Volatility drag and leverage

Why a 3× fund doesn't return 3× over time, and how swings quietly eat compound growth.

Average yearly return of the index10.0%

The arithmetic mean: the average of the daily returns, scaled to a year.

Index volatility15% a year

Leverage, reset every day2×

Negative leverage is a short fund: it aims to return the opposite of the index each day.

Years held5

Index, average return11.0%What the dial promised

Index, actual growth a year+9.8%Formula: +9.3%

2× fund, growth a year+17.8%2× the index would be +19.6%

2× fund lags its promise38%of simulated markets

One typical market, chosen because the index ends close to its median outcome.

The index averaged 11.0% a year but actually compounded at +9.8%: volatility drag cost about 1.1% a year. The 2× fund doesn't earn 2× that. It grew at a typical +17.8% a year, because its drag is 4 times as big.

Growth for every amount of leverage

Same index, same shocks, leverage from 3× short to 3× long. The curve is a hill, not a straight line.

Two days, one lesson

The index rises 10%, then falls 10%. It ends at $99.00 from $100. A 2× fund moves 20% each day and ends at $96.00.

The best leverage here

Growth peaks at 4.44×, where adding more return stops paying for the extra drag. Past roughly 8.9×, a leveraged fund grows slower than holding no leverage at all. Real funds also pay borrowing costs and fees, so their peak sits lower still.

Try this

Pick choppy sideways. The index ends roughly flat. Where does the 3× fund end, and why?
In the steady bull, move leverage from 1× to 3×. Does growth triple?
Set leverage to −1×. Can a short fund lose money even when the index ends flat?

The math

g_L \approx L\mu - \tfrac{1}{2}L^{2}\sigma^{2}, \qquad L^{*} = \frac{\mu}{\sigma^{2}}

$g_L$: compound growth rate with leverage L
$\mu$: average (arithmetic) return of the index
$\sigma$: volatility of the index
$L$: leverage, reset daily
$L^{*}$: leverage with the fastest long-run growth

Ask the Loop about this Next: Correlation and covariance