## Monday, October 6, 2014

### Retrenchment under Limits

The graphs in my previous post looked a bit off: the long-term balance could grow without limit. Once the long-term balance grows large enough that the interest earned at each step exceeds the penalty for transfers to the short-term balance, then collapse becomes impossible.

The long-term balance corresponds roughly to altitude in the flight power curve model, while the short-term balance corresponds roughly to speed. One good flight strategy is to stay at a high enough altitude that recovery is possible from a stall. More altitude makes it possible to recover even from a series of stalls. But ultimately altitude is limited by the reduced air pressure. In a real economic situation there is a similar problem. The more one invests, the more one is pushed out on the risk-return frontier. The easy pickings are exhausted.

To model such limits, I tweaked the structure of my simple model. When the long-term balance is low, it earns a reliable 5 percent. But as the balance grows, there is an increasing probability of a smaller return, to the point where it becomes possible to lose up to 15 percent of the balance in a single step. So the average return gradually declines, as the long-term balance increases, from 5 percent to -5 percent.

Here are four different runs, all with the same threshold. Most likely there are smarter strategies that can outperform this simple threshold strategy. But this new model does seem to be more realistic.

#### 1 comment:

1. With the parameter values I used for these runs, the average transfer each time unit is -20. This has to be paid out of the interest on the long-term balance. Interest payments increase as the long-term balance increases, up to a balance of about 2100, where the average interest is almost 43. The average interest first exceeds the average transfer at a long-term balance of about 500. As the long-term balance grows above 2100, the average interest payments start to decline - this is the growth limit being felt. The average interest falls below the average transfer at about 4100.

Presumably a strategy that targeted the long-term balance sweet spot of 2100 would have the best results. But in a realistic scenario, the strategy would not have access to the probability distributions and their dependence on the long-term balance, etc.

I imagine folks already set up competitions for these sorts of problems, in settings of control theory or approximate dynamic programming.