How do we find the remaining principal at the end of each year with an amortized loan repayment schedule? We need to Calculate how much of each annual payment is for interest and then apply the remaining amount against the principal. Each succeeding year starts with the new lower principal, and the interest owed for that year is simply the **interest rate** times this new lower principal amount.

**Definition**: *We call the listing of the annual interest expense, the reduction of principal each year, and the ending balance or remaining principal an amortization schedule of the payoff of the loan.*

A primary characteristic of an amortization schedule is that it indicates the remaining principal after each payment.

**Amortization Schedule Example**

Let’s look at an amortization schedule for the $25,000 loan payoff at 8% annual interest with six equal annual payments. Recall that each year the payment is $5,407.88. What is the remaining principal of the loan at the end of the first year?

- Calculate the interest expense for one year when the principal is $25,000 and the interest rate is 8%:

Interest expense first year=$25,000×0.08=$2,000

- Calculate the amount available for reducing the principal after you have subtracted the interest expense from the payment:

Available for principal reduction first year=$5,407.88−$2,000=$3,407.88

- Calculate the new lower principal at the end of year one:

End-of-year remaining principal=$25,000−$3,407.88=$21,592.12

In the second year, the beginning outstanding principal is now $21,592.12, and it is this lower amount that is earning interest for the lender. This ending balance is also the present value of the remaining payments. So for the **second year** we have the following steps:

- Calculate the interest expense for the second year by multiplying the remaining principal by the interest rate (8%):

Interest expense second year=$21,592.12×0.08=$1,727.37

- Calculate the amount available for decreasing the principal in the second year by subtracting the second year’s interest expense from the annual payment:

Available for principal reduction second year=$5,407.88−$1,727.37=$3,680.51

- Calculate the new lower principal at the end of year two. It is the principal at the start of the year minus the principal reduction amount:

End-of-year remaining principal=$21,592.12−$3,680.51=$17,911.61

This process continues for the entire six years, and at the final payment of $5,407.88, the principal is entirely paid off. **Table 1** presents the complete amortization schedule.

**Table 1 **Amortization Schedule for a $25,000 Loan at 8% with Six Annual Payments

Year |
Beginning Principal |
Annual Payment |
Interest Expense |
Principal Reduction |
Remaining Principal |
---|---|---|---|---|---|

1 | $25,000.00 | $ 5,407.88 | $2,000.00 | $ 3,407.88 | $21,592.12 |

2 | $21,592.12 | $ 5,407.88 | $1,727.37 | $ 3,680.51 | $17,911.61 |

3 | $17,911.61 | $ 5,407.88 | $1,432.93 | $ 3,974.95 | $13,936.66 |

4 | $13,936.66 | $ 5,407.88 | $1,114.93 | $ 4,292.95 | $ 9,643.71 |

5 | $ 9,643.71 | $ 5,407.88 | $ 771.50 | $ 4,636.38 | $ 5,007.33 |

6 | $ 5,007.33 | $ 5,407.92 | $ 400.59 | $ 5,007.33 | $ 0 |

Total | $32,447.32 | $7,447.32 | $25,000.00 |

The last payment is four cents higher to make the remaining principal come out to exactly zero at the end of the six years because we rounded the interest calculations and the payments to the nearest whole cent. If you were to calculate the annual payment beyond two decimals, you would get $5,407.884656.

Amortization schedules are very common, and we use them on many loans, such as those for cars, mortgages, and consumer products. If you want to pay off a loan early, the present value of the remaining payments is the outstanding balance on the loan at the end of each period.

**Example**

Several home loans are for thirty years, but the mean time an individual stays in the same home is approximately seven years. When people sell their current home to move to another home, they must pay back their loan from the proceeds of the sale of their current home. Precisely how much do they require to pay? The present value of the leftover payments specifies that amount and is the current principal balance of the loan. You can easily discover it by looking at the amortization schedule.