Monday, January 14, 2013

Betting/trading strategies- Sizing

One of the mysteries facing finance professionals is how to reconcile the quantitative with the qualitative/discretionary. I actually think most gambling professionals do this quite well (eg. bet sizing on poker)- this may be because the risk/reward is much more well-defined (vs investing). I would like to propose a system to conduct position sizing that integrates the qualitative with the quantitative.

this is dependent on 3 things. 

(1) how risky the thing is on a daily basis (ie. can go up/down by $1 vs $100)
(2) how much conviction you have (ie. i would do something like, 3 conviction levels, lowest = believe can get 5% Return on Risk, mid = 10%, high = 20%)
(3) what is your intended time horizon (or alternative way to say this is take profit/stop loss level)

Then (4) plug into kelly's criterion and take 1/2 kelly as position size

So taking aapl as an example:
(1) daily range is say $15
(2) say you have high conviction (ie. you believe you can make $0.2 vs every $1 you risk, as an average of many bets with this level of conviction. this could mean you make $1.2 half the time vs lose $1 half the time, or that 60% of the time you make $1, and 40% of time you lose $1). I think these conviction levels make sense. 5% = any lower and you should definitely just put it in cash/ST bonds. 20% = anything higher and this is a once in a lifetime/decade type trade, where you really just plunge as much as possible (and sizing is to make sure you can maintain exposure in face of MTM losses)
(3) let's say your intended time horizon is 1yr. then yearly vol is $15 *sqrt(252) ~= 240. this sounds about right (eg; this yr aapl range was from 380-700) 
(4) so every share of aapl (550), you may make +290 (240*1.2) vs lose -240. kelly's = EV/win = 50/290 = 0.17. which means that you should risk 17% of your portfolio.
taking half kelly to be conservative, that is 8.5% of portfolio. which means amt of AAPL shares to buy = your total portfolio value * 8.5% / 240

so eg: on 1mil portfolio, you should buy 355 shares of aapl (195k) if you intended to hold it for 1yr+ and have medium conviction on it. this is about 20% of your portfolio, which is very aggressive sizing already. for long/short equity, anything 10%+ would be considered concentrated. the reason why it is high here is because you have super high conviction assumptions. 

note that
(1) we havn't looked at portfolio correlation yet, which would involve dialing down sizing if you have similar exposures.
(2) this # that we got is the MAX risk you should ever take. ie. anything more is theoretically bad for you (ie. your LT returns will be lower than if you just took less risk). so depending on how risk averse you are, you should be sizing significantly less than kelly. (eg: you could always size 1/4 kelly)
(3) can play around with the skew/kurtosis of returns to get a different sizing. In fact, all the steps above are actually asking you to describe a probability distribution of your return for this trade. (1) is asking for stdev, (2) is asking for mean, (3) is looking at how returns scale with time (is there autocorrelation?) which is also going to be related to kurtosis in this case (+ve autocorrelation = higher kurtosis compared to standard assumptions when scaled up with time) (4) is asking about the skew (are you 50% to win 1.2 and 50% to lose $1, or are you 60% to make $1 and 40% to lose $1)


  1. 1. What's the relationship between Kelly and Sharpe ratio?

    2. If your daily range is $15, isn't your daily vol $7.5 and your annual vol $120?

    3. I didn't know that high autocorrelation = high kurtosis. Very interesting.

  2. lol blogger needs to have a way for me to get emailed when someone comments.

    1) apparently kelly bet amt = excess_return / stdev^2 = sharpe / stdev

    2) this is interesting. empirically, i know that vol calculated on daily returns vs average daily range is maybe 1:1.2 (ie. much less than 1:2 and close to 1:1). theoretically if stock opens, chooses one direction and only moves in that direction then you would get 1:1. vs you are saying mkt does a complete reversal every day

    3) i'm pretty sure autocorrelation on daily basis will lead to kurtosis on a monthly/yearly basis