**Summary**

- A one-third UPRO, two-thirds cash portfolio behaves almost exactly like the S&P 500, but with one-third the investment.
- Which means if you have the funds to do it, you could use UPRO to boost your effective annual Roth IRA contribution from $5,500 to as high as $16,500.
- This article lays out the theory, implementation, and historical performance of the UPRO/cash Roth IRA strategy.
- It's perfectly legal. Trust me - I have 55 SA followers and a Twitter.

**One-Third UPRO + Two-Thirds Cash = 100% S&P 500**

In a recent article, I presented statistical properties of several two-fund portfolios involving leveraged ETFs. I showed that you can combine a c-times daily leveraged ETF with cash to achieve any daily multiple of the index between 0 and c. For a leveraged ETF that closely tracks the index, the expected daily return of the portfolio is (c x d) times that of the underlying index, where d is the proportion allocated to the ETF; the variance of daily returns is (c x d)^{2} times that of the underlying index. You can see that the ratio of expected daily return to standard deviation (the square root of variance, i.e. "volatility") is the same as the underlying index (and the leveraged ETF itself) regardless of c and d. In that sense, leveraged ETFs offer an opportunity to achieve a certain level of aggression without taking on disproportionate risk.

An interesting special case is an allocation of (1/c x 100)% in the leveraged ETF, and [(1 - 1/c) x 100]% in cash. Outside of tracking error, this portfolio behaves exactly the same as if 100% of the funds were in the index itself rather than split between the ETF and cash. To give a simple example, suppose we had $1,000 in ProShares UltraPro S&P 500 (NYSEARCA:UPRO), a 3x daily S&P 500 ETF, and $2,000 in cash. If the S&P 500 increased by 1% on the first day, the 3x ETF should increase by 3% and gain $30. Of course, if the full $3,000 was in the S&P 500, a 1% gain would also translate to a $30 gain.

**Rebalancing Costs**

As the balance in the leveraged ETF changes, the ETF/cash allocations deviates from the target. You have to rebalance occasionally to keep the allocations where you want them. In my opinion, a reasonable strategy is to reallocate whenever the effective multiple of the portfolio falls outside of 0.9-1.1. That translates to (c x d) falling outside of 0.9-1.1, or, for the UPRO/cash scenario, the proportion in UPRO falling outside of 0.300-0.367.

Rebalancing means transaction fees and, if you have to rebalance more than once a year (and aren't operating inside a Roth IRA), short-term capital gains taxes. Therefore, in most cases you would be much better off just investing in the index rather than maintaining a balanced portfolio of leveraged ETF and cash to mimic the index.

**UPRO** **+ Cash in a Roth IRA**

In the last section, we saw that a portfolio of $1,000 in UPRO, $2,000 in cash behaves almost exactly like $3,000 in the S&P 500 ("almost" because UPRO daily gains might not be exactly 3x the S&P 500's). If we can make $1,000 in UPRO act like $3,000 in the S&P 500, then we can also make $5,500 in UPRO act like a $16,500 investment in the S&P 500.

In other words, if we have the resources available, we can use 3x leveraged ETFs to boost our effective annual contribution to our Roth IRA from $5,500 up to as high as $16,500. That means more tax-free growth, which I think many Roth IRA investors would be excited about.

Leveraged ETFs based on the S&P 500 seem like natural choices here, but you could use any index that has 3x leveraged ETFs attached to it: Russell 2000, Russell 3000, NASDAQ 100, Dow Jones Industrial Average, etc.

**Implementation**

Suppose we have the resources to invest $16,500 over the course of the year, and plan to make equal contributions on the first trading day of each month.

One-twelfth of $16,500 is $1,375, so let's say we have $1,375 on hand on the first trading day of January. We invest one-third of this amount, $458.33, in UPRO. That's about three shares of UPRO (which closed at $137.02 on March 30, 2015). For now let's ignore the problem of whole shares, and imagine we could get the exact proportions we wanted. We have $458.33 in UPRO and the remaining $916.67 in cash outside the Roth IRA.

After our first contribution, our portfolio of UPRO and cash has a balance of $1,375, and the one-third UPRO, two-thirds cash allocation makes it operate much like $1,375 in the S&P 500.

On the first trading day of February, we have an additional $1,375 available. Our total portfolio balance is the value of the UPRO holdings, plus $916.67 in leftover cash from the first month, plus $1,375 in new cash. At this point, we calculate one-third of the total balance (UPRO balance + $916.67 + $1,375), and buy or sell UPRO in an amount that gets us back to one-third UPRO, two-thirds cash. Most likely this would involve buying more UPRO, although if UPRO did extremely well in January, it is possible that we would need to sell some UPRO to get back to the one-third allocation. In that scenario, we would sell UPRO within the Roth IRA, and hold the freed-up funds inside the Roth IRA, e.g. in a money market fund.

We repeat the process from March to November, keeping track of our total contributions. On the first day of December, we rebalance while ensuring that we reach the $5,500 Roth IRA maximum. This might mean moving some extra cash into the Roth IRA.

As for the "whole share" problem, a natural solution is to get to the number of UPRO shares that makes the UPRO allocation as close to one-third of the portfolio balance as possible. As the portfolio balance increases, it becomes easier to find a whole number of shares that keeps the allocation near one-third.

If we're really set on having our portfolio mimic the S&P 500 throughout each month, we could rebalance anytime the UPRO allocation deviates a certain amount from one-third. As mentioned earlier, I would suggest rebalancing when the effective S&P 500 multiple falls below 0.9 or above 1.1, which translates to the UPRO allocation falling below 30% or above 36.7%.

**Proof of Concept:** **2010**

UPRO was introduced in June of 2009, so let's start by looking at how the "game the Roth IRA maximum" strategy would have played out in 2010.

I implemented the method in R following the approach described in the previous section. $7 fees were charged for each trade, and whole shares of UPRO were bought or sold at the closing price on the first trading day of each month. (Feel free to ask if anything is unclear, or take a look at the R code on my website.)

The left plot shows the balance in UPRO, cash outside the IRA, and cash inside the IRA throughout the year. We see that cash outside the Roth IRA was the biggest component, followed by UPRO and then cash inside the IRA. Contributions reached the maximum of $5,500 by the end of the year.

The center plot shows that we were able to maintain the one-third UPRO allocation pretty well throughout the year.

The right plot compares the UPRO/cash portfolio to $16,500 invested in the S&P 500 in monthly intervals (with no transaction fees). We see that the UPRO/cash strategy worked extremely well. You can hardly distinguish it from $1,375 monthly investments in the S&P 500.

**Performance from 2010 to 2014**

The next figure compares the UPRO/cash Roth IRA strategy to $16,500 invested annually in the S&P 500 from 2010 to 2014. The UPRO/cash portfolio does a great job mimicking $16,500 annual contributions to the S&P 500.

**Performance During a Market Crash**

It would be nice to see how the UPRO/cash strategy performs during down markets as well, e.g. during the market crash of 2008. UPRO has only been around since mid-2009, but it's easy enough to simulate a perfect 3x S&P 500 ETF by multiplying daily S&P 500 gains by 3. The next figure shows how the 3x ETF/cash Roth IRA strategy would have fared from 2006 to 2014.

We had a problem in 2008. Looking at the middle graph, we see that the 3x ETF's losses were such that we couldn't get back to the one-third allocation without going over the $5,500 annual maximum. Interestingly, investing the full $5,500 on the first day of 2009 wasn't even enough to get back to one-third 3x ETF.

Another $5,500 on the first day of 2010 got us close but still not quite back to one-third. We got back on target in 2011, but another dip rendered us underexposed to the 3x ETF again. Finally in 2012 we got back on track.

Looking at the right graph, interestingly enough the 3x ETF/cash combination actually protected us from further losses in 2008 (compared to 100% invested in the S&P 500). This makes sense because in periods of underexposure to the 3x ETF the portfolio will move less than the S&P 500. Anyway, the benefit was short-lived. As the market rebounded, our 2-3 year period of underexposure to the 3x ETF cost us quite a bit of growth. Therefore the 3x ETF/cash portfolio didn't do quite as well as the hypothetical $16,500 annual investment in the S&P 500.

**Closing Comments**

A lot of people contribute $5,500 to their Roth IRAs every year and would contribute more if they could. In this article, I demonstrated how you can use UPRO and cash to mimic $16,500 in annual Roth IRA contributions into an S&P 500 index fund. You can use the strategy with other leveraged S&P 500 ETFs, or with leveraged ETFs attached to other indexes.

While I don't think it would make a huge difference, a monthly rather than daily 3x leveraged ETF would be preferable. That would ensure that a properly weighted 3x ETF/cash portfolio would achieve the same monthly gains as the full balance in the S&P 500, without worrying about allocations drifting over the course of a month. There are a few 2x monthly S&P 500 ETFs available (DXSLX, SPLX), but no 3x monthly funds as far as I know.

The 2010-2014 period was obviously a bull market with very strong and steady growth. In down markets, the UPRO/cash strategy can get off track and temporarily not behave all that much like the S&P 500. This happens when market losses make the UPRO allocation fall below one-third, and you can't buy enough UPRO to get back to the target without exceeding the Roth IRA maximum. Such periods should be temporary if you continue to contribute to the Roth IRA. In periods of underexposure to UPRO, the UPRO/cash portfolio will realize lower gains (or losses) compared to the same total balance in the S&P 500.

One thing to consider is what to do with the growing "cash outside Roth IRA" balance (the orange lines on the figures). In the 2006-2014 analysis, this balance eventually grew to about $100,000. The vast majority of it can't be moved into the IRA due to the annual $5,500 limit. But it's still a necessary part of the portfolio, to make it behave like the S&P 500. You could invest this money into the market, e.g. an S&P 500 index fund outside of the Roth IRA. But then your overall portfolio would really be operating like a leveraged S&P 500 ETF rather than the S&P 500. One alternative is to invest the excess cash in a long-term T-bill. As of April 1, 2015, a 10-year T-bill yields 1.87% annually. You may as well earn 1.87% annually on the excess cash that you can't move into the Roth IRA anyway. Those earnings could offset transaction costs and UPRO's expense ratio and maybe make your UPRO + cash + T-bill portfolio perform better than 100% S&P 500.

All in all, I don't see any major drawbacks of this strategy. You do have some extra costs. First, you'll generally have to make one trade per month, which shouldn't be too damaging - $84 a year if you have $7 trades. Second, you'll pay a little more in expense ratios compared to putting the full balance in an S&P 500 index fund. For example, SPY has an expense ratio of 0.09%, while the effective expense ratio for a one-third UPRO, two-thirds cash portfolio is (0.95%/3) = 0.32%. But over time you would have to expect these costs to be worth the benefit of essentially tripling your contributions to a tax-free account. And you could probably recoup these expenses by investing extra outside cash in T-bills.

**Disclosure:** The author is long UPRO.

**Additional disclosure:** The author used Yahoo! Finance to obtain historical prices for the S&P 500 and UPRO, and used R to analyze the data and generate figures. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.