You have two investment options. You can invest in cash which returns 0% each day, and you can invest in a stock which on any given day has a 50% random chance of losing one-half of its value and a 50% chance of doubling. How should you allocate your money between the two? To be clear, over time if you buy and hold the stock, you expect a zero growth because ½* 2 = 1

It may seem that there is no way to make money, but that wouldn’t make for an interesting post. In fact, you could expect to make about 6% per day on average if you could trade without transaction costs.

Put ½ your money in cash and ½ in the stock and rebalance to that proportion every day. You would expect that on half of the days the stock will go down and your portfolio will be worth 0.75 of its previous value. This is because you would have the half that you put in cash plus one-half of the half you put into stocks. On the days the stock goes up, your portfolio will be worth 1.5x as much because you will have the half you invested in cash plus 2x the half you put in stocks.

Thus, every two days we’d expect the portfolio to grow by a factor of 0.75 * 1.5 = ¾ * 3/2 = 9/8. Over n days, the growth factor would be (9/8)^(n/2).

Thus if you started with $1, you’d expect to have $361 over 100 days.

Even if both your investment choices are lousy, there still might be a way to make money. Imagine your investment choices are stocks A and B. On any given day there is a 50% random chance that A will only be worth 10 cents on the dollar – it loses 90% of its value. There is a 50% chance it will double. Thus over time, we’d expect to lose money if we only held that stock. In fact, after two days, we’d expect to have twenty cents on the dollar. That doesn’t sound good. Stock B is exactly the same except when A loses 90%, B will double and when A doubles, B loses 90%. So now we have 2 lousy individual investments (but the good news is they move out of synch). Let’s split our money between the two and see what happens.

Each day our portfolio will grow by a factor of (0.5 * 0.1) + (0.5 * 2) = 1.05. Yes, even though both stocks on average lose money each day, we would make 5% each and every day.

In the absence of transaction costs, one can make money on volatility just by rebalancing. You can even make more than the best performing investment. The tough question is why aren’t we all rich?

*While this is not investment advice and is for entertainment / educational purposes, it does suggest the benefit of diversifying and rebalancing.*

If you are fascinated by this and you like math, you may want to read the paper “Universal Portfolios” by Thomas Cover.