Statistical Modelling For Actuarial Studies Using PCA Assignment Sample

Introduction: Summary Statistics, Share Price Plots and PCA Results

This assessment is going to discuss the statistical modelling in which various statistical calculations will be performed, such as Summary statistics, plots and principal component analysis (PCA). For the calculation, RStudio software will be used.

Question 1: Summary Statistics

Figure 1: Summary statistics

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Some findings that come out of the summary statistics for the share price returns of different companies include the following: For CSL the share prices vary from a minimum of 255.6 to a maximum of 306.4 with a mean of 281.9 and a median of 286.2. This means that the share prices in CSL’s are not very dispersed but they are also not volatile or far away from the mean value (Dege and Brüggemann 2024). The first quarter at 268.7 and the third quarter at 294.7 indicate that 50% of share prices are confined within this range indicating moderate and fairly stable price shareholders volatility with moderate deviations around the middle (C = 26.50, Fluctuation = 2.77 of NAB is a little lower than CSL, which varies between 25.06 and 30.83. Similarly, the median, more precisely 28.93, is also slightly higher and the IQR 26.88- 29.88 clearly illustrates that the variation or dispersion in the case of NAB’s share price is relatively low and not very dispersed as compared to CSL where the variation is comparatively wider between the first and third quartiles.

The same is true for ANZ shares, as there is a mean, median and range of 23.19, 23.54, 20.68 to 25.22 respectively. Thus, the IQR between 22.17 and 24.00 cautiously suggests that the central 50% of ANZ’s share prices are relatively tighter than CSL and NAB in terms of their volatility.

 The shares of WES have a mean price of 46. 80 Additionally, it has a pricing range of 40.32–51.97. The IQR of 45.12 to 48.50 lies closer to the midpoint of the range, signifying moderate variability in share prices while the share prices do not significantly differ, they do not stick to a very stable range either (Bhat et al. 2023).

The average of respondents’ ratings towards GMG is 19.71 on the scale with the range of 15.64 to 23.59 which is lower than the other companies and the distribution is somewhat narrow. The performance of 18.79 to 20.52, aided by the IQR, also induces moderate variation to contend that the performance of this team is fairly constant.

Question 2: Plot of Share Price

Figure 2: Plot of share price

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CSL can be observed as having a general upward movement over the period in analysis. From January 2022, when it is around 256, the share price of CSL continues to go up with some fluctuations and peaks at 306.39 in May 2023 (Tucker et al. 2023). These movements demonstrate that CSL has high growth and investor confidence as the stock prices continue to rise. Although there are variations, especially in the second semester of 2022, the performance is still relatively upward, indicating a strong performance.

NAB used to show more fluctuation than JB Hi-Fi in its share price. Starting from the beginning of the year, 25.06 in 2022, the price of NAB records a varying moving average, showing both the high and the low points. The highest value recorded here is 30.83 and the lowest is 20.68. This volatility points to the fact that NAB’s share price depends on market fluctuations or organisational events and hence is volatile.

ANZ shows a better consistent level in comparison with NAB. The share price experiences small variations depending on the market and can be between 20.68 and 25.22. While the price of ANZ indicated that it was on an upward slope from early 2022 to mid-2023, the rise was not near the magnitude observed in CSL (Kania and Kusumah 2023). This stability may be due to the company’s financial performance efficiency or confidence in the market.

WES shares started at 49.91 in January 2022 and the general tendency is upward, with the maximum value by the indicator at 51.97. It can be summarised that the pattern is quite steady with certain oscillations, which means that there is continuous growth with minor market changes.

In the period under consideration, GMG has a declining pattern. The share price is assumed to begin at 22.81 and fall to approximately 19.63 by May 2023 (Bilder and Loughin 2024). This will imply that GMG may be facing some problems or else a conventional market situation that may have an impact on its performance.

ALL has a moderate variation, with these shares having been oscillating between 30.28 and 39.43. However, it shows that there is no progressive tendency, but it oscillates depending on the investors’ attitude and the market conditions.

It is evident from QBE that there is an upward trend in the share price from 10.63 to 15.33 implying growth and/or positive performance; hence, acceptance of the hypothesis. This sort of steady rise usually signals good financial health or improving market conditions (Wen et al. 2023). The performance of JHX is 47.06 - 47.6 with a stable increase, which indicates that the company’s share price has also been rising from 47.

Question 3: PCA

Figure 3: PCA

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The area to the right of the vertical line is 50.95% and represents the amount of variance in the data accounted for by the first principal component or as it is commonly referred to, the PC1. Such a considerable proportion suggests that PC1 explains the overall direction in all share prices (Kamel and Abonazel 2023). Thus, originating from a High standard deviation (2.2573), this PC1 could be more inclined towards capturing a broad-based market factor or some factor that has a similar impact to most of the shares. Its cumulative proportion gets to 50.95%, which implies that this component only accounts for half of the variation in the data.

Figure 4: Screen plot

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The second component of principle components analysis (PCA2) has a standard deviation of 1.1723 and it explains that 13.74% of variability is not explained by PCA1. The cumulative proportion rises to 64.70%, which implies that by using both PC1 and PC2, the variance of the data is accounted for a significant amount (Alfaris et al. 2023). This component may be specific to the sector increase or any other influential factor apart from the market trend.

Figure 5: Principle component

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 Extensions of the PCA from components PC3 through to PC8 reveal smaller additional proportions of variance, with cumulative percentages totalling 97.55% at the PC7 level. These components help to explain even more subtle likes and dislikes of share prices. Such as 11.31% is seen in PC3 which is a considerable but smaller influence compared to PC1 and PC2.

The remaining two factors, or the minor components specifically PC9 and PC10 account for the minute proportion of variance (0.727% and 0.473%, respectively), and therefore provide a minimal explanation. These components are affiliated with noise or with extremely specialised influences governing a minute portion of information.

Question 4

The study on share price returns using the tool of PCA gives insights that can be used for strategic management inside a firm. Awareness of the major sources of variation that are used in the variation of share price data can also go a long way in risk management, investment discretion and overall understanding of a market.

Firstly, the PCA provides information about major trends that act on the share prices. This is an expected result, as the first principal component (PC1) should capture 50.95% of the variance and most probably accounts for broad market trends or factors that affect all stocks in the same manner. About this component, the company is in a position to correlate the investments with more general market moves so that the portfolio is in a position to be sensitive to major trends in the market (Nasution et al. 2023). The knowledge assists in the identification of probable trends in the market and therefore the investment decisions.

Secondly, the second component (PC2) portrays 13.74% of the variance, showing the sector-wise variation or some other uncontrollable factor which has not been covered by PC1. It can also be applied to sharpen investment tools by selecting certain types of sectors or adjusting the investment portfolios by the rates of specific sectors. Knowledge of these extra parts enables the firm to invest on a more diversified basis and hedge against sector-specific volatilities.

Conclusion

In conclusion, according to the PCA, most of the variations in share price returns are explained by the first few numbers of PCs: PC1 and PC2 dominate the picture. The following components serve to reveal deviations that are smaller and more localised and thus offer a global rather than local view of the arrangement of the data. The evidence suggests that there are some firms where share prices are less volatile and, hence, the interquartile range is relatively narrow, while in other cases there is substantial variation. This is because the performance and market conditions of these companies vary, and so is the case with investors’ sentiment.