Scalper1 News
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. Scalper1 News
Scalper1 News