Summary A 50/50 stocks and bonds portfolio typically generates better risk-adjusted returns than a stocks-only portfolio. This is because bond funds generate positive alpha. For an S&P 500 index fund paired with an uncorrelated bond fund, the net beta is 0.5 and the net alpha is one-half the bond fund’s alpha. An easy way to improve raw and risk-adjusted returns is to allocate one-sixth to a 3x S&P 500 fund, and five-sixths to the bond fund. The portfolio beta is still 0.5, but portfolio alpha is five-sixths rather than one-half of the bond fund’s alpha. The strategy generalizes to asset allocations other than 50/50 and allows for non-zero correlation between the bond fund and the S&P 500. Fixed Stock/Bond Portfolios Personal investors typically increase exposure to bonds as they get closer to retirement, reducing risk and drawdown potential while also sacrificing raw returns. Consider a 50% stocks, 50% bonds portfolio based on a simple S&P 500 index fund and a total bond mutual fund or ETF. The beta for such a portfolio is simply the average beta of the two funds. If there is no correlation between the two funds, the portfolio beta is 0.5. That means it tends to move 0.5% for every 1% the S&P 500 moves, which of course reduces both growth potential and drawdown potential. The portfolio alpha for a 50/50 strategy is one-half the bond fund’s alpha. So if the bond fund has positive alpha due to maturing bonds and/or falling interest rates, the portfolio will have positive alpha. This is unlike a 100% S&P 500 portfolio, which by definition has zero alpha. Notably, net positive alpha is the reason that portfolios with both stocks and bonds generally have better risk-adjusted returns than portfolios with only stocks. If bond funds didn’t generate positive alpha, you’d be better off allocating a fixed percentage to cash rather than bonds to reduce your portfolio’s beta. The same logic here applies to asset allocations other than 50/50. For example, the net alpha and beta for a 20% S&P 500, 80% total bond fund would be four-fifths the bond fund’s alpha and 0.2, respectively. Again, we are assuming zero correlation between the two funds for the moment. Fixed Stock/Bond Portfolios With Leverage A common approach to achieve a net beta of 0.5 is to allocate 50% of assets to an S&P 500 fund, and 50% to a bond fund. But we can gain a notable advantage by using a leveraged S&P 500 fund to achieve the same beta. If we used a 3x daily S&P 500 fund, we would need to allocate 16.67% of assets to the 3x fund and the remaining 83.33% to the bond fund. Our portfolio beta is still 0.5, but our portfolio alpha is now five-sixths (rather than one-half) the bond fund’s alpha. A higher alpha for the same 0.5 beta translates to better raw and risk-adjusted returns. A More General Framework Suppose you wish to achieve some target beta by combining a leveraged S&P 500 fund and a particular bond fund. Let a represent the allocation to the leveraged S&P 500 fund. This is what we want to calculate. Let b represent the bond fund’s beta. Let c represent the leveraged fund’s target multiple. Let beta represent your desired portfolio beta. The necessary allocation to the leveraged fund is given by: a = ( b eta – b ) / ( c – b ) For a concrete example, suppose we wanted to use ProShares UltraPro S&P 500 (NYSEARCA: UPRO ), a 3x daily S&P 500 ETF, and Vanguard Total Bond Market ETF (NYSEARCA: BND ), to achieve a portfolio beta of 0.75. For b , I’ll use BND’s beta since inception, which is -0.035. Our target beta is 0.75 and c is UPRO’s leverage multiple, which is 3. a = (0.75 – -0.035) / (3 – -0.035) = 0.259. So we need to allocate 25.9% of our assets to UPRO, and the remaining 74.1% to BND. By doing so, we’ll retain 74.1% of BND’s alpha (which is 0.0191%). If we had used SPY rather than UPRO, we would have retained only 24.1% of BND’s alpha. The portfolio alphas would be 0.0142% and 0.0046%, respectively. Practical Considerations The main drawback of my approach is that it requires more frequent re-balancing to maintain a target asset allocation. This translates to more trading fees and possibly more short-term capital gains taxes. Also, leveraged funds have negative alpha due to their expense ratios. For example, UPRO’s expense ratio of 0.95% translates to a daily alpha of -0.0038%. For the above example, a 25.9% allocation to UPRO would contribute an alpha of -0.00098% (25.9% of -0.0038%), which is very small compared to BND’s alpha contribution of 0.0142% (74.1% of 0.0191%). An Illustration With UPRO and BND Time to put my money where my mouth is. Let’s look at growth of $100k for various target betas achieved by combining SPY with BND, and by combining UPRO with BND. For beta of 0.1, I rebalance whenever the effective beta goes outside 0.075-0.125; for beta of 0.25, 0.2-0.3; for beta of 0.5, 0.45-0.55; for beta of 0.75, 0.7-0.8; and for beta of 0.9, 0.85-0.95. I deduct $7 for each trade (i.e. $14 per rebalance) and assume BND has a beta of -0.035 throughout. (click to enlarge) Performance metrics are given below. Table 1. Performance metrics for SPY/BND and UPRO/BND portfolios with various target betas. Beta Funds Trades Final Bal. ($1k) CAGR (%) Sharpe MDD (%) Alpha (%) 0.10 SPY/BND 3 140.7 5.6 0.116 4.2 0.00018 UPRO/BND 33 143.2 5.8 0.113 4.7 0.00019 0.25 SPY/BND 3 156.4 7.3 0.109 4.2 0.00015 UPRO/BND 35 162.7 8.0 0.110 5.1 0.00018 0.50 SPY/BND 3 183.1 10.1 0.080 8.1 0.00009 UPRO/BND 82 197.5 11.4 0.090 7.7 0.00015 0.75 SPY/BND 2 214.8 12.9 0.068 12.9 0.00005 UPRO/BND 125 237.4 14.7 0.078 12.0 0.00013 0.90 SPY/BND 0 236.7 14.6 0.064 16.9 0.00001 UPRO/BND 153 264.2 16.6 0.074 14.9 0.00011 It makes sense that we see better performance with UPRO/BND with increasing target beta. The greater the target beta, the more we have to allocate to SPY in the SPY/BND portfolio, and the less alpha we retain from the BND allocation. UPRO allows us to allocate more to BND and thus utilize more of its alpha. Risks There are some reasons for caution when trading leveraged funds. I want to briefly re-iterate similar points as in my recent article, A Simple SPY Top-Off Portfolio . If SPY has an intraday loss greater than 33.33%, you could lose your entire balance in the leveraged ETF. UPRO and other leveraged S&P 500 ETFs have historically done an excellent job achieving their target multiple, but there is no guarantee they will continue to do so going forward. In between rebalancing periods, you can suffer some irrecoverable losses due to volatility decay. I would add that the strategy presented in this article uses leveraged funds, but only to achieve a net portfolio beta somewhere between 0 and 1. In that sense, some of the concerns normally associated with leveraged funds do not apply here (e.g. extreme volatility and potentially catastrophic drawdowns). Conclusions The “bonds” part of a stocks and bonds portfolio reduces risk. But so would cash. The reason we prefer bonds is that they generate positive alpha, which improves risk-adjusted returns. Typically, a stocks and bonds portfolio utilizes only a fraction of the bond fund’s alpha. An easy way to increase that fraction is to use leverage. Historical data for UPRO and BND support the notion that using a leveraged fund in place of SPY allows you to capture more a bond fund’s alpha, thus improving both raw and risk-adjusted returns.