principal component
The outstanding balance at the end of the first payment
In general, the interest component of the ‘t’th installment is
The principal component of the ‘t’th installment is
and the outstanding balance at the end of the ‘t’th payment is
EXAMPLE 2.21
Sylvie has borrowed $25,000 from First National Bank and has to pay it back in eight equal annual installments. If the interest rate is 8% per annum on the outstanding balance, what is the installment amount, and what will the amortization schedule look like?
Let us denote the unknown installment amount by A. We know that
We will analyze the first few entries in Table 2.6 in order to clarify the principles involved. At time 0, the outstanding principal is $25,000. After one period, a payment of $4,350.37 will be made. The interest due for the first period is 8% of $25,000, which is $2,000. Consequently, the excess payment of $2,350.37 represents a partial repayment of principal. Once this amount is repaid and adjusted toward the principal, the outstanding balance at the end of the first period will become $22,649.63. At the end of the second period, the second installment of $4,350.37 will be paid. The interest due for this period is 8% of the outstanding balance at the start of the period, which is $22,649.63. Thus the interest component of the second installment is $1,811.97. The balance, which is $2,538.40, constitutes a partial repayment of principal. The value of the outstanding principal at the end should be zero. As can be seen, the outstanding principal declines after each installment payment. Because the payments themselves are constant, the interest component will steadily decline while the principal component will steadily increase.
TABLE 2.6 An Amortization Schedule
| Year | Payment | Interest | Principal Repayment | Outstanding Principal |
|---|---|---|---|---|
| 0 | 25,000 | |||
| 1 | 4,350.37 | 2,000.00 | 2,350.37 | 22,649.63 |
| 2 | 4,350.37 | 1,811.97 | 2,538.40 | 20,111.23 |
| 3 | 4,350.37 | 1,608.90 | 2,741.47 | 17,369.76 |
| 4 | 4,350.37 | 1,389.58 | 2,960.79 | 14,408.97 |
| 5 | 4,350.37 | 1,152.72 | 3,197.65 | 11,211.32 |
| 6 | 4,350.37 | 896.91 | 3,453.46 | 7,757.86 |
| 7 | 4,350.37 | 620.63 | 3,729.74 | 4,028.12 |
| 8 | 4,350.37 | 322.25 | 4,028.12 | 0.00 |
Thus, in the case of such loans, the initial installments will consist largely of interest payments. As we approach the maturity of the loan, however, the corresponding installments will consist largely of principal repayments.
AMORTIZATION WITH A BALLOON PAYMENT
Julie Tate has taken a loan of $25,000 from First National Bank. The loan requires her to pay in eight equal annual installments along with a terminal payment of $5,000. This terminal payment that has to be made over and above the scheduled installment in year eight is termed as a balloon payment. The interest rate is 8% per annum on the outstanding principal. The annual installment may be calculated as follows.
