# 13.2 Calculating the Final Payment

# Calculating the Final Payment

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 installments, 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 the Final Payment Different?

Section 11.4 introduced the calculations to determine the annuity payment. Observe that you always needed to round a non-terminating annuity payment to two decimals. It is rare for a calculated annuity payment not to require rounding. The rounding up or down of the annuity payment forms the basis for adjusting the final payment.

### How It Works

The following six steps are needed to calculate the final payment. These steps are designed to integrate with the next section, where the principal and interest components on a series of payments involving the final payment are calculated.

**Step 1**: Identify all seven time value of money variables. If all are known, proceed to step 2. Most commonly, PMT is unknown. Solve for it using the appropriate formula and round the PMT to two decimals.

**Step 2**: Calculate the future value of the original principal at n−1 payments. For example, if your final payment is the 24th payment, you need the balance remaining after the 23rd payment.

**Step 3**: Calculate the future value of all annuity payments (n-1) already made. Remember that if the final payment is the 24th payment, then only 23 payments have already occurred.

**Step 4**: Subtract the future value of the payments from the future value of the original principal (step 2 − step 3) to arrive at the principal balance remaining immediately prior to the last payment. This is the principal owing on the account and therefore is the principal portion (PRN) for the final payment. The final payment must reduce the annuity balance to zero!

**Step 5**: Calculate the interest portion (INT) of the last payment using Formula 13.1D on the remaining principal.

**Step 6**: Add the principal portion from step 4 to the interest portion from step 5. The sum is the amount of the final payment.

## Your BAII Plus Calculator

The calculator determines the final payment amount using the AMORT function described in Section 13.1. To calculate the final payment:

- You must accurately enter all seven time value of money variables (N, I/Y, PV, PMT, FV, P/Y and C/Y). If PMT was calculated, you must re-enter it with only two decimals while retaining the correct cash flow sign convention.
- Press 2nd AMORT.
- Enter the payment number for the final payment into P1 and press Enter followed by ↓.
- Enter the same payment number for P2 and press Enter followed by ↓.
- In the BAL window, note the balance remaining in the account after the last payment is made. Watch the cash flow sign to properly interpret what to do with it! The sign matches the sign of your PV. The next table summarizes how to handle this balance.

Type of Transaction | Positive BAL | Negative BAL |
---|---|---|

Loan | Increase final payment | Decrease final payment |

Investment Annuity | Decrease final payment | Increase final payment |

**Loans**. For a loan for which PV is entered as a positive cash flow and hence PMT is a negative cash flow, a positive balance means you are borrowing it. Thus, you need to increase the final payment by this amount to pay off the loan. A negative balance means you overpaid and the bank owes you. Thus, you need to decrease the final payment by this amount.**Investment Annuities**. For an investment annuity where PV is entered as a negative cash flow and hence PMT is a positive cash flow, a negative balance means you still have money invested, so you should add it to your final payment to get it back. A positive balance means you have been paid too much, so you need to decrease your final payment by this amount.- A helpful key sequence shortcut to arrive at the final payment is to have BAL on your display and then press − RCL PMT =
- This sequence automatically adjusts the payment accordingly for both loans and investment annuities. Manually round the answer to two decimals when the calculation is complete.

- If you are interested in the PRN or INT portions of the final payment, the INT output is correct. However, the PRN output is incorrect since the calculator has not adjusted the final payment. You must adjust the PRN output in the same manner and amount as the final payment (by adding or subtracting the BAL remaining).

**Concept Check**

#### Example 13.2.1: Final Payment on a Loan

Recall Example 13.1.1, in which Nichols and Burnt borrowed $10,000 at 8% compounded quarterly with month-end payments of $452.03 for two years. The accountant now needs to record the final payment on the loan with correct portions assigned to principal and interest.

**Solution:**

**Step 1:** Given information:

PV_{ORD} = $10,000; I/Y = 8%; C/Y = 4; PMT = $452.03; P/Y = 12; Years = 2; n = 12 × 2 = 24; FV = $0

**Step 2:** Calculate the future value of the loan principal at the time of the 23rd payment using Formula 9.2B.

[latex]i=\frac{I/Y}{C/Y}=\frac{8\%}{4}=2\%[/latex]

[latex]i_{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.02)^{\frac{4}{12}}-1=0.00662271\; \text{per month}[/latex]

[latex]\begin{align} FV&=PV(1+i_{eq})^n\\ &=\$10,\!000(1+0.00662271)^{23}\\ &=\$11,\!639.50884 \end{align}[/latex]

**Step 3:** Calculate the future value of the first 23 payments using Formula 11.2A

[latex]\begin{align} FV_{ORD}&=PMT \left[\frac{(1+i_{eq})^n-1}{i_{eq}}\right]\\ &=\$452.03\left[\frac{(1+0.00662271)^{23}-1}{0.00662271}\right]\\ &=\$11,\!190.39157 \end{align}[/latex]

**Step 4**: Calculate the principal balance remaining after 23 payments.

BAL = FV − FV_{ORD} = $11,639.50884 – $11,190.39157 = $449.11727

**Step 5**: Calculate the interest portion by using Formula 13.1A.

INT = BAL × i_{eq}= $449.11727 × 0.00662271 = $2.97437

**Step 6**: Calculate the final payment by totaling steps 4 and 5 above.

Final PMT = $449.11727 + $2.97437 = $452.09

**Calculator instructions:**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

24 | 8 | 10,000 | -452.03 | 0 | 12 | 4 |

P1 | P2 | BAL (output) | PRN (output) | INT (output) |
---|---|---|---|---|

24 | 24 |
0.061582 *added to payment of $452.03=$452.09 |
449.055627 *added BAL=$449.12 |
2.974372 |

The accountant for Nichols and Burnt should record a final payment of $452.09, which consists of a principal portion of $449.12 and an interest portion of $2.97.

#### Example: 13.2.2: Final Payment on an Investment Annuity

Recall Example 13.1.2, in which Baxter has $50,000 invested into a five-year annuity that earns 5% compounded quarterly and makes regular end-of-quarter payments of $2,841.02 to him. He needs to know the amount of his final payment, along with the principal and interest components.

**Solution:**

Calculate the principal portion (PRN) and the interest portion (INT) of the final payment on the five-year investment annuity, along with the amount of the final payment itself (PMT).

**Step 1**: Given information:

PV_{ORD} = $50,000; I/Y = 5%; C/Y = 4; PMT = $2,841.02; P/Y = 4; Years = 5; n = 4 × 5 = 12; FV = $0

**Step 2:** Calculate the future value of the investment at the time of the 19th quarterly payment using Formula 9.2B.

[latex]\begin{align} FV&=PV(1+i)^n\\ &=\$50,\!000(1+0.0125)^{19}\\ &=\$63,\!310.48058 \end{align}[/latex]

**Step 3:** Calculate the future value of the first 19 payments using Formula 11.2A.

[latex]\begin{align} FV_{ORD}&=PMT \left[\frac{(1+i)^n-1}{i_{eq}}\right]\\ &=\$2,\!841.02\left[\frac{(1+0.0125)^{19}-1}{0.0125}\right]\\ &=\$60,\!504.54645 \end{align}[/latex]

**Step 4**: Calculate the principal balance remaining after 19 payments.

BAL = FV − FV_{ORD} = $63,310.48058 – $60,504.54645 = $2,805.934127

**Step 5**: Calculate the interest portion by using Formula 13.1A.

INT=BAL × i_{eq}= $2,805.934127 × 0.0125 = $35.074176

**Step 6**: Calculate the final payment by totaling steps 4 and 5 above.

Final PMT = $2,805.934127 + $35.074176 = $2,841.01

**Calculator instructions:**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

20 | 5 | -50,000 | 2841.02 | 0 | 4 | 4 |

P1 | P2 | BAL (output) | PRN (output) | INT (output) |
---|---|---|---|---|

20 | 20 |
0.011696 *subtracted from payment of $2,841.02=$2841.01 |
2,805.945823 *subtract BAL=$2,805.93 |
35.074176 |

Baxter will receive a final payment of $2,841.01 consisting of $2,805.93 in principal plus $35.08 in interest.

## Calculating Principal and Interest Portions for a Series Involving the Final Payment

Now that you know how to calculate the last payment along with its interest and principal components, it is time to extend this knowledge to calculating the principal and interest portions for a series of payments that involve the final payment.

### How It Works

For a series of payments, you follow essentially the same steps as in Section 13.1; however, you need a few minor modifications and interpretations:

**Step 1**: Draw a timeline. Identify the known time value of money variables, including I/Y, C/Y, P/Y, Years, and one of PV_{ORD} or TV_{ORD}. The annuity payment amount may or may not be known.

**Step 2**: If the annuity payment amount is known, proceed to step 3. If it is unknown, then solve for the annuity payment using Formulas 9.1 (Periodic Interest Rate) and 11.1 (Number of Annuity Payments) and by rearranging Formula 11.4 (Ordinary Annuity Present Value). Round this payment to two decimals.

**Step 3**: Calculate the future value of the original principal immediately prior to the series of payments being made.

**Step 4**: Calculate the future value of all annuity payments already made prior to the first payment in the series.

**Step 5**: Calculate the balance (BAL) prior to the series of payments by subtracting step 4 (the future value of the payments) from step 3 (the future value of the original principal). The result of this step determines the amount of principal remaining in the account. This is the PRN for the series of payments, since the remaining payments must reduce the principal to zero.

**Step 6**: Calculate the future value of the original principal immediately at the end of the timeline.

**Step 7**: Calculate the future value of all annuity payments, including the unadjusted final payment.

**Step 8**: Calculate the balance (BAL) after the series of payments by subtracting step 7 (the future value of the payments) from Step 6 (the future value of the original principal). The fundamental concept of time value of money allows you to combine these two numbers on the same focal date. Do not round this number. The result of this step determines the amount of overpayment or underpayment, which you must then adjust in the next step.

**Step 9**: Calculate the interest portion using Formula 13.1D, but modify the final amount by the result from step 8. Hence, Formula 13.1D looks like

INT = N × PMT – PRN + (balance from step 8)

#### Example 13.2.3: Principal and Interest for a Series Involving the Final Payment

Revisit Example 13.1.1. The accountant at the accounting firm of Nichols and Burnt is completing the tax returns for the company and needs to know the total principal portion and interest expense paid during the tax year encompassing payments 13 through 24 inclusively. Recall that the company borrowed $10,000 at 8% compounded quarterly, with month-end payments of $452.03 for two years.

**Solution:**

Calculate the total principal portion (PRN) and the total interest portion (INT) of the 13th to 24th payments on the two-year loan. This involves the final payment since the 24th payment is the last payment, requiring usage of the adapted steps discussed in this section.

**Step 1**: Given information:

PV_{ORD} = $10,000; I/Y = 8%; C/Y = 4; PMT = $452.03; P/Y = 12; Years = 2; n = 12 × 2 = 24; FV = $0

**Step 2**: Skip this step, since PMT is known.

**Step 3:** Calculate the future value of the loan principal after the 12th monthly payment using Formula 9.2B.

[latex]i=\frac{I/Y}{C/Y}=\frac{8\%}{4}=2\%[/latex]

[latex]i_{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.02)^{\frac{4}{12}}-1=0.00662271\; \text{per month}[/latex]

[latex]\begin{align} FV&=PV(1+i_{eq})^n\\ &=\$10,\!000(1+0.00662271)^{12}\\ &=\$10,\!824.32166 \end{align}[/latex]

**Step 4:** Calculate the future value of the first 12 payments using Formula 11.2A

[latex]\begin{align} FV_{ORD}&=PMT \left[\frac{(1+i_{eq})^n-1}{i_{eq}}\right]\\ &=\$452.03\left[\frac{(1+0.00662271)^{12}-1}{0.00662271}\right]\\ &=\$5,\!626.369243 \end{align}[/latex]

**Step 5**: Calculate the principal balance after 12 payments. Note that for payments 13 through 24, PRN = BAL_{P1.}

BAL _{P1}= FV − FV_{ORD} = $10,824.32166 – $5,626.369243 = $5,197.952417

**Step 6**: Calculate the future value of the loan principal after the 24th monthly payment using Formula 9.2B.

Recall: i_{eq}=0.00662271 per month

[latex]\begin{align} FV&=PV(1+i_{eq})^n\\ &=\$10,\!000(1+0.00662271)^{24}\\ &=\$11,\!716.59393 \end{align}[/latex]

**Step 7**: Calculate the future value of all 24 payments using Formula 11.2A.

[latex]\begin{align} FV_{ORD}&=PMT \left[\frac{(1+i_{eq})^n-1}{i_{eq}}\right]\\ &=\$452.03\left[\frac{(1+0.00662271)^{24}-1}{0.00662271}\right]\\ &=\$11,\!716.53229 \end{align}[/latex]

**Step 8**: Calculate the balance on the loan after all payments.

BAL_{P2 }= FV − FV_{ORD }= $11,716.59393 – $11,716.53229 = 0.06164

Note that this amount is used to adjust step 9.

**Step 9**: Calculate the interest portion by using the adjusted Formula 13.4.

n = 13th through 24th payment inclusive = 12 payments;

INT = 12 × $452.03 – $5,197.952417 + 0.06164 = $226.47

**Calculator instructions:**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

24 | 8 | 10,000 | -452.03 | 0 | 12 | 4 |

P1 | P2 | BAL (output) | PRN (output) | INT (output) |
---|---|---|---|---|

13 | 24 | 0.0615825 |
5,197.890788 *add BAL=$5,197.90 |
226.469212 |

## Exercises

In each of the exercises that follow, try them on your own. Full solutions are available should you get stuck.

- Semi-annual payments are to be made against a $97,500 loan at 7.5% compounded semi-annually with a 10-year amortization.

a) What is the amount of the final payment? (**Answer**: $7,016.43)

b) Calculate the principal and interest portions of the payments in the final two years. (**Answers:**$2,446.01)

- A $65,000 trust fund is set up to make end-of-year payments for 15 years while earning 3.5% compounded quarterly.

a) What is the amount of the final payment? (**Answer**: $5,662.21)

b) Calculate the principal and interest portion of the payments in the final three years. (**Answer:**$1,137.16)

- Mirabel Wholesale has a retail client that is struggling and wants to make instalments against its most recent invoice for $133,465.32. Mirabel works out a plan at 12.5% compounded monthly with beginning-of-month payments for two years.

a) What will be the amount of the final payment? (**Answer**: $6,248.88)

b) Calculate the principal and interest portions of the payments for the entire agreement. (**Answer:**$16,505.73)

**Note**: Solution to exercises are demonstrated using the calculator only.