The power of Cash position, Part 1: A toy model with known probabilities.

05Nov12

A long time ago MarketSci had some strategies called Scotty and YK.

These strategies are currently retired, but audited performance result can be obtained from here and here.

There were ‘by and large’ daily MR strategies. To be honest, that is a crude simplification, because YK  was a learning algorithm, but because daily MR was very successful in 2008, the YK strategy played mostly that.

A debate was risen that how to play correctly a Mean Reversion signal.

If SPY on a given day drops -5%, it is clear that a Mean Reversion strategy would go long at the end of the day to prepare for the bounce tomorrow.

However, if SPY drops only 0.10% our human intuition says that it is not a strong signal to go long tomorrow.

Should we go to Cash if the MR signal is weak?

Michael admitted that with a close to zero, like 0.1% daily change weak signal, the expected profit next day is quite small, but he insisted that we should go long even in this case, because the Expected Value of the profit is still positive.

In this post, we construct an artificial SPY example with known probabilities. Our mathematical model will be a stochastic process, so the outcome of every simulation is not deterministic, but random. However all probabilities are known.

We will play a daily Mean Reversion (MR) on this artificial SPY stock that we construct.

Our aim is to refute that claim: we show that even if the expected profit is positive, we better stay in Cash on a weak MR signal.

At first let’s construct our SPY stock.

Assuming Gaussian distribution, the real world SPY has a mean 0.000384, that is 0.0384% and a Standard Deviation (SD) of 0.0124, that is 1.24%.

In our mockup SPY, we create an SPY with SD = 3%. (assume Beta = 2 compared to the real world SPY, and YK strategy played the Ultra ETFs anyway), and with a mean, that is not fixed, but varies.

We want to play daily MR on this stock, so we construct an SPY that has the following mean as %change for the next day:

 

 

Note that the mean is known every day, but the proper next day outcome is not known, so we still have a random process. After the Mean is determined by this function, the actual next day %change is determined by a Gaussian process with this Mean.

Note the strong Mean Reversion feature of the generated time series: when previous day %change is negative, we generate a new day with a positive expected value.

Let be f() a stochastic function that transform the %change of the previous day to the %change of next day for SPY, we can say that:

  E(f(x) | for all x, where x < 0) > 0      , which means negative days imply that the next day change has a positive Expected Value

and

  E(f(x) | for all x, where x > 0) < 0               , which means positive days imply that the next day change has a negative Expected Value

Perfect instrument for playing daily MR.

Our next day SPY price is calculated by

SPY = SPY * (1 + f(x)),

where f(x) = N(mean for next day, SD);

where N(mean for next day, SD) is a Normal distribution process with the specified mean and SD.

We have a threshold of -1% for the strong mean reversion. If today %change is less than that, we have a strong MR on the next day, because our Expected %change for next day is +0.3%.

The whole f(x) function is similar to a -X function, except that between -0.5% and +0.5% it is very, very close to zero.

We say we have weak MR signal when the %change of the previous day is in this range.

Michael can argue that even in this region, the expected profit for next day is positive with MR strategy, but we will show it is not the case.

The expected profit is positive; yes; but it doesn’t imply we should take a position other than Cash.

(the Expected profit of the next day is positive, yes, but the Expected profit of the MR strategy will not be)

We generate two strategies: one (Strategy1) is a pure MR that goes long if the previous day was negative and goes short otherwise.

The other strategy (Strategy2) goes to cash in case of week MR signal, between -0.5% and 0.5%.

We generated 100,000 days of data that would equal to 400 years of stock market days.

One run of the simulation is charted here: (click for better image)

 

 

The outcome of the stochastic simulation is random; therefore we repeated the simulation 100 times to got reliable (not too random) results. The presented statistics is the average of those 100 simulations.

The MR+Cash position strategy is in cash 13% of the time. This is with the default 3% SD of the SPY generation.

(In case we use 2% SD for SPY generation, we are in cash 20% of the time.)

 

Conclusion:

– Because of the cash position, we are not surprised too much that the 2.8% SD of the MR+Cash strategy is less than the 3.0% for the pure MR strategy.

What really surprising is that the profit, the annual CAGR is higher too (37.01% instead of the 35.04%).

It is higher in spite of the fact that the Expected Profit was positive on those days that we replaced by cash position.

So we missed some positive profit in the MR+Cash strategy compared to the MR, still got better profit.

How is it possible that even with those missed profit, the CAGR is higher in the MR+Cash strategy?

The answer lies in the volatility. And the fact that we are talking about a time series, a.k.a. a sequence of discrete days.

If our job would be to bet on the MR outcome on a weak MR signal just once, only on a single day in our lifetime, we would bet on playing the MR strategy (and no Cash), because the expected outcome is always positive if we play the MR.

However, now we are dealing with a time series, and the daily aggregation (multiplication) of every day simulations. In this case, not only the expected value, but the SD of the time series matters too.

In this case, we would omit using the MR strategy on weak MR signals, and we would stay in cash.

As the simulation shows this decreases the volatility and increases the profit too.

– With both better CAGR and better SD, no wonder the Sharpe is increased from 11.68 to 13.23.

– The answer to the mystery is that the time series profit is decreased by the volatility drag of the time series.

– When we bet on the outcome of the next day, the expected profit should be higher than a threshold to compensate the volatility drag. And this threshold for the expected profit should be significantly higher than zero.

– Our world is not simple black and white. If it doesn’t worth to short SPY, it doesn’t imply we should long SPY. There is a fine line between the two, where no short, no long positions worth taking. We better stay in cash. The higher the volatility the larger is the region around the decision boundary of the expected profit where we should stay in cash.

– the cash position is generally preferred when we are uncertain about the outcome. Cash position is a good risk mitigation tool too.

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One Response to “The power of Cash position, Part 1: A toy model with known probabilities.”


  1. 1 The power of Cash position, Part 2: A toy model with known probabilities. « Neural Sniffer

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