Application of the Extended Gompertz Model in Analyzing Insurance Growth
Abstract
The South African insurance sector is experiencing a positive growth as the nation is on high quality economic growth and development. However, there is little attention with regards to research on the growth analysis, hence the researchers aim to bridge the gap by analyzing the growth using a mathematically based approach. To verify the wide spread phenomenon behind insurance growth an extended Gompertz model (EGM) which is a member of the unified Richards family was used. The quantitative approach by means of functional limits, the cumulative distribution approach, initial value problem (IVP) and the qualitative derivative approach were used to fully analyze the model. We managed to derive a cumulative function was derived which can be used to estimate the number of insurance growth indicators.
The maximum carrying capacity of an insurance industry was estimated using the IVP which in our case is time dependent hence does not concur with other Gompertz related works. Using both the qualitative and derivative approach, a growth function which produced the same pattern with the original Gompertz curve with K(t) as the asymptotically stable and non-constant growth limit were deduced. Hence we can conclude that the growth of insurance sectors does follow a sigmoid shape with non-constant maturity levels. Lastly, we performed a statistical analysis of the nexus between insurance sector growth and economic development using GDP and insurance indicators (net premiums) data. From the statistical analysis done the results showed a positive relationship between the two. This showed that, insurance sector indeed plays a significant role towards economic development and as such their growth patterns should be well attended.
References
[2] Athenia, B.S., and Ntwanano, J.G. 2014. Insurance sector development and economic growth: Evidence from South Africa. Corporate Ownership and Control, 11 (4): 530-538.
[3] Begall, S. 1997. The application of the Gompertz model to describe body growth. Growth, Development and Aging, 61(2): 61-7.
[4] Gompertz, B. 1825. 0n the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London B: Biological Sciences, 115: 513-583. Available at: http://www.med.mcgill.ca/epidemiology/hanley/ c609/material/Gompertz-1825.pdf
[5] Grzegorz, R., Iwona, G., and Katarzyna, S. 2015. The Gompertz function and its applications in management. Foundations of Management, 7(1):185-190. DOI: https://doi.org/10.1515/fman-2015-0035. Available at: https://content.sciendo.com/view/journals/fman/7/1/article-p185.xml
[6] Han, L., Li, D., Moshirian, F., and Tian, Y.2010. Insurance development and economic growth. The Geneva Papers on Risk and Insurance – Issues and Practice, 35 (2): 183–199. Available at: https://link.springer.com/article/10.1057/gpp.2010.4
[7] Jana, S., and Kar, A. 2013. Mathematical study of a prey–predator model in relevance to pest control. Nonlinear Dynamic, 74: 667–683. DOI 10.1007/s11071-013-0996-3.
[8] Joachim, K. 2011. Evolutionary model of an anonymous consumer durable market. Physica A: Statistical Mechanics and its Applications, 390(14): 2692-2715.
[9] Laired, A.K. 1964. Dynamics of tumor growth. British Journal of Cancer, 18:490-502.
[10] Makeham, W.M. 1873. On the integral of the Gompertz’s function for expressing the values of sums depending upon the contingency of life. Journal of the Institute of Actuaries and Assurance Magazine, 17(5):305-327.
[11] Norton, L.A. 1988. Gompertzian model of human breast cancer growth. Cancer Research, 48: 7067-7141.
[12] Piotrowska. J.M., and Forys. U. 2011. The nature of Hopf bifurcation for the Gompertz model with delays. Mathematical and Computer Modelling, 54 (9–10): 2183-2198.
[13] Ricker, W.E. 1954. Stock and recruitment. Journal of Fisheries Research Board of Canada, 11: 559–623.
[14] Rogers, S., Pesti, G.M., Marks, H.L. 1987. Comparison of three nonlinear regression models for describing broiler growth curves. Growth, 51(2):229-39.
[15] Tjorve, M.C.K., and Tjorve, E. 2017. The use of Gompertz models in growth analysis and new Gompertz-model approach: An addition to the Unified-Richards family. PLoS ONE 12(6). DOI: https://doi.org/10.1371/journal.pone.0178691
[16] Von Bertalanffy, L. 1934. Untersuchungen über die Gesetzlichkeit des Wachstums. I. Allgemeine Grundlagen der Theorie; mathematische und physiologische Gesetzlichkeiten des Wachstums bei Wassertieren. Archiv für Entwicklungs mechanik, 131: 613–652. DOI: 10.1007/BF00650112
[17] Ward, D., and Zurburegg, R. 2000. Does insurance promote economic growth? Evidence form OECD Countries. Journal of Risk and Insurance, 67(4): 489-506.
[18] Zwiefel, J.R., and Lasker, R. 1976. Prehatch and posthatch growth of fishes-a general model. Fishery Bulletin, 74(3): 609-621. Available at: https://swfsc.noaa.gov/publications/CR/1976/7646.PDF
[19] Zwietering, M.H., Jongnburger, I., Rombouts, F.M., and Van’T Riet, K. 1990. Modelling of the bacterial growth curve. Appl. Environ. Microbial; 56(6):1875-1881.
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