Optimal investment in the presence of intangible assets and collateralized optimal debt ratio in jump-diffusion models


This paper studies an investor’s optimal investment, optimal net debt ratio with collateral security and optimal consumption plan in an economy that faces both diffusion and jump risks. The underlying assets are tangible and intangible assets. There have been major challenges of quantifying intangible assets. In this paper, we assume that the price of our intangible asset follows a jump-diffusion process. The tangible asset of the investor is divided into two parts: financial and nonfinancial (fixed) assets. Financial assets are invested into a market that is made up of a riskless asset and multiple risky assets (stocks). The fixed asset and stock prices follow jump-diffusion processes. This paper aim at (1) maximizing the total expected discounted utility of consumption in the infinite time horizon in the presence of both tangible and intangible assets, (2) determining the optimal net debt ratio for an investor in the presence of jumps, (3) determining the effect of intangible assets on the investor’s portfolio and (4) determining the optimal portfolio strategies of an investor who invest in an economy that is exposed to both diffusion and jump risks. This paper obtained the optimal consumption, optimal portfolio strategies and optimal net debt ratio of an investor under some suitable utility functions. We found that the value of the intangible assets depends explicitly on all the market parameters including the production rate, optimal wealth and optimal debt ratio of the investor over time. This implies that any change in the market parameter values, optimal investment, optimal net debt ratio and the production rate will lead to changes in the values of the intangible assets.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Damodaran, A.: The Value of Intangibles (2017). http://people.stern.nyu.edu/adamodar/pdfiles/ovhds/dam2ed/intangibles.pdf

  2. 2.

    Millar, A.: How to Calculate Intangible Assets in Company Valuation. Investing Research, Education (2014). https://www.seeitmarket.com/calculate-intangible-assets-company-valuation-13632/

  3. 3.

    Holloway, B.P., Reilly, R.F.: Intangible Asset Valuation Approaches and Methods. Intangible Asset Valuation Insights (2012). http://www.willamette.com/insights_journal/12/autumn_2012_2.pdf

  4. 4.

    Nkeki, C.I.: Optimal investment risks and debt management with backup security in a financial crisis. J. Comput. Appl. Math. 338, 129–152 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Zou, B., Cadenillas, A.: Optimal investment and liability ratio policies in a multidimensional regime switching model. Working paper (2015)

  6. 6.

    Ait-Sahalia, Y., Cacho-Diaz, J., Laeven, R.J.A.: Modeling financial contagion using mutually exiting jump processes. J. Financ. Econ. (2015). https://doi.org/10.1016/j.jfineco.2015.03.002

    Article  Google Scholar 

  7. 7.

    Nguyen, T.H.: Optimal investment and consumption with downside risk constraint in jump-diffusion models (2016). arXiv:1604.05584v1 [q-fin.MF]

  8. 8.

    Hong, Y., Jin, X.: Explicit solutions for dynamic portfolio choice in jump-diffusion models with multiple risky assets and state variables and their application. Working paper, International Business School Suzhou, Xi’an Jiaotong-Liverpool University (2016)

  9. 9.

    Junca, M., Serrano, R.: Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics. arXiv:1411.1103

  10. 10.

    Nkeki, C.I.: Optimal investment and optimal additional voluntary contribution rate of a DC pension fund in a jump-diffusion environment. Ann. Financ. Econ. (2017). https://doi.org/10.1142/S2010495217500178

    Article  Google Scholar 

  11. 11.

    Nkeki, C.I.: Optimal pension fund management in a jump-diffusion environment: theoretical and empirical studies. J. Comput. Appl. Math. 330, 228–252 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Nkeki, C.I.: Optimal investment strategy with dividend paying and proportional transaction costs. Ann. Financ. Econ. (2018). https://doi.org/10.1142/S201049521850001X

    Article  Google Scholar 

  13. 13.

    Jin, Z.: Optimal debt ratio and consumption strategies in financial crisis. J. Optim. Theory Appl. (2014). https://doi.org/10.1007/s10957-014-0629-0

    Article  MATH  Google Scholar 

  14. 14.

    Liu, W., Jin, Z.: Analysis of optimal debt ratio in a Markov regime-switching model. Working paper, Department of Finance, La Trobe University, Melbourne, Australia (2014)

  15. 15.

    Bank, P., Riedel, F.: Optimal consumption choice with intertemporal substitution. Ann. Appl. Probab. 11(3), 750–788 (2001)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Nkeki, C.I.: Optimal investment risks management strategies of an economy in a financial crisis. Int. J. Financ. Eng. (2018). https://doi.org/10.1142/S2424786318500032

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Stein J.L.: Stochastic optimal control modeling of debt crises. CESifo working paper no. 1043 (2003)

  18. 18.

    Stein J.L.: Application of stochastic optimal control to financial market debt crises. CESifo working paper no. 2539 (2009)

  19. 19.

    Krouglov, A.: Simplified mathematical model of financial crisis. MPRA paper no. 44021, pp. 2–11 (2013)

  20. 20.

    Maurer, R.H., Schlag, C., Stamos, M.Z.: Optimal life-cycle strategies in the presence of interest rate and inflation risk. Working paper, Department of Finance, Goethe University, Frankfurt, Germany (2008)

  21. 21.

    Zou, J., Zhang, Z., Zhang, J.: Optimal dividend payouts under jump-diffusion risk processes. Stoch. Models 25, 32–347 (2009). https://doi.org/10.1080/15326340902870133

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ait-Sahalia, Y., Cacho-Daiz, J., Hurd, T.R.: Portfolio choice with jumps: a closed-form solution. Ann. Appl. Probab. 19(2), 556–584 (2009). https://doi.org/10.1214/08-AAP552

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Liua, T, Zhaob, J., Zhaoc, P.: Portfolio problems based on jump-diffusion models. 26(3), 573–583. Faculty of Sciences and Mathematics, University of Ni\({\hat{s}}\), Serbia (2012). https://doi.org/10.2298/FIL1203573L

  24. 24.

    Thakoor, N., Tangman, Y., Bhuruth, M.: Numerical pricing of financial derivatives using Jain’s high-order compact scheme. Math. Sci. 6, 72 (2012). https://doi.org/10.1186/2251-7456-6-72

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Vajargah, F.K., Shoghi, M.: Simulation of stochastic differential equation of geometric Brownian motion by quasi-Monte Carlo method and its application in prediction of total index of stock market and value at risk. Math. Sci. 9, 115–125 (2015). https://doi.org/10.1007/s40096-015-0158-5

    Article  MATH  Google Scholar 

  26. 26.

    Farnoosh, R., Rezazadeh, H., Sobhani, A., Behboudi, M.: Analytical solutions for stochastic differential equations via Martingale processes. Math. Sci. 9, 87–92 (2015). https://doi.org/10.1007/s40096-015-0153-x

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Nkeki, C.I., Nwozo, R.C.: Variational form of classical portfolio strategy and expected wealth for a defined contributory pension scheme. J. Math. Finance 2, 132–139 (2012). https://doi.org/10.4236/jmf.2012.21015

    Article  Google Scholar 

  28. 28.

    Nkeki, C.I., Nwozo, R.C.: Optimal investment under inflation protection and optimal portfolios with stochastic cash flows strategy. IAENG Int. J. Appl. Math. 43(2), 54–63 (2013)

    MathSciNet  Google Scholar 

  29. 29.

    Nkeki, C.I.: Mean-variance portfolio selection problem with time-dependent salary for defined contribution pension scheme. Financ. Math. Appl. 2(1), 1–26 (2013)

    MATH  Google Scholar 

  30. 30.

    Nkeki, C.I.: Stochastic funding of a defined contribution pension plan with proportional administrative costs and taxation under mean-variance optimization approach. Stat. Optim. Inf. Comput. 2, 323–338 (2014)

    MathSciNet  Google Scholar 

  31. 31.

    Nkeki, C.I.: Optimal surplus, minimum pension benefits and consumption plans in a mean-variance portfolio approach for a defined contribution pension scheme. Konuralp J. Math. 3(2), 219–244 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Charles I. Nkeki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nkeki, C.I., Modugu, K.P. Optimal investment in the presence of intangible assets and collateralized optimal debt ratio in jump-diffusion models. Math Sci (2020). https://doi.org/10.1007/s40096-020-00343-8

Download citation


  • Net debt ratio
  • Optimal investment
  • Collateral security
  • Intangible assets
  • Consumption plan
  • Jump-diffusion risks
  • Power utility

JEL Classification

  • G11
  • G12
  • C02
  • C22
  • C61