Is this game unfair? Is it dangerous? How much should you bet each game? What is your expected value?

It's the same deal, right?

Not quite.

If you bet all of your money, you could double your money, or you could lose all your money. If we iterate, and I have enough dollars to pay off your winning streak, you are almost certain to eventually flip a tail and lose. Now I have all of your dollars, and I keep them, because you can't afford to play.

The Second Law of Averting Ruin

Don't Bet the Farm

I surreptitiously gave myself a massive advantage in the second game. I said "I have enough dollars to pay off your winning streak". The player with more money has a massive advantage. But if you are betting against "the market", you had best assume that "the market" has a lot more dollars than you.

Kelly's Optimal Strategy

I will not derive the formula here, but the optimal betting strategy is to bet a certain fraction of your bankroll.

For an even-money bet, the formula is simply

The fraction is only 1 if the probability of winning is also 1.

The Kelly formula becomes more complex if it's not an even-money bet. It must be computed from the distribution of payoffs (including negative payoffs).

It's important that you understand this formula exists for the insights it provides, that the optimal strategy is to bet an optimal fraction, that underbetting will underperform, overbetting will underperform, that double the fraction has an expected growth rate of zero, and for anything bigger than that it will be negative, so Betting the Farm is a sure path to ruin, but we can't actually use it, because in practice we can't know our payoff distribution!

We can only estimate it from past data, and the market conditions shift over time.

You Are not Using Enough Leverage

Depending on the distribution of payoffs, it is possible for the optimal Kelly fraction to be over unity. This tends to happen when losing your bet doesn't mean losing all of your bet.

Now how can you possibly bet more than your bankroll?

Borrow money.

If you sign up for a margin account with your stockbroker, your broker will loan you the money to buy more shares of stock, using your shares as collateral for the loan. You can obtain twice the growth as the price goes up, because you can afford twice as many shares for the money. This is one example of a core trading concept called leverage.

Double the returns sounds great, until you realize that returns can also be negative. Your losses will double too. Leverage is a double-edged sword.

Individual stocks can and do drop to zero (or close enough to it that it makes little difference to you), but an index like the S&P 500 can only drop to zero if all 500 of America's biggest companies drop to zero at the same time. Not likely.

Historically, we can see that the optimal leverage factor for U.S. stocks would have been about 2x. Nobody can know what the future payoff distribution will be, but 2x seems like a good prior.

A word of caution: some brokers charge outrageous amounts of interest for margin loans. Consider a more reasonable broker. (You can sometimes circumvent this by using box spreads to borrow from the market instead at a better rate. These are dangerous if you don't know what you are doing.)

I Can Beat the Index

Remember the Very Basic trading system from last time? It was a stupid-simple balanced portfolio of SPY and TLT. (Tracking the S&P 500, and 20-year U.S. Treasury Bonds, respectively.)

Well, here's a slightly better trading system.

One reason I picked those two is because these ETFs have 2x leveraged versions: SSO and UBT.

Leveraged ETFs are another way to obtain leverage that do not require a margin account. These are not as flexible as a margin account, because you can buy almost anything on margin, but leveraged ETFs are only available for certain things.

The backtested performance of a portfolio of 100% SPY (the S&P 500 index) from February 2010 through July 2020 was a compound annual growth rate of 12.7% with a volatility of 13.11%, and a maximum drawdown of -19.43%.

The backtested performance of 50% SSO and 50% UBT (rebalanced at 10% absolute deviation) over the same period was a CAGR of 21.75%, with a volatility of 14.41% and a max drawdown of -13.46%.

That's superior to the index by almost every measure. Bonds zig when stocks zag, so the volatility of their sum is barely more than SPY alone, but the returns have almost doubled due to the 2x leverage.

That's a Sharpe ratio of 0.93 for SPY and 1.4 for SSO/UBT. Sharpe ratio is a measure of risk-adjusted returns. Two portfolios with the same Sharpe would have about the same return if leveraged to the same volatility. You can approximate the Sharpe ratio by dividing the annual return by the volatility. (The real formula also subtracts out the risk-free rate.)

The astute reader may notice that this time frame conveniently avoided the 2008 crisis. Unfortunately, these leveraged ETFs weren't available back then, so there's no data. But it does include the recent corona crash, and did fine. We can, however, look at how a portfolio of SPY and TLT would have done over that period if leveraged to 2x using margin. It turns out to still have about 17% CAGR, and a superior Sharpe ratio vs 1x SPY alone over the same period. Past performance cannot guarantee future results. But I have to wonder, why isn't everyone doing something like this?

If your Sharpe ratio is over 1.0, you're doing well. As you can see, this system is still stupid-simple and it got all the way up to 1.4. Trading does not have to take a lot of effort to be worth your while.

But we can do better. Aim for 2.0.

You Must Know When to Leverage Down


This is a backtest showing the growth of a $10,000 portfolio of 100% SVXY. Look at that ride. Years of high-powered heartburn, and then in 2018, following a race to almost $200,000, in a moment of pure terror, it falls off a cliff. For a final CAGR of just 3.09%. What was it all for?

Doesn't that look like fun? Is this something you'd like to buy and hold? No? DON'T PANIC! Sarcasm aside, that is exactly what makes it go up so hard: risk premium.

And that is why it's a great investment. No, I am serious, look at this chart compared to 100% SPY:

SVXY versus SPY

It was doing so much better until it wasn't. Nevertheless, 3% is not a great return.

SVXY is a great illustration of what we call negative skew. If you plot its distribution of daily returns (the difference between daily close to daily close prices) as a histogram, you'll see a distribution that kind of looks like a bell curve, but has a long tail on the left representing those times when it suddenly drops. You Do Not Bet the Farm when trading, especially when there's a negative skew. Stocks and bonds also tend to skew a bit negative, but SVXY's return distribution is especially dramatic.

Remember the Kelly fraction is computed from the payoff distribution. The optimal leverage for any given trade is unlikely to be 1x. As we saw for our stock portfolio, 2x was much better. But in this case, even 1x is too much.

How do you leverage less than 1x? Don't use the whole bankroll.

Here's 50/50 SVXY and cash, rebalanced at 10% absolute deviation, with 100% SPY as a benchmark again.

half SVXY versus SPY

We can see it's outperforming the SPY portfolio almost the entire time, until the corona crash, but even now, it's got a very respectable 12% CAGR.

Just by leveraging down to 0.5x, SVXY has become a valid investment it its own right, but it's not a great fit for our simple rebalancing system. You don't have to leave cash in your portfolio to leverage down. You just need something to balance it with. Some other uncorrelated asset would work. Equal weights of SVXY, SSO, and UBT does perform a bit better (in a backtest), but given the levels of volatility involved, how well that might do in the future is really uncertain. And SVXY has a .68 correlation with SSO, so it doesn't diversify all that much.

It's probably not worth it, but neither is it insane. If we could somehow avoid or mitigate even a few of those drops, SVXY would be a powerful addition to our system. And if our avoidance or mitigation fails, we're probably doing no worse than SPY. More on that idea later in the sequence.

Don't Put All Your Nest Egg in One Asset. Use a Basket.

"Don't Bet the Farm" is a warning about overallocating your trades. Leveraging too high is not the only way this can happen.

Diversification is not about the number of tickers in your portfolio. Trading multiple assets that are too correlated is another way to Bet the Farm. The stock market as a whole shows a great deal of correlation with the S&P 500. Just because assets seem different, doesn't mean they are. Don't rely on intuition here. Run the numbers. Look at the correlations. Correlations also shift over time. That you were diversified before doesn't mean you are now.

A Somewhat Safer System

Our last system had only two assets: SSO, and UBT. Diversification is not about the number of tickers. SSO tracks an index of 500 different stocks. And the bonds are nicely anticorrelated. This is already a lot more diversified than a portfolio of two stocks.

How hard would it be to diversify more? We would need an asset that isn't bad to buy and hold and not very correlated with either.

I have a candidate in mind. UGL is a 2x leveraged ETF tracking the price of gold.

Our 50/50 allocation was rather simplistic. Why that instead of some other number?

Equal weight of one-third each would work OK, but I think we can do better.

Correlations can shift over time, but so can payoff distributions. We can't really know what those are, which makes it hard to use Kelly sizing, but do keep the principles we learned from Kelly in mind.

We can compute the average volatility of each asset over the last month, which turns out to be a pretty good estimate of tomorrow's volatility for that asset. This may be a good enough proxy for the payoff distribution to be useful in sizing. We know that more volatile assets can drop faster, so we'll give them a lower allocation to reduce risk. The less volatile assets need more leverage to produce enough return, so we'll give them higher allocation.

This type of allocation is called a risk parity portfolio. There are different ways to calculate these weightings, but here's one. Decide on a maximum acceptable volatility for the whole portfolio. 12% is probably reasonable. That's close to the volatility of SPY. Don't Bet the Farm. I would not go over 20%. Remember we don't know where the Kelly size is. Leveraging too much will actually reduce returns.

Then your target dollar exposure for an asset is Vol is inversely proportional to target weight. More vol, less exposure. Less vol, more exposure. In practice, this formula will tend to undershoot your max vol, since you get the diversification benefits of these assets, their net vol will be less than the sum of their vols. There are other ways of computing the weightings that account for this effect.

If the total exposure is less than your bankroll, you will be leveraging down by keeping some cash in reserve. If the total exposure exceeds your bankroll, you will have to use more leverage to hit your target allocations. These ETFs are already leveraged 2x, but there are some 3x ETFs available that you might use instead. TMF for the bonds, and UPRO for the S&P.

Diversify Even More. And Then Diversify Again.

There are only so many uncorrelated exchange-traded asset classes to choose from. To really diversify, you'll have to get creative. Consider more exotic investments like cryptocurrency, prediction markets, and peer-to-peer lending.

But eventually, you'll run out of asset classes. To diversify more, you have to diversify your strategies.

There will be more (and more advanced) strategies later in the sequence. An active strategy can have a very different payoff distribution than the underlying assets it trades. You can think of a trading system like another uncorrelated asset class. Similar to the way you can allocate a risk parity portfolio of assets, you can also create a portfolio of strategies weighted by risk.


You will lose the game, even if it's completely fair, by oversizing your bets.

"Don't Bet the Farm" means you don't put all your nest egg in one asset class, you don't leverage too high (but do leverage enough), and you must diversify as much as you can manage, not just among asset classes, but among strategies.

The Third Law is up next.


New Comment
15 comments, sorted by Click to highlight new comments since: Today at 4:12 PM

If I understand correctly your analysis shows that the strategy with leveraged ETFs outperforms the strategy without leverage (and is indeed comparable to leverage on margin). Is there an underlying frequent rebalancing or other effect that I am missing?

My understanding was that leveraged ETFs are only useful for day trading and not for long-term buy-and-hold because of the internal re-adjustment to the index each day. If that holds then I would expect the better returns you show to be an artefact of the chosen time period, but find it still surprising!

Is there an underlying frequent rebalancing or other effect that I am missing?

Rebalancing is required. My example rebalanced after a 10% deviation, but you'd get very similar performance by rebalancing monthly.

My understanding was that leveraged ETFs are only useful for day trading and not for long-term buy-and-hold

That does seem to be a common misconception, but it's easily shown to be false. A (e.g.) 3x daily ETF tracks at 3x per day. Over longer periods, due to compounding, this accelerates your returns (even your negative returns), so don't expect performance to be 3x over periods longer than a day. Results can be even higher than 3x for instruments that tend to trend up in the long term, which stocks and bonds do.

(This is not financial advice.)

I'm not sure the linked article shows it to be false, exactly. If you look at the first graph, you can see that over 135 years of US data, 1x leverage returns about 4% annually, and 2x leverage about 5%, with double the risk. That's pretty bad, unless you're desperate for risk.

Now, that plot doesn't include dividends, which are an important part of the calculation. And using different countries or time periods will give different results, as they demonstrate later on. Still, if you're discussing a type of investment the SEC specifically warns people about, there might be more to say here?

Leveraging up, on its own, is never going to improve your Sharpe ratio. But you can use leverage to get a better return if your portfolio is below your risk tolerance, at least until you hit the optimal bet size.

Volatility does have bad effects on leverage, which is why my somewhat safer system reduces the leverage for an asset when its vol is high, even below 1x, if necessary, by holding cash. The Kelly strategy means there is always an optimal amount of leverage, and it's unlikely to be exactly 1x.

(Still not financial advice.)

Well, Kelly is optimal if you have infinite time, or logarithmic utility. In practice we all have finite time, and many of us are more risk-averse. Plus, as you mentioned earlier, Kelly is only optimal if you know the payoff distribution, which you don't.

I'm not saying leverage can't be a useful addition to a portfolio; just that there are also reasonable concerns about it. Yes, a leveraged mix of equities and bonds has done pretty well the past forty years. But the thirty before that, it was a disaster. Sure, the macroeconomic regime was different then; but in a world of negative interest rates, are you sure it won't change again?

Sure, the macroeconomic regime was different then; but in a world of negative interest rates, are you sure it won't change again?

Nope. We are not trying to avoid all risk. We're trying to get exposed to risk so we can get paid for it. The right side is uncomfortable. Taking on the risk of bonds crashing, in the appropriate amount, so we can get paid for it, is exactly the point of adding leveraged bonds to a risk premium portfolio. If it weren't risky, there wouldn't be a risk premium for holding it. People on the margins have been forecasting the end of the bond bull market for years. If you had listened to them then, you'd have given up the returns up till now.

If you play the game long enough, risks will eventually bite you. You will have drawdowns. Don't Bet the Farm. But the market pays you extra for it. You'll still eventually come out ahead if you size your exposure appropriately.

Thanks for the link and the insights, I just did a small toy calculation to check and my intuition was wrong: There seems to be much more to leveraged ETFs, like you say!

For an even-money bet, the formula is simply


This article says that while this is often quoted, it's only true for an even-money bet where you lose everything. Which is a shame because this would have been a fairly easy heuristic.

In particular, for a single asset, the formula becomes

Where is the drift, is the risk-free rate, and is the volatility. This is the better heuristic, and is perhaps the formula I should have included in the first place.

But in practice, we don't know our payoff distribution. Note that the drift is not simply the observed returns. It's what's left over when you subtract out the stationary noise. Which part of the return is the drift and which part is the noise? We don't know! And we don't really want the past drift and volatility, but the future drift and vol for the bet we're about to make. These are not constants. The past can help us estimate the future, but it's very uncertain.

Furthermore, this is only for a single asset. If you apply this formula to each asset in a portfolio, and they're at all positively correlated, this will tend to overallocate.

Don't Bet the Farm. If you're going to use it as a heuristic, then due to these uncertainties, it's probably best to think of this as an upper bound, rather than an exact target.

Right. Sorry if that wasn't clear. The Wikipedia article I linked to right below the formula explains other cases.

There is a very extensive discussion of a UPRO/TMF strategy here. One thing to note is taxes severely decrease the returns of strategies which require frequent re-balancing.

Taxes are going to be a problem for any active strategy. "DON'T PANIC" from part 2 is mainly a warning about trading too often. In the U.S. at least, you can avoid some taxes by trading in a Roth IRA, but it's problematic for early retirement. There are limits to how much you can contribute. If you mess up and bet the farm that limits how quickly it can recover. You are allowed to withdraw your contributions early without penalty (but not your returns, with a few exceptions). You also can't trade on margin, but can use ETFs with leverage.

I came across a Mad Fientist article suggesting that a traditional IRA is worth it, even for early retirement, even if that means paying a 10% penalty on some of it.

Important points to add

1. Make sure you have enough data that the result is not due to chance. 10 years of monthly data is laughably insufficient IMHO.

2. Make sure that you did not let your subconscious brain conveniently cherry pick a place and a time of high performance as your backtest playground. That is, make sure the data is representative of at least the breadth of past scenarios. Protip: it is quite unusual for stock markets have such high returns. They rarely last. Have a look at the Japanese Stock market over the last 50 years. Or the Chinese stock market in recent years. Or the 1930s in the US.

3. Make sure your optimal leverage is not right next to a 'cliff' in the parameter space where even slightly more leverage would have wiped you out.

I did a similar analysis going back to 1926 for the US alone and found that the optimal leverage was a very fragile 10%. 20% did far worse.

Many people tell me they are bullish on America and so the USA outperformance (best in the world over many periods in living memory) will continue. OK, you are bullish on a country that will pick either Donald Trump or Joe Biden as the president for the next 4 years. Good luck!! Would you have been bullish on the USA when its outperformance started, shortly after a ruinous civil war, with rampant corruption and civil disturbances, only barely emerging from developing country status?

10 years of monthly data is laughably insufficient IMHO.

Not sure where you got the "monthly" (I'd be using daily bars at the coarsest), but I quite agree. Market data is extremely noisy. But going too far back in history doesn't necessarily help because market conditions change. There were a lot of exploitable patterns historically that no longer hold. So rather than going deep, go broad. Diversify and look at patterns that hold for a basket of assets. And be cautious of old patterns that don't appear in recent data.

Make sure that you did not let your subconscious brain conveniently cherry pick a place and a time of high performance as your backtest playground

It's the recent history of what I'm actually investing in. I will have more to say about overfitting later in the sequence.

Make sure your optimal leverage is not right next to a 'cliff'

A fractional-Kelly bet should not behave like that, especially if you're diversified. Even in the Nikkei when it had a return of zero (1984-2009), the optimal leverage of about 0.5x would have produced positive growth.

Of course, I can't guarantee that a nuclear war won't destroy the world and your portfolio with it. For some level of black-swan risk, you've got more pressing concerns than your rate of return.

I did a similar analysis going back to 1926 for the US alone and found that the optimal leverage was a very fragile 10%. 20% did far worse.

Details matter. How were you rebalancing? If you still have the data handy, I'm curious how the risk-parity allocation would have done.

I'm interested in what you think of this paper.

Or at least the summary here.

OK, you are bullish on a country that will pick either Donald Trump or Joe Biden as the president for the next 4 years.

I know, right? On the other hand, America's institutions have shown some remarkable resilience, despite Trump's efforts to dismantle them (the current pandemic fiasco notwithstanding).