# 3.3 Calculating the Final Payment

## Learning Objectives

• Calculate the final payment for a loan

## Formula & Symbol Hub

#### Symbols Used

• $BAL$ = Principal balance immediately after a loan payment
• $INT$ = Interest portion of a loan payment or a series of payments
• $PRN$ = Principal portion of a loan payment or a series of payments
• $PMT$ = Annuity payment amount
• $I/Y$ = Nominal interest rate
• $P/Y$ = Number of payments per year or payment frequency
• $C/Y$ = Number of compounds per year or compounding frequency
• $N$ = Total number of annuity payments

## Introduction

If you have ever paid off a loan you may have noticed that your last payment was a slightly different amount than your other payments. Whether you are making monthly insurance premium payments, paying municipal property tax instalments, financing your vehicle, paying your mortgage, receiving monies from an investment annuity, or dealing with any other situation where an annuity is extinguished through equal payments, the last payment typically differs from the rest, by as little as one penny or up to a few dollars. This difference can be much larger if you arbitrarily chose an annuity payment as opposed to determining an accurate payment through time value of money calculations.

Why is it important for this final payment to differ from all of the previous payments? From a consumer perspective, you do not want to pay a cent more toward a debt than you have to. In $2011$, the average Canadian is more than $\100,000$ in debt across various financial tools such as car loans, consumer debt, and mortgages. Imagine if you overpaid every one of those debts by a dollar. Over the course of your lifetime those overpayments would add up to hundreds or even thousands of dollars.

## Calculating the Final Payment

When you calculate out the payment for a loan, the actual payment is rounded to two decimal places. It is rare for a calculated loan payment not to require rounding. The rounding up or down of the payment forms the basis for adjusting the final payment.

Consider the situation where the loan payment is rounded up. Suppose the calculated loan payment is $\999.995$. Rounding to two decimals, the actual payment made is $\1,000$. So with each $\1,000$ loan payment, you are overpaying the debt by $\1,000− \999.995 = \0.005$. If you make $20$ such payments, you end up overpaying the debt by $20\times \0.005=\0.10$. Therefore, when it comes to the final payment you need to compensate for all of the overpayments made, reducing the final payment by $\0.10$. Because the principal is slightly smaller at all times as a result of the overpayment, an additional adjustment may be needed because of less interest being calculated.

Now, consider the situation where the loan payment is rounded down. Suppose the calculated loan payment is $\1,000.0025$. Rounding to two decimals, the actual payment made is $\1,000$. So, with each $\1,000$ loan payment, you are underpaying the debt by $\1,000.0025-\1,000=\0.0025$. If you make $20$ such payments, you end up underpaying by $20\times \0.0025=\0.05$. Therefore, when it comes to the final payment, you need to increase the final payment by this amount. Because the principal is slightly larger at all times as a result of the underpayment, an additional adjustment may be needed because of more interest being calculated.

We have already learned how to calculate the final payment by completing the last row of the amortization schedule. This method gives the following formula for calculating the final payment.

$\mbox{Final Payment} = \mbox{Balance from Second Last Row}+\mbox{Interest from Last Row}$

Alternatively, we can calculate the final payment without constructing the last row of the amortization schedule. This method is based on the assumption that all of the payments, including the final payment are the same. The final payment is found by adjusting the regular payment ($PMT$) by the amount overpaid or underpaid.

$\begin{eqnarray*} \mbox{Final Payment} & = & PMT-\mbox{Amount Overpaid} \\ \\ & \mbox{or} & \\ \\ \mbox{Final Payment} & = & PMT+\mbox{Amount Underpaid} \end{eqnarray*}$

### Key Takeaways

An overpayment means that you have paid more through the regular payments then is necessary. The total amount overpaid is subtracted from the payment to find the final payment. Consequently, the final payment is smaller than the other payments.

An underpayment means that you have not paid enough through the regular payments than is necessary. The total amount underpaid is added to the payment to find the final payment. Consequently, the final payment is larger than the other payments.

## Using the TI BAII Plus Calculator to Find the Final Payment

To use the amortization worksheet to find the final payment:

1. Solve for any unknown quantities about the loan. You need to know all of the information about the loan first before you can use the amortization worksheet.
2. Enter the values of all seven time value of money variables into the calculator ($N$, $PV$, $FV$, $PMT$, $I/Y$, $P/Y$, $C/Y$). If you calculated PMT in the first step, you must re-enter it rounded to two decimals and with the correct cash flow sign. Make sure the payment setting is set to END, and obey the cash flow sign convention. Because this is a loan, $PV$ (the loan amount) is positive and $PMT$ is negative.
3. Go to the amortization worksheet by pressing 2nd AMORT (the $PV$ button).
4. Enter the payment number for the final payment into $P_1$ and $P_2$.
5. Find the $BAL$ entry. Watch the cash flow sign of the $BAL$ entry to properly interpret what to do with it!
6. Calculate the final payment:
• A negative $BAL$ entry indicates an overpayment. Then

$\mbox{Final Payment} = PMT-\mbox{Overpayment}$

• A positive $BAL$ entry indicates an underpayment. Then

$\mbox{Final Payment} = PMT+\mbox{Underpayment}$

### Example 3.3.1

A $\10,000$ loan at $8\%$ compounded quarterly is repaid with month-end payments of $\200$. Calculate the final payment.

Solution

Step 1: Calculate the number of payments.

 PMT Setting END $N$ $?$ $PV$ $10,000$ $FV$ $0$ $PMT$ $-200$ $I/Y$ $8$ $P/Y$ $12$ $C/Y$ $4$

$N=60.9273...\rightarrow 61 \mbox{ payments}$

Step 2: Calculate the balance for payment $61$ (the last payment).

To find the balance for payment $61$, set $P_1=61$ and $P_2=61$.

 PMT Setting END $N$ $61$ $PV$ $10,000$ $FV$ $0$ $PMT$ $-200$ $I/Y$ $9$ $P/Y$ $12$ $C/Y$ $4$ $P_1$ $61$ $P_2$ $61$

$BAL=-\14.49$

Step 3: Calculate the final payment.

Because the BAL entry from the previous step is negative, the $\14.49$ is an overpayment and must be subtracted from the $\200$ payment.

$\begin{eqnarray*} \mbox{Final Payment} & = & 200 -14.49 \\ & = & \185.11 \end{eqnarray*}$

Step 4: Write as a statement.

The final payment for the loan is $\185.11$.

### Example 3.3.2

A $\50,000$ loan at $5\%$ compounded quarterly is repaid with quarterly payments for $6.5$ years. Calculate the final payment.

Solution

Step 1: Calculate the payment.

 PMT Setting END $N$ $26$ $PV$ $50,000$ $FV$ $0$ $PMT$ $?$ $I/Y$ $5$ $P/Y$ $4$ $C/Y$ $4$

$PMT=\2,264.36$

Step 2: Calculate the balance for payment $26$ (the last payment).

To find the balance for payment $26$, set $P_1=26$ and $P_2=26$. Remember to re-enter the payment rounded to two decimal places.

 PMT Setting END $N$ $26$ $PV$ $50,000$ $FV$ $0$ $PMT$ $-2,264.36$ $I/Y$ $5$ $P/Y$ $4$ $C/Y$ $4$ $P_1$ $26$ $P_2$ $26$

$BAL=\0.13$

Step 3: Calculate the final payment.

Because the $BAL$ entry from the previous step is positive, the $\0.13$ is an underpayment and must be added to the $\2,264.36$ payment.

$\begin{eqnarray*} \mbox{Final Payment} & = & 2,264.36 +0.13 \\ & = & \2,264.49 \end{eqnarray*}$

Step 4: Write as a statement.

The final payment for the loan is $\2,264.49$.

### Try It

1) Semi-annual payments are made against a $\97,500$ loan at $7.5\%$ compounded semi-annually with a 10-year amortization. Calculate the final payment.

Solution
 PMT Setting END $N$ $20$ $PV$ $97,500$ $FV$ $0$ $PMT$ $?$ $I/Y$ $7.5$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\7,016.30$

 PMT Setting END $N$ $20$ $PV$ $97,500$ $FV$ $0$ $PMT$ $-7,016.30$ $I/Y$ $7.5$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $20$ $P_2$ $20$

$BAL=\0.13$

$\begin{eqnarray*} \mbox{Final Payment} & = & 7,016.30+0.13 \\ & = & \7,016.43 \end{eqnarray*}$

### Try It

2) A $\65,000$ loan at $3.5\%$ compounded quarterly is repaid with $\600$ monthly payments. Calculate the final payment.

Solution
 PMT Setting END $N$ $?$ $PV$ $65,000$ $FV$ $0$ $PMT$ $-600$ $I/Y$ $3.5$ $P/Y$ $12$ $C/Y$ $4$

$N=130.309...\rightarrow 131 \mbox{ payments}$

 PMT Setting END $N$ $131$ $PV$ $65,000$ $FV$ $0$ $PMT$ $-600$ $I/Y$ $3.5$ $P/Y$ $12$ $C/Y$ $4$ $P_1$ $131$ $P_2$ $131$

$BAL=-\413.98$

$\begin{eqnarray*} \mbox{Final Payment} & = & 600-413.98 \\ & = & \186.02 \end{eqnarray*}$

## Section 3.3 Exercises

1. A $\15,000$ loan at $10\%$ compounded quarterly is repaid with quarterly payments for three years. Calculate the final payment.
Solution

$\1,462.27$

2. A $\85,000$ loan at $6.75\%$ compounded monthly is repaid with monthly payments for seven years. Calculate the final payment.
Solution

$\1,273.03$

3. A $\32,000$ loan at $8.25\%$ effective is repaid with annual payments for $10$ years. Calculate the final payment.
Solution

$\4,822.82$

4. A $\250,000$ loan at $5.9\%$ compounded semi-annually is repaid with monthly payments of $\2,400$. Calculate the final payment.
Solution

$\1,265.95$

5. A $\28,250$ loan at $9\%$ compounded quarterly is repaid by monthly payments over five years. Calculate the final payment.
Solution

$\585.50$

6. You took out a $\90,000$ home renovation loan at $2.7\%$ compounded semi-annually. You make quarterly payments of $\3,000$ to repay the loan. Calculate your final payment.
Solution

$\1,864.93$

7. Stuart and Shelley just purchased a new $\65,871.88$ Nissan Titan Crew Cab SL at $8.99\%$ compounded monthly for a seven-year term. What is the amount of the final payment?
Solution

$\1,059.89$

8. Wile E. Coyote owes the ACME Corporation $\75,000$ for various purchased goods. Wile agrees to make $\1,000$ payments at the end of every month at $10\%$ compounded quarterly until the debt is repaid in full. What is the amount of the final payment?
Solution

$\512.66$