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

Maximizing Compounded Charge Per Unit Of Measurement Of Return

A elementary formula that few traders utilize

Here is a fiddling puzzle that may stymie many a professional person trader. Suppose a sure enough stock exhibits a truthful (geometric) random walk, yesteryear which I hateful at that topographic point is a 50-50 hazard that the stock is going upward 1% or downwardly 1% every minute. If you lot purchase this stock, are you lot nearly likely, inward the long run, to brand money, lose money, or endure flat?

Most traders volition blurt out the respond “Flat!”, in addition to that is wrong. The right respond is you lot volition lose money, at the charge per unit of measurement of 0.5% every minute! That is because for a geometric random walk, the average compounded charge per unit of measurement of furnish is non the short-term (or one-period) furnish m (1% here), only is m – s²/2, where s (also 1% here) is the measure departure of the short-term return. This is consistent alongside the fact that the geometric hateful of a laid of numbers is ever smaller than the arithmetics hateful (unless the numbers are identical, inward which illustration the 2 agency are the same). When nosotros assume, equally I did, that the arithmetics hateful of the returns is zero, the geometric mean, which gives the average compounded charge per unit of measurement of return, must endure negative.

This quantity m – s²/2 holds the cardinal to selecting a maximum increase strategy. In a previous article (“How much leverage should you lot use?”), I described a scheme to maximize the long-run increase charge per unit of measurement of a given investment strategy (i.e., a strategy alongside a fixed m in addition to s) yesteryear leveraging. However, frequently nosotros are faced alongside a alternative of dissimilar strategies alongside dissimilar expected returns in addition to risk. How produce nosotros direct betwixt them? Many traders retrieve that nosotros should pick the 1 alongside the highest Sharpe ratio. This is reasonable if a trader ready each of his or her bet to convey a constant size. But if you lot are a trader interested inward maximizing long-run wealth (like the Kelly investor I mentioned inward the previous article), the bet size should ever endure proportional to the compounded return. Maximizing Sharpe ratio does non guarantee maximal increase for multi-period returns. Maximizing m – s²/2 does.

For farther reading:

Miller, Stephen J. The Arithmetic in addition to Geometric Mean Inequality. ArithMeanGeoMean.pdf

Sharpe, William. Multi-period Returns. http://www.stanford.edu/ wfsharpe/mia/rr/mia_rr3.htm

Poundstone, William. (2005). Fortune’s Formula. New York: Hill in addition to Wang.

Universal Portfolios

Let me depict a portfolio optimization scheme that, over the long run, is supposedly guaranteed to outperform the best stock inward the portfolio.

Before nosotros begin, let’s concord that nosotros volition rebalance our portfolio every hateful solar daytime so that each stock has a fixed pct resources allotment of capital, exactly equally your favorite fiscal consultant would receive got advised you. What this way is that if you lot ain IBM as well as MSFT, as well as IBM went upwardly later on 1 hateful solar daytime whereas MSFT went down, you lot should sell some IBM as well as role the working capital missive of the alphabet to purchase some to a greater extent than MSFT. There is a technical term for such portfolios: they are called “constant rebalanced portfolios”. Notice also the similarity amongst the Kelly criterion which I wrote about before: Kelly criterion asks you lot to hold a constant leverage, which is similar maintaining a fixed pct resources allotment betwixt cash (debt) as well as stock.

But what should the fixed pct resources allotment be? Here is where the scheme gets interesting. Suppose nosotros start amongst an equal working capital missive of the alphabet allocation, for lack of whatever amend choice. At the cease of the day, your portfolio has a sure enough internet worth. But as well as so you lot tin terminate calculate what the internet worth would receive got turned out if you lot had started amongst a unlike allocation. Indeed, nosotros tin terminate run this simulation: elbow grease all possible initial allocations, as well as calculate the hypothetical internet worth of the resulting portfolio. Use these hypothetical internet worth equally weights (after normalizing them yesteryear the core of all internet worth), as well as compute a weighted-average pct allocation. Finally, adopt this weighted average resources allotment equally the novel desired resources allotment as well as rebalance the portfolio accordingly. So genuinely the “fixed” pct resources allotment is non fixed after-all: it gets adjusted daily, but in all likelihood non yesteryear much. Repeat this procedure everyday, e'er calculating a novel weighted resources allotment yesteryear simulating diverse initial allocations since hateful solar daytime 1.

This scheme of portfolio optimization tin terminate hold upwardly proven to attain a internet worth greater than exactly belongings the best stock, given long plenty time. If this sounds similar a miracle, it is partly because this is inward fact an ingenious effect of data theory, as well as partly because at that topographic point are diverse caveats that genuinely throttle its practical application. The proof that it plant (at to the lowest degree inward theory) is rather technical as well as I volition allow the interested reader peruse the master copy articles): nosotros acquire by to lose coin fifty-fifty when a toll serial exhibits a geometric random walk. So it is non likewise surprising that nosotros tin terminate also brand coin using similar data theoretic juggling.

Now for the caveats. Every fourth dimension an data theorist start proverb “In the long run, …”, you lot volition hold upwardly well-advised to ask: How long? In my geometric random walk example where the volatility (standard deviation) of returns every menses is 1%, nosotros detect that the compounded charge per unit of measurement of furnish is an agonizingly pocket-size -0.005% per period. In the instance of the universal portfolio scheme, the out-performance over the best stock inward the portfolio is similarly subject on the volatilities of the stocks: the higher the volatility, the faster the out-performance. Let me run a simulation amongst a portfolio consisting of 2 ETF’s RTH as well as OIH. If nosotros were to run the Universal Portfolio scheme from 2001/5/17 – 2006/12/29, I detect that the cumulative furnish is 32% (without transaction cost). Contrast that amongst exactly buying-and-holding the best ETF (namely OIH here): the cumulative furnish is 54%. The Universal Portfolio loses. Does this hateful the theory is wrong? Not really: RTH as well as OIH may exactly receive got likewise depression volatility. Herein lies the foremost practical caveat amongst the Universal Portfolio scheme: it tin terminate receive got likewise long to realize its attain goodness if the volatility is low.

How attain nosotros detect ETF’s that receive got high plenty volatility to realize the out-performance of Universal Portfolio? Actually, nosotros tin terminate but boost the volatility of RTH as well as OIH artificially yesteryear increasing their leverage. So let’s tell nosotros leverage both of them 2x. This way their daily returns as well as volatilities are both doubled. Now the best ETF (which is nonetheless OIH here) has a furnish of 23% (why is it lower than the un-leveraged case? Remember the formula m-s2/2 inward my previous article.) , but the Universal Portfolio has a furnish of 45%. So at 1 time the Universal Portfolio wins. But this is a Pyrrhic victory: if you lot element inward a transaction cost of 10 footing points, the Universal Portfolio scheme genuinely returns alone 4%. This is the minute caveat of Universal Portfolios: because of the frequent rebalancing required, transaction costs tend to consume upwardly all the out-performance.

Now at that topographic point is a concluding caveat. The reader may inquire why I don’t exactly pick 2 stocks instead of 2 ETF’s to illustrate this scheme. Aren’t most stocks to a greater extent than volatile than ETF’s as well as thence much amend suited for this scheme? Indeed, most academic papers, including Prof. Cover’s master copy paper, role a yoke of stocks for illustration. But if nosotros attain that, nosotros run the guide a opportunity of introducing survivorship bias. Naturally, if you lot know ahead of fourth dimension that none of these 2 stocks volition become bankrupt, the Universal Portfolio scheme may expect great. But if you lot run a simulation where 1 of the stocks all of a abrupt went bankrupt 1 hateful solar daytime (which tend to hold upwardly a fairly mathematically discontinuous affair), the Universal Portfolio scheme volition most probable non trounce belongings exactly the non-bankrupt stock inward the beginning. Using ETF’s eliminated this problem. But as well as so ETF’s are far less volatile.

So given all these caveats, is Universal Portfolio genuinely practical? Prof. Cover seems to holler back so. That’s why he has started a hedge fund to essay it.