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Hello Gerry,
Gee you are putting me under the pump now.
It is easy to get lost amongst the various definitions that float
around.
For expectancy I have:
van Tharp
expectancy == (probability of a win * ave Win) - (prob loss * ave
Loss)
My understanding is that the ave W or L is measured in $ values.
This definition is consistent with 'mathematical expectation"
>From Ralph Vince
Published by John Wiley & Sons, Inc.
Library of Congress Cataloging-in-Publication Data
Vince. Ralph. 1958-The mathematics of money management: risk analysis
techniques for traders / by Ralph Vince
MATHEMATICAL EXPECTATION
By the same token, you are better off not to trade unless there is ab-
solutely overwhelming evidence that the market system you are con-
templating trading will be profitable-that is, unless you fully
expect the market system in question to have a positive mathematical
expectation when you trade it realtime.
Mathematical expectation is the amount you expect to make or lose, on
average, each bet. In gambling parlance this is sometimes known as
the player's edge (if positive to the player) or the house's
advantage (if negative to the player):
(1.03) Mathematical Expectation = ?"[i = 1,N](Pi*Ai)
where
P = Probability of winning or losing.
A = Amount won or lost.
N = Number of possible outcomes.
The mathematical expectation is computed by multiplying each pos-
sible gain or loss by the probability of that gain or loss and then
sum-ming these products together.
Let's look at the mathematical expectation for a game where you have
a 50% chance of winning $2 and a 50% chance of losing $1 under this
formula:
Mathematical Expectation = (.5*2)+(.5*(-1)) = 1+(-5) = .5
In such an instance, of course, your mathematical expectation is to
win 50 cents per toss on average.
Consider betting on one number in roulette, where your mathemati-cal
expectation is:
ME = ((1/38)*35)+((37/38)*(-1))
= (.02631578947*35)+(.9736842105*(-1))
= (9210526315)+(-.9736842105)
= -.05263157903
Here, if you bet $1 on one number in roulette (American double-zero)
you would expect to lose, on average, 5.26 cents per roll. If you bet
$5, you would expect to lose, on average, 26.3 cents per roll. Notice
that different amounts bet have different mathematical expectations
in terms of amounts, but the expectation as a percentage of the
amount bet is always the same. The player's expectation for a series
of bets is the total of the expectations for the individual bets. So
if you go play $1 on a number in roulette, then $10 on a number, then
$5 on a number, your total expectation is:
ME = (-.0526*1)+(-.0526*10)+(-.0526*5) = -.0526-.526 .263 = -.8416
You would therefore expect to lose, on average, 84.16 cents.
This principle explains why systems that try to change the sizes of
their bets relative to how many wins or losses have been seen
(assuming an independent trials process) are doomed to fail. The
summation of negative expectation bets is always a negative
expectation!
The most fundamental point that you must understand in terms of money
management is that in a negative expectation game, there is no money-
management scheme that will make you a winner. If you con-tinue to
bet, regardless of how you manage your money, it is almost certain
that you will be a loser, losing your entire stake no matter how
large it was to start.
This axiom is not only true of a negative expectation game, it is
true of an even-money game as well. Therefore, the only game you have
a chance at winning in the long run is a positive arithmetic
expectation game.
van Tharps expectation is a restatement of Vinces
Mathematical Expectation = (.5*2)+(.5*(-1)) = 1+(-5) = .5 since the
loss confers negative sign to the second part of the equation.
I had to use a simple example to sort out my comments:
trade 3 times
win twice * $2 each
ave$ won == 2
lose once * $1
ave $ loss = 1
gross win == $4
gross loss == $1
gross $won/gross $ lost == 4 == ProfitFactor (definition used by AB)
nett $gain = 4 - 1 == 3
expectancy $ = net $gain/#trades in total == 3/3 == $1
PF (also) == W/L * ave$W/ave$L = 2/1 * 2/1 == 4/1
van Tharp expectancy
probality of win == 2/3 == 0.666
prob of loss == 1/3 == 0.3333
expectancy == (0.666 * 2) - (0.333 * 1) == 1.332 - 0.333
van Tharp expectancy is approx == $1
this is the same as PF expectancy
I went off Profit Factor as a metric because of the errors introduced
when adjusted data is used (using unadjusted data biases backtesting
for EOD trades).
I use expectancy in a different way - I am not a mathematician but
the way I use it is a lot closer to geometrical mean than anything
else - I use expectancy in % terms - for my own use I changed the
name to PowerFactor - it also has a better mathematical relationship
to some other useful metrics, including some used in portfolio
management.
Thanks for your post - important topics that are always worth the
discussion.
brian_z
--- In amibroker@xxxxxxxxxxxxxxx, "gerryjoz" <geraldj@xxx> wrote:
>
> In an earlier post, expectancy was associated with profit factor.
> It is more closely related to payoff ratio.
> In Van Tharp's book, 2nd edition, "Trade your way...", page 204 et
> seq, he calculates
> Expectancy = average profit/ # trades
> divided by average loss.
> Payoff ratio is average profit/average loss,
> so
> Expectancy = payoff ratio/# trades.
> --which can give very low numbers, and makes the concept rather
> dubious if you are using it as an absolute value for comparing
systems
> with different numbers of trades. It might be better to use trades
per
> annum.
> To be fair Van Tharp only gives that way of calculating expectancy
as
> a default if the risk of a trade isn't able to be calculated taking
> into account a pre-determined proportion of equity. For that, you
need
> to read the whole chapter.
> Personally i find CAR/MDD, RRR more relevant, along with the raw
> Payoff ratio.
>
> The K-ratio isn't worth the space it takes up: RRR is simpler.
>
> regards
> Gerry
>
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