Can topology predict a financial crisis?

Final year project

Chen Xilin
NTU/SPMS student

Data

##      REAL   PHRM    ELMH  RETL   FOOD    TRAN
## 4  833.22 1098.6 1728.10 904.3 873.42 1591.91
## 5  827.92 1077.8 1712.61 897.4 856.62 1557.82
## 6  817.92 1070.9 1703.17 885.2 846.85 1542.76
## 7  763.56 1051.9 1672.58 865.4 821.61 1510.62
## 8  765.31 1026.1 1631.15 852.9 788.20 1456.11
## 9  796.50 1029.8 1646.94 865.0 819.48 1489.35
## 10 819.00 1041.7 1647.54 870.7 815.41 1515.23
## 11 819.00 1041.7 1647.54 870.7 815.41 1515.23
## 12 824.88 1047.9 1667.27 878.7 819.66 1528.74
## 13 818.24 1049.3 1658.89 880.5 821.53 1519.20
## 14 798.86 1041.5 1628.81 871.8 806.29 1489.68
## 15 793.00 1035.4 1626.85 866.3 796.01 1498.43

Correlation coefficients

The correlation coefficient between two industries \(i\) and \(j\) over a period from day \(d_1\) to day \(d_2\) is \[C_{d_1,d_2}(i,j)= \frac{\sum_{i=d_1}^{d_2}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=d_1}^{d_2}(x_i - \bar{x})^2} \sqrt{\sum_{i=d_1}^{d_2}(y_i - \bar{y})^2}},\qquad \mathrm{dist}(i,j)=1-C_{*,*}(i,j)\]

Distance Matrix

##            BNK       AIR      REAL      PHRM
## BNK  0.0000000 0.1030363 0.1071617 0.5644072
## AIR  0.1030363 0.0000000 0.2616577 0.5933658
## REAL 0.1071617 0.2616577 0.0000000 0.6044045
## PHRM 0.5644072 0.5933658 0.6044045 0.0000000

Persistent homology

Point cloud \(\longrightarrow\) Distances \(\longrightarrow\) Chain of embedded simplicial complexes \(\longrightarrow\) barcode

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Threshold = 0.4. Loops, i.e., (METL - REAL - ELEC - RUBB).

Barcode

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32 connected components, 8 loops and 2 cavities.

Method 1

Existence of Cavities

\(d_2-d_1=30\)

Observation

  1. Cavity appears in non-financial crisis periods.

  2. Cavity begins to disappear from the end of a non-financial crisis period.

  3. Cavity begins to appear from the end of a financial crisis period.

  4. The frequency of cavities has a downward trend.

cavities Non-financial Crisis Financial Crisis
Begin Appear Disappear
End Disappear Appear

Method 2

\(C(B)\) = the number of connected components corresponding to the smallest threshold value for which the simplicial complex has a loop.

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\(C(B)=21\)

\(d_2-d_1=10\)

Mean and variance

Growth Crisis Crisis Growth Crisis Crisis
Mean 13.46154 13.75758 15.50633 13.57143 14.14085 13.70312
Variance 25.10 25.48 30.97 19.29 28.04 23.86

Observations

  • The plot is not informative
  • C(B) is not visibly related to macroeconomic periods
  • Variance of C(B) may be, indeed, related to macroeconomic periods

Conclusion

Although we are not able to predict an economic crisis with a reasonable degree of certainty, we still clearly see that topology is able to capture something about economics. Most notably, the simplicial complex associated with a growth period has a cavity but the simplicial complex associated with a crisis period does not.

Directions of further research

  1. Hourly data

  2. Economies of countries other than Japan

  3. Other topological data analysis techniques, i.g., cluster analysis, persistence landscape, Mapper etc.