# 6.3 Calculating the Final Payment

## LEARNING OBJECTIVES

- Calculate the final payment for a loan.

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 × $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 × $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.

[latex]\mbox{Final Payment} = \mbox{Balance from Second Last Row}+\mbox{Interest from Last Row}[/latex]

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.

[latex]\begin{eqnarray*} \mbox{Final Payment} & = & PMT-\mbox{Amount Overpaid} \\ \\ & \mbox{or} & \\ \\ \mbox{Final Payment} & = & PMT+\mbox{Amount Underpaid} \end{eqnarray*}[/latex]

### NOTE

- 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:

- 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.
- 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. - Go to the amortization worksheet by pressing 2nd AMORT (the PV button).
- Enter the payment number for the final payment into P1 and P2.
- Find the BAL entry. Watch the cash flow sign of the BAL entry to properly interpret what to do with it!
- Calculate the final payment:
- A
**negative**BAL entry indicates an**overpayment**. Then[latex]\mbox{Final Payment} = PMT-\mbox{Overpayment}[/latex]

- A
**positive**BAL entry indicates an**underpayment**. Then[latex]\mbox{Final Payment} = PMT+\mbox{Underpayment}[/latex]

- A

## Example

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 |
[latex]10,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-200[/latex] |

I/Y |
[latex]8[/latex] |

P/Y |
[latex]12[/latex] |

C/Y |
[latex]4[/latex] |

[latex]N=60.9273...\rightarrow 61 \mbox{ payments}[/latex]

**Step 2:** Calculate the balance for payment 61 (the last payment). To find the balance for payment 61, set P1=61 and P2=61.

PMT Setting |
END |

N |
[latex]61[/latex] |

PV |
[latex]10,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-200[/latex] |

I/Y |
[latex]9[/latex] |

P/Y |
[latex]12[/latex] |

C/Y |
[latex]4[/latex] |

P1 |
[latex]61[/latex] |

P2 |
[latex]61[/latex] |

[latex]BAL=-\$14.49[/latex]

**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.

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

The final payment for the loan is $185.11.

## EXAMPLE

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 |
[latex]26[/latex] |

PV |
[latex]50,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
? |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]4[/latex] |

C/Y |
[latex]4[/latex] |

[latex]PMT=\$2,264.36[/latex]

**Step 2:** Calculate the balance for payment 26 (the last payment). To find the balance for payment 26, set P1=26 and P2=26. Remember to re-enter the payment rounded to two decimal places.

PMT Setting |
END |

N |
[latex]26[/latex] |

PV |
[latex]50,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-2,264.36[/latex] |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]4[/latex] |

C/Y |
[latex]4[/latex] |

P1 |
[latex]26[/latex] |

P2 |
[latex]26[/latex] |

[latex]BAL=\$0.13[/latex]

**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.

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

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

## TRY IT

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.

**Click to see Solution**

PMT Setting |
END |

N |
[latex]20[/latex] |

PV |
[latex]97,500[/latex] |

FV |
[latex]0[/latex] |

PMT |
? |

I/Y |
[latex]7.5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PMT=\$7,016.30[/latex]

PMT Setting |
END |

N |
[latex]20[/latex] |

PV |
[latex]97,500[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-7,016.30[/latex] |

I/Y |
[latex]7.5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

P1 |
[latex]20[/latex] |

P2 |
[latex]20[/latex] |

[latex]BAL=\$0.13[/latex]

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

## TRY IT

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

**Click to see Solution**

PMT Setting |
END |

N |
? |

PV |
[latex]65,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-600[/latex] |

I/Y |
[latex]3.5[/latex] |

P/Y |
[latex]12[/latex] |

C/Y |
[latex]4[/latex] |

[latex]N=130.309...\rightarrow 131 \mbox{ payments}[/latex]

PMT Setting |
END |

N |
[latex]131[/latex] |

PV |
[latex]65,000[/latex] |

FV |
[latex]0[/latex] |

PMT |
[latex]-600[/latex] |

I/Y |
[latex]3.5[/latex] |

P/Y |
[latex]12[/latex] |

C/Y |
[latex]4[/latex] |

P1 |
[latex]131[/latex] |

P2 |
[latex]131[/latex] |

[latex]BAL=-\$413.98[/latex]

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

**Exercises**

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

**Click to see Answer**$1,462.27

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

**Click to see Answer**$1,273.03

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

**Click to see Answer**$4,822.82

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

**Click to see Answer**$1,265.95

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

**Click to see Answer**$585.50

- 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.

**Click to see Answer**$1,8,64.93

- 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?

**Click to see Answer**$1,059.89

- 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?

**Click to see Answer**$512.66

#### Attribution

“13.2: Calculating the Final Payment” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“13.2: Calculating the Final Payment” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.