## Can Expectancy be positive and CAGR negative?

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bobsyd
Roundtable Knight
Posts: 376
Joined: Wed Nov 05, 2008 2:49 pm

### Can Expectancy be positive and CAGR negative?

Noticed some test results with negative CAGR but slightly positive expectancy - initially thought formula/calculation must be wrong but now believe it is possible as shown in the attached simple example.

Any thoughts on why it is possible?
Attachments
CURIOUS EXPECTANCY RESULT.xls

sluggo
Roundtable Knight
Posts: 2986
Joined: Fri Jun 11, 2004 2:50 pm
Your math is correct. Your hypothesis, "the quantity Van Tharp calls Expectancy is useful," perhaps not.

For another perspective, take a quick look at the last 3 sentences on page 34 of Ralph Vince's The Mathematics of Money Management. The decision variable that R.V. suggests you rely upon, is negative for your spreadsheet example. As you had hoped.

bobsyd
Roundtable Knight
Posts: 376
Joined: Wed Nov 05, 2008 2:49 pm
Unfortunately p34 wasn't one of the preview pages and from what I was able to preview I'm afraid that book is beyond me!

DeanoT
Roundtable Fellow
Posts: 52
Joined: Sat Apr 15, 2006 7:37 am
Location: Melbourne, Australia
According to the Kelly formula, your optimal risk for the scenario you demonstrated is:

Kelly = Win % - ((1 - Win %) / (Avg Win/Avg Loss))
= .5 - (( 1 - .5) /(1.02/1.00))
= 0.0098 or 0.98%

Your example bet more than twice the optimal bet size (2%), hence the negative overall result.

It might help to understand this by running your example again, but this time risking a much larger amount, say 50% of your account:

i.e.

Start with \$250,000 and win 1.02 of the amount risked (\$125,000) and you end up with \$377,500.

Risk 50% of your \$377,500 and lose, and you are down to \$188,750.

Also re-run your simulation on a bet size less than 0.98% and you should see a profit.

http://en.wikipedia.org/wiki/Kelly_formula

added by Moderator: the Kelly formula gives the betsize which maximizes growth in a Bernoulli Trials (2 outcomes, fixed probability) game. Betting less than Kelly gives a slower growth rate, as does betting more than Kelly. If you increase betsize very very slightly above Kelly, bankroll growth diminishes very very slightly. If you continue to increase betsize above Kelly, bankroll growth continues to fall; eventually, growth becomes negative. In the example discussed here, the Critical Overbetting Point, where bankroll growth falls to zero and begins to turn negative, occurs at a betsize of 0.0196078. If the trader in this example bet 1.96078 percent of his bankroll on every trade, his bankroll growth would be zero. Plug in some numbers and try it.