Multiple Selections and Sharpe Ratio

[Rushian]

I would be interested in the fellow members' thoughts on using the Sharpe ratio (or return-to-risk ratio) in selecting an "optimal" allocation of stakes when there are multiple simultaneous bets. I've tried to summarize my thoughts in an entry in my blog.

Multiple Selections and Sharpe Ratio Full or fractional Kelly staking has been in the foremost of betting staking plans. Using the available odds, and our estimate of the "fair" probability of an event happening, we can calculate what proportion of our bank should be staked in each bet. However, what happens if there are more than one, say n, concurrent selections? How much should one stake in total on the n selections, and how much on each of the selections? What if we looked at the concept of expected reward-to-risk ratio? In finance, this is also known as Sharpe Ratio and it measures the excess return of an investment per unit of risk. Let's denote a bet with a letter i. The estimated probability of the bet winning is p_i while the available odds are o_i. The bet would represent value if p_i > 1/o_i. Now, if we invest one unit in this bet the expected profit becomes: E[Profit_i] = (p_i * o_i - 1) The standard deviation (a measure of the risk) of the bet's profit can be shown to be: SD[Profit_i] = o_i * Sqrt[ p_i * (1 - p_i) ] Denoting the expected profit and its standard deviation as E_i and SD_i respectively, the Sharpe ratio for a unit stake in a single bet is given by: SR_i = E_i / SD_i When we have more than one selections, i = 1, 2, ..., n, and we invest x_i units in bet i, the total expected profit is given by: ExpTotProf = x₁ * E₁ + x₂ * E₂ + ... + x_n * E_n = Sum(x_i * E_i) whereas its standard deviation (because of independence between bets) now becomes: SDTotProf = Sqrt ( (x₁ * SD₁)² + (x₂ * SD₂)² + ... + (x_n * SD_n)² ) The Sharpe Ratio is given by: SR = ExpTotProf / SDTotProf This means that SR is now simply a function of the available odds, the estimated probabilities and the stake in each bet. We can therefore select those stakes (x₁, x₂, ..., x_n) which maximize SR with respect to some constraint e.g Sum(x_i) = 1. It turns out that the solution to this optimization problem is simply selecting stakes x_i proportional to E_i / SD_i² or more specifically: x_{i =} (E_i / SD_i²) / Sum[ E_i / SD_i² ] Although this may sound heavy, it's quite simple in practical terms. Consider 4 bets, available at odds of 2.05, 1.50, 5.00 and 2.50. Let the estimated probabilities for these bets be 60%, 68%, 22% and 45% respectively. If you follow the steps above you should get the following figures: Note that with the calculated allocation we have achieved a reward-to-risk ratio of 0.256, which is higher than the ratio that we would have achieved had we invested all of our stake on Bet 1 (which had a Sharpe ratio of 0.229).

Any views/suggestions/thoughts?

[Rushian]

Re: Multiple Selections and Sharpe Ratio Anyone ?? :unsure

[Kanga]

Re: Multiple Selections and Sharpe Ratio Hi Rushian, I think to be fair this sort of complex formulae is way over the top of most of us average morsels ... I can follow this to some extent but I'm still unsure as to what you are ultimately asking. If it helps I can offer some simple advice I have picked up over time and one of those things is that kelly staking requires you to know your advantage/edge very accurately and is disproportionally risky. The ansell staking plan maybe of use to you :- http://www.p2pbetting.com/Articles/2005/Nov/TheAnsellStakingPlan/tabid/851/Default.aspx Personally i've found that staking according to the calculated probability of the outcome has been the best way forward for me.

[slapdash]

Re: Multiple Selections and Sharpe Ratio I hadn't heard the term "Sharpe Ratio" before. It looks like a nice simple measure of risk vs. reward, but I don't think it's clear that it's precisely this that you should be trying to optimize. I suspect that comparing the ratios of the usual Kelly stakes for the individual bets will give a slightly "better" answer to the question that you're trying to answer (though it's not going to make a huge difference). Also, and I'm sure you realize this, knowing the ratio of bet-sizes doesn't tell you how much you should be betting in total. Here's a simple example, with just two bets. Suppose Bet A is at odds of 2.00 with a 55% chance of success. Bet B is at odds of 11.00 with a 10% chance of success. As single bets, Kelly would say that you should bet 10% of your bank on Bet A and 1% on Bet B. A ratio of 10:1. Your scheme gives a ratio of 11:1. But it's possible to numerically calculate the exact amount you should bet (if you're putting both bets on simultaneously) that has the same properties as Kelly staking for one bet at a time (i.e., maximize the expected value of log(bank)). It turns out to be 9.9909% of your bank on Bet A and 0.9908% on Bet B (i.e., almost identical to the simple Kelly values). A ratio of about 10.08:1. A while ago I looked at some examples of betting on multiple selections simultaneously, and what "generalized Kelly" said, and it turns out that until you get close to staking your entire bank, it isn't that different from Kelly. For example, suppose you have a bunch of bets at odds of 2.00 and 55% chance of success, so for individual bets Kelly says you should stake 10% of your bank. What if you're betting on several such bets at the same time? If there are 5 bets, you should stake 9.591% of your bank on each. Even if there are 10 such bets (so that if you used the individual Kelly value of 10% on each you would use your entire bank), you should stake 8.967% of your bank on each. I.e., you should use almost 90% of your total bank. So until you get up to using your entire bank, the "generalized Kelly" stake depends rather little on the number of bets. For practical purposes, if (like me) you stake conservatively anyway compared to your estimate of the "Kelly stake", and don't make enough bets that you'd come close to using your entire bank, I think it's fairly safe just to ignore the effect of putting on several bets at the same time.

[Rushian]

Re: Multiple Selections and Sharpe Ratio Kanga, I agree that for this to work, similar to the single-bet Kelly staking plan, one has to be able to accurately estimate the probabilities of events occurring, a tough task in itself. Even then, it's better to err on the side of caution as the loss function of overestimating and underestimating these probabilities is not symmetric, and heavily penalizes in terms of short-term volatility any over-estimation of the event probabilities. As for the ansell staking plan, I haven't had it in mind. I'll read up on it. slapdash, Thanks for the analysis. If I understand this correctly you are saying that this approach would yield very similar "weights" on each bet to the allocation that one would get by calculating single-bet Kelly stakes and normalizing them so that they add to 1. So even in the case where one has a large number of bets which according to single-bet Kelly stakes would need more than 100% of the bank to be invested, by normalizing them so that they add to a number, say 1, you would still do pretty well and, in fact, similar to what I suggested in the opening post. However, an additional benefit of using the approach described in the opening post (which could yet be handled by your suggestion, although it's not immediately obvious to me how) emerges when bets are not necessarily independent. If there was a way of quantifying any correlation between the profits from the different bets (say Bet 1 and 2 were on the same match, and their profits are not independent) then one could build a variance-covariance matrix of the profits from each bet (call this S) and using the column vector of expected profits (let's call this mu) the allocation which maximizes the Sharpe Ratio would be proportional to: inv(S) * mu Using this notation the (i,j)-element of S would denote the covariance between profit from bet i and profit from bet j whereas the ith diagonal element in S would denote the variance of profit from bet i. Finally, you are right in the fact that this method still does not allow us to determine an absolute value of the total stake. I was wondering whether there was a way of coming to that result by looking at the whole portfolio of bets as 1 bet (which of course has more than one possible outcomes - not just win/lose) and perhaps replacing in some way the inputs of the original Kelly formula (i.e. prob and odds) with their corresponding averages from our 1 complex bet... Maybe this needs further work...