The Caveman Forecaster wrote an interesting post comparing a portfolio invested in the Russell 1000 ETF, IWM, and one that was 2/3 in cash and 1/3 in BGU which is an ETF that returns three times what the Russell 100 does daily. The idea is that the leveraged ETF gives you roughly the return of the index on the upside and there is downside protection since most of the portfolio is cash.
This is the table he published:
| Portfolio | Up 50% | Down 20% | Down 50% | Down 75% |
| $100 IWM | $150 | $80 | $50 | $25 |
| $33.33 BGU: $66.67 cash | $150 | $80 | $66.67 | $66.67 |
One feature of this portfolio is that a large move down, 50% or greater, will wipe out the BGU part of the portfolio, leaving the cash. But now, the portfolio is all cash, so it doesn’t participate in the market anymore.
The table is correct if you make the move in one step. If you get there in two steps the second portfolio lags to the upside and does better to the downside, if the two steps in the downward move are both down. If one step is up and the other down, then the BGU / cash portfolio lags. The following table illustrates the point for a net 20% down move.
| Portfolio | Down 20% | Down 10%; Down 11.1% | Up 7%; Down 25.23% |
| $100 IWM | $80 | $80 | $80 |
| $33.33 BGU: $66.67 cash | $80 | $82.23 | $76.47 |
The first column is a one step move down 20%. The second column is a step down 10% and then another step down 11.1% which is a total loss of 20%. The third column is a step up of 7%, then a step down of 25.23%. Why does the BGU portfolio have this seemingly erratic behavior?
It has to do with compounding. The reason compounding grows faster than an arithmentic growth is the multiplying of terms. So the IWM portfolio grows in two steps to
$100 * (1 + r1) * (1 + r2) = $100 * [1 + (r1 + r2) + r1 * r2]
The first term is the principal, $100. The second term is the arithmetic return, r1 + r2, and the third term is what compounds the returns, r1 * r2. Usually, r1*r2 is small. If they are both one percent, multiplying them gives one hundredth of one percent, tiny. If both returns are ten percent, then r1*r2 is one percent. Now, it is beginning to matter. In the BGU portfolio, this term is multiplied by 3. So two ten percent moves will contribute 3% to the overall return.
It is this term, and all higher order terms, that is different between the two portfolios. When there are several big moves, these terms are large because they have the powers of 3 multiplying them.
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