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.