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What To Do When Your Stocks And Bonds Portfolio Reaches Minimum Volatility

Summary Investors typically increase exposure to bonds as they near retirement, hoping to reduce volatility and drawdown risk. It is very possible to reach a point where further increasing exposure to bonds will increase rather than decrease volatility. This phenomenon is more likely to occur with longer duration bond funds. Once you reach minimum volatility for a two-fund stocks and bonds portfolio, you can further reduce risk by (1) buying treasuries or (2) switching to a shorter term bond fund. There is no general result for which strategy is preferred, but (2) tends to give better returns and may be easier to implement. Expected Returns and Volatility as you Increase Bond Exposure Suppose you are implementing a basic stocks and bonds portfolio comprised of two Vanguard mutual funds: Vanguard 500 Index Fund Investor Shares (MUTF: VFINX ) and Vanguard Long-Term Bond Index Fund (MUTF: VBLTX ). Using historical data going back to Feb. 28, 1994, here is how expected returns and volatility of the VFINX/VBLTX portfolio vary with asset allocation. (click to enlarge) Here the top-right point represents 100% VFINX/0% VBLTX; the next data point is 90% VFINX/10% VBLTX; and so on until the bottom-most point, which is 0% VFINX/100% VBLTX. As you near retirement, you may increase your VBLTX allocation to reduce risk. If you go from 90% VFINX/10% VBLTX to 60% VFINX/40% VBLTX, for example, you reduce your expected returns a little (0.041% to 0.037%), while reducing volatility considerably (1.06% to 0.70%). Further increasing the VBLTX allocation reduces volatility, but only to a point. At 25.8% VFINX/74.2% VBLTX, you reach the leftmost point on the curve, and further increasing VBLTX allocation actually increases volatility while reducing expected returns. Of course, there is never a good reason to increase volatility and decrease expected returns. So looking back at the past 21.5 years, you would never have wanted to allocate more than 74.2% to VBLTX in a VFINX/VBLTX portfolio. Longer Duration Bond Funds Have Lower Critical Points The expected returns vs. volatility curve doesn’t always have a clear critical point like we saw for VFINX/VBLTX. In general, longer duration bond funds are more likely to exhibit this phenomenon. You can see this when you compare the curve for VFINX paired with VBLTX to VFINX paired with Vanguard’s short-term and intermediate-term bond funds, VBISX and VBIIX . (click to enlarge) Looking at the blue curve, VFINX/VBISX does have a minimum volatility point, but it’s at a very high VBISX allocation (4.3% VFINX/95.7% VBISX). Note however that if you’re using VFINX and VBISX you probably wouldn’t want to go higher than 90% VBISX, as doing so sacrifices considerable expected returns while reducing volatility very little (if at all). The green curve is in between the first two, with minimum volatility at 12.7% VFINX/87.3% VBIIX. I would not recommend going any higher than 80% VBIIX, though, from an expected returns/volatility standpoint. Reducing Volatility Beyond the Critical Point What do you do if you want to further reduce volatility after reaching your portfolio’s critical point? I see two reasonable options: Allocate some of your portfolio to treasuries (e.g. 10-year US treasury bonds). Swap for a shorter duration bond fund. Let’s go back to the first two-fund portfolio, VFINX/VBLTX. Suppose we’re at 25.8% VFINX/74.2% VBLTX and we recognize that we’ve reached minimum volatility. We would like to reduce volatility to one-fourth that of VFINX (the leftmost dotted line in the previous figures, at 0.298). We can’t do it with all of our assets allocated to VFINX or VBLTX. Let’s consider option (1). Allocating some of your portfolio to cash would pull the red curve down and to the left. But if you’re going to have cash, you may as well get some interest on it. So instead of cash let’s say we generate risk-free returns on whatever percentage we pull out of our VFINX/VBLTX portfolio, from investing those assets in US treasuries for example. The next figure shows the expected returns vs. volatility curves for various allocations to a risk-free investment that returns 1.5% annually. (click to enlarge) To clarify, the highest curve the same as we saw before; the next highest is 10% receiving risk-free 1.5% annual returns, and the remaining 90% split to VFINX/VBLTX in 10% increments; and so on until the lowest curve (which you can barely see), which is 90% risk-free 1.5% annual returns, and the remaining 10% split to VFINX/VBLTX in 10% increments. The first curve to extend to a volatility of 0.298 is the one with 40% allocated to the risk-free investment. For this portfolio, we would have to allocate the remaining 60% of our assets to 30% VFINX/70% VBLTX, to achieve an expected return of 0.0226% with volatility of 0.298%. Now let’s consider option (2). The next figure is the same as the last one, but with the curves for VFINX/VBIIX and VFINX/VBISX included. (click to enlarge) Interestingly, swapping VBLTX for VBISX lets us reach a volatility of 0.298 with a mean daily return slightly higher than that reached with VFINX/VBLTX and 40% risk-free. A 24.7% VFINX/75.3% VBISX portfolio has means returns of 0.0232%. A natural question is how the risk-free rate affects whether strategy (1) or (2) is better. For the Vanguard funds examined here, strategy (1) would always outperform strategy (2) if the risk-free rate was 4% or higher (i.e. rarely or never). Strategy (2) would always outperform strategy (1) if the risk-free rate was 0% (i.e. you held cash rather than treasuries). For risk-free rates between 0% and 4%, it really depends on the particular level of volatility you’re trying to achieve. Conclusions I think a lot of investors operate under the assumption that increasing exposure to bonds reduces volatility. But in fact there is often a point where further increasing exposure to bonds increases volatility and reduces expected returns. You don’t want to go past that point. To reduce volatility further than your two-fund portfolio allows, you can either allocate some of your assets to a risk-free investment, say US treasuries, or you can switch to a shorter duration bond fund. I favor the second strategy, as it tends to allow for greater expected returns and seems logistically easier to implement. More generally, I think it is very important to know where your portfolio is at in terms of the expected returns vs. volatility curve. You should have a good idea of how any potential change in asset allocation or choice of funds affects your portfolio’s characteristics.

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.