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Lets say that I have an imaginary model that always gives me a 10% edge over straight 50/50 odds, one day in advance, for an index (i.e. 60% chance of winning / 40% chance of losing).

How would I calculate the ideal position size to maximize long term portfolio gain, given a chosen risk threshold?

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3 Answers

up vote 4 down vote accepted

The optimal position size can be determined with the Kelly criterion. In your specific case, the long term growth rate of the capital X is maximized by betting $$(0.6-0.4)X=0.2X$$ at each opportunity.

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How do I calculate the proportion of the total assets under management to commit to the trade? Would I be correct in assuming some relationship with Historical Volatility (HV) or Implied Volatility (IV)? –  Contango May 23 '11 at 22:28
    
Have you looked at the Wikipedia article I had linked to? The position size is determined by the expected payoff $\mu$ and its volatility $\sigma$. Roughly speaking, a Kelly player bets $\mu/\sigma^2$ of their capital at each opportunity. –  olaker May 23 '11 at 22:34
    
I'm not sure that this it the whole story. There is a big difference between buying $100 of IBM (which currently has 16% At-The-Money Implied Volatility) or SLV (which currently has 60% At-The-Money Implied Volatility). –  Contango May 23 '11 at 22:47
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As stated, the problem has a unique and precisely determined solution. Now, the crucial part of the story is to estimate $\mu$ and $\sigma$ for the specific gambling or investing game one is interested in. –  olaker May 23 '11 at 22:57
    
The Kelly Criterion is for a two outcome bet (not a continuous distribution of possible outcomes). At even money (50%/50%) and 1-to-1 odds, the amount to bet is 0% of your stash. At 1-to-1 odds and probabilities of 60%win/40%lose, the bet is 20% of your stash. At 1-to-1 odds and 80%win/20%lose, the bet is 60% of your stash. And at 1-to-1 odds and 100%win/0%lose, the bet is 100% of your stash. –  bill_080 May 24 '11 at 2:59
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If you want to use the Kelly Criterion you might find this link, particularly part III, useful.

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Now that is a very nice article - informal, but more informative than a splatter of opaque math symbols. –  Contango May 25 '11 at 10:44
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Use the Kelly Criterion (as suggested by @olaker). For the amount of money to put into each transaction, use Implied Volatility to calculate the amount you are risking to within a VAR (Value At Risk) of 99% (i.e. +/- 3 standard deviations of the underlying).

See How to calculate future distribution of price using volatility?

Caveats:

  • Markets price distributions have fat tails. When markets crash, they really crash.
  • Take into account skew on Implied Volatility (IV). For stocks, the IV skew is negative, so there is a greater chance of a rapid drop in the price. For commodities, the IV skew could be positive, there is a greater chance of a rapid increase in the price.
  • The Kelly Criterion means the greatest long term increase in the value in your portfolio, at the expense of huge drawdowns and large swings in the value of your portfolio. In practice, you'll have to dial down the risk until the expected drawdown is acceptable. You might have to do some Monte Carlo simulations to work out the actual risk level at any point in time.

Then again, you could always use OpenGamma to handle your risk management for you.

Update:

See How to estimate the probability of drawdown / ruin?

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