# Leverage, from Philip McDonnell

July 13, 2007 |

There is a kind of trap that is implicit in maintaining a constant leverage ratio. Note that when the market falls one must sell to rebalance the ratio. After the market has risen one must buy more to rebalance. In a negative daily autocorrelation environment, this is exactly the wrong thing to do from a trading perspective.

An example: Suppose a stock is \$100 and we buy 2x for \$200. The stock drops to \$90 we have lost \$10 x 2 = \$20. We now have equity of only \$80. So we rebalance to own only \$160 worth of stock or 1.778 shares. The stock recovers to \$100. We profit by 17.78 so our equity is now \$177.78. We have lost \$22.22.

In contrast, if we had bought for cash and the stock recovered to \$100 we would break even. If we do not rebalance we also break even. But by not rebalancing we run the added risk that during a decline we are implicitly taking on a higher leverage ratio than the 2x originally intended.

Rebalancing frequency at higher leverage ratios such as 5x is fraught with danger. The best way to calculate the ratio at any given time is to find the leverage ratio and rebalance frequency which will optimize the log of the expected relative portfolio returns. Ideally one should use the empirical distribution of returns as the basis for this calculation.

## Charles Pennington writes:

I've tried to answer this question using daily total return data for SPY since 2/1/1993. I tried a series of leverage ratios from 100% to 1400%.

I assume that you pay margin interest of 6% annually (0.024% per day) on your borrowings. Of course one could do a little better on the calculation by including the time dependence of the margin interest rate.

Column labels (in order):
leverage ratio
\$1 grew to
compound annual % return

100%, \$4.45, 10.8%
150%, \$5.20, 12.0%
200%, \$5.50, 12.4%
250%, \$5.24, 12.0%
300%, \$4.51, 10.9%
400%, \$2.44,  6.3%
500%, \$0.87, -1.0%
600%, \$0.20,-10.5%
700%, \$0.03,-21.5%
800%, \$0.003, -33.3%
900%, \$0.00015, -45.3%
1000%, \$(5*10^-6), -56%
1400%, wipeout (lose more than everything)

– 200% leverage had the highest total return, but it was not much higher than the return for 100% or 150% leverage, and of course the risk and volatility was much higher.

– Wipeout occurred at 1400% leverage. However, this assumes that one rebalanced daily. If you only rebalanced once per year, then 500% leverage would have more than wiped you out during the 2002 S&P decline of 22.2%.

These are sobering findings that suggest you should not have steady-state S&P exposure exceeding about 150%. Possibly higher leverage ratios can be occasionally useful if you have some reason to think that the expectation is higher than average in the near future, but it doesn't look good to go above, say, 400% unless you have a real live crystal ball.

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