Mathematical divergence model and basic design of simple interest formula and compound interest formula
We will explain the differences in the formula structures of the future value calculation formula, which is a simple interest operation formula, and the compound interest operation formula.
The simple interest operation formula is a mechanism in which interest is granted at a fixed rate only on the principal, and if the initial principal is set as the present value, the future value is expressed as the product of the present value and the sum of the sum of R times the number in parentheses.
On the other hand, the compound interest formula includes a reinvestment process in which the accrued interest is incorporated into the next principal amount, and the future value is the product of the gift value and the sum of R in parentheses to the N power.
This simulator is equipped with an engine that evaluates these formulas in parallel, and calculates the difference in asset amounts in milliseconds when inputting the same initial principal and assumed annual interest rate.
Underlying these two formulas is a fundamental difference in the mathematical properties of linear and exponential growth. A graph of simple interest always draws a straight line with a constant slope, but a graph of compound interest draws a curve whose tangent line becomes steeper over time.
In order to faithfully reproduce this mathematical dissociation model within the simulator, we have implemented local calculation logic that can be completed within the browser, achieving high-speed calculation processing without server communication.
Exponential deviation as the number of years of operation increases and the number of years of doubling using the rule of 72
We will analyze in detail how the accelerating expansion of the compound interest effect occurs as the operating years N parameter increases.
Due to the nature of the exponential function known as the n-th power, the difference between simple interest and compound interest is minute at the initial stage, but immediately after a certain period of time, which is the break-even point, the difference widens dramatically.
As an auxiliary indicator to visually and intuitively understand this exponential deviation, this simulator has a built-in function to estimate the number of doubling years based on the rule of 72.
The rule of 72 is an empirical rule for calculating the approximate number of years it will take for the principal to double, and is calculated by dividing the number of years by the estimated annual interest rate.
When the expected annual interest rate parameters are manipulated on the simulator, this doubling period is recalculated in real time, and the user is presented with a concrete time axis guideline.
If a long-term setting is made such that the number of years of investment N exceeds this number of doubling years multiple times, it is possible to directly check on the simulator how the difference in future value from simple interest investment diverges to an astronomical value.
Calculation of real yield after tax and model considering 20.315% withholding tax
In order to accurately simulate asset management in the real financial market, we will detail the logic for calculating the final asset amount difference based on the after-tax real yield.
Under the Japanese tax system, a withholding tax of 20.315% is normally imposed on investment profits from stocks and investment trusts.
In addition to calculating the pre-tax future value, this simulator also performs a parallel calculation of the after-tax future value using this tax rate.
Specifically, we use a strict after-tax compounding model that deducts 20.315% from each period's fixed interest amount and reinvests only the remaining amount into the next principal amount.
In the case of simple interest investment, calculations are similarly made on the assumption that the same rate of tax will be deducted at maturity or when interest is received each period.
The presence or absence of this tax deduction process has a significant impact on the compound interest effect, especially when the investment period is long.
The after-tax compounding curve will be under significant downward pressure compared to the theoretical pre-tax compounding curve, as taxes reduce the amount of principal that can be reinvested.
On the simulator, you can switch between these two curves and compare them, making it possible to accurately measure the negative impact that taxes have on long-term performance.
Fully local in-browser calculation of initial principal and assumed annual interest rate parameters
We will explain the internal structure of the fully local in-browser computing architecture, which is the technical basis of the simulator.
The three main input values: initial principal parameter, assumed annual interest rate parameter, and investment period parameter are passed to the calculation engine through the virtual document object model the moment the user interacts with a slider or text box on the interface.
The calculation engine does not issue any asynchronous communication requests to external servers, and completes future value calculations only within the client-side JavaScript execution environment.
The main reason for adopting this completely local calculation method is to maximize the responsiveness of the user interface to parameter changes.
A complete calculation cycle on the client side that eliminates server-side rendering was essential in order to provide users with feedback without delay about how small changes in the assumed annual interest rate or yearly increases or decreases in the investment period would affect the final asset amount.
The entire process, from validating input values to correcting rounding errors in floating-point operations to formatting calculated numbers into comma-separated strings, runs optimally on the browser's main thread.
Superimposed graph display of management asset amount trends and real-time redrawing mechanism
We will explain the operating principle of the superimposed graph display system that visualizes the calculated future value for each year.
The simulator retains annual asset value trends from the start of operation to the final year in memory as array data, and converts this as an input data set for the graph drawing library.
The trend data for simple interest investment and the trend data for compound interest investment are drawn as line segments of different colors superimposed on a coordinate plane with the same horizontal and vertical axis scales.
This superimposed display is designed to help you understand at a glance how the gap between the two is increasing over time.
Furthermore, in conjunction with the real-time calculation engine mentioned in the previous chapter, a redraw process of the entire graph is triggered every time a parameter changes.
During the redrawing process, visual continuity is maintained by updating the dataset and applying smooth path deformation using transition animations, rather than repeatedly destroying existing graph objects and creating new ones.
This allows users to interactively explore how the straight line of simple interest and the curve of compound interest change shape, and at what point they intersect or diverge, while changing the assumed annual interest rate.
Long-term asset formation and profitability comparison guide for new NISA and fixed interest deposits
Finally, we will present specific usage guidelines for applying this simulator to practical financial decision making. One system to maximize the benefits of compound interest is to use the new NISA account, where all profits are tax-free.
By adjusting the simulator's tax rate setting to zero percent, you can draw a pure tax-free compound interest curve when using the new NISA system.
By comparing this tax-free compound interest curve with the 20.315% after-tax compound interest curve explained in Chapter 3: , it is possible to understand the tax saving effect of the new NISA on a concrete monetary basis.
Furthermore, if the assumed annual interest rate parameter is set to an extremely low value, such as the very low annual interest rate that is the current level of general fixed interest deposits, you will see stagnant asset trends with no real difference between compound interest investment and simple interest investment.
Conversely, if you enter the expected annual interest rate from an index fund, etc., the power of compound interest in building long-term assets will be clearly shown.
In this way, the ultimate role of this simple interest compound interest comparison simulator is to accurately simulate the long-term return difference between low-yielding safe assets and managed assets with a certain level of risk, and to provide individual users with information on how to make optimal portfolio construction decisions.