Why does rebalancing leveraged portfolios buy winners and sell losers?

Question

From "Diversification Return and Leveraged Portfolios" by EDWARD QIAN at http://www.master272.com/finance/memoire_2016/qian_diversification_return.pdf

Consider again a two-asset 200/100 portfolio with 200% long in Asset 1 and 100% short in Asset 2. Suppose Asset 1 returns 50% and Asset 2 returns 0%. At the end of the period. Asset 1 grows to 300% and Asset 2 remains at —100%, so that the net value of the portfolio doubles. As a result, the portfolio weights shrink to 150/50 (300/200 = 150% and 100/200 = 50%). To rebalance the portfolio to the original 200/100 target weights, we would buy an additional 50% of Asset 1 (the winner) and short an additional 50% of Asset 2 (the loser). It is easy to prove mathematically that when a leveraged portfolio has positive returns, gross leverage declines; thus, leverage would have to increase to get back to the original weights.

The way I see it, we start with a portfolio of assets at a 2:1 ratio and after a period it becomes 3:1, so to rebalance to the original allocation you would sell the asset that appreciated. I don't understand the math he puts forth here.

Answer 1

Portfolio as it stands:

 300% A
-100% B
Total: +200%. 
Ratio 1.5:0.5

Short additional 50% B. Now:

 300% A
-150% B
  50% Cash
Total: +200%.
Ratio 1.5:0.75

The problem is that you have to rebalance vs the total value of the portfolio. To get back to having the amount of A being 2x the portfolio total, you need to buy more A, to get it up to 400%.

I suspect part of the confusion is expressing both position sizes and ratios in percent. Consider it in millions of dollars: Originally you had $1m thus:

+$2m A
-$1m B
Total: +$1m
Ratio: 2:-1

Then

+$3m A
-$1m B
Total: +$2m
Ratio: 3:-1

Need to rebalance vs total value of $2m.

$a + b = 2,000,000 \\ a/(a+b)=2 \\ a = 4,000,000 \\ b = -2,000,000$

Change is:

+$1m A
-$1m B

i.e. short \$1m B, and buy $1m A.

+$4m A
-$2m B
Total: $2m
Ratio: -2:1

The point of the article is that whereas a long-only portfolio would always sell winners and buy losers when returns are positive, thus creating a mean-reversion profile, a portfolio containing short positions like this will end up following the trends; the more the performance of A and B diverges, the greater the divergence in the portfolio.

If, in the next investment period, A went back to its original value (1->1.5->1, i.e. returned -33.3%) and B continued to hold its original value (1->1->1), then the portfolio would lose \$1.33m, and have a total value of \$0.66m. By following the trend and creating a path-dependence, we lose 1/3 of our original portfolio value. A long-only portfolio of 2:1 would have gained only \$0.33m, rebalanced, and lost \$0.296m, and ended up at \$1.037m.

The thesis demonstrated here is that a portfolio containing only long positions is long vol - volatility without a resulting drift resulted in a positive return. By comparison, a portfolio with a short position was implicitly also short vol, which is perhaps something investors might not be aware of.