**Background on Strategy**

I believe Harry Long wrote the first S&P 500/20+ Year Treasury article on SA. His article, A Weird All-Long Strategy, was published in August 2014. Mr. Long didn't mess around with the SPDR S&P 500 Trust ETF (NYSEARCA:SPY) and the iShares 20+ Year Treasury Bond ETF (NYSEARCA:TLT), but rather went straight for the 3x leveraged approach of pairing the Direxion Daily S&P 500 Bull 3X Shares ETF (NYSEARCA:SPXL) with the Direxion Daily 30-Year Treasury Bull 3x Shares ETF (NYSEARCA:TMF). He recommended 50% SPXL, 45% TMF, and 5% the iPath S&P 500 VIX Short-Term Futures ETN (NYSEARCA:VXX), rebalanced annually.

In a second article, Mr. Long moved to 50% SPXL, 40% TMF, and 10% with the VelocityShares Daily 2x VIX Short-Term ETN (NASDAQ:TVIX). In a third article, he arrived at 30% SPXL, 30% with the VelocityShares Daily Inverse VIX Medium-Term ETN (NASDAQ:ZIV), 30% TMF, and 10% TVIX.

Frank Grossmann wrote an article in November 2014 titled The SPY-TLT Universal Investment Strategy. He discussed several strategies involving SPY and TLT, which basically used different rules for choosing allocations to each fund. Mr. Grossmann favors a "trailing modified Sharpe ratio" method, where your SPY-TLT allocation is the one that would have minimized (a version of) the Sharpe ratio over the last 63 or 72 days.

A recent article by Mr. Grossmann introduced a similar strategy that used 3x leveraged versions of the two funds, SPXL and TMF.

Mr. Long's constant allocation approach takes the view that an S&P 500/20+ Year Treasury portfolio has good properties in general, and makes no attempt to predict the market. Mr. Grossmann's approach uses the idea that the two funds perform well together, but also operates under the assumption that recent market conditions will tend to continue into the near future.

**Why SPY and TLT?**

If you're thinking about trying a SPY/TLT or SPXL/TMF portfolio, you should have a feel for why the two funds work well together.

Mr. Long and Mr. Grossmann focus their articles mainly on historical performance of the strategies and don't spend too much time discussing why the two funds should complement each other. However, I did find a few snippets.

In his third article, Mr. Long classifies the four funds into "return-generating components" (SPXL, ZIV) and "hedging components" (TMF, TVIX). The idea is that the portfolio's return-generating components foster growth in a strong market, and its hedging components limit losses in a down-market.

Classifying TMF as a hedging component is reasonable in the sense that TMF has negative beta. However, TMF has gained over 200% in the past 5 years. Certainly, historical performance of the four-fund combination has benefited from TMF's remarkable performance in recent years, and TMF is not simply acting as a hedge.

In his second article, he notes that "we have succeeded in dramatically reducing the correlation to the SPY from 0.39 to 0.19."

But low correlation with the market does not generally imply a good portfolio. You can reduce correlation with the market by combining any positive beta fund with an inverse S&P 500 ETF. But doing so will not generally boost returns or improve risk-adjusted returns.

Mr. Grossmann emphasizes the negative correlation between TLT and SPY (or equivalently TMF and SPXL). But negative correlation with the market simply means that TLT and TMF have negative beta. Many funds have negative beta; that doesn't mean that if you pair them with SPY you end up with good performance.

In the next section, I argue mathematically that strong historical performance is primarily due to TLT and TMF's positive alpha.

**The Math of Two-Fund Portfolios**

Suppose we have two funds, fund 1 and fund 2. To get an idea for how they tend to move with the market, you can regress each fund's daily gains, Y_{1} and Y_{2}, against daily gains of the S&P 500, X:

Y_{1} = alpha_{1} + beta_{1} X + e_{1}, e_{1} ~ (0, sigma_{1}^{2})

Y_{2} = alpha_{2} + beta_{2} X + e_{2}, e_{2} ~ (0, sigma_{2}^{2})

Here e_{1} and e_{2} represent random errors around the regression line, which are assumed to have mean 0 and some constant variance (sigma_{1}^{2} and sigma_{2}^{2}).

If we allocate some proportion of our money, say c_{1}, to the first fund, and the remaining (1 - c_{1}) to the second fund, then daily gains for the portfolio, Z, are given by Z = c_{1} Y_{1} + (1 - c_{1}) Y_{2}.

Based on the regression models for Y_{1} and Y_{2}, the regression model for Z is as follows:

Z = (c_{1} alpha_{1} + c_{2} alpha_{2}) + (c_{1} beta_{1} + c_{2} beta_{2}) X + e, e ~ (0, c_{1}^{2} sigma_{1}^{2} + c_{2}^{2} sigma_{2}^{2} + Cov(e_{1}, e_{2}))

A few things to note here:

- Alpha for the two-fund portfolio is (c
_{1}alpha_{1}+ c_{2}alpha_{2}), and beta is (c_{1}beta_{1}+ c_{2}beta_{2}). - In other words, alpha and beta for the two-fund portfolio are somewhere between the individual alphas and betas for each fund.
- Cov(e
_{1}, e_{2}) represents additional covariance between Y_{1}and Y_{2}not simply due to having betas with opposite signs.

The third point is particularly relevant for SPY/TLT or SPXL/TMF combinations. The negative correlation here is simply due to the fact that TLT has a negative beta and SPY has a beta of 1. There is no "extra" correlation beyond what you always have with one positive beta fund and one negative beta fund.

An interesting observation is that it is literally impossible to improve upon the market's expected daily return-to-volatility ratio without having positive alpha. Let's consider just the first fund. If daily gains are linearly related to the market's then we have E[Y_{1}] = alpha_{1} + beta_{1} E[X]. The variance of daily gains is V[Y_{1}] = beta_{2}^{2} V[X] + sigma_{2}^{2}. The ratio of expected return to standard deviation is (alpha_{1} + beta_{1} E[X]) / sqrt(beta_{1}^{2} V[X] + sigma_{2}^{2}). The error variance term sigma1^{2} is non-negative, so if alpha_{1} is 0, then the ratio is less than E[X]/sqrt(V[X]). In other words, you're not getting any better risk-adjusted returns than SPY.

What about negative correlation? Doesn't matter.

If alpha_{1} and alpha_{2} are both zero, then the expected daily return of our two-fund portfolio is E[Z] = (c_{1} beta_{1} + c_{2} beta_{2}) E[X]. The variance is (c_{1} beta_{1} + c_{2} beta_{2})^{2} V[X] + V[e_{1} + e_{2}]. Even if Y_{1} and Y_{2} are perfectly negatively correlated, and we choose weights such that Z's error variance is 0, we can only match the expected return-to-volatility ratio of the S&P 500. All our two-fund portfolio would accomplish is a certain (unplanned) beta. But there are easier ways to get a certain beta (leveraged ETFs + cash as described in this article).

Think about it this way. Suppose you have two funds with perfect negative correlation, but each with zero alpha. You could allocate cash to each fund in order to completely remove dependence on the market. What would you have? A portfolio that gains 0% every day regardless of market movement. Cash under your mattress.

If you choose some other combination of weights, you end up with a leveraged ETF - a portfolio with zero alpha and some beta, which moves deterministically with the market. But if that's what you're after, just buy a leveraged ETF. And it's extremely rare to have perfect negative correlation, so more likely you'll end up with a poor man's leveraged ETF - a portfolio with zero alpha and some beta, with greater volatility than a leveraged ETF with the same beta.

**We Need Alpha. TLT, TMF, and ZIV Have It**

The point of going through all of this math is to show that TLT/SPY, TMF/SPXL, and TMF/SPXL/ZIV/TVIX portfolios are performing well because they use funds with positive alpha.

The positive-alpha funds are TLT, TMF, and ZIV, as seen below.

Alpha and beta estimates for TLT, TMF, and ZIV, using daily data from each fund's inception through April 10, 2015.

Ticker | Alpha | P-value | Beta | P-value |

TLT | 0.0445 | 0.001 | -0.2978 | <0.001 |

TMF | 0.1841 | 0.005 | -1.4966 | <0.001 |

ZIV | 0.0555 | 0.040 | 1.3909 | <0.001 |

Particularly impressive is TMF's alpha of 0.1841, meaning it gains on average 0.1841% on days when the S&P 500 is unchanged (which averages up to 58.7% annually). That's some good alpha.

TLT and TMF have negative beta. Equivalently, they are negatively correlated with the S&P 500.

The strong performance of TMF/SPXL in recent years makes perfect sense considering TMF's positive alpha. A 50/50 split with SPXL should have alpha one-half that of TMF alone, or 0.0921, and beta right in the middle of -1.4966 and 3, or about 0.75. It's almost like a slightly deleveraged S&P 500 ETF, but with daily gains uniformly shifted up by 0.092%.

**Can TLT/TMF Maintain Positive Alpha Going Forward?**

There's no arguing the fact that TLT and TMF have been high-alpha funds in recent years. The real question is whether they will continue to have high alpha going forward.

Based simply on the fact that TLT has had positive alpha pretty consistently since 2003 (11 out of 13 years, 2 significantly greater than 0), I would lean towards predicting that TLT and TMF will continue to have positive alpha going forward. But I doubt TMF will continue to triple every 5 years. Anyway, the strategy's performance, as I see it, completely depends on positive alpha going forward. So I recommend really thinking about this issue and only attempting a TLT/SPY or TMF/SPXL strategy if you have confidence that TLT and TMF will continue to have positive alpha.

**SPY Works, But We Can Do Better**

The way I see it, the S&P 500 part of TLT/SPY and TMF/SPXL doesn't add much to the strategy. It basically allows us to adjust beta while pulling alpha down towards zero. We'd be better off using a positive beta fund that also has positive alpha. XIV and ZIV come to mind, and indeed Mr. Long has written several articles on a 40% XIV, 60% TMF strategy (e.g. this one). An interesting issue with XIV is that it is non-linearly related to the S&P 500. That makes it a bit harder to choose weights to achieve a certain alpha and beta, since portfolio gains won't be linearly related to S&P 500 gains.

**Don't Get Attached to TLT/TMF**

I would argue that there's nothing really special about the TLT/SPY or TMF/SPXL combination. It basically takes a positive alpha fund and pairs it with the market to produce a positive alpha, shallow beta portfolio. You can do the same thing with any positive alpha fund. (Not that it's easy to find such a fund).

My view is that "all" you need to get rich fast is two positive alpha funds: one with negative beta, and one with positive beta. You can combine two such funds in extremely profitable ways. For example, a natural choice is to weight them such that you get zero beta and positive alpha - a portfolio that grows (on average) every day regardless of market movement. You can theoretically do this with TLT and SPY, but using SPY's zero alpha hurts the net alpha of the portfolio. Pairing TMF with ZIV or XIV (e.g. Mr. Long's 60% TMF, 40% XIV) seems very promising.

**Conclusions**

If you're thinking about implementing a strategy with good historical performance, you need to have a feel for what makes the strategy work. Only then can you make an informed guess at whether it will continue working in the future.

For the reasons described in this article, I believe that TLT/SPY and TMF/SPXL strategies work primarily because TLT and TMF have positive alpha. Negative correlation is not as important, for several reasons. First, in the absence of positive alpha, all you can do with negative correlation is adjust your portfolio's beta and essentially create a poor man's leveraged ETF. Second, if TLT and SPY happened to have positive correlation, you could just use an inverse S&P 500 ETF in place of SPY to achieve the same result.

If positive alpha is the driving force behind strong performance of TLT/SPY and TMF/SPXL, then it is only wise to implement one of these strategies if you are confident that TLT and TMF will continue to have positive alpha.

Finally, I should point out that Mr. Grossmann's articles do *not* advocate for a fixed allocation to TLT and SPY, or to TMF and SPXL. He uses an adaptive allocation method that has a "predict the coming market" component. Similarly, Mr. Long's articles on the topic involve three or four funds, not just the two. My goal is not at all to discredit these authors' ideas, but to better understand performance of TLT/SPY strategies in general.

**Disclosure:** The author is long XIV.

The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **The author used Yahoo! Finance to obtain historical prices for the S&P 500, TLT, TMF, and ZIV, and used R to analyze data. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.