# INVESTMENT RETURNS WITH AN N-STEP GENERALIZED UTILITY FUNCTIONS

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In this paper, we formulated a general way of generating optimal wealth of investment returns with N-step utility functions through a martingale approach. The problem of maximization is solved via Lagrange method. The performance of any investment
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ICASTOR Journal of Mathematical Sciences Vol. 9, No. 2 (2015) 51 - 55   __________________________________________________________________________ Correspondence: Eghwerido Joseph Thomas,  Department of Mathematics and Computer Science, Federal University of Petroleum Resources Effurun, P.M.B 1221, Effurun, Delta State, Nigeria.  Email: eghwerido.joseph@fupre.edu.ng INVESTMENT RETURNS WITH AN N-STEP GENERALIZED UTILITY FUNCTIONS E. J. THOMAS, E. EFE-EYEFIA & E. EKUMA-OKEREKE, Department of Mathematics and Computer Science, Federal University of Petroleum Resources Effurun, Effurun, Delta State, Nigeria.  ABSTRACT In this paper, we formulated a general way of generating optimal wealth of investment returns with N-step utility functions through a martingale approach. The problem of maximization is solved via Lagrange method. The performance of any investment from day-to-day is determined by the ratio of the probability of rising to falling and the neutral probability. KEYWORDS AND PHRASES:    Martingale, Risk Neutral Probability, Utility Function, Optimal Wealth. INTRODUCTION Investors usually incur loss as a result of wrong investment options and could thus be confronted with inability to quickly determine the best allocation of wealth in their choice of investment. The amount and period of investment in achieving a particular goal in relation to probability of doubling or halving of returns can thus be of particular importance, hence, one has got a certain amount of money and tries to invest it in such a way to draw maximum or optimal returns. With payment     returns an amount       after one period. The rate of return is that value     that makes the present value of the return equal to the initial payment.           (1) where;   = amount invested in security        = investment returns  J. T. EGHWERIDO, E. EFE-EYEFIA & E. EKUMA-OKEREKE   52     = a non-negative random variable   = the rate of return from investment    1. RISK NEUTRAL MEASURE: Given a probability space ,,  Let us consider a single-period binomial model; then a probability measure     is called risk neutral if for all 0,1,…,       !  "# \$  Suppose we have a two-state economy. Let the initial stock price be S which can either go up to % &   or down to % '   with interest rate R > 0 and % ' (1)%(% & *  Then, the risk neutral probability of an upward stock movement is given by  +  !"!  ! - "!   (2) Let the derivative with pay-off be  . &   when the stock price moves up and  . '   when it goes down, then we can price the derivative via  . /   - 2"/  3    (3) Hence, the pay-off for  N   period 4    at time step   is given by 4  5 67"  with probability 58\$19 67"    :0,1,…,58 for an even step or 4  5 67""  with probability 5891\$19 67""    :0,…,5891 for an odd step and for initial capital  x  with probability of increase and decrease q  and  p  respectively and risk neutral probability measure ; . The dynamic optimization problem stated can be represented as a static optimization problem over terminal wealth: < = >?@ A BCDE  (4) subject to BDE F   (5) where  H = denotes state,  U = the utility function, and  Q = risk neutral probability   x = initial stake.  INVESTMENT RETURNS WITH AN N-STEP GENERALIZED UTILITY FUNCTIONS 53 2. GENERALIZED N-STEP UTILITY FUNCTIONS Let GH   8 G  H   utility function, such that IJIK    Suppose L & M58\$ 67N=  67"   G   (6) and, L O L & 9PM58\$; 67"67N= ;  H  9\$  (7) applying the Lagrangian method, it is easy to see that Q IR S IJ  0 8 IR S IJ 0 we have G;  PT;U 67" T;U   and for G;  PT;U 67" T;U    V   G; W PT;U 67"W T;U W *  Hence, G; = PT;U 67  Clearly, G;  G; = X;Y  T;U  G; = T;;U   Also, ZL O ZP[\58\$ 67N= ; 67" ;  G9]0  which implies that, GB + );E 67    J. T. EGHWERIDO, E. EFE-EYEFIA & E. EKUMA-OKEREKE   54   Suppose  + ) + 1, then GH    CONCLUSION This study concluded that our initial stake is maximized by the ratio of the probability of rising to falling together with the risk neutral probability measure. REFERENCE 1. P. Battocchio, and F. Menoncin, Optimal Pension Management in a Stochastic Framework, Insurance: Mathematics and Economics, 34, pp.79-95, 2004. 2. G. Deelstra, M. Grasselli, and P.-F. Koehl, Optimal Investment Strategies in the Presence of a Minimum Guarantee, Insurance: Mathematics and Economics, 33, pp. 189-207, 2003. 3.   G. Deelstra, M. Grasselli, and P.-F. Koehl, Optimal Design of the Guarantee for De_ned Contribution Funds, Journal of Economic Dynamics and Control, 28, pp. 2239-2260, 2004. 4. L.Delong, R. Gerrard, and S. Haberman, MeanVariance Optimization Problems for an Accumulation Phase in a De_ned Bene_t Plan, Insurance: Mathematics and Economics, 42, pp. 107-118, 2008. 5. P. Emms, and S. Haberman, Asymptotic and Numerical Analysis of the Optimal Investment Strategy for An Insurer, Insurance: Mathematics and Economics, 40, pp. 113-134, 2007. 6. J. T. Eghwerido and T. O. Obilade, Optimization of Investment Returns with N- Step Utility Functions, Journal of the Nigerian Mathematical Society 33, pp. 311-320, 2014. 7. H. Hong-Chih Optimal Multiperiod Asset Allocation:Matching Assets to Liabilities in a Discrete Model, Journal of Risk and Insurance,77(2), pp. 451-472, 2010. 8. R. Korn, Worst-case scenario investment for insurers. Insur Math Econ 36, pp. 1-11, 2005. 9. R. Korn, and E. Korn, Option pricing and portfolio optimization. AMS, Providence, 2001. 10. R. Korn and H. Kraft, Optimal portfolios with default able securities: a _rms value approach. Int J Theory Appl Financ 6, pp. 793-819, 2003. 11. R. Korn and O. Menkens, Worst-case scenario portfolio optimization: a new stochastic control approach. Math Methods Oper Res 62 (1), pp. 123-140, 2005.  INVESTMENT RETURNS WITH AN N-STEP GENERALIZED UTILITY FUNCTIONS 55 12. H. Kraft and M. Steffensen, Portfolio problems stopping at first hitting time with application to default risk. Math Methods Oper Res 63, pp. 123-150, 2006. 13. H. Kraft and M. Steffensen, Portfolio problems stopping at first hitting time with application to default risk. Math Methods Oper Res 63, pp. 123-150, 2006. 14. MacDonald, B.-J., and A. J.G.Cairns, Getting Feedback on De_ned Contribution Pension Plans, Journal of Risk and Insurance, 76(2), pp. 385-417, 2009. 15. H. Martin, Martingale pricing applied to dynamic portfolio optimization and real options, International Journal of Theoretical and Applied Finance, Vol 11, Issue 2005. 16. J. Mulvey and B. Shetty, Financial Planning via Multi-Stage Stochastic Optimiza- tion, Computers and Operations Research, 31(1), pp. 1-20, 2004. 17. H. Pham, On some recent aspects of stochastic control and their applications, Probability Surveys 2, pp. 506-549, 2005. 18. L.C.G. Rogers, The relaxed investor and parameter uncertainty Finance Stochastic 5, pp. 131-154, 2001. 19. W. Schachermayer, Utility maximization in incomplete markets, In: Stochastic methods in finance, Lectures given at the CIME-EMS Summer in Bressanone/Brixen, Italy, (M.Fritelli, W. Runggladier, eds.), Springer Lecture notes in Mathematics, 1856, 2003. 20. M. Schweizer, 'A guided tour through quadratic hedging approaches', in option pricing interest rates, and risk management, eds. Jouini E., Museiela M., Cvitanic J., Cambridge university Press, pp. 538-574, 2000.

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