11 – M4 L4 17 Path Dependency 1 V3

Why will two portfolios employing the same underlying strategy run on the same as at universe during the same period yield a different portfolio if started at different points in time? The reason is that transitions between different sets of portfolio weights can be more or less costly. Consider two portfolios trying to make the same transition to the same ideal portfolio. A starting portfolio close to this portfolio might be directed here, but one starting from farther away might be directed elsewhere. For example; say I start my strategy on January 1 and the signal says, go long Apple 10 percent. So, I enter this 10 percent long position, then later on June 1, the ideal position is to go short Apple 10 percent. If I’ve imposed a turnover constraint or penalty in the problem, the optimizer may not conclude that it’s actually ideal to enter this 10 percent short position because it’s too far away from the position I’m already in. However, if my way on Apple was previously zero, it may tell me that the best next step is to go 10 percent short on Apple. This could even happen for portfolio started at the same time but holding different amounts of capital if they have a liquidity constraint. A liquidity constraint means that we restrict our positions to be below some percent of each asset’s daily trade volume. If two portfolios have different capital amounts than the smaller capital portfolio is less constrained by a liquidity constraint. For example, if you have some small stock that trades one million dollars per day, and you have a small portfolio that wants to take a 10 percent or 100k position in that stock, which is 10 percent of its volume, that would be okay. But if you had a $100 million portfolio and the same Alpha vector, then that portfolio would want to take a $10 million position in the stock, which would be 10 times its daily volume, and so you’d only be able to take a very tiny position. So, ultimately, due to these trading constraints, the final portfolio weights of these portfolios would be very different. In practice, you may see this effect if you run the same strategy for different clients. Their portfolios may net different returns. This is the surprising but very common feature. It also affects backtesting. We’ve talked about backtesting only a little, but you’ll recall that it involves rigorously simulating, trading a portfolio by testing on historical data. This type of simulation involves a lot of data and a lot of steps. You may want to speed it up by running parts of the computation in parallel across many computers. However, this is difficult because each step of the computation where we calculate what the portfolio would be at each point in time is in practice dependent on the previous stage. We call this path dependency. However, for portfolios of the same size over long periods of time, like years, you would expect the paths of the portfolios to rejoin each other. People have attempted creative solutions to this difficulty. They may run simulations of 18-month windows of time that have six months of overlap on different machines with the whole simulation spanning 10 years. Then, they chop off three months on either end of each simulation and stitch these together.

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