Discounted Kelly Formula

The Kelly Formula, or Fortune's Formula, was originally developed for applications where the time value of money was not a consideration, such as horse racing.  I'm encountering problems in CASTrader where the time horizon of the "bet" is significant.  If you happen to use the Kelly Formula in prediction markets (i.e. - event derivatives markets) with events having long time horizons, then you probably need to adjust the Kelly Formula.   The caveat here is that without numerous repeated applications, the Kelly Formula can be hazardous to your wealth. CASTrader will have diverse set of multiple thousands of traders applying the formula, this is less of an issue.  After searching around, I found nothing that deals with a Discounted Kelly Formula, so I thought I'd derive it.  Recall from before that the Kelly Formula is:

fraction of bankroll to bet = f = edge / odds = (W*o-L) / o

where:

W = Actual probability of winning

L = Actual probability of losing = 1 - W

o = posted odds

Note that for 5:1 odds, o would be 5, and for 3:2 odds (1.5:1), o would be 1.5.  Recall also that the way say 4:1 odds work, if you bet $1 and win, you get $4 back plus your original bet.   The house or market usually gets your $1 bet up front, and pays you the $5 back if and when you win and nothing if you lose. [Confused? Ask Dr. Math] What if the bet doesn't pay off for several years?  Enter Present Value (PV).  The posted odds are illusory, because they are mixing a future payoff worth less than it appears with a bet in the present.  We need to adjust the odds, and at the risk of making up a new term:

Time value adjusted odds = Present Odds = PO = PV(o+1) - 1

In other words, the Present Odds are the Present Value of the future payoff (o+1) minus 1 to convert back to odds.  Thus, the Discounted Kelly Formula becomes:

fraction of bankroll to bet = f = discounted edge / discounted odds = (W*PO-L) / PO

Present value can be tailored to virtually any payoff scheme, but for the simple case of one payoff in the future:

PO = PV(o+1) - 1 = (o+1)/((1+dr)^n) - 1   where: dr = discount rate; n=number of compounding periods

Example.  Using the horse racing example from before:  posted odds = 5:1, but you think the horse has a 1/3 chance of winning.  the only difference is, this race takes place 2 years in the future, and you expect you could make 6% in alternative investments each year (one possible measure of your discount rate).  Thus:

PO = (5+1)/((1+.06)^2) - 1 = 4.34;  and the Discounted Kelly Formula is:

f = (W*PO-L) / PO  = ((1/3)*4.34 - 2/3) / 4.34 = 0.1797

Compare this with the Kelly Fraction (f) = 0.2 we calculated before.  All other things equal, the Discounted Kelly Formula says you should bet less the more distant the future payoff is, which makes sense.  It just so happens in this case that if the race were 12 years or more in the future, Kelly would say not to bet at all, because after discounting the odds, you have no edge left.

Comments

[cpptrader]

Great post and nice alteration to the formula.

[Alan J]

Thanks, C++. Nice blog you have there. I'll definately be checking it out.

[MikeR]

A question. To use Kelly one assumes a probability of winning and a fixed loss and a fixed gain. Many trading situations have an expected loss and an expected gain, but with a standard deviation attached to each. Does anyone know how to adapt Kelly to this situation? I assume it has to be different than when there are known fixed payoffs such as horse racing etc.

[Jeff Paul Internet Business]

I am running a small internet marketing business as my side business. This has helped me cut time when I’m free from my busy schedule. I really get satisfaction when I am my own boss.

[Radon Mitigation]

Air Quality Control is saving lives in Colorado,Wisconsin, Michigan, Indiana, Pennsylvania, and Connecticut by installing radon gas mitigation, remediation, and abatement systems. They are America's Radon Reduction Specialists.

[3d displays]

Magnetic is the industry-leader in auto-stereoscopic 3D LCD displays, 3D content creation, 3D digital signage solutions and 3D software. See 3D without the glasses with Magnetic’s Enabl3d 3D displays.

[penis enlargement]

penis enlargement products and penis enlargement reviews that discover top penis size enlargement pills, patches, oils and devices that gain 1-3 inches gain in your penis size at www.penissizeenlargement.net

[penis enlargement]

Are you ashamed of your small size penis? here is the top penis enlargement methods and penis enlargement products that enlarge penis size naturally in weeks at www.Naturalherbalz.com

[Acomplia]

This is gonna huge, i just cant belive it that i am standing at a great blog of my life, i am really glad to have my comment here in very decent topic. thanks to webmaster.

[penis enhancement]

This is one of the best blog, i have ever seen in my life, really impressive work by webmatser, i must wanna share my views about this posting, that very low rang of people talk about this in public forum so hats off to blogger. thnx.

[Penis enlargement]

I am Very thank full the owner of this blog. Becouse of this blog is very imformative for me.. And I ask u some thiing You make more this type blog where we can get more knowledge. and any one tell me how can I find this type blog. http://www.penisenlargementz.com

[enlarge penis]

hello webmaster! i am really appreciating you for your kindness that you have made a really interesting blog and pick all the great topics for which peoples are always seeking to get information and share their ideas. Great work Man and keep it going. God bless you!

[Acomplia]

Rimonabant (also known as Zimulti, Acomplia, Slimona or Obitone) is an anorectic anti obesity drug. It is a CB1 cannabinoid receptor antagonist. Rimonabant has been found to stop food craving enough to help people lose weight, and could also help curb other unhealthy urges, such as smoking. www.genericacomplia.us

[Hair loss]

Hair Loss Treatments - Products Cures and Solutions Guide. Read the Hair Loss Guide - Know about Causes, Treatments, Solutions and hairstyles for Men and Women at www.Managehairloss.com

[penis enlargement pills]

That's really true, i have also searched and came to know that there are many methods for internet marketing, one can find many web marketing ways in social networking sites like face book and twitter. www.male-sexual-styles.com

[vigrx plus]

I LOVE YOU FULL! vigrx plus

[vimax pills]

I LOVE YOU FULL! does vimax work?

[santana]

Nice blog, its great article informative post, thanks for sharing it. Thanks for the information!

[Penis Enlargement]

VigRx, Prosolution, Natural Gain Plus, Vigrx Plus, MagnaRx, Size Genetics & More Penis enlargement and Breast Enlargement Products at http://www.naturalherbalsinc.com/

[Penis Enhancement]

penis enlargement facts that aware you about penis size, and gives you complete solution to have a bigger, larger and harder penis size without having surgery and other scams, check it out www.penissizeenlargement.net

[youtube music]

Nice blog, its great article informative post, thanks for sharing it. Thanks for the information!

[Home improvements]

Home, improvements, Vancouver renovations, Renovation, Reno's, north, shore, commercial finishing, additions, remodeling, kitchen renovations, bathroom renovation, plumbing, ceramic tile setter, flooring, North Vancouver, Vancouver, restorations
URL: www.looking4trades.com www.bcrestorations.ca

[buy viagra online]

Hi, I think your article its very important and interesting,good work, thanks for sharing!! Have a nice day!