Calculating Loan Payments: An Introduction to Annuity Payment Formulas

When you take out a loan, it's crucial to understand the monthly payment amount you'll be required to make. This article covers the process of calculating loan payments using the annuity payment formula, focusing on common scenarios such as a 6% annual interest rate with quarterly compounding, and a 6% annual interest rate compounded annually over 15 years.

Understanding the Annuity Payment Formula

The formula for calculating the annuity payment can be expressed as:

Formula: P (frac{r cdot PV}{1 - 1 cdot r^{-n}})

tP Annual payment tPV Present value of the loan, i.e., the amount borrowed tr Annual interest rate as a decimal tn Total number of payment periods

Example Calculation for a 200,000 Loan at 6% Interest Rate

Let's walk through the calculation for a loan of 200,000 at an annual interest rate of 6%, to be repaid over 15 years with annual compounding.

Numerical Steps

tConvert the annual interest rate to a decimal: 0.06 tIdentify the PV (Present Value): 200,000 tIdentify the n (number of payment periods): 15

Now, plug these values into the formula:

P (frac{0.06 cdot 200000}{1 - 1 cdot (0.06)^{-15}})

Step-by-Step Calculation

tCalculate the value of (0.06)^{-15}: t tt(0.06)^{-15} approx 0.397 t tCalculate the numerator: 0.06 cdot 200000 12000 tCalculate the denominator: 1 - 0.397 approx 0.603 tDivide the numerator by the denominator: t ttP frac{12000}{0.603} approx 19895.51 t

Therefore, the annual payment amount for the loan is approximately 19895.51.

Interest Compounding Scenarios

Reducing Balance Basis

In many cases, especially for loans like mortgages, the interest is charged on a reducing balance. This means that the interest for each period is calculated on the remaining loan balance after each payment. Here's how it works for the given example:

tFor the first 12 months, the interest is charged on the full 200,000: t ttInterest for 12 months 0.06 cdot 200000 12000 t tAfter 12 months, the interest for the next 3 months is calculated on the remaining balance of 200,000 12000: t ttInterest for next 3 months 0.06 cdot 212000 cdot frac{3}{12} 3180 t tTotal repayable 200,000 12,000 3,180 215,180

To find the annual payment, divide the total repayable by 15:

Annual payment frac{215180}{15} approx 14345.33

Car Hire Purchase Scenario

In scenarios like car hire purchase, the interest might be calculated first and then added to the loan before determining the monthly repayments. This approach ensures a clear representation of the total cost.

Spreadsheet Method for Complex Situations

If you're unsure about using complex mathematical formulas, you can create a simple spreadsheet. Here are the steps:

tCreate a row for each compounding period tInclude columns for starting balance, interest for the period, and balance at the end of the period tCarry the last column (balance at end of period) back to the next row in the first column tCopy the second row down the page

This method helps in visualizing the compounding process and making the calculation easier.