Optimal investment strategy

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btilly
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Joined: Fri Sep 26, 2008 6:45 am

Optimal investment strategy

Post by btilly » Mon Oct 27, 2008 11:59 pm

(I've noticed a lack of Calculus in the P.E. problems. I don't know if that is deliberate, but solving this problem does require some Calculus. Though substantially less than you might think.)

In investing we are faced with many different choices for what to do with our money, each of which has its own risk/reward profile. Investment books suggest that we diversify our investments - don't put all of your eggs in one basket. But what is the best diversification strategy?

This can be studied in simple cases. For example we can look at an investor in a simplified model by giving him the same choice at time t0, t1, t2, and so on. At each step the investor divides his money between two options. One choice is to keep the money as cash, in which case it survives to the next time point unchanged. The other is a risky investment that 1/3 of the time will lose all of its value, and 2/3 of the time will double in value. The investor will invest a fixed fraction of his money in each option at each point in time. The goal is to find the investment strategy that will, with 100% odds, eventually beat any other investment strategy.

Obviously you want at least some of your money to be in the risky investment, because it has better than even returns. But equally obviously if you put all of your money in the risky investment then at some point you will lose all of your money, which loses to every other investment strategy. Somewhere between there is a happy medium.

In this case the happy medium is to risk 1/3 of your money, and to keep 2/3 of it in the form of straight cash.

Let us make this more complicated. At each step let us give our investor a choice of 4 investment options. The first is cash. Investment A will, as before, 2/3 of the time double your money, and 1/3 of the time will lose it. Investment B is riskier with higher rewards still. 2/3 of the time when A doubles your money, B multiplies it by 4 instead. Otherwise B loses your money. And investment C has even odds of tripling your money.

What is the optimal investment strategy now? That is, what division of money between Cash, option A, option B, and option C will, in the long run, beat any other possible way of dividing your money at each step?

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stijn263
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Re: Optimal investment strategy

Post by stijn263 » Tue Oct 28, 2008 10:35 am

In this case the happy medium is to risk 1/3 of your money, and to keep 2/3 of it in the form of straight cash.
I think this depends on your risk preferences. Without specifying your risk utility function any amount could be a good strategy

btilly
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Re: Optimal investment strategy

Post by btilly » Tue Oct 28, 2008 2:11 pm

stijn263 wrote:
In this case the happy medium is to risk 1/3 of your money, and to keep 2/3 of it in the form of straight cash.
I think this depends on your risk preferences. Without specifying your risk utility function any amount could be a good strategy
In the problem I said, "The goal is to find the investment strategy that will, with 100% odds, eventually beat any other investment strategy." This condition is enough to uniquely specify the optimal investment strategy.

In a real world investing scenario this condition is similar to, "You're a young person with a very long investing horizon." Which means that you (should) have very high risk tolerance.

btilly
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Joined: Fri Sep 26, 2008 6:45 am

Re: Optimal investment strategy

Post by btilly » Wed Oct 29, 2008 6:04 pm

Since nobody seems to be nibbling on this, I'll drop a hint.
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Once you have selected an investment strategy, at each step your net worth is multiplied by a random number. Repeated multiplication is too hard to reason about.

btilly
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Re: Optimal investment strategy

Post by btilly » Wed Dec 17, 2008 12:11 am

In case anyone is curious, I'll outline the solution.
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Your net worth after n steps is the product of n independent random variables. That is too complicated. Take logs and the log of your net worth is the sum of n independent random variables. Much better. The strong law of large numbers says that the log of your net worth divided by n will converge, with 100% odds, to the expected value of the change in the log of your net worth after one step. Therefore the strategy that maximizes that will, with 100% odds, eventually beat every other strategy.

With one possible risky investment this max-min problem is easy to solve and gives rise to the Kelly criterion which all gamblers know.

If x, y, and z are the fractions of your wealth that go into the three risky investments, this becomes a max-min problem in three variables. Solve that numerically and you're done.

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