Notes from the seminar on Mathematical Finance, held at University of Chicago, April 2007.

  1. The Kelly or capital growth criteria maximizes the expected logarithm as its utility function period by period.
  2. It has many desirable properties such as being myopic in that today’s optimal decision does not depend upon yesterday’s or tomorrow’s data,

    Success in investments has two key pillars: (1) devising a strategy with positive expectation and (2) betting the right amount to balance growth of one’s fortune against the risk of losses

  3. It asymptotically maximizes long run wealth almost surely and it attains arbitrarily large wealth goals faster than any other strategy.
  4. Also in an economy with one log bettor and all other essentially different strategy wagers, the log bettor will eventually get all the economy’s wealth.
  5. The drawback of log with its essentially zero Arrow-Pratt absolute risk aversion is that in the short run it is the most risky utility function one would ever consider.
  6. Since there is essentially no risk aversion, the wagers it suggests are very large and typically undiversified.
  7. Simulations show that log bettors have much more final wealth most of the time than those using other strategies but can essentially go bankrupt a small percentage of the time, even facing very favorable investment choices.
  8. One way to modify the growth-security profile is to use either ad hoc or scientifically computed fractional Kelly strategies that blend the log optimal portfolio with cash. to keep one above the highest possible wealth path with high probability or to risk adjust the wealth with convex penalties for being below the path.

    Additionally legendary investors such as John Maynard Keynes (0.8 Kelly) running the King’s College Cambridge endowment, George Soros (? Kelly) running the Quantum funds and Warren Buffett (full Kelly) running Berkshire Hathaway had similarly good results but had much more variable wealth paths

  9. For log normally distributed assets this simply means using a negative power utility function whose risk aversion coefficient is 1:1 determined by the fraction and vice versa.
  10. For other asset returns this is an approximate solution.
  11. Thus one moves the risk aversion away from zero to a higher level.
  12. This results in a smoother wealth path but usually has less growth.
  13. This seminar is a review of the good and bad properties of the Kelly and fractional Kelly strategies and a discussion of their use in practice by great investors and speculators most of whom have become centi-millionaires or billionaires by isolating profitable anomalies and betting on them well with these strategies.
  14. The latter include Bill Bentor the Hong Kong racing guru, Ed Thorp , the inventor of blackjack card counting who compiled one of the finest hedge fund records.
  15. Both of these gamblers had very smooth, low variance wealth paths.
  16. Additionally legendary investors such as John Maynard Keynes (0.8 Kelly) running the King’s College Cambridge endowment, George Soros (? Kelly) running the Quantum funds and Warren Buffett (full Kelly) running Berkshire Hathaway had similarly good results but had much more variable wealth paths.
  17. The difference seems to be in the choice of fraction and other risk control measures that relate to true diversification and position size relative to liquid assets under management.
  18. Success in investments has two key pillars: (1) devising a strategy with positive expectation and (2) betting the right amount to balance growth of one’s fortune against the risk of losses
  19. A strategy which has wonderful asymptotic long run properties (1) the log bettor will dominate other strategies with probability one and (2) accumulate unbounded amount more wealth.  But in the short run the strategy can be very risky since it has very low Arrow-Pratt risk aversion.  Z.PNG
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