If I have a portfolio of stocks and want to add treasuries would it be better to add very long duration treasuries or a levered position in shorter duration treasuries? Why?
If you are looking to construct an optimum portfolio, you would do neither. Assuming your portfolio of stocks, long duration treasuries, and short duration treasuries are the universe of investable assets, one would construct a portfolio of these assets utilizing a mean-variance optimization. The most efficient, highest sharp ratio portfolio would be a combination of these assets and would represent the capital market portfolio. If one desired to take leverage, one would leverage this portfolio (rather than just the shorter duration treasuries). Just adding long duration treasuries, or short duration treasuries, would most likely result in a portfolio in an inefficient portfolio (one not on the efficient frontier).
If one were to just look for a diversifying asset, without looking at actual return, risk, and correlation numbers, I would guess that the shorter duration treasuries would be less correlated to stock returns and would be a better diversifier than longer duration treasuries. One would have to question the rationale of levering shorter duration treasuries. At the extreme, levering 4 week Treasuries, would probably end up costing you money. Your borrowing costs, due to your credit risk, would most likely be higher than that of the US Treasury.
Another rationale I have heard of adding Treasuries to a portfolio of equities is as a hedge against an equity market crash. The rationale being that US Treasuries would rally in such an environment and would offset losses in the equity portfolio. If this is the rationale for adding Treasuries to your portfolio of stocks, you would want longer duration Treasuries as the duration would provide more protection for a given move in rates.
Seen differently to Altracoon's perfectly correct answer, if your investable universe is stocks (think "benchmark"), then the bond decision is not so simple. You are 100% invested in stocks, and just want to know what flavour of bonds to add to this is likely to have a better vs worse risk-reward.
In this world, you can overlay a few long-bond futures, some 10s, a lot of 5s, or a ton of 2s on top of your equity.
Which begs the question of the correlation of different tenors to equities. That is, seen against the "carry and roll" of holding different bonds. Broadly speaking, short-dated bonds are more responsive to changes in monetary policy expectations. Long-dated to long-term nominal expectations. The "beĺly" (ie 5s) typically captures a mix of the two, but are often the most generous to hold for holding' s sake.
So you have to ask yourself whether it's the Fed or the US economy you're really worried about. Is it the damage from potential hikes or a slowdown that policy cannot prevent that you're worried about? That pretty much answers which side of 5s/10s you'd want to pivot on.
The belly vs pure proxy decision always says more about bond market conditions than anything about stock:bond dynamics. Curve dynamics rarely make much of a difference at the long-end. But the slope at 10s can make a huge difference. Think of this as the running cost/benefit of getting this duration hedge for your equities. 5s is typically the "best" duration to hold to get duration exposure.
You just have to be confident that as you move thus along the spectrum between policy and nominal growth, that this is the risk you really wish to hedge for your stocks.
Given that 2s often cease to represent anything more than a bet on Fed timing... and Long/Ultras can be subject to weird pension hedging dynamics... it's often worth turning the original question on its head. What merit is there bothering with 2s or 30s if 5s and 10s capture the essence of both without the funnies?