John Kelly was a mathematician and scientist from Texas who worked for AT&T in the mid-20th century. He's famous among professional gamblers and hedge fund managers but relatively unknown among the general public.

Kelly is best known for his formula called the Kelly Criterion (mathematical proof here, if you're so inclined). He died of a stroke in 1965 at the age of 41 but has a cult following among professional traders hedge fund managers today.

Around the same time, author and mathematics professor Ed Thorp popularized card counting in blackjack, using Kelly's theories to determine the size of his bets depending on whether the deck was hot or cold. He later applied the theories to his hedge fund, with massive success.

While the Kelly Criterion is somewhat well known in trading circles, what is less well known is that you can use algebra and calculus to rearrange Kelly's theories to answer more complex questions, like how much leverage to use in your portfolio and how much of your portfolio to allocate to different strategies.

I've used Kelly's theories, along with formulas developed by Harry Markowitz (the inventor of the efficient frontier), and the current consensus on asset allocation theory to design portfolios that deliver superior absolute and risk-adjusted returns compared to the S&P 500.

The idea behind the strategy is to use ETFs to build a core allocation of quality small-cap and value stocks, then overlay it with a satellite momentum strategy that takes more risk when the volatility environment is most conducive, the same way that card counters do with their bets when the deck gets hot or cold.

Stock returns are notoriously hard to forecast, but volatility levels are much easier to forecast. **Since leveraged ETFs are implicitly short volatility by their structure, being able to predict volatility predicts their returns.**

If you can predict volatility, you can make a ton of money.

## How much leverage should you use in your portfolio?

Using calculus on Kelly's original formula, we arrive at this:

For a portfolio with one asset (K is the leverage to use):

**K= Mean geometric return – borrow cost / standard deviation ^{2}.**

This is an approximation as far as leveraged ETFs are concerned (since they rebalance daily and I'm using annualized returns), but the model is pretty close to the actual historical returns.

For the S&P 500:

^{Expected Return= 9.5 percent - 2.7 percent borrow cost (~current LIBOR cost on financed portion +0.95 percent expense ratio) / }^{15 percent standard deviation^2}

**I get an optimal leverage ratio of a little over 3 using this formula.**

If you use more leverage, you are likely to underperform in the long run. If you use less, you will also underperform, but with less risk.

This simple model almost exactly hits the bulls-eye with the historical data.

**Historically Optimal Leverage Ratio of the S&P 500**

Source: *Double-Digit Numerics*

There's some interesting math behind this. Leveraged ETFs rebalance daily to their target leverage, so when the market is trending upwards, the ETFs are able to use their collateral to acquire exponentially larger positions. In turn, this exponentially increases their returns in an up-trending market.

When there's high volatility, daily rebalancing forces the fund to buy high and sell low, which exponentially decreases returns. If you use a small amount of leverage, the first force strongly outweighs the second force. However, if you use too much leverage, your returns start to go down.

This means that there's a quadratic relationship between leverage and returns over time. It isn't shown on the graph, but 5 times leverage would have wiped you out in 1987 when markets declined over 20 percent in one day. After 1987, the NYSE and Nasdaq now have various circuit breakers to protect against violent market declines. Even with this, 3x still wins the race, even when 1987 and 2008 are taken into account.

*But even if we can't predict the numerator, what if we could predict the denominator?* The answer is that we can, albeit imperfectly! More on this later.

## Implications of the optimal leverage ratio

It is not an accident that under Reg T, the margin allowed to buy stocks starts at 2x leverage and is allowed to float to 4x before margin calls are issued. The Fed designed this to maximize the wealth of margin borrowers (assuming they're smart enough to borrow cheaply).

You can also use this same formula to prove that anyone borrowing from Etrade or TDAmeritrade at their current rate of 11 percent is a fool.

Mathematics disproves the popular line that if you hold leveraged ETFs for longer than a day, they inevitably march to zero. Before I studied the math behind it, I also heavily criticized them. However, the popular narrative is wrong.

Leveraged ETFs are still risky little instruments, but it turns out that they are not doomed to go to zero. In fact, leveraged ETFs that use LIBOR-for-index total return swaps (UPRO, TQQQ) do much, much better than 3X the index in bull markets and/or low-interest rate environments. By isolating these environments, we do better than 3x the index on this potion of our portfolio.

The daily rebalancing that the media maligns leveraged ETFs for is a source of outperformance, not underperformance. The upshot to this, however, is that leveraged ETFs are implicitly short volatility, which can be a benefit or problem, depending on the level of volatility realized.

This means that a position in a 3x leveraged ETFs 1/3 the size of a position in the underlying index tends to beat it over time. What this means for the model is that a small position in leveraged Nasdaq combines incredibly well with value stocks, small-cap stocks, and long-term government bonds.

To show this, I'm going to use Markowitz's efficient frontier model. John Kelly also developed a model for multiasset portfolios but my software uses Markowitz's model. I believe the models are substantially similar if you measure geometric returns rather than arithmetic returns. So, in the spirit of Kelly, that's just what I've done. I believe the results are more accurate when using geometric mean returns (also known as the compounded annual growth rate).

Now that we've established that the optimal leverage for broad market indices is roughly 3x, let's let the 3x levered asset compete for portfolio space in an efficient frontier model. This isn't the final portfolio but is intended to ballpark the optimal allocation to apply to the volatility forecasting model.

Also, note that due to the fact that correlations tend to rise in times of high volatility, we shouldn't do this for the whole portfolio unless we have an extremely high risk tolerance. I'm choosing to apply leverage on the asset that I believe has the highest return over the risk-free rate, which the model shows as the Nasdaq.

You can also apply leverage to the bond portion of your portfolio using Treasury futures on shorter-term bonds to replicate long-duration ones at a higher return and similar risk profiles, but most people don't understand how to trade them, so I'm going to use TLT for the model demonstration. TLT also has a nice bonus in that it's constantly in demand by short sellers looking to hedge, which we can make money off of. More on this later.

## Leveraged ETFs change the efficient frontier

From the time I started writing for Seeking Alpha, I've been building a list of market anomalies, most of which are small edges that compound over time. However, as the list of anomalies has grown, I've found that you can optimize portfolios to exploit the constraints and biases of other investors and end up with a lot more money.

While each edge is individually small, I've built it up to the point where you can beat the S&P 500 by a large margin without a ton of effort on your part. My work on factor investing shows that you can beat the S&P 500 by about 2 percent per year by tilting your portfolio towards momentum, value, and small-cap stocks with a quality factor overlay on each without resorting to any fancy strategies. I'd like to update this model plus show you some of my work on volatility forecasting models, volatility targeting, and the provocative leveraged ETF strategy I developed using the Nasdaq.

Right away, when you introduce a leveraged Nasdaq ETF to the model, it changes the efficient frontier. At higher risk allocations, the model tends to take positions in value stocks, small-cap stocks, and government bonds, coupled with relatively small (but leveraged) positions in the 3x Nasdaq fund. This works really well because value stocks and momentum (as defined by the Nasdaq 100) tend to combine well in concert, far better than, for example, value stocks and leveraged S&P 500. In turn, stocks (with or without leverage) tend to combine well with long-term government bonds. The recent selloff has confirmed the benefit of long-term Treasuries, which are up about 6 percent since the selloff began.

*Source: Portfolio Visualizer*

A simple implementation of this portfolio model beats the S&P 500 by 4.2 percent annualized with the same level of risk.

*Source: Portfolio Visualizer*

However, astute readers will see this and note that the model portfolio has only seen a bull market.

This was a big problem to overcome at first. However, I was able to address it by using two methods. The first is by using simple volatility forecasting models to adjust the exposure to the leveraged ETF based on the volatility environment (just like blackjack card counters use the count of the deck).

It sounds crazy, but there's actually mountains of research showing that volatility can, to some extent, be forecasted. If you can forecast volatility, you still can't forecast total stock returns, but you *can* forecast the returns of leveraged portfolios. Basically, if you can build an accurate volatility forecasting model, you can predict the future returns of leveraged ETFs, which are implicitly short volatility.

The second method I use here is called a core-satellite portfolio, which I'll explain in a second. It's designed to harvest the gains from the leveraged ETF strategy in good times and replenish the satellite portfolio in bad times. That way you're not betting the farm, and your returns for the satellite portfolio tend to be closer than the arithmetic mean than the geometric mean. This reduces your drawdowns and increases your overall returns.

## Volatility forecasting and the core-satellite portfolio

When I was starting college, I took a class on meteorology. It was honestly my favorite class I've ever taken. When I was there they taught us various models for forecasting. None were perfect, but they allowed us to make educated guesses as to how the weather would be. What I found particularly interesting was that they used to use these models called CAPE models to predict severe thunderstorms.

It turns out that predicting volatility isn't much different than predicting the weather. While we can't predict volatility with 100 percent accuracy, we can predict it enough to make money. Research shows that predicting volatility is much, much easier than predicting future stock returns.

There a couple of stylized facts that help us predict volatility in the stock market.

*This first is that volatility today tends to correlate with volatility tomorrow.*

*Source: Bloomberg via Lazard Research *

The R-squared number means that you can account for 35 percent of next month's volatility on average just based on this month's volatility. By contrast, the paper shows that the R-squared of next month's returns based on this month's returns is roughly 0.

*The second fact is that volatility tends to revert to the mean over time, particularly when volatility is high. *

After the storm passes, we can safely releverage. When you combine persistence and mean reversion, volatility forecasts are actually not that hard to pin down.

Award-winning research from Seeking Alpha contributor Charlie Bilello shows that the 200-day moving average is a great indicator of volatility.

*Source: Leverage for the Long Run*

A logical place to start a volatility forecasting model is by using the 200-day moving average in the S&P 500 as a signal. There's a very clear split on volatility environments going back to 1928 on this.

Additionally, Bilello's research shows that market crashes tend to cluster during times of high volatility. The way you can implement this is to have a factor-driven core portfolio that's invested at all times, combined with a satellite portfolio that takes positions in TQQQ or TLT depending on whether the volatility environment is predicted to be sunny or stormy.

In my testing, I looked at volatility targeting models and at 200-day moving average models. I found that a modified 200-day moving average model worked best for simplicity and effectiveness.

The core portfolio is a 60/40 portfolio. 45 percent of the portfolio is invested in VYM (value stocks/quality overlay), 15 percent is invested in IJR (small caps/ quality overlay), 33 percent is invested in TLT (long-term Treasuries), and 7 percent is invested in gold (hedges against inflation with roughly the same expected return as Treasuries). You could safely invest anywhere from 5-25 percent of your portfolio in the satellite portfolio without messing up your risk profile too much. The model uses 20 percent in the satellite portfolio and 80 in the core portfolio, rebalanced annually.

The satellite portfolio is invested either in TQQQ or TLT, depending on the volatility environment. During periods of risk-on, TQQQ performs strongly, and during periods of risk-off, TLT tends to shine.

Here's the full algorithm for the core satellite model.

The model results are based on a core-satellite model, where the long-term strategic core allocation is combined with a timing model based satellite portfolio. 80% of assets are allocated to the core portfolio with 4 assets and 20% of assets are allocated to the satellite portfolio. Market timing results from 2011 to 2018 for ProShares UltraPro QQQ (TQQQ) are based on 200 trading day simple moving average of SPDR S&P 500 ETF (SPY). The timing portfolio is invested in the asset when the signal asset adjusted close price is greater than or equal to the moving average, otherwise the portfolio is invested in iShares 20+ Year Treasury Bond ETF (TLT). Timing model trades are executed using the end of month close price each month based on the end of month signals. The portfolio is rebalanced annually. The selected year range for the timing test was automatically adjusted based on the available data for ProShares UltraPro QQQ (TQQQ) (Mar 2010-Nov 2018).

The results are pretty nice.

Portfolio Growth: Core Satellite

*Source: Portfolio Visualizer*

**Returns and Standard Deviation: Core Satellite**

*Source: Portfolio Visualizer*

The 80-core/20-satellite portfolio registered a CAGR of 17.8 percent and a standard deviation of 11.6 percent, compared to the S&P 500 with a CAGR of 12.6 percent and a standard deviation of 10.9 percent.

Astute readers are going to notice that I ran this model during a bull market. The TQQQ didn't exist prior to 2010 but SSO, a 2x leveraged S&P 500 fund did exist back then.

Here's how the model did through the 2008 crash and recovery.

**Portfolio Growth: 200-day MA in S&P 500**

*Source: Portfolio Visualizer*

You should note that volatility tends to cluster around the 200-day moving average and tends to remain elevated for extended periods of time after a spike. We shouldn't blindly put the Range Rover on autopilot on the highway and hop in the back seat, but if we intelligently design the autopilot and use discretion to get out when volatility spikes, we do very, very well.

There are certain areas where we need to either override the autopilot or design the algorithms to trade against the constraints of other investors.

**1. I think investors should rebalance their holdings on either the last few days of the month or use the 29.5-day lunar calendar (i.e. rebalance every full or new moon). **It's cryptic, but it works.

Tons of institutional investors rebalance on the first of the month, and there aren't many new ideas in finance, so everyone ends up buying and selling the same assets. Every rebalancing strategy I test tends to do worse than it otherwise would when executed on the first day of the month rather than at any other time.

**2. You need to be careful when the index is around the 200-day moving average.** The market tends to trade weirdly around the 200-day moving average, and volatility tends to cluster around it in my experience.

There are a couple ways to avoid problems around this. The first is that I would require the index to be 1 percent above or below for at least 3 days to be able to accurately forecast volatility. To avoid getting whipsawed, the model is also designed to trade once per month based on the simple volatility forecast. This is a good practice. The second is that if you go to TLT from TQQQ, you need to stay in TLT for 60 days minimum. Avoiding losses is more important than running up the score.

Also, just to prove that I didn't torture the data, here's a classic volatility targeting strategy invested in the same assets. You could run this too if you so chose to.

It returned a 17.5 percent CAGR with a Sharpe of 1.62. Those are really good numbers.

**Portfolio Growth and Returns: Volatility Targeting**

*Source: Portfolio Visualizer*

Here's the algorithm for the other model.

Market timing results from 2011 to 2018 are based on the annual target volatility of 12.00%. The timing portfolio adjusts the equity allocation monthly based on realized historical volatility and the specified target volatility using 1 calendar month volatility period. Timing model trades are executed using the end of month close price each month based on the end of month signals. The selected year range for the timing test was automatically adjusted based on the available data for ProShares UltraPro QQQ (TQQQ) (Mar 2010-Nov 2018).

The point to this isn't that you necessarily should try to time the market (even though buying dips isn't a bad strategy). The point is you can profit by forecasting volatility with the use of a leveraged portfolio.

By doing this, you're applying risk when the odds are most in your favor. When you account for all the angles, the core-satellite strategy beats the S&P 500 by roughly 6 percent per year at the same level of risk. The volatility targeting strategy beats it by a little less but takes less risk than the S&P.

## Securities lending is the cherry on top

Where things start to get really interesting is when you look at the borrow rates for TQQQ and TLT. For those of you who aren't familiar with securities lending, it is the practice of shareholders lending their shares to short sellers (often market makers and hedgers) for a fee. Fidelity has a securities lending program, as does Interactive Brokers. The typical borrow rate is 0.3 percent per year, but borrow rates can be much higher when a stock or ETF is in demand.

Most stocks and ETFs don't have a lot of demand from short sellers, but some are considered hard-to-borrow. When this happens, you can make anywhere 5 to 100 times the average borrow rate on stocks, depending on supply and demand. Tesla is a good example of a stock that was persistently hard to borrow.

The thing about TLT and TQQQ (and UPRO), however, is that there's persistent demand from short sellers for them.

*TQQQ's borrow rate fluctuated between 1 and 4 percent over the last year.*

Granted, your broker will take half the profit from securities lending, but assuming TQQQ averaged 2.5 percent for the year, you would pocket 1.25 percent of your invested capital in free money. This reduces your risk and increases your return, rain or shine. There is simply more demand from shorts for these assets than there is a supply of people willing to lend them. I'm almost reluctant to write about this for fear that people will dump supply onto the market, but here it is.

Note that as the market has fallen, short demand has fallen as short sellers have covered their hedges.

**TQQQ borrow rate on Interactive Brokers**

*Source: IB Borrow Desk*

**Borrow rates for TLT**

*Source: IB Borrow Desk*

TLT isn't typically quite as hard to borrow but still is well above average. However, adding 0.5 percent per year to your bond returns makes a big difference, boosting your long run returns from roughly 3.5 percent to roughly 4.

Securities lending isn't a fringe practice but is practiced behind the scenes by the Vanguards and Fidelity's of the world on a large scale. Short sellers are required to post daily collateral, and their brokers are on the hook if they don't pay.

Securities lending is a really nice cherry on top of the satellite strategy. It generates cash rain or shine and tilts the odds even further in our favor. The model runs are actually too low when you account for securities lending. The Sharpe ratio of the strategy inches towards 2 when you account for securities lending.

I've kicked around starting a fund to take advantage of these strategies with an old friend who's getting his master's in finance in the spring. If we had a strong prime brokerage relationship and sufficient capital, instead of getting half the money from the securities lending, we could hope to get like 90 percent of it. The economies of scale from securities lending could pay a lot of the fee we would charge for managing the fund. It would be a hedge fund strategy with a mutual fund price/liquidity.

It's just an idea at this point, but feel free message me what you think.

## Conclusion

You can combine strategies originally invented by card counters with asset allocation theory to make a lot of money. By combining strategies like factor investing, volatility targeting, and core-satellite rebalancing, you can achieve much higher returns than would otherwise be possible. If you have feedback or questions about the strategy, feel free to use the comment section.

*Good luck to all!*

-Logan C. Kane

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.