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Summary I recently wrote about VFINX/VBLTX portfolios, and how to choose an asset allocation to maximize returns for the level of volatility you can tolerate. Swapping VFINX for a leveraged S&P 500 ETF makes the maximization game much more profitable. You can achieve a greater expected return for any particular level of volatility. You lose the benefit of completely free trades in a Vanguard account, but the improvement in expected returns is definitely worth it. Mathematically, using a leveraged version of VFINX allows you to increase your allocation to VBLTX, capturing a greater percentage of its alpha. I believe UPRO/VBLTX (or SPXL/VBLTX) can be an excellent core portfolio for many investors. VFINX and VBLTX In a recent article, I looked at the performance of various two-fund “stocks and bonds” portfolios comprised of Vanguard mutual funds. I paired the Vanguard 500 Index Fund Investor Shares (MUTF: VFINX ) with Vanguard bond funds of various durations, and found that the long-term bond fund, the Vanguard Long-Term Bond Index Fund (MUTF: VBLTX ), was generally the best choice in terms of maximizing expected returns for a particular level of volatility. Here is a slightly modified version of a graph from that article (curves for the other bond funds removed): (click to enlarge) To get you up to speed, the upper-right point on the curve shows that for a portfolio comprised of 100% VFINX, and 0% VBLTX, the mean and standard deviation of daily gains going back to 1994 are 0.042% and 1.192%, respectively. The next point, which represents 90% VFINX and 10% VBLTX, results in a slightly lower mean (0.041%) and considerably lower standard deviation (1.061%), making it arguably the better portfolio. You can see how mean and standard deviation vary as VFINX allocation increases in 10% increments all the way to 0% VFINX, 100% VBLTX. Notably, standard deviation is minimized for 25.8% VFINX, 74.2% VBLTX. So if you were a relatively conservative investor who wanted to take on no more than 75% of the S&P 500’s volatility, you would look at the second-from-the-right vertical line, and see that to maximize expected return you would need to be just below the 3rd data point from the right, or a VFINX allocation slightly below 80%. A nice aspect of a two-fund strategy based on Vanguard mutual funds is that trading costs are very low. The mutual funds have very low expense ratios and can be traded commission-free in a Vanguard account. 3x VFINX and VBLTX Something magical happens when you swap VFINX for a hypothetical 3x daily version of it: you get a drastically better expected returns for any given level of volatility. Take a look: (click to enlarge) (Note: Data points represent 10% allocation steps for VFINX/VBLTX, and 5% allocation steps for 3x VFINX/VBLTX. Also, daily gains for the hypothetical 3x VFINX fund were calculated by simply multiply VFINX gains by 3 and then subtracting a fixed value corresponding to a 1% annual expense ratio.) You can see that the blue curve offers drastically better mean returns than the red curve. For example, 90% VFINX/10% VBLTX (second point from the right on the red curve) has a standard deviation of 1.061% and a mean of 0.042%; 30% 3x VFINX/70% VBLTX (7th point from the bottom on the blue curve) has a very similar standard deviation of 1.064, with a much greater mean of 0.058%. In addition, with 3x VFINX/VBLTX you have the option of taking on more volatility than the S&P 500, and getting an excellent additional return. For example, if you can tolerate up to 50% more volatility than the S&P 500, you can achieve an 84.3% greater mean return (51.2% 3x VFINX/48.8% VBLTX: standard deviation 1.788%, mean 0.077%). CAGR vs. MDD I think the mean vs. SD plot best describes the performance of various VFINX/VBLTX portfolios. But CAGR vs. MDD is also very interesting, and highlights the huge improvement you get with 3x VFINX. (click to enlarge) You see drastically better raw returns for various maximum drawdowns with 3x VFINX/VBLTX compared to VFINX/VBLTX. One interesting special case, 35% 3x VFINX/65% VBLTX has about the same MDD as VFINX (55.4% vs. 55.3%), but with a much greater CAGR (14.7% vs. 9.1%). Also noteworthy, the CAGR for 3x VFINX/VBLTX portfolios starts to decrease once the allocation to 3x VFINX reaches about 70%. How to Invest in 3x VFINX Vanguard does not offer a leveraged version of VFINX (or any leveraged funds for that matter), but there are several 3x daily S&P 500 ETFs to choose from. The ProShares UltraPro S&P 500 ETF (NYSEARCA: UPRO ) and the Direxion Daily S&P 500 Bull 3x Shares ETF (NYSEARCA: SPXL ) are two options. They both have expense ratios right around 1%, and both have done an excellent job tracking 3x daily S&P 500 gains over their 6-7 year lifetimes. I know some readers will take issue with the fact that my results are based on sort of “fake” data, as I just multiplied daily VFINX gains by 3 to simulate a leveraged version of the fund (or, equivalently, the performance of UPRO or SPXL before they were around). I wouldn’t worry about this too much. All signs indicate that daily leveraged ETFs like UPRO and SPXL have very minimal tracking error. Mathematical Basis Intuitively, the reason 3x VFINX/VBLTX provides better mean returns for a given level of volatility is that it allows for a greater allocation to the alpha-generating VBLTX. Suppose you can achieve a volatility of 1% with either 90% VFINX/10% VBLTX or 40% 3x VFINX/60% VBLTX. Which will have greater expected returns? The second, because it retains 40% of VBLTX’s alpha rather than only 10%. Now for a more mathematical approach (feel free to skip). Consider a VFINX/VBLTX portfolio where C represents the proportion allocated to VFINX, and (1-C) the allocation to VBLTX; and a 3x VFINX/VBLTX portfolio where D represents the proportion allocated to 3x VFINX, and (1-D) the allocation to VBLTX. Suppose we start at the top-right part of the first figure (i.e. C = D = 1) and decrease both C and D to the point where both portfolios have the same volatility. It is easy to see that D will be less than C, i.e. you will have to allocate less to 3x VFINX than to VFINX to achieve a certain portfolio volatility. So the two portfolios have the same volatility, and D < C. Let's compare their expected returns. Let X = daily VFINX return and Y = daily VBLTX return. The first portfolio's daily return, say Z 1 , is given by Z 1 = C X + (1-C) Y. The second portfolio's daily return, say Z 2 , is given by Z 2 = 3D X + (1-D) Y. How do Z 1 and Z 2 compare? Let's subtract their expected values, and see if we can figure out if the difference favors one or the other. E(Z 2 ) - E(Z 1 ) = [3D E(X) + (1-D) E(Y)] - [C E(X) + (1-C) E(Y)] = [3D - C] E(X) + [(1-D) - (1-C)] E(Y) We know E(X) and E(Y) are both positive (otherwise we wouldn't invest in stocks or bonds). The coefficient [(1-D) - (1-C)] is also positive since D < C. Thus the entire expression will be positive as long as 3D > C, or equivalently D is greater than one-third of C. I’m sure there’s some way to prove this is true under certain circumstances. But it’s good enough to just look at a plot of C and D vs. volatility, and observe that indeed 3D > C (i.e. dotted black line is above blue line), except at the very left side of the graph. (click to enlarge) Conclusions The more I think about leveraged ETFs, the more valuable I realize they are. Here, I show that you can drastically improve performance of a S&P 500/long-term bonds portfolio by simply replacing the S&P 500 fund with a 3x version. Whatever level of volatility you are willing to tolerate, you can achieve higher expected returns by simply using a leveraged S&P 500 fund. The reason is positive alpha. Using a leveraged stocks fund lets you achieve a particular level of volatility while allocating a greater percentage of your assets to an alpha-generating bond fund. More capital generating more alpha means greater returns. The results here are shown for VBLTX, but the main points should also hold for other long-term bond mutual funds or ETFs. Additionally, for those wary of investing in long-term bonds given that interest rates are about to rise, I would suggest considering a similar approach with a short or intermediate-term bond funds. Scalper1 News
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