# 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 = (WPO-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.