1. Introduction

Utility Maximization emerged from the early theories of economics and is an essential concept in modern finance mathematics. The basis of the idea is the assumption that individuals seek to derive the highest level of utility or satisfaction out of the monetary resources and goods they possess. The social concept of utility was formalized in the 20th century by economists such as John von Neumann and Oskar Morgenstern, who proposed the utility theory. Their work set the basis for understanding of decision-making under risk conditions where it was assumed that people would behave in such a way as to give themselves the highest expected utility. Utility maximization is also central to portfolio optimization in financial mathematics [1]. People allocate capital to various securities to achieve the highest possible return when investing. Investor risk preferences are collected via utility functions and integrated into the concept. These functions, which are the quadratic, exponential, power, and logarithmic functions, assess the proportional impact of diverse wealth levels on an investor's pleasure. Different degrees of risk are associated with each type of utility function that can affect investing decisions. In the intertemporal settings where investments and consumption decisions pertain over different periods, utility maximization extends beyond the single-period models. These include stochastic differential equations in continuous-time models, among others. When considering initial wealth, stock prices, interest rates, consumption rates, and other variables, maximizing the expected utility of terminal wealth and intermediate consumption remains the goal [2]. Over time, utility maximization has evolved along with progression in mathematical and computational methods, constituting crucial components of risk management, insurance and financial planning.

This Paper examines the several aspects of utility maximisation in financial mathematics including the background of the utility maximization model, utility functions, risk attitudes, and the application of the utility maximization model in finance over the years. This work endeavours to enhance the understanding of financing mathematics on utility maximization. In addition, this essay focuses on how investors employ these concepts in their risk management and financial schedules. By integrating knowledge from core concepts and recent findings, this essay presents a coherent view of how and why utility maximization impacts portfolio selections, investment choices, and risk management of financial uncertainties.

2. Function

Previous research that has been carried out on this subject has enhanced a clear understanding of the process of decision-making in financial matters about the concept of utility maximization. Several scholars have examined different forms of utility functions and their effects on risk preference. Additional emphasis has also been laid on the temporal dimension of the concept, along with its operational usage for risk management and financial planning horizons. Utility functions determine investors' risk tolerance levels, and in this respect, they directly influence the strategies adopted to manage portfolios. As for the utility functions, logarithmic, power, exponential, and quadratic utilities, as well as other functions, can be used to consider the level of risk aversion. Under uncertainty, each function defines different investor behaviors and preferences [3]. For example, some utility functions, such as logarithmic, indicate diminishing marginal utility to money, as people's satisfaction decreases as they earn more. Such behavior aligns with the risk aversion of some people who prefer preserving their wealth in the present rather than aiming for higher growth [3]. This utility function is most commonly applied to retirement and long-term investment decisions. However, it can be deemed simpler despite disfavoring the risk for constant incremental return. It is commonly employed in finance to model risk aversion where the degree is low.

On the other hand, power utility functions offer more flexibility as they consider the varying relative risk aversion. This kind of utility function explains how investors' risk preferences vary as their wealth changes. An example of this would be an investor who has continuous relative risk aversion (CRRA), which simply means that no matter the amount of wealth in their possession, they will always maintain that same proportion of their total wealth invested in risky securities. Perhaps this continuity will be beneficial in the creation of such dynamic investment strategies. These strategies can be adjusted to the conditions of the market, but they do not require a stark shift in the investor profile. These models are most suitable for individuals or entities having a large extent of liquidity or for otherwise institutional use who are not concerned with variations that are registered within periods of comparatively brief timeframe and who desire to have a steady growth pattern. That said, there is another perspective on risk behavior with regard to exponential utility functions. From this perspective, wealth is said not to affect an investor’s risk level, which is called Constant Absolute Risk Aversion (CARA). If investors are given an ample amount of risk and want to maintain a course at all times, even when their value fluctuates, this function is used most often. CARA's shareholders are generally conservative and often prefer the preservation of capital over the accumulation of wealth [3]. Because of this method, exponential utility functions are a standard selection in risk management and insurance where preventing an adverse state of affairs is considered more desirable than achieving gains. These functions are derived from the utility theory advanced by von Neumann and Morgenstern, where the individuals focus on utility generated from riches.

To contribute to this debate, Harel, Francis, and Harpaz propose comparing traditional utility functions with behavioral reflections from the Prospect Theory by Kahneman and Tversky [3]. Real investors are aligned with Prospect Theory, which establishes that average citizens often experience loss aversion. The former models are built upon rational investor behavior and include the CRRA and CARA models. This disparity helps one understand why some investors may exaggerate when the market is down or avoid some high-risk assets even when the returns are massive. Thus, in contrast to the classical view of economic rationality, Prospect Theory can be considered more valuable as a practical tool for defining financial decisions due to psychological factors affecting investor's behavior [4]. This comparison underlines the main dissimilarities between more behavioral factors and traditional economic models, indicating that both should be considered when evaluating investment proposals. Hence, it becomes pertinent to appreciate the implications of these utility functions because they pertain directly to the decisions that investors make, especially in risky or less risky contexts. However, Kassimatis raises concerns regarding whether and how M-V portfolios are suitable substitute measures for CRRA utility functions [5]. Although M-V portfolios are part of the portfolio theory, Kassimatis argues that the ideal CRRA investor should not use them, especially when addressing downside risk. From the findings, Kassimatis pointed out that low-risk CRRA portfolios diverge enormously from M-V portfolios, implying that investors emphasizing minimum variance would be better off investing in CRRA portfolios [5]. Choosing the proper utility function according to investors' attitudes and risk preferences is necessary to consider and optimize appropriate utility. This stresses the need to develop investment plans that reflect investors' risk preferences to achieve high satisfaction with investment decisions among investors.

3. Time Span

Regarding the choice of consumption and investment across time, multi-period intertemporal utility models apply to finance. These models help make the best decision by explaining how utility evolves in time and other factors such as investment returns, risk tolerance and consumption needs. When outlining the definition of a limited number of effective long-term savings plans for an individual, Gerrard et al. describe probability hedging as a way to match investment returns to the risk tolerance of a saver [6]. Their work advocates for probability hedging under the logarithmic utility function, which outperforms traditional terminal wealth distribution under one of the many preferred utility functions due to its simplicity. This method can assist financial advisors in managing the flow of communication and conveyance of decision-making to non-proficient investors regarding complex investment plans. Through probability hedging, investors are in a position to balance the risk tolerance to their long-term financial goals and hence ensure that investments made are secure and have a focus on growth.

Adjustment between risk preferences and time horizons is essential in capturing intertemporal benefits. As much as investors have different risk-taking abilities, they must change how they wish to invest at different periods in their lives. For example, a young investor who has many years of investment in front of them may not be very risk averse and, as a result, prefer stocks among other similar securities. On the other hand, an older investor, especially one who is already nearing retirement, might prefer investing in relatively safer securities that will generate good and relatively stable returns in the near future. The ensured variation in the risk profile depicted in this paper underlines the need to include intertemporal models that have the flexibility to respond to changes known to occur in the tolerance for risks and market situations. To ensure the chosen strategy is the best possible at any point in time and generally achieves improved optimization over time, these models often employ stochastic dynamic programming. Aycinena et al., assess the applicability of intertemporal choice tasks for assessing risky behavior in financial management [7]. They emphasize the role of pursuing the right intertemporal preferences by relating them to temporal discount rates and consumption smoothing. In the study including Guatemalan Conditional Cash Transfer (CCT) beneficiaries, Aycinena et al. argue that predicting preferences over high-stakes payment plans is possible based on the experimental measurements [7]. The discovery above underlines the need for precise intertemporal choice experiments to consider these variables and ensure credible representation and considerable insight into financial decision-making over long periods. Since the intertemporal utility function affects behavior, any financial planner can build plans that optimally satisfy current and future demands based on utility across timeframes. The intertemporal preferences are mainly due to diminishing marginal utility since people prefer present consumption to future consumption. Future financial planning models must incorporate this preference to ensure optimal wealth and consumption usage is achieved over time.

4. Risk Management

Moreover, the concepts of utility maximization play a significant role in risk management and financial management because they are critical determinants of sound investment decisions and efficient and effective risk minimization strategies. Concerning the limitations of MPT (modern portfolio theory), Rodríguez, Gómez, and Contreras proposed a diversified behavioral portfolio model that considers the various distribution characteristics of returns, including skewness and kurtosis [8]. Fuzzy theory is used to assess investor preferences as an alternative to the applied MV approach to their model. This approach offers an improved estimation of risk made possible by incorporating more moments of return that yield a higher level portfolio in line with the desires of the investors. Portfolio models that are more accurate and flexible are required due to inflationary forces, market fluctuations resulting from occurrences such as the COVID-19 pandemic, and downturns [4]. External forces may profoundly shift strategies or relations in a market, often in ways that traditional theoretical frames, including MPT, cannot capture. Shifts in the supply chain or geopolitical events that could lead to abrupt, significant market responses are some of the factors that affect the distribution of fat tails in return. In such situations, the skewness and kurtosis-incorporated behavioral model of Rodríguez et al. can offer a more realistic approach, stating the higher probability of gaining more or losing more [8].

Grable, Rabbani, and Heo also pointed out that financial risk tolerance and financial risk aversion are most impactful in the context of household investment [9]. These works show that, in other words, these notions, viewed as mutually exclusive and antagonistic, work in tandem to determine the level of portfolio risk. The two main concepts include financial risk tolerance, which reflects an investor's willingness to engage in risky business based on self-estimations, and financial risk aversion, which indicates an investor's unwillingness to engage in risk-taking [10]. The inclusion of these aspects into investment models enhances the accuracy of portfolio risk forecasts, as well as minimizes bias in financial analysis. Risk aversion and tolerance help investors to view different prospects as necessary when planning investment management plans [11]. With these parameters in place, investors are more likely to move through the existing and perhaps even financial labyrinth and get profound and precise recommendations that are theoretical and practical. These sophisticated techniques assist financial planners in enhancing risk-return optimisation for uncertainty.

Because of today’s complex markets, such plans must include behavioral and psychological factors in addition to statistical risk models. This is important because macroeconomic factors from all over the world, such as interest rates, trade wars, and political instabilities, might change the investor’s risk tolerance levels [12]. In constructing different portfolios for different people, one needs to be cautious because risk-taking ability and financial risk-taking capacity are unique. Risk diversification in planning, besides risk-taking, enables investors to develop fresh growth initiatives while lowering business risk [9]. High-risk tolerance investors and very high-risk aversion investors can invest in stocks or equities and bonds. This ensures that the portfolio aligns with the investor's financial objectives and psychological risk tolerance, enhancing long-term financial viability. Furthermore, there is evidence that the application of utility maximization is more than just determined by tenets of the existing legislation and advanced technologies. Fan and Chatterjee highlighted that robo-advisory services are among the most crucial factors shaping investment and risk decisions [13]. Technologies that support models of determining investment portfolios tailored to an individual's attitude to risk and expected utility help to design personal trading profiles that could be less sensitive to market shifts [14]. For these developments to occur, it is possible to manage them appropriately to protect investors' interests in rapidly transforming contexts. Hypothetically, such developments could potentially generate problems.

5. Conclusion

Utility maximisation is vital in financial mathematics and helps explain the financial choices in the uncertain financial world. Utility maximisation captures something genuinely human, beyond theories and models: the never-ending struggle of wanting more and wanting security, of investing something in a proposition and possibly being rewarded with a considerable quantum. This theory is beneficial when the economic conditions are unpredictable or in scenarios such as fluctuations in the market or a financial event that could compromise traditional investment practices. The set of utility functions, which can be power, exponential, or logarithmic, has to be reconsidered, taking into account the real risk aversion, and this one depends on extra factors as well. Intertemporal utility maximisation exposes long-run budgeting and consumption patterns when spread out over time. It forms part of modern psychological risk management theories and financial decision-making, as seen in diversified behavioural portfolios and probability hedges. In this context, these methods ensure that the risk-return profiles of investment strategies match individual risk tolerance levels and investment horizons. It should be believed that the utility maximisation remains one of the most reliable theories that can still be applied to studying investors’ behaviour in the context of somewhat more complex financial markets. However, as the world becomes more connected and riskier, these behavioural aspects need to be introduced in these models further and further to ensure the models depict the abstract theories and the specific difficulties investors face.