lee1026 wrote:Incidentally I'm not sure where or how risk, and risk-adjusted reward, make their appearance in the Kelly criterion framing. What exactly is the measure of risk in the Kelly world, and what is the counterpart to the Sharpe ratio?

Sharpe ratio is independent of leverage, so all of the bets that we are discussing have the same Sharpe ratio. Similarly, the concept of risk adjusted returns doesn't apply. Traditionally, the argument is that you can leverage up or down for your desired risk as long as the risk adjusted returns are good. But if we are talking about how much leverage to use, that become a moot point very quickly.

Sharpe ratio is independent of leverage in the quasi-single period CAPM world. But in the CAPM world the return also increases toward infinity with more and more leverage.

Seems this conversation can be ambiguous as to whether we're talking about coin flip game for which Kelly math is well explained in original paper, or adapting it to stocks. Latter is the more practical discussion of course, but stuff like ratio of return/(std dev of return) is also different between the two. I have the same question as Nispirus as to the correct analog of Sharpe ratio in coin flip game, but taking a pseudo-Sharpe ratio of just the (log rtn)/(std dev of log rtn) among 2000 flip games betting heads at even odds on a coin which comes up heads 53% of the time, that ratio strictly declines the more you wager, though the max expected log return is at 6% of capital wagered.

Back testing for SPTR leveraged at 3 mo LIBOR+25 quaterly rebalance, see above, the Sharpe Ratio is also not a constant with leverage.

June 1 1988>Dec 1 2014

Quasi Riskless (ie 3mo LIBOR flat) return: 4.04%

SPTR at 100%: 10.42%, annualized quarterly std dev of return: 16.33%, Sharpe Ratio, as in (stock-'riskless')/(std dev of risky)= .390

SPTR at 117%: 11.18%, std dev 19.35%, SR .369

SPTR at 140%: 12.03%, std dev 23.68%, SR .337

SPTR at 190%: 13%, std 34.72%, SR .258

SPTR at 200%: 12.99%, std 37.39%, SR .239

SPTR at 275%: goes bust 12/2008

For 6/1/200>12/1/2014:

LIBOR rtn: 2.28%

SPTR at 100%: 5.37%, std dev 19.08%, SR .161

SPTR at 117%: 5.25%, std dev 22.77%, SR .130

SPTR at 140%: 4.80%, std 28.12%, SR .090

SPTR at 190%: 2.28%, std 42.14%, SR 0

SPTR at 275%: goes bust 12/2008

I know graphs are more entertaining, sorry.

Anyway these results like any historical results don't 'prove' anything about the future, nor would collecting them further back, whereas theoretical results out to infinity are meaningless (the usual dilemma). However I'd say the *general indication* is that a stock allocation somewhere in the low to mid 100%'s may be reasonable for an investor highly focused on return and not particularly concerned with annualized std dev noise. The later shorter period isn't the worst imaginable for leverage but pretty bad and the moderately 100%+ investor doesn't come out *that* much worse than 100% investor, and enough better in the longer period to possibly be worthwhile, depending on preference.