Mathematical derivation of the calculation formula for equal principal and interest repayment and equal principal repayment
We will explain in detail the mathematical background of the formula for calculating the monthly repayment amount for equal principal and interest repayments.
If the borrowing principal is P, the monthly interest rate is r, and the total number of repayments is n, the monthly payment amount PMT is precisely derived by applying the formula of the sum of geometric progressions.
The post-payment balance for each cycle is defined as the previous month's balance plus the accrued interest for the period, minus the scheduled repayment amount.
By recursively expanding this recurrence formula up to the nth time and constructing an equation with the boundary condition that the final residual principal becomes completely zero, an unchanging relational expression between PMT and P is determined.
Specifically, it converges to the calculation formula \text{PMT} = P \times \frac{r(1+r)^n}{(1+r)^n - 1}, which is an extremely sophisticated function structure.
On the other hand, in the equal principal repayment architecture, the monthly principal amount is fixed as a constant obtained by dividing the total loan amount P by the total number of borrowings n, and dependent interest on the decreasing remaining principal is added each time.
For this reason, a linear function-like decreasing curve is drawn mathematically, with the total payment amount being the largest at the beginning of repayment and the interest burden decreasing over time.
The calculation engine internally converts these two logarithmic calculation models into parallel algorithms, and has the ability to instantly determine the exact repayment amount to the nearest 1 yen using floating point arithmetic based on the initial conditions entered by the user.
Combined weighting logic of twice-yearly bonus addition amount and principal repayment schedule for monthly repayments
In the internal calculation process when combined bonus payment is selected, a method is adopted in which a dual principal repayment schedule is constructed in which the monthly repayment portion and the specific bonus addition portion are completely independent, and these are integrated on the time axis.
First, we extract the percentage of the principal to be used for bonus repayments out of the total loan amount as a parameter, and then apply compound interest every six months independently to the separated principal portion.
The required payment amount for the bonus month is a combination of the principal and interest for the basic monthly repayments, the principal redemption amount exclusively for bonus repayments, and the unpaid interest accumulated over the past six months, causing highly discontinuous fluctuations in cash flow.
In this system, in order to track this irregular compound weighting logic with millisecond accuracy, a basic index function that is incremented every month and a bonus trigger function that fires every six months are operated in a synchronized manner to simulate the decreasing trajectory of the principal balance.
The final redemption schedule data output from this two-layer calculation loop statistically reflects the difference between the fixed level payment amount in the regular month and the outstandingly high payment amount in the bonus month, and creates a detailed payment trajectory in virtual space that is completely synchronized with the user's actual annual income and expenditure balance.
Calculation of the difference in the total interest reduction amount at the time of early repayment in short term type vs. reduced repayment type
In measuring the financial impact of voluntary prepayments, this tool uses a rigorous formula to calculate the difference in total interest reduction resulting from two completely different approaches: term shortening and repayment reduction.
If you select the shortened period algorithm, the invested special repayment funds will be applied in full to the most recent unredeemed principal, and the future interest debt that would have been accrued to the disappeared principal block will be completely waived.
This process truncates the end of the repayment schedule ahead of schedule, resulting in a dramatic interest compression effect by reversing the compounding effect.
In contrast, in the repayment reduction type calculation model, the future monthly repayment amount is recalculated on a zero basis, based on the new remaining principal reduced by the injection of special funds, while maintaining the originally set remaining repayment period as an absolute constant.
Although this method is directly linked to improving monthly liquidity on hand, since the principal itself remains for a long period of time, the total amount of interest reduction when compared to the shorter term method will be mathematically subordinated.
When the calculation engine receives the specified execution month and input amount as input, it executes both of these redemption processes in parallel in separate threads, and presents the final difference in total interest expense to the user with resolution in yen.
Conversion from annual interest rate to monthly interest rate and dynamic generation mechanism of redemption table
In processing the interest rate parameters that are a prerequisite for various loan products, we will detail the application of the conversion formula r = \text{annual interest rate} / 12 from annual interest rate to monthly interest rate, and the dynamic generation mechanism of the accompanying amortization schedule, the amortization schedule.
Since the base interest rate offered by financial institutions to the market is usually expressed as an annual percentage, the core module runs a preprocessor that unconditionally divides the input annual interest rate into 12 equal parts and converts it into an effective monthly interest rate r for monthly loop calculations.
Using this minimized monthly interest coefficient, we start a recursive iterative process that separates and calculates the amount applied to principal and the amount applied to interest for all time periods from the first payment to the final payment.
Within this iteration, the Amortization Schedule is constructed by sequentially storing four main variables in memory as a multidimensional array: the total monthly repayment amount, the interest paid for the month, the actual principal reduction amount, and the principal balance after settlement.
The moment a parameter on the screen is manipulated using a slider, such as a slight change in interest rates or an extension of the repayment period, the system immediately discards the existing array data and re-executes thousands of floating point operations in the background based on the new boundary conditions, realizing reactive data binding that dynamically redraws the latest complete redemption table on the virtual DOM tree.
Complete local calculation of loan amount, annual income, and interest rate parameters in your browser
The biggest factor that determines the technical superiority of this system is its architecture, which allows all financial calculations using the loan amount, user's annual income, and subdivided interest rate parameters to be completed only in the local environment within the browser, without any API communication with external servers.
By making full use of the highly optimized JavaScript numerical calculation library implemented in the front-end layer, complex exponential and logarithmic function calls required every time a user input value change event occurs in a zero-latency environment that is completely unaffected by network latency.
In real-time calculation of the repayment burden ratio, which indicates the proportion of the total annual repayment amount to the input tax-included annual income, and in the execution of stress tests that assume stress interest rates, which are the risk of future interest rate increases, we generate safe and reliable calculation results by consuming only the client terminal's CPU resources, without transmitting any highly confidential personal information over the Internet.
This serverless, completely local calculation model structurally guarantees the highest level of information security and privacy protection, and at the same time provides an overwhelming user experience that allows detailed scenario verification to be repeated infinitely without any functional deterioration even in an offline environment.
Procedures for creating a repayment plan when purchasing a home and comparing interest rates and refinancing
The system systematizes the steps to create a practical repayment plan according to the user's specific goals, from the phase of raising funds to purchase a new home to comparing and considering the economic rationality of refinancing an existing home loan.
In the new purchase simulation module, the total amount of borrowing required is calculated backwards from the target property price and the down payment that can be prepared from one's own funds, and multiple financing scenarios are virtually constructed, such as a mixed rate plan with variable interest rates and long-term fixed interest rates.
On the other hand, in the refinancing simulation execution mode, the current loan balance, remaining repayment period, and currently applied interest rate are defined as input variables, and the future cash flow if payments are continued until maturity at the existing financial institution is established as the baseline for comparison.
Based on this baseline, the algorithm automatically identifies the repayment period that is the break-even point when the total payment amount reverses and becomes positive by including the initial costs associated with switching, such as the interest rate offered by the new financial institution, registration fees, administrative fees, and credit guarantee fees, as a negative cash flow.
By visually comparing and examining the differences between multiple simulation results displayed side-by-side on the screen, users will be able to derive the optimal repayment strategy based on mathematical grounds that best fits their future life plans and predictions of macroeconomic interest rate trends.