Introduction

In ^1,2, Economic fluctuation refers to the variations and unpredictable changes over a stochastic period in economic activities which are influenced by a complex interplay of economic factors such as turbulence in inflation rate, human development index, volatility in gross domestic product (GDP) of a country due to fire-brigade approach in new policy implementations and currency-exchange rates. These economic fluctuations can occur over various time frames, ranging from short-term to long term shifts in economic trends of insurance industry in Nigeria. ³ Insurance is defined as a contract in which the policyholder pays a certain amount of money called premium to the insurance company for insurance coverage against any potential loss. According to ⁴, the live wire of an insurance industry depends on the premium payment by its policyholders. One notable aspect of economic fluctuations in the insurance industry which encompasses the alternating periods of expansion and contraction in its economic activities is the level of patronage by their policyholders. In ^5,6, one major macroeconomic factor that determines the level of patronage insurance industry received for its services and products depends on the protection of policyholders’ trust which influences its investment returns. During an expansionary phase, the investment of an insurance industry experiences growth in return which reveals the high level of trust and patronage shown by its policyholders in purchasing the insurance policies and payment of premiums. Conversely, during a contraction or recession, the investment returns of the insurance industry declines resulting in inadequate claims resolution by the insurance industry which affects the policyholders trust. The expansionary and contraction phase for alternating investment returns of the insurance industry depends on the policyholders trust and patronage which result to advanced stochastic economic movements or volatility. ^7,8 revealed that macroeconomic indicators such as policyholders lack of trust and patronage, inflation rates, currency exchange rates, gross domestic product (GDP) growth rates, stock market indices, human development index, unemployment rates and fiscal policy measures influences insurance industry investment returns which result to future adverse effects of low premium payments by policyholders, decrease in the insurance industry profitability, delay in claims resolution, regulatory penalties and fines, loss of policyholders trust and loyalty. The existing literatures in this concept revealed that no research has been carried-out mathematically in addressing the adverse effects of economic fluctuations on the Investment Returns of Insurance Industry in Nigeria which left a huge research gap. There is an urgent need to understand, evaluate and manage these economic fluctuations which are very crucial for policymakers, policyholders and for businesses to navigate the dynamic economic landscape effectively and also to recommend best ways to reduce its adverse effects on investment returns of insurance industry in Nigeria.

To tackle these adverse effects of advanced stochastic movements in key macroeconomic indicators which causes economic fluctuations and influences the investment returns of Nigerian insurance industry resulting into adverse effects of future delay and volatility-noise in the financial market, this study applied two-step Hybrid Block Adams Moulton Methods (2HBAMM) in solving some Advanced Stochastic Time-Delay Differential Equation (ASTDDE) with new sequence for delay and noise terms computations. ⁹ defined ASTDDE as a stochastic process that depends on the current state and the future uncertainties. A volatility term is an uncertainty event of any probability variables {X_t,t _E T} where X_t is the volatility-time and T is the time variations of various instabilities in different fields of study. ¹⁰ defined fluctuation as a probability process for collection of random variables on set of discrete time points controlled by probabilistic laws. ¹¹ formulated a structural model equation of Advanced Stochastic Time-Delay Differential Equation (ASTDDE) which contains the present state and the future uncertainties as presented below;

Adapting equation (1) the constructs of this study were taken into consideration, then the modeled equation for this study becomes;

where u(t) is the initial function a,B are drift and stochastic coefficients representing the expected rate of returns on investment, IR(t) represents the investment returns of the Insurance Industry, t is the time of delay in months, is called the lag, is called the future term and is the future lag term of investment returns. The variable AEEF(t) represents the economic fluctuations with its differential equivalence dAEEF(t) as the noise term together with the solution of the future delay term for investment returns IR(t+?)dAEEF(t) on the stochastic or diffusion part of (2). The drift part of the equation (2) dIR(t)=?(IR(t),IR(t+r),t)dt is deterministic and takes care of the average time rate of the investment returns without any risk. The stochastic or diffusion part dIR(t)=?(IR(t),IR(t+r),t)dAEEF(t) is the volatility risk factor of economic fluctuation on investment returns of policymakers in the modeled equation (2).

In the quest of obtaining the numerical solution of ASTDDE, most scholars such as ^{12, 14} used interpolation approach in computing the lag and volatility terms of ASTDDE for approximate solutions and suffered difficulties in obtaining the Minimum Absolute Random Errors (MAREs) at the Lowest Computer Processing Unit Time (LCPUT) which affected the accuracy of their numerical solution. ¹⁵ obtained the analytic solution of a deterministic and stochastic model of a time-varying investment returns with random parameters and revealed that the lower the absolute random errors of a stochastic process, the lower the risk effect of economic fluctuations in any financial model. The difficulty detected by the above researchers and adverse effects of economic fluctuations on the investment returns of insurance industry caused by macroeconomic indicators left a huge research gap which needs to be filled which is the motivation behind this study. To reduce these adverse effects of economic fluctuations and to overcome the difficulty experienced by the researchers in the application of interpolation approach for computation of the lag and volatility term, new sequences modelled by ¹⁶ were used. This can be done by solving numerically some examples of the modeled equation using the proposed method.

Research Method

Formulation of the Method

The discrete schemes of the two-step Hybrid Block Adams Moulton Methods (2HBAMM) were derived by ¹⁷ through matrix inversion techniques on the k-step multistep collocation method developed by ¹⁸ and presented as;

Evaluation of Fundamental Properties of the Proposed Method

Following the steps developed by ^{19, 20} the fundamental properties for the convergence and stability of the proposed method are examined.

Order and Error Constant

As the proposed method is one of the families of Linear Multistep Method, ¹⁹analyzed that LMM is said to be of order p if C₀ = C₁ = 0,….., C_p = 0 but C_p+1 = 0 and C_p+1 is the error constant.

The order and error constants for (3) are analyzed and presented as follows;

Therefore, (10) has an order, o = 5 and error constants

Consistency

According to ¹⁹, a Linear Multistep Method is said to be consistent if the order . Since the order of the method (2HBAMM) as computed is p > 1 , hence the consistency of the method is satisfied.

Zero Stability Analysis

In ²⁰, the zero-stability of a LMM is satisfied if no roots of u_i,I = 1,2,3,….,n the initial characteristic polynomial A(u) developed as A(u) = det(uM⁽¹⁾₂ – M⁽¹⁾₁) are greater than 1 which satisfy |u_i| < 1 and the roots |u_i| are simple or distinct.

The zero stability for (3) is determined as follows:

where

and

Now we have,

Using Maple (18) software, we obtain:

Since |u_i| < 1, i = 1,2,3,4 the discrete schemes in (3) is zero stable.

Convergence

Following that the discrete schemes (3) of the proposed method are both consistent and zero stable, the method is convergent.

Region of Absolute Stability

The P - and Q - regions of absolute stability of the proposed method for discrete schemes (3) are presented in figure 1 to 2 below:

Figure 1: Region of -stability Click here to view Figure

Figure 2: Region of -stability Click here to view Figure

Numerical Implementation and Computations