Question for MathProf, David Spence, Francis Salmon, ETF

Hi guys,

I would like to ask you a question regarding an article MathProf posted on this website. Here is a link to the article.

http://www.bj21.com/bj_reference/pages/overheadeffectonriskofruin.shtml

Here is a summary of the relevant passage from MP's article:

Expectation/Hr = $30
Expenses/Hr = $15
Kelly-factor = 1.0
BR = $3k

After accounting for hourly expenses, MP implies that the Kelly-factor changes from 1.0 to 2.0 (ROR of 36.8%). Unfortunately, he does not show the math he uses to calculate the 36.8% ROR.

Can you please explain how to determine the new ROR (adjusted k-f)which takes expenses into account? I don't think the math is too complicated, I just don't know how to do it.

If you guys don't mind, I have one more question for good measure:

Expectation/Hr = $100
Expenses/Hr = $20
Kelly-factor = 0.25
BR = $100k

What does the Kelly-factor become AFTER taking expenses into account? Please show the math you use to figure it out.

Thanks to whomever can answer,
MJ

MJ,

I really thought I had answered you, but I'll try again. You rarely seem satisfied with initial answers that you get.

"Here is a summary of the relevant passage from MP's article:

Expectation/Hr = $30

Expenses/Hr = $15

Kelly-factor = 1.0

BR = $3k

"After accounting for hourly expenses, MP implies that the Kelly-factor changes from 1.0 to 2.0 (ROR of 36.8%). Unfortunately, he does not show the math he uses to calculate the 36.8% ROR."

If the e.v. halves, and you don't change the bank, it's as if the Kelly fraction doubles. The ROR is simply the square root of the 0.135 ROR for full-Kelly, without resizing. The formula for ROR of various Kelly fractions is to raise 0.135 to the reciprocal of the Kelly fraction, or multiple, you're changing to. Since you're now betting double Kelly, the ROR is 0.135^1/2 = 0.368 = 36.8%.

"Can you please explain how to determine the new ROR (adjusted k-f)which takes expenses into account? I don't think the math is too complicated, I just don't know how to do it."

See above.

"If you guys don't mind, I have one more question for good measure:

Expectation/Hr = $100

Expenses/Hr = $20

Kelly-factor = 0.25

BR = $100k

"What does the Kelly-factor become AFTER taking expenses into account? Please show the math you use to figure it out."

You've changed your e.v. from $100 to $80, so it is now four-fifths of the original. If you don't change your bank, you're now betting 5/4 (the reciprocal) of the original Kelly fraction, so (5/4) x (1/4) = 5/16.

"Thanks to whomever can answer,"

Quick grammar lesson, free of charge: Thanks to whoever can answer! :-)

Don

accurate to say that you have increased your risk by 25% (4/16 to 5/16), by reducing your e.v. by 20% ($100 to $80)?

Thanks in advance for response.

AB

"Since the Kelly fraction increased from 4/16 (1/4) to 5/16, is it... accurate to say that you have increased your risk by 25% (4/16 to 5/16), by reducing your e.v. by 20% ($100 to $80)?"

No, it's not accurate. In our previous example, when we doubled our Kelly fraction, from 1 to 2, did we double our ROR in going from 13.5% to 36.8%? Clearly, no. You can see that the ROR function is not a linear one; it involves an exponent.

The ROR for using 0.25 (that is, 1/4) Kelly (without resizing) is 0.135^4 = 0.033%. Then the new ROR, for using 0.3125 (5/16) is 0.135^(16/5) = 0.135^3.2 = 0.165%.

Don

Thanks Don, I get it now. In your example on the other board, you were adding bank to try and offset expenses, which I just found confusing. I like to work with a fixed bank, as MP did.

One more question on this topic pertaining to standard deviation.

Expectation/Hr = $100
Expenses/Hr = $20
Kelly-factor = 0.25
BR = $100k
SD = $2000/Hr

Suppose our counter plays for 10,000 hours, what would be the standard deviation of this player after adjusting for expenses?

His net EV/Hr = $80, so if he plays for 10,000 hrs then the Net EV for his play should be $80k. In so far as SD is concerned, square rt (10,000) = 100. 100 x $2000/Hr = $200k.

Now, would it be correct to say that 1SD of play for our counter after 10,000 hours is $80k +/- $200k?

Thanks again,
MJ

Minor point! :-) Thanks.

Don

Just to clarify, we are not talking about Kelly betting here. We are talking about the RoR for fixed betting. These are two different issues.