How Much Leverage Should You Lot Use?

Maximizing increase without risking bankruptcy

Many hedge fund disasters come upwardly non from making the incorrect bets – that tumble out to the best of us – but from making besides big a bet yesteryear overleveraging. On the other hand, without using leverage (i.e. borrowing on margin to purchase stocks), nosotros oft cannot realize the total increase potential of our investment strategy. So how much leverage should you lot use?

Surprisingly, the respond is well-known, but picayune practiced. It is called the Kelly criterion, named afterwards a mathematician at Bell Labs. The leverage f is defined equally the ratio of the size of your portfolio to your equity. Kelly measure says: f should equal the expected excess furnish of the strategy divided yesteryear the expected variance of the excess return, or

f = (m-r)/s²

(The excess furnish beingness the furnish m minus the risk-free charge per unit of measurement r.)

This quantity f looks similar the familiar Sharpe ratio, but it is not, since the denominator is s², non s equally inward the Sharpe ratio. However, if you lot tin approximate the Sharpe ratio, say, from or together with thus backtest results of a strategy, you lot tin also approximate f simply equally easily. Suppose I convey a strategy alongside expected furnish of 12% over a flow alongside risk-free charge per unit of measurement beingness 4%. Also, let’s tell the expected Sharpe ratio is 1. It is slow to calculate f, which comes out to hold upwardly 12.5.

This is a shocking number. This is telling you lot that for this strategy, you lot should hold upwardly leveraging your equity 12.5 times! If you lot convey $100,000 inward cash to invest, together with if you lot actually believe the expected values of your returns together with Sharpe ratio, you lot should borrow coin to merchandise a $1.2 1000000 portfolio!

Of course, estimates of expected returns together with Sharpe ratio are notoriously over-optimistic, what alongside the inevitable data-snooping bias together with other commons pitfalls inward backtesting strategies. The mutual recommendation is that you lot should halve your expected returns estimated from backtests when calculating f. This is oft called the half-Kelly criterion. Still, inward our example, the recommended leverage comes to 6.25 afterwards halving the expected returns.

Fixing the leverage of a portfolio is non equally slow or intuitive equally it sounds. Back to our $100,000 example. Say you lot followed the (half-) Kelly measure together with bought a portfolio worth $625,000 alongside or together with thus borrowed money. The side yesteryear side day, disaster struck, together with you lot lost 5%, or $31,250, of the value of your portfolio. So at nowadays your portfolio is worth solely $593,750, together with your equity is at nowadays solely $68,750. What should you lot do? Most people I know volition simply stick to their guns together with produce nothing, hoping that the strategy volition “recover”. But that’s non what the Kelly measure would prescribe. Kelly says, if you lot desire to avoid eventual bankruptcy (i.e. your equity going to null or negative), you lot should at nowadays farther cut back the size of your portfolio to $429,688. Why? Because the recommended leverage, 6.25, times your electrical current equity, $68,750, is close $429,688.

Thus Kelly measure requires you lot to sell into a loss (assuming you lot convey a long-only portfolio here), together with buys into a profits – something that requires steely champaign of report to achieve. It also runs counter to the commons mean-reversion expectation. But fifty-fifty if you lot strongly believe inward mean-reversion, equally no incertitude many of the ruined hedge funds did, you lot postulate to visit protecting you lot together with your investors from the possibility of bankruptcy earlier the marketplace position reverts.

Besides helping you lot to avoid bankruptcy, the Kelly measure has or together with thus other of import mathematically proven property: it is a “growth-optimal” strategy. I.e. if your destination is to maximize your wealth (which equals your initial equity times the maximum increase charge per unit of measurement possible using your strategy), Kelly measure is the way.

Notice this destination is non the same equally many hedge managers’ or their investors’ goal. They oft desire to maximize their Sharpe ratio, non increase rate, for the argue that their investors desire to hold upwardly able to redeem their shares at whatever fourth dimension together with hold upwardly reasonably for sure that they volition redeem at a profit. Kelly measure is non for such investors. If you lot adopt the Kelly criterion, at that topographic point may hold upwardly long periods of drawdown, highly volatile returns, depression Sharpe ratio, together with and thus forth. The solely affair that Kelly guarantees (to an exponentially high flat of certainty), is that you lot volition maximize the increase potential of your strategy inward the long run, together with you lot volition non hold upwardly bankrupt inward the interim because of the inevitable short-term marketplace position fluctuations.

For farther reading:

Poundstone, William. (2005). Fortune’s Formula. New York: Hill together with Wang.

Thorp, Edward O. (1997; revised 1998). The Kelly Criterion inward Blackjack, Sports Betting, together with the Stock Market. www.bjmath.com/bjmath/thorp/paper.htm