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Towards A Zero-Beta Stocks And Bonds Portfolio

Summary A low-risk investor may want to completely remove systematic risk associated with stock market trends (i.e. achieve portfolio beta of 0). You can do this by pairing an S&P 500 mutual fund or ETF with any negative-beta bond fund. The necessary allocation to the S&P 500 fund is given by c = beta / (beta – 1), where beta is the bond fund’s beta. The beta of a bond fund changes over time. One approach is to use a trailing 50-day moving average to estimate your bond fund’s current beta. Backtested performance of a zero-beta SPY/TLT strategy suggest very good raw and risk-adjusted returns since mid-2002 (CAGR 7.2%, MDD 21.4%, Sharpe ratio 0.049). Alpha and Beta of a Two-Fund Portfolio Alpha and beta are the intercept and slope, respectively, when you regress a fund or portfolio’s daily gains vs. daily gains for a standard index. In this article, I use the SPDR S&P 500 Trust ETF ( SPY) as the standard index. For a portfolio with some allocation to two different funds, the portfolio alpha is simply the weighted average of the two funds’ alphas, and the portfolio beta is the weighted average of the two funds’ betas. For example, suppose you pair SPY, which has alpha of 0 and beta of 1 by definition, with a bond fund that has alpha of 0.002% and beta of -0.1. If you allocated 25% to SPY and 75% to the bond fund, your portfolio alpha would be 0.25 (0%) + 0.75 (0.002%) = 0.0015%, and your portfolio beta would be 0.25 (1) + 0.75 (-0.1) = 0.175. One can show that when pairing SPY with a bond fund with some particular beta, the necessary SPY allocation for portfolio beta of 0 is given by c = beta / (beta – 1). Why Target Zero Beta? It may sound strange, but a portfolio with net beta of 0 on average moves 0% for every 1% change in the S&P 500. In other words, it has no dependence on market movement. Generally when investors add exposure to bonds they retain some positive net beta, but much smaller than 1. By reducing beta, they shield themselves from major portfolio losses in the event that the stock market takes a big dip, while also sacrificing raw returns if the stock market performs well and gains, say, 8% a year. With beta of 0, you theoretically completely shield your portfolio from any market movement. Does that mean 0% portfolio gain every day? Thankfully, no. A zero-beta portfolio comprised of a stocks fund and a bond fund has positive alpha due to the bond allocation, which gives the portfolio growth potential. SPY and TLT Consider a two-fund stocks and bonds portfolio comprised of SPY and the iShares 20+ Year Treasury Bond ETF (NYSEARCA: TLT ). If you pool together all daily gains going back to TLT’s inception in July 2002, TLT has alpha of 0.043% and beta of -0.297. That means that various allocations to SPY and TLT can achieve portfolio alphas between 0% and 0.043%, and portfolio betas between -0.297 and 1. The figure below illustrates this. (click to enlarge) We see that 22.9% SPY/77.1% TLT achieves a portfolio beta of 0, with a nice portfolio alpha of 0.033%. Note that 22.9% agrees with our formula for SPY allocation to achieve zero beta: c = beta / (beta – 1) = -0.297 / (-0.297 – 1) = 0.229. In terms of Sharpe ratio, we’re doing pretty well at 22.9% SPY, although Sharpe ratio is maximized at 40.7%. But our goal here is zero beta, so we stick with 22.9% SPY. Note that alpha decreases uniformly with increasing beta in this scenario, since increasing beta requires decreasing the TLT allocation and capturing a lower percentage of its alpha. Historical Performance of 22.9% SPY/77.1% TLT Performance of the zero-beta SPY/TLT portfolio (with free daily rebalancing) since inception is shown below. (click to enlarge) The zero-beta portfolio ended above TLT and slightly below SPY, but had much better risk-adjusted performance, as you can see below. Table 1. Performance metrics from July 30, 2002, to Nov. 3, 2015. Fund CAGR (%) Max Drawdown (%) Sharpe ratio SPY 8.6% 55.2% 0.033 TLT 7.2% 26.6% 0.036 22.9% SPY/77.1% TLT 8.3% 19.3% 0.055 Issues With 22.9% SPY/77.1% TLT Portfolio Two issues with the zero-beta SPY/TLT portfolio come to mind: Actual beta changes over time, because TLT’s beta changes. There is no way we could have predicted that the SPY allocation to achieve average beta of 0 from 2002-2015 was 22.9%. Issue (1) means our zero-beta portfolio’s beta is not always 0. For example, here is how the TLT’s beta, and the 22.9% SPY/77.1% TLT portfolio’s beta, vary over the backtested period, using a 50-day moving average. (click to enlarge) We see that TLT’s beta varies quite a bit (-1.05 to 0.45). The 22.9% SPY/77.1% TLT portfolio’s beta range is smaller (-0.58 to 0.58), but still too great for a supposed zero-beta portfolio. A First Crack at a Truly Zero Beta SPY/TLT Portfolio A natural solution to both issues (1) and (2) is to monitor TLT’s beta over time, and update the asset allocation accordingly. For a first attempt I’ll arbitrarily choose a 50-day trailing moving average. Every day, I’ll calculate TLT’s beta according to the previous 50 daily gains, and re-allocate if the current portfolio beta based on the SPY and TLT balance and TLT’s current beta is outside of (-0.15, 0.15). But what happens when TLT’s beta turns positive? In that case there is no way to achieve zero beta with SPY and TLT. Three options come to mind: Hold cash until TLT’s beta turns negative again. Allocate 100% to TLT, since that is the closest to zero beta we can achieve with SPY/TLT and we utilize all of TLT’s alpha. Swap SPY for an inverse S&P 500 ETF (e.g. SH) to achieve zero beta. I think this is an important topic for future work. The third seems most defensible, but for simplicity I’ll use (2) here. TLT’s beta was only positive about 16% of the time, so it may not make a huge difference. The next figure shows portfolio beta for the adaptive zero-beta SPY/TLT strategy based on 50-day trailing moving averages. (click to enlarge) Much better. The 22.9% SPY/77.1% TLT portfolio and the adaptive zero-beta SPY/TLT portfolio had actual betas outside of (-0.1, 0.1) 63.3% and 43.1% of the time, respectively; outside of (-0.2, 0.2) 38.2% and 18.2% of the time; and outside of (-0.3, 0.3) 19.0% and 6.9% of the time. However, the adaptive strategy did require a whopping 1,264 trades, or an average of about 97 trades per year. I didn’t incorporate trading costs into this backtest, but they would be substantial unless your portfolio balance was very high. In terms of the usual performance metrics, the adaptive strategy had CAGR of 7.2%, MDD of 21.4%, and Sharpe ratio of 0.049. Note that if you only rebalance when portfolio beta goes outside of (-0.3, 0.3) rather than (-0.15, 0.15), you “only” need 618 trades (48 per year), but your portfolio beta deviates more from 0. That portfolio had a backtested CAGR of 8.6%, MDD of 26.6%, and Sharpe ratio of 0.052. Implementation Details Implementing this strategy takes a little work. Every day, you would have to download SPY and TLT’s closing prices for the past 50 days, calculate daily gains, and estimate TLT’s beta. You would then have to calculate your portfolio’s effective beta, and adjust your allocations if necessary. It isn’t actually too hard to do this. You can estimate TLT’s trailing 50-day beta in a few lines of R code using my “stocks” package. First install the package (you only have to do this once): > install.packages(“stocks”) Then load it and call the beta.trailing50 function: > library(“stocks”) > beta.trailing50(“TLT”) Then you’d have to log into your investments account, get your current SPY and TLT allocation, and calculate your effective beta (SPY allocation * 1 + TLT allocation * current beta). If effective beta is out of range, calculate the target SPY allocation (c = beta / (beta – 1)) and rebalance accordingly. It’s not ideal, but it really only takes a few minutes. My sense is that you could monitor TLT’s beta and your portfolio’s beta a little less stringently (e.g. once a month rather than every day) and still do all right. I plan to test this in future work. Conclusions I really like the idea of having a portfolio with considerable growth potential but no systematic dependence on stock market trends. TLT is a good candidate to pair with SPY for this purpose, because it is has positive alpha and negative beta. TLT’s average beta since inception is -0.297, which means you need to allocate 22.9% to SPY and 77.1% to TLT to achieve zero beta. Such a portfolio had excellent performance since 2002, but wasn’t entirely satisfactory because the actual beta often deviated far from 0, and you couldn’t have known to allocate 22.9% to SPY during that 13-year period to achieve average zero beta. While it may not be the optimal solution, I found that you could keep the portfolio beta much closer to 0 by monitoring TLT’s beta using a trailing 50-day moving average. Future work will focus on comparing the three approaches mentioned for when TLT’s beta turns positive, and on adjustments to keep the portfolio beta as close to zero as possible without suffering excessive trading costs.

Portfolio Optimization With Leveraged Bond Funds

Summary Bond funds are great because they generate alpha and usually have negative correlation with stocks. Using the leveraged version of a bond fund can drastically improve portfolio optimization (i.e. produce greater expected returns for a given level of volatility). I use SPY/TLT and SPY/TMF to illustrate. SPY/TLT Portfolio Optimization Consider a two-fund portfolio optimizaton problem based on the SPDR S&P 500 ETF Trust (NYSEARCA: SPY ) and the iShares 20+ Year Treasury Bond ETF (NYSEARCA: TLT ). Often the goal is to maximize the ratio of expected returns to volatility (Sharpe ratio). I don’t like that approach, because when you maximize Sharpe ratio, you tend to get a portfolio with great risk-adjusted returns but relatively small raw returns. Instead, let’s say the goal is to choose an asset allocation that maximizes expected returns for some level of volatility that you can tolerate. A good way to do that is to look at a plot of mean vs. standard deviation of daily returns for various asset allocations. Here is that plot using SPY and TLT data going back to 2002. (click to enlarge) The red curve shows mean and standard deviation of daily portfolio gains for various asset allocations. The points represent 10% asset allocation increments. The top-right point is 100% SPY, 0% TLT; the next point is 90% SPY, 0% TLT; and so on until the bottom-most point on the other end of the curve, which is 0% SPY, 100% TLT. Suppose you want no more than three-fourths the volatility of SPY, or a standard deviation no greater than 0.93%. Looking at the graph, we want to be right around the third data point from the upper-right end of the curve. That data point represents 80% SPY, 20% TLT. This is the optimal allocation for an investor who wants to maximize returns at three-fourths the volatility of SPY. SPY/3x TLT Portfolio Optimization Let’s see how replacing TLT with a perfect 3x daily TLT fund (no expense ratio, no tracking error) affects the portfolio optimization problem. (click to enlarge) The red curve shows the same data as in the first figure, it just looks different because I had to zoom out to accommodate the SPY/3x TLT curve. Here I show asset allocations in 5% increments for the blue curve. The lowest point on the blue curve is 100% SPY, 0% 3x TLT; the next point is 95% SPY, 5% 3x TLT; and so on until the rightmost point, which is 0% SPY, 100% 3x TLT. Interestingly, increasing 3x TLT exposure from 0% reduces volatility and increases mean returns up until about 25% 3x TLT. Over the volatility range 0.884%-1.235%, you can do substantially better in terms of maximizing mean returns for a given level of volatility with SPY/3x TLT compared to SPY/TLT. Going back to the first example, at a volatility of 0.93%, or three-fourths the volatility of SPY, the best mean return you can achieve with SPY/TLT is 0.039%, with 80.1% SPY and 19.9% TLT. The best you can do with SPY/3x TLT is 0.059%, with 65.5% SPY and 34.5% 3x TLT. Daily returns of 0.059% and 0.039% correspond to CAGRs of 16.0% and 10.3%, respectively. For another interesting special case, suppose you can tolerate the volatility of SPY. With SPY/TLT, the optimal portfolio is 100% SPY and 0% TLT, with a mean daily return of 0.040%. With SPY/3x TLT, the optimal portfolio is 48.4% SPY and 51.6% 3x TLT, with a mean daily return of 0.069%. Also noteworthy is the fact that SPY/3x TLT portfolios are capable of achieving volatility greater than SPY, while SPY/TLT portfolios are not. This could be appealing to aggressive investors. A Real 3x Bond Fund: TMF So far, I’ve shown that a perfect 3x daily TLT fund would be extremely useful for portfolio optimization. The next question is whether such a fund exists, and how “perfect” it is in regard to expense ratio and tracking error. There are a few options, but I think the most relevant is the Direxion Daily 20+ Year Treasury Bull 3x Shares (NYSEARCA: TMF ). TMF was introduced on April 16, 2009, and has a net expense ratio of 0.95%. The next figure shows that indeed TMF effectively multiplies daily TLT gains by a factor of 3. The correlation between actual TMF gains and 3x TLT gains over TMF’s 6.5-year lifetime is 0.996. (click to enlarge) I realize that TMF does not attempt to track 3x TLT, but rather 3x the NYSE 20 Year Plus Treasury Bond Index (AXTWEN). But practically speaking TMF operates very much like a 3x TLT ETF. Let’s go ahead and re-examine the mean vs. standard deviation plot for SPY/TLT, SPY/3x TLT, and SPY/TMF over TMF’s lifetime. (click to enlarge) This is interesting, and slightly disappointing. As in the previous plot, we see that SPY/3x TLT achieves drastically better mean returns for particular levels of volatility compared to SPY/TLT. The orange curve for SPY/TMF is also higher than SPY/TLT, but not as much so as SPY/3x TLT. It seems that TMF’s reasonable expense ratio and tiny tracking error do detract somewhat from the optimization problem. But we still see that increasing exposure to TMF from 0% to about 20% reduces volatility and increases expected returns, and SPY/TMF does much better than SPY/TLT for those who can tolerate volatility between 0.722% and 1.022%. Leveraged Bond Funds Multiply Alpha and Beta As I’ve argued in other articles (e.g. SPY/TLT and SPXL/TMF Strategies Work Because of Positive Alpha, not Negative Correlation ), the reason bond funds compliment stocks so well is that they generate positive alpha. A bond fund with zero or negative alpha has no place in any portfolio; you would be better off using cash to adjust volatility and expected returns. Anyway, bond funds are special because they generate alpha. Ignoring tracking error and expense ratio, a leveraged version of a bond fund multiples both the alpha and beta of the underlying bond index. We can see this with TLT and TMF. Over TMF’s lifetime, their alphas are 0.061 and 0.173, and their betas are -0.492 and -1.493, respectively. TMF’s alpha is 2.84 times that of TLT’s, and its beta is 3.03 times that of TLT’s. 3x greater alpha does not immediately render 3x TLT the better choice for portfolio optimization. You have to look at the effect on both expected returns and volatility, which are both functions of alpha and beta. Suppose you can achieve the same portfolio volatility with c allocated to SPY and (1-c) to TLT, or with d allocated to SPY and (1-d) to 3x TLT. If you subtract the expected return of the SPY/TLT portfolio from the expected return of the SPY/3x TLT portfolio, you get: (d-c) E[X] + [3(1-d) – (1-c)] E[Y] where X represents the daily return of SPY, and Y the daily return of TLT. Whether this expression is positive or negative depends on d, c, E[X], and E[Y] (which can also be expressed as alpha + beta E[X]). For SPY and TLT, the expression is always positive, which means that SPY/3x TLT provides better expected returns than SPY/TLT for any level of volatility that both can achieve. Conclusions Leveraged bond funds appear to be extremely useful for portfolio optimization. In the case of SPY and TLT, we saw that using a 3x version of TLT, like TMF, allows us to: Improve expected returns for a particular level of volatility. Achieve the same volatility as SPY, but with drastically better expected returns. Take on extra volatility beyond SPY’s in pursuit of greater raw returns. In practice, TMF’s expense ratio and tracking error detract somewhat from the performance of an ideal SPY/3x TLT portfolio. But SPY/TMF still allows for substantial improvements over SPY/TLT in terms of maximizing returns for a given level of volatility.

Be Careful Holding ETFs Long Term

Friday happy hour conversation in the Village reminds us why holding levered ETFs more than a day isn’t a good idea. Leveraged ETFs can suffer from disproportionate downside. Risks are added from levered ETFs taking on derivatives and exposure to debt markets.. Always consult your personal financial advisor before holding ETFs over the course of the long term. By Parke Shall Today we wanted to go over a topic that we were asked about on Friday at happy hour. Thom and I were having a conversation with someone who was talking about their portfolio to us. This person commented that they had been holding several leveraged ETFs over the course of months, and he did not understand why the moves that the ETFs were making did not seem congruent with the moves in the individual sectors that they represented. This brings us to a topic that we don’t think enough people know about or understand. Not all ETFs are created equal. Some ETFs are designed specifically to be held over the course of the long term. Good examples of these are ETFs like the iShares 20+ Year Treasury Bond ETF (NYSEARCA: TLT ) or the iShares Nasdaq Biotechnology ETF (NASDAQ: IBB ), two different style unlevered ETFs that we have talked about in our last four or five articles. TLT tracks the yield on treasuries, and IBB is an unlevered ETF that tracks the biotech sector. Each sector has an ETF, or several ETFs, similar to IBB for biotech. We have heard a lot about IBB over the last month because biotech has crashed, so we’re using that as an example. ETFs like IBB are helpful in showing sector moves proportionate to the broader market, like you can see in the below chart. IBB data by YCharts TLT tracks 20 year treasuries and provides a dividend according to their yield. TLT joins a host of other ETFs, like the Vanguards High Dividend Yield ETF (NYSEARCA: VYM ) which are meant to and designed to be hold for the longer-term, and have minimal fees. They take a small management fee, but they can be good to hold for conservative investors over the course of long-term. Any type of ETF for bonds especially makes bond investing a little bit easier, as sometimes buying individual bonds can be too costly for retail investors. So let’s look at what makes leveraged ETFs difficult to hold for more than a day or two, and why they should not be traded over the course of weeks or months. A simple example is this. If you buy a $50 2x levered ETF that goes up 10% you’re going to see a return of 20%, and that ETF will be priced at $60. The next day, the ETF falls back from $60-$50, you would expect the underlying to be the same as it was prior to the first day. The problem is that the drop from 60 to 50 is only about a 17% drop, meaning the underlying would only need to fall about 8 1/2% for you to lose the same amount that you made when the market grew 10% in the day prior. This type of attrition makes these instruments difficult to hold over the course of weeks or months. This is why it is not uncommon to see splits of different natures, including reverse splits, take place in these instruments. Like the gentleman we were speaking to yesterday, one needs to be aware of the mechanics of how leveraged instruments work before making what we believe to be a terrible mistake in buying them and letting them sit in your portfolio unwatched. The same goes for ETNs (exchange traded notes) that have major risk to the debt that’s been issued by a bank (or other institution) that presents counterparty risk. Sometimes with ETFs or ETNs that have these characteristics, you wind up seeing charts like this. UWTI data by YCharts In addition a lot of levered instruments will rebalance or reset on a daily basis, meaning that if the markets are volatile and not moving in one set direction, you could wind up taking losses on a day where the sector or underlying appears neutral. Finally, one needs to realize that these type of instruments may achieve their leverage from utilizing derivatives like options and sometimes debt instruments. These types of risks are not suitable for those looking to buy and hold or those investing for the long term. Before picking up an ETF to hold for the long term, make sure to check with your personal financial advisor.