We present the approach developed in the section above. The sample is constituted of 516 stocks each year from 1995 to 2012.
Table 2.4-1 Performance of all the 516 stocks from the migration over 1995 to 2012
1995- 2012 0 1 2 3 4 5 6 7 8 9 0 25.0% 8.3% 8.3% 33.3% 0.0% 8.3% 0.0% 8.3% 8.3% 0.0% 1 5.7% 2.9% 8.6% 25.7% 20.0% 11.4% 11.4% 5.7% 8.6% 0.0% 2 1.5% 0.8% 8.3% 11.3% 13.5% 15.8% 22.6% 18.0% 8.3% 0.0% 3 0.0% 1.8% 4.5% 8.3% 14.4% 23.2% 24.3% 16.8% 6.7% 0.0% 4 0.1% 0.5% 1.7% 5.9% 16.6% 26.1% 27.7% 16.0% 5.0% 0.4% 5 0.0% 0.3% 1.2% 5.6% 14.9% 27.8% 29.0% 15.6% 5.2% 0.4% 6 0.0% 0.2% 0.7% 5.0% 16.5% 28.8% 26.7% 16.4% 5.3% 0.4% 7 0.1% 0.1% 1.2% 4.2% 16.5% 28.1% 28.1% 15.6% 5.4% 0.7% 8 0.0% 0.0% 1.3% 7.1% 17.6% 30.2% 29.6% 9.2% 4.5% 0.6% 9 0.0% 0.0% 0.0% 5.1% 28.2% 28.2% 25.6% 7.7% 5.1% 0.0%
Table 2.4-1 shows the entire migration probability sample for 516 stocks; this is simply the number of times each F-score appeared; the probability is expressed as a percentage and is calculated by dividing a frequency by the total frequency and multiplying by 100. Please note that we let F-scores 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 be described as “F-1, F-2, F-3, F-4, F-5, F-6, F-7, F-8, and F-9” when it comes to giving an explanation of the graphs or charts.
This gives us a record of past outcome and an understanding of possible trend – a trend being the general movement in the data over time. If someone can understand past changes over time in the F-score then that same person/investor can consider ways of projecting these forwards and using such a projection as a measure for future forecasts in the likelihood of the F- score. P io tro ski sc o re in y ear “t ”
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An investment company might be particularly interested in the change of the F-scores when it comes to buy one of them. We can describe the relationship using a straight line of probabilities. Consider the following example: where an investor is interested in the volatility over time of an F-score of F-9. If we bought a stock with a score of F-9 the chances that it stays an F-9 the following year are null; however, the investor has an 8% chance that the score becomes an F- 7 and therefore stays in his portfolio. Of course, when analyzing those figures, the investor should be careful as those F-scores might be influenced by a range of factors including the economic wisdom. The figures do, however, provide a useful guide. Overall, it can be expected that the high F-score decreases over time whilst an increase can be viewed in the bottom F-score. Taking another example, an investor who buys a stock rated an F-7 in the F-score is likely to have a 16% chance of this stock remaining an F-7; however, he has a 0% chance of the stock becoming an F-0.
Figure 2.4-1 High F-score change. The histogram displays the distribution of the F-score by showing for each corresponding F-score the probability expressed in percentage
This histogram in Figure 2.4-1 appears to have almost a perfect bell-shaped pattern: there are more data in the middle and less towards the two extremes, suggesting that the assumption of normality in the future can be tested using t-test.
This histogram displays the change in a high F-score from 1995 to 2012 for a high F-score to become an F-5 or an F-6. In this way the histogram displays the entire distribution of the high F-scores. From this histogram we can see as well the low probability for a change in a low F-score (i.e., from F-0 to F-2). When we look at the shape of the F-7 the distribution appears to be almost normally distributed with the same probability for an F-score of F-7 to become an F-6 or F-5, as figures show the probability is 28% respectively. Also, an investor has an approximately 16%
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chance for the F-7 to stay the same or 17% that it becomes an F-4. However, when looking at the F-8 we cannot draw the same conclusion as there is a 9% chance for the F-8 to become an F-7 and an approximately 18% or double chance for the F-8 to become an F-4. Overall, interesting conclusions can be draw out of this histogram. Also, please note the likelihood of the F-9 to become an F-4, an F-5, and an F-6 is approximately the same, around a 28% chance, meaning there might be some stability before the stock loses all its value.
Figure 2.4-2 Low F-score change. The histogram displays the distribution of the F-score by showing for each corresponding F-score the probability expressed in percentage
This histogram in Figure 2.4-2 displays the change in a low F-score from 1995 to 2012, for an F-0 to become an F-3. From this histogram we can see the absent probability for an F-0, F-1, F- 2, and F-3 to become an F-9, meaning that the F-score is quite stable. The ability of the F-2 and F- 3 to grow gradually should also be noted; a better picture will be displayed and discussed later. The linear upgrade in percentage chance can be seen in F-3, F-4, F-5, and F-6 when looking at those two scores; also, the same gradual decline is apparent in F-7 and F-8.
In the histogram, it is also worth noticing the constant 8% chance for F-0 to become F-1, F-2, F-5, F-7, and F-8. As well, interestingly, the histogram is showing the absence of probabilities for F-0 to become either an F-4 or F-6, which might signify something perhaps when we run some econometric tests – those two scores might reveal another picture. Also, it is worth looking at the likelihood of F-3; indeed, the null probability chance for an F-3 to become F-0 can be seen, implying the idea that an investor interested in using an F-3 strategy has to screen in stocks rated F-1 to benefit from the potential upgrade over time.
0% 5% 10% 15% 20% 25% 30% 35% 0 1 2 3 4 5 6 7 8 9 0 1 2 3
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Additionally, when looking at F-1 an investor has the same chance for F-1 to become either F-0 or F-7 with a 6% chance for each of them; the same observation can be applied for F-1 to become F-2 or F-8 with a 9% chance.
Figure 2.4-3 shows the migration of an F-score of 2 and 3 for the period 1995 to 2012; as discussed in the previous paragraph, the idea is to get a clearer picture. The y-axis represents the percentage chance that this occurrence appears. The percentage change for both F-scores grows gradually for an F-score of 2, 3, 4, and 6 with a corresponding fall in F-7 and F-8; this growth can be explained by some predominance for those two scores to gradually change over time, particularly the ability for those two scores previously rated as poor investment, due to them showing some ability to continuously improve the prospects of the firms. Some investment strategy might be interesting to develop out of this figure. The percentage chance for F-2 and F-3 to become an F-7 or an F-8 is respectively 18% (8%) for F-2 and 17% (7%) for F-3. In contrast, the ability to become an F-6 is respectively 23% for F-2 and 24% for F-3. The decline after F-6 might be seen as a signal for a sell when buying an F-2 or F-3. Also, the ability to become F-9 for both of those two scores is null, meaning that buying an F-2 or F-3 will lose all its power after being upgraded as an F-8. As spotted earlier, the chance for F-3 to become F-0 is null.
Figure 2.4-3 F-score of 2 and 3 change. The histogram displays the distribution of the F-score by showing for each corresponding F-score the probability expressed in percentage
As described earlier, here in Figure 2.4-3 we have a clearer picture of the outcome of an F-score of F-2 or F-3 in our migration stability, highlighting some interesting pictures, as buying both of those two scores could lead gradually to a potential upgrade of the score, implying an eventual investment opportunity.
0% 5% 10% 15% 20% 25% 30% 0 1 2 3 4 5 6 7 8 9 2 3
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Figure 2.4-4– Medium F-score change. The histogram displays the distribution of the F-score by showing for each corresponding F-score the probability expressed in percentage
When examining Figure 2.4-4, it can be seen that the histogram is centred around an F- score of F-5 to F-6, the values vary from 15% to 30% and the shape of the distribution is quite normally distributed, which means that when an investor buys a score of F-5 to F-6 it has approximately the same chance to go either to the right or the left of the tail.
The centres and spreads are not too different from the high F-score histogram in Figure 2.4-4 but the shape appears to be slightly right skewed, while the high F-score appears to be left skewed with less probability on the F-9 and F-8. Please note the ability for F-4, F-5, and F-6 to become either F-3 or F-8 or F-4 or F-7, describing a bell shape. Also noticeable is the absence for F-4, F-5, F-6 to become F-0.
Overall, these histograms provide an easy to understand summary of the distribution of the F-score migration.