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\[\pi(p)=0,\, p \lt 1/2,\]  \[\pi(p)=\pi,\, p \geq 1.\]  I give the calculations below, but the next two graphs illustrate the results. As is clear from the graphs (and intuitively obvious), obvious given our training),  the trading policy is contrarian for \(p \geq 1\) and momentum for \(p \le 1\). Downside protection forces to sell the asset when its price has gone down, and to buy it when it has gone up. This closes the illustrations. I'll now move on to investigate the way contrarian and momentum trading interact with discrete time rebalancing. In the context of discrete time rebalancing, price cycles hurt momentum trading while it benefits contrarian trading. It is clear from all the examples we have set that continuous trading, in the ideal situation of smooth price trajectories, is completely neutral vis-à-vis price cycles. Price cycles don't create nor destroy value in such a context. This opens the door to a certain type of performance attribution which precisely measures the contribution of price cycles to discretely rebalanced portfolios.