Optimization Strategy Report : Based on Expectation Error-Fixing Method

Optimization Strategy Report : Based on Expectation Error-Fixing Method

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Optimization Strategy Report : Based on Expectation Error-Fixing Method


Securities are never perfectly priced in the market. The value at which the security trades in the market is driven by investors’ desire to invest in the security. This difference in investors’ expectation of the price, and the implied price based on the fundamentals data is the Expectation Error.

Another example shown below is Tesla. At the beginning of 2020, the company successfully built a production facility in Shanghai, China, with a goal to produce 150,000 units annually of their Model 3.

This changed the investors’ expectation and caused an increase in stock price despite the fundamentals not changing.

Another example regarding Tesla (not pictured) involved one of CEO Elon Musk’s tweets. Musk had tweeted “Tesla’s stock price is too high,” which changed investors’ expectation and caused the price to take a sharp dip, despite the fundamentals not changing.

It is also observed that small cap stocks outperform large cap stocks; however, considering that large cap value stocks outperform small-cap growth stocks, it can be assumed that the value-glamor effect seems to be stronger than the market cap effect.

This is not to suggest that growth stocks should always be ignored in favor of value stocks. There are periods where investing into growth produces better returns than value stocks.

As suggested by the image below, growth stocks outperform value stocks in a bear market while value stocks outperform growth stocks in a bull market. This was the fundamental groundwork in developing the trading strategy in this paper.


To take advantage of expectation error, a model must be defined.  High price to book ratio firms (also called “growth” or “glamor”) are defined as those above the 70th percentile. Middle PB firms are the ones with values between the 30th and 70th percentiles, and low (colloquially known as “value”) are the ones below the 30th percentile.

According to the Stanford study by Joseph D. Piotroski and Eric C. So, prices for growth firms are overly optimistic and value stocks are priced overly pessimistically.  The error is realized with news or earnings, so by holding a value firm with strong fundamentals or shorting a growth firm with weak fundamentals, the expectation is to outperform. 

The model scores on 8 metrics over 3 dimensions.  Consideration of a firm’s strength of fundamentals is determined by its profitability, operating efficiency, and leverage/liquidity.  All metrics are binary; if a firm’s statistics do not earn them a 1, the default value is 0. These fundamental statistics are aggregated into an “F-score” with the formula:

FSCORE = F(ROA) + F(CFO) + F(ΔROA) + F(Accrued Profit) +F(Δ Current Ratio) + F Δ(Long-term Debt) + F(Δ Gross Margin)+ F Δ(ATO)

Where the F () for each term is defined in the following paragraphs, and the F-scores defined in the matrix as high if 7 or greater, middle if 4-6 and low if 3 or below.

Profitability measures comprise the majority of the F-score. 

A company receives a 1 for having a positive return on assets, or ROA. 

This is a binary indicator of whether the firm generates a positive profit or loss. 

ROA is net income divided by total assets, so any positive net income firm is granted a 1 in this category.  ROA is also compared year over year.  If ROA increases, it signals that a greater net income, or profit, was generated using the same, or less, assets. 

Firms that improve their ROA receive another 1.  Accrued profit earns a 1 by being smaller than 0.  

         The third metric under profitability is current operating cash flow, a measure that takes net income and accounts for non-cash expenses and changes in working capital. Positive current operating cash flow adds 1 to the F-score. Lastly, accrued profit is counterproductive for profitability since profits earned but not realized have an opportunity cost associated.

The metrics for leverage and liquidity are change in long term debt and change in current ratio year over year. 

A decrease in long-term debt is indicative of a company not reliant on external funding. 

A firm is fundamentally stronger when it is deleveraging so a value of 1 is assigned to this component if there is a net decrease in long-term debt.  Current ratio is defined as current assets divided by current liabilities, whereas it is a measure of whether a firm has capital on hand to fulfil its short-term obligations. 

An increase in current ratio shows a smaller chance of a firm defaulting on its obligations and thus a value of 1 is assigned for when the current ratio increases.

Optimization Strategy Report : Based on Expectation Error-Fixing Method

The two metrics to capture operating efficiency are year over year changes in gross margin and asset turnover. 

Either component adds a 1 if the year over year increase is positive. 

A positive change in gross margin signifies the ability for the firm to cut costs or to sell more for the same cost, either being a positive fundamental signal. 

Asset turnover is net sales divided by average total assets over the period.  There is an opportunity cost of capital for assets, so to improve the ratio by selling more, or selling the same but holding less assets, the firm would benefit.

 The results in the above regression within the Piotroski/So report are consistent with the strategy.  Dummy variables for a stock’s classification based on PB and F-score were regressed against returns. 

In the first column, it is evident by the coefficients that even before the F-score classification, value stocks outperform middle PB stocks which in turn outperform growth stocks. In column 2, the interaction dummy variable for the F-score classification is included.  All coefficients are statistically significantly different from 0.  The expected returns for the regression are calculated in the following table.

From the marginal effects determined from the regression interaction coefficients which were all significant, it is conclusive that higher F-score dominates lower F score absolutely. 

Conversely, for a given F-score, the returns of low PB firms absolutely dominate the returns of higher PB firms. 

When comparing only the PB classification of a firm versus returns this may not be as significant, but when producing an arbitrage portfolio of shorting high PB, low F-score stocks with longing low PB, high F-score stocks one would expect to generate returns significantly different from 0. Alternatively, long short positions could be applied based on broader market trends to outperform the benchmark.


Rule books:

After we have finished the F-Scoring system, we want to apply it on our decision.

First step to build the model is to have some rules. We considered the trend of our macro economics and decided a bear market, thus, we would focus on short strategy more than on long strategy.

Second, based on the theorem, we received quarterly data and did not do too much of high frequency trading. The following are the reasons: as financial statements of these big companies were updated quarterly, so volatility, compared to weekly change,  is relatively small.  Last but not least of our rule, we want to find expectation errors and take advantage of it.

Measurement of expectation errors:

Measurement of expectation errors let us decide whether the companies have expectation errors and how severe that is. We will use companies’  price to book ratio as reference with an F-score.

For short:  We use p/b against last years’. If this year’s p/b is bigger, and its F-score is less than 5, we believe this company is overestimated, so we will short it.

For long: We use p/b against the last two quarters average, if this quarter’s is smaller with an F-score bigger than 6, then we believe this company is underestimated, so we will long it.

Model Application:

Based on rule books and information we gathered, we do our trading allocation like the following: 70% asset goes to short, 30% asset goes to long, and we will perform a hedge strategy for long since we considered a bear market.

Following is the stock picks suggested by model: (Long at left, Short at right)

As we can see, for the short part, the model selected IBM,PFE and TRV  in January 2020.  For the long part, the model selected MSFT and PG.


It may raise a concern that they have their resources allocated to these big companies. So based on their own model, they will invest a lot on these overvalued stocks, and they don’t care much about the short term fundamentals of companies.

So the price of the stocks still remain high during the period, which may cause loss on our short strategy.

For long: Let’s say the stock has been overvalued, then its fundamental keeps performing well recently, which means the intrinsic value increases and we can long to get profit, but a lot of times the price may not continue to increase since it was extremely overvalued before.

A good example is Tesla, as investors no longer believe Tesla’s stock can be valued over $900, even if the company is performing well, the price won’t increase.


Long-only strategy

When the entire market is performing well, the strategy can also work very well and have a positive alpha return according to the backtesting result.

Then the quarterly return for the long-only strategy was calculated. We can see that the number of winning quarters for the long-only strategy is 13, and the losing number is 8.

The success rate is relatively high. (Winning quarter means the quarter our portfolio outperforms the benchmark ). The underlying cause for it may be that the U.S. stock market also performed very well in the past 5 years and it is easy to find the public’s investment preferences.

Optimization Strategy Report : Based on Expectation Error-Fixing Method

However, the long only strategy  has some disadvantages. When the equity market is a bear market or there is a systemic risk such as the outbreak of the COVID-19.

Almost every stock price will fall. Under these circumstances our strategy will not work every well. The market will not have an expectation error opportunity for a long strategy. So we need some method to manage the risk of our strategy.

Optimization Strategy Report : Based on Expectation Error-Fixing Method

Short-only strategy

Then we back-tested the short only strategy. We can see that the front end of the short strategy outperforms the benchmark. The reason is that 2015 is another volatile year for oil prices and its continued weakness highlighted concerns about global growth.

As the U.S. market continued to improve economically in 2015, markets were dogged by the realization that much of the rest of the world isn’t faring so well. It has become very clear that Europe, China, and many emerging markets are struggling with protracted economic weakness. The US Equity is pessimistic during that time. So it’s a good time for us to short overvalued companies. This is the reason why our strategy outperformed the benchmark during that time.

Hedging strategy

So in order to make our portfolio have lower volatility when the market is poor, we will combine the short-only strategy and long only strategy together.

Average-MA-Score = Score MA 15 +Score MA 20 + Score MA 25

   + Score MA 30 + Score MA 35

At first A scoring system was formed relying on  the 15days, 20days, 25 days, 30days and 35 days moving average lines. If the direction of the MA is upward, then the score for this item will be 1, otherwise its score will be -1. By summing the score of these 5 items.

We get the final score at the current time. if the average-score at current point is smaller than zero, then we need to do some short to hedge the portfolio.

From the figure above we can see that the green signal means that we need to do the shorting. The red signal means we only need to use the long strategy. According to this back-testing performance the hedging portfolio will have lower volatility than the long-only portfolio and will have  relatively higher total return.

Summary Statistics of Simulated Trading 

According to the performance of the portfolio, its mean return is 0.12% (daily) and the standard deviation is 0.85% (daily). The t-ratio of mean return is 1.1263 and P-value for it is 0.2645, so we can say that our daily return is significantly different from zero.

The excess return over the T-bill is 5.92% (for the test period). According to the t-test the P-value will be 0.313, which means the excess return is not significantly different from zero.

The excess return over the DJIA return is 15.8% the main reason for the outperformance is that the DJIA had a great drawdown caused by COVID-19, while during this time we had short more than $800,000 during that time.

However according to the T-test the p-ratio for the excess return is 50% which means that our excess return is not significantly different from zero. 

The best weekly return is 8.8% (the week started from 2/24), while the worst weekly return is -7.04%. The maximum consecutive losing week is the week started from 3/23 to 3/27 because we held a lot of short positions, however the market rebounded a lot during that week.


From our perspective, the strategy relying on the stock scoring system and the definition of expectation error could outperform the market when the market is rising steadily.

However, when there is systemic risk in the market, the long-only strategy couldn’t resolve risks very well. So the introduction of shorting strategy helps reduce the volatility of the strategy when the market sentiment is pessimistic and most stocks’ prices drop a lot. 

However, there is still room for improvement. The time-series based trading signal is too sensitive to short-term market fluctuations. This will result in an increase in transaction fees. Therefore, we need to develop a trading signal that is more effective to capture changes over long periods. In this way our strategy will perform better when the market declines.

Optimization Strategy Report : Based on Expectation Error-Fixing Method


Piotroski, J. D., & So, E. C. (2012). Identifying Expectation Errors in Value/Glamour Strategies: A Fundamental Analysis Approach. Review of Financial Studies, 25(9), 2841-2875. doi:10.1093/rfs/hhs061

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Optimization Strategy Report : Based on Expectation Error-Fixing Method

Written by Dawei Xu, Junyi Yuan & Alexander Fleiss

Research – Rebellion Research

Optimization Strategy Report : Based on Expectation Error-Fixing Method

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