Home / Regular Issue / JSSH Vol. 31 (5) Aug. 2023 / JST-3700-2022


On Estimating the Parameters of the Generalised Gamma Distribution based on the Modified Internal Rate of Return for Long-Term Investment Strategy

Amani Idris Ahmed Sayed and Shamsul Rijal Muhammad Sabri

Pertanika Journal of Social Science and Humanities, Volume 31, Issue 5, August 2023

DOI: https://doi.org/10.47836/pjst.31.5.07

Keywords: Generalised gamma distribution, modified internal rate of return, moment methods, simulated annealing algorithm

Published on: 31 July 2023

The generalised gamma distribution (GGD) is one of the most widely used statistical distributions used extensively in several scientific and engineering application areas due to its high adaptability with the normal and exponential, lognormal distributions, among others. However, the estimation of the unknown parameters of the model is a challenging task. Many algorithms were developed for parameter estimation, but none can find the best solution. In this study, a simulated annealing (SA) algorithm is proposed for the assessment of effectiveness in determining the parameters for the GDD using modified internal rate of return (MIRR) data extracted from the financial report of the publicly traded Malaysian property companies for long term investment periods (2010–2019). The performance of the SA is compared to the moment method (MM) based on mean absolute error (MAE) and root mean squares errors (RMSE) based on the MIRR data set. The performance of this study reveals that the SA algorithm has a better estimate with the increases in sample size (long-term investment periods) compared to MM, which reveals a better estimate with a small sample size (short-time investment periods). The results show that the SA algorithm approach provides better estimates for GGD parameters based on the MIRR data set for the long-term investment period.

  • Abubakar, H., & Sabri, S. R. M. (2021a). Incorporating simulated annealing algorithm in the Weibull distribution for valuation of investment return of Malaysian property development sector. International Journal for Simulation and Multidisciplinary Design Optimization, 12, Article 22. https://doi.org/10.1051-/smdo/2021023

  • Abubakar, H., & Sabri, S. R. M. (2021b). Simulation study on modified weibull distribution for modelling of investment return. Pertanika Journal of Science and Technology, 29(4), 2767-2790. https://doi.org/10.47836/pjst.29.4.29

  • Ahmad, A. G. (2015). Comparative study of bisection and Newton-Rhapson methods of root-finding problems. International Journal of Mathematics Trends and Technology, 19(2), 121-129. https://doi.org/10.14445/22315373/ijmtt-v19p516

  • Baldwin, R. H. (1959). How to assess investment proposals. Harvard Business Review, 37(3), 98-104.

  • Besley, S., & Brigham, E. F. (2015). CFIN4 (with Finance CourseMate). Cengage Learning.

  • Bílková, D. (2012). Lognormal distribution and using L-moment method for estimating its parameters. International Journal of Mathematical Models and Methods in Applied Sciences, 6(1), 30-44.

  • Biondi, Y. (2006). The double emergence of the modified internal rate of return: The neglected financial work of Duvillard (1755 - 1832) in a comparative perspective. The European Journal of the History of Economic Thought, 13(3), 311-335. https://doi.org/10.1080/09672560600875281

  • Bonazzi, G., & Iotti, M. (2016). Evaluation of investment in renovation to increase the quality of buildings: A specific Discounted Cash Flow (DCF) approach of appraisal. Sustainability, 8(3), Article 268. https://doi.org/10.3390/su8030268

  • Brealey, R. A., Myers, S. C., & Allen, F. (2006). Principles of Corporate Finance. IrwinMcGrawHill.

  • Černý, V. (1985). Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications, 45(1), 41-51.

  • Chang, T. P. (2011). Performance comparison of six numerical methods in estimating Weibull parameters for wind energy application. Applied Energy, 88(1), 272-282.

  • Chaurasiya, P. K., Ahmed, S., & Warudkar, V. (2018). Study of different parameters estimation methods of Weibull distribution to determine wind power density using ground based Doppler SODAR instrument. Alexandria Engineering Journal, 57(4), 2299-2311. https://doi.org/10.1016/j.aej.2017.08.008

  • Cont, R. (2001). Empirical properties of asset returns: Stylised facts and statistical issues. Quantitative Finance, 1(2), 223-236. https://doi.org/10.1080/713-665670

  • Crama, Y., & Schyns, M. (2003). Simulated annealing for complex portfolio selection problems. European Journal of Operational Research, 150(3), 546-571. https://doi.org/10.1016/S0377-2217(02)00784-1

  • Du, K. L., & Swamy, M. N. S. (2016). Simulated annealing. In Search and Optimisation by Metaheuristics (pp. 29-36). Birkhauser. https://doi.org/10.1007/978-3-319-41192-7_2

  • Eric, U., Olusola, O. M. O., & Eze, F. C. (2021). A study of properties and applications of gamma distribution. African Journal of Mathematics and Statistics Studies, 4(2), 52-65. https://doi.org/10.52589/ajmss-mr0dq1dg

  • Fama, E. F. (1963). Mandelbrot and the stable Paretian hypothesis. The Journal of Business, 36(4), 420-429. https://doi.org/10.1086/294633

  • Franzin, A., & Stützle, T. (2019). Revisiting simulated annealing: A component-based analysis. Computers and Operations Research, 104, 191-206. https://doi.org/10.1016/j.cor.2018.12.015

  • Gomes, O., Combes, C., & Dussauchoy, A. (2008). Parameter estimation of the generalized gamma distribution. Mathematics and Computers in Simulation, 79 (4), 955-963. https://doi.org/10.1016/j.matcom.2008.02.006

  • Greenstein, L. J., Michelson, D. G., & Erceg, V. (1999). Moment-method estimation of the Ricean K-factor. IEEE Communications Letters, 3(6), 175-176. https://doi.org/10.1109/4234.769521

  • Honore, B., Jørgensen, T., & de Paula, A. (2020). The informativeness of estimation moments. Journal of Applied Econometrics, 35(7), 797-813. https://doi.org/10.1002/jae.2779

  • Hosking, J. R. M., Wallis, J. R., & Wood, E. F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27(3), 251-261.

  • Idris, A. A., & Muhammad, S. S. R (2022). A simulation study on the simulated annealing algorithm in estimating the parameters of generalized gamma distribution. Science and Technology Indonesia, 7(1), 84-90. https://doi.org/10.26554/sti.2022.7.1.84-90

  • Kellison, S. G. (2009). The Theory of Interest. McGraw Hill Education.

  • Khodabina, M., & Ahmadabadi, A. (2010). Some properties of generalized gamma distribution. Mathematical Sciences, 4(1), 9-28.

  • Kiche, J., Ngesa, O., & Orwa, G. (2019). On generalized gamma distribution and its application to survival data. International Journal of Statistics and Probability, 8(5), 1927-7040. https://doi.org/10.5539/ijsp.v8n5p85

  • Kierulff, H. (2008). MIRR: A better measure. Business Horizons, 51(4), 321-329. https://doi.org/10.1016/j.bushor.2008.02.005

  • Kim, S., Lee, J. Y., & Sung, D. K., (2003). A shifted gamma distribution model for long-range dependent internet traffic. IEEE Communications Letters, 7(3), 124-126. https://doi.org/10.1109/lcomm.2002.808400

  • Kirkpatrick, S., Gelatt Jr, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671-680. https://doi.org/10.1126/science.220.4598.67

  • Lakshmi, R. V., & Vaidyanathan, V. S. (2016). Three-parameter gamma distribution: Estimation using likelihood, spacings and least squares approach. Journal of Statistics and Management Systems, 19(1), 37-53. https://doi.org/10.1080/09720510.2014.986927

  • Malá, I., Sládek, V., & Habarta, F. (2022). Comparison of estimates using L and TL moments and other robust characteristics of distributional shape and tail heaviness. REVSTAT-Statistical Journal, 20(5), 529-546. https://doi.org/10.57805/revstat.v20i5.386

  • Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x

  • Munkhammar, J., Mattsson, L., & Rydén, J. (2017). Polynomial probability distribution estimation using the method of moments. PloS One, 12(4), Article e0174573. https://doi.org/10.1371/journal.pone.0174573

  • Naji, L. F., & Rasheed, H. A. (2019). Estimate the two parameters of gamma distribution under entropy loss function. Iraqi Journal of Science, 60(1), 127-134. https://doi.org/10.24996/ijs.2019.60.1.14

  • Orús, R., Mugel, S., & Lizaso, E. (2019). Quantum computing for finance: Overview and prospects. Reviews in Physics, 4, Article 100028. https://doi.org/10.1016-/j.revip.-2019.100028

  • Osborne, M. J. (2010). A resolution to the NPV–IRR debate? The Quarterly Review of Economics and Finance, 50(2), 234-239. https://doi.org/10.1016/j.-qref.2010.01.002

  • Özsoy, V. S., Ünsal, M. G., & Örkcü, H. H. (2020). Use of the heuristic optimization in the parameter estimation of generalized gamma distribution: Comparison of GA, DE, PSO and SA methods. Computational Statistics, 35(4), 1895-1925. https://doi.org/10.1007/s00180-020 00966-4

  • Pascual, N., Sison, A. M., Gerardo, B. D., & Medina, R. (2018). Calculating internal rate of return (IRR) in practice using improved newton-raphson algorithm. Philippine Computing Journal, 13(2), 17-21. https://pcj.csp.org.ph/index.php-/pcj/issue/view/28

  • Quiry, P., Dallocchio, M., LeFur, Y., & Salvi, A. (2005). Corporate Finance: Theory and Practice (6th Ed). John Wiley & Sons Ltd.

  • Rocha, P. A. C., de Sousa, R. C., de Andrade, C. F., & da Silva, M. E. V. (2012). Comparison of seven numerical methods for determining Weibull parameters for wind energy generation in the northeast region of Brazil. Applied Energy, 89(1), 395-400. https://doi.org/10.1016/j.apenergy.2011.08.003

  • Ross, A. S., Westerfield, R. W., & Jordan, B. D. (2010). Fundamentals of Corporate Finance. The McGraw-Hill Companies, Inc.

  • Sabri, S. R. M., & Sarsour, W. M. (2019). Modelling on stock investment valuation for long-term strategy. Journal of Investment and Management, 8(3), 60-66. https://doi.org/10.11648/j.jim.20190803.11

  • Satyasai, K. J. S. (2009). Application of modified internal rate of return method for watershed evaluation. Agricultural Economics Research Review, 22, 401-406.

  • Sayed, A. I. A., & Sabri, S. R. M. (2022). Transformed modified internal rate of return on gamma distribution for long term stock investment. Journal of Management Information and Decision Sciences, 25(S2), 1-17

  • Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x

  • Stacy, E. W., & Mihram, G. A. (1965). Parameter estimation for a generalized gamma distribution. Technometrics, 7(3), 349-358. https://doi.org/10.1080/00401706.1965.10268

  • Tizgui, I., El Guezar, F., Bouzahir, H., & Benaid, B. (2017). Comparison of methods in estimating Weibull parameters for wind energy applications. International Journal of Energy Sector Management, 11(4), 650-663. https://doi.org/10.1108/IJESM-06-2017-0002