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Hi Brian,
Because this is rather long, I'll net it all out at the beginning.
van Tharp expectancy as a dollar amount=average profit per all trades.
He gets there by starting out with the concept of what you are risking
per trade.
Expectancy=e*R,
where R is the the amount at risk, empirically the average loss of
losers. "e" takes a little calculation but its easier to just take the
average profit per trade and divide by the average loss of losers.
e=(net profit/number of trades)/(average loss per losing trade)
you are right that
Mathematical Expectation = ?"[i = 1,N](Pi*Ai)
> where
> P = Probability of winning or losing.
> A = Amount won or lost.
> N = Number of possible outcomes.
In my copy of his book i didn't see that as his explicit definition of
expectancy, nor did i see
expectancy == (probability of a win * ave Win)- (prob loss * ave Loss)
explicitly either.
He uses expectancy as a measure against the amount at risk.
i copied how the formula read, not what Van Tharp actually used in his
calculations. So i got his ideas wrong in my previous note (or being
cheeky, his text and his numbers didn't quite line up, but i should
have noticed that).
So please discard my earlier calculation of expectancy.
My apologies for that.
What he wrote on page 204 was
Expectancy=average profit/per trade
divided by
average loss.
His actual calculation was
average profit = net profit/total number of trades
it is net profit that is the numerator not average profit, and what it
is not, is the average profit of winners. The average loss though is
the loss of losers divided by the number of losers.
So how it really should have looked in his book to be consistent with
his calculations is:
expectancy= average profit over all trades/average loss per losing
trade as a factor to be applied to average loss.
For van Tharp the average loss is the amount at risk, in effect what
you expect to lose on a losing bet. His expectancy tells you how much
in dollars you expect to win as a proportion of that, which as you say
comes out as a dollar amount.
In my copy of his book, 2nd ed page 198 he writes "expectancy is your
average gain or loss stated in terms of R" where R is empirically
approximated by the average loss, the sum of the losing trade values
divided by the number of losers. BUT his average gain in the
calculation is net profit divided by all trades.
All that appears to reduce to van Tharp saying (In effect)
expectancy = (net profit (or loss)/total trades) times average loss of
losers.
Let's take a simple example and follow van Tharp:
100 trades, 60 losers 40 winners, the total profit of winners is 80
($2 each), and of losers is 30 ($.5 each), net profit=50.
Van Tharp according to the book, has
average profit/per trade= net profit/total number of trades=
50/100=.5
average loss= 30/60=.5.
expectancy=(average profit/average loss)* average loss
expectancy= (.5/.5) *R
r=.5.
But mathematical expectation is 50. The difference is the need to
divide by the number of trades to get expectancy.
here are the numbers from the book
net profit=10843
number of trades=103
average profit (of all trades)=105.27
average loss (of losers)=721.73=R
expectancy =.15 R
my calculation
e=105.27/721.73
e*R=105.27= net profit per trade.
Or to put it simply:
Net profit per trade is expectancy.
That's a whole chapter for him to tell you that you need to make a net
profit on average per trade, and you should be comfortable with the
amount at risk per trade.
--- In amibroker@xxxxxxxxxxxxxxx, "brian_z111" <brian_z111@xxx> wrote:
>
> 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@> 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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