13.1: Calculating Interest and Principal Components
 Page ID
 22145
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)How much of the principal do you pay off when you make a loan payment? One year ago you purchased your $250,000 dream home on a 25year mortgage at a fixed 5% compounded semiannually interest rate. With monthly contributions of $1,454.01, or $17,448.12 in total for the past year, you figure you must have put a serious dent in the balance owing.
You get a rude shock when you inspect your mortgage statement, as you realize the balance owing is not what you expected. Your remaining balance is $244,806.89, reflecting a principal reduction of only $5,193.11! The other 70% of your hardearned money, amounting to $12,255.01, went solely toward the bank's interest charges.
Many people do not fully understand how their loan payments are portioned out. Over the full course of the 25year mortgage you will pay $186,204.46 in interest charges at 5% compounded semiannually, or approximately 74.5% of the home's price tag. That is a total of $436,204.46 paid on a $250,000 home. What if interest rates rise? At a more typical 7% semiannual rate, you would owe $275,311.51 in total interest, or 110% of the value of your home.
Knowing these numbers, what can you do about them? Term, interest rates, payment amounts, and payment frequency all affect the amount of interest you pay. What if you made one extra mortgage payment per year? Did you know that it would only take approximately 21 years instead of 25 years to own your home? Instead of paying $186,204.46 in interest you would pay only $156,789.33, a savings of almost $30,000!
These calculations should make it clear that both businesses and consumers need to understand the interest and principal components of annuity payments. This section shows you how to calculate principal and interest components both for single payments and for a series of payments.
What Is Amortization?
Amortization is a process by which the principal of a loan is extinguished over the course of an agreedupon time period through a series of regular payments that go toward both the accruing interest and principal reduction. Two components make up the agreedupon time component:
 Amortization Term. The amortization term is the length of time for which the interest rate and payment agreement between the borrower and the lender will remain unchanged. Thus, if the agreement is for monthly payments at a 5% fixed rate over five years, it is binding for the entire five years. Or if the agreement is for quarterly payments at a variable rate of prime plus 2% for three years, then interest is calculated on this basis throughout the three years.
 Amortization Period. The amortization period is the length of time it will take for the principal to be reduced to zero. For example, if you agree to pay back your car loan over six years, then after six years you reduce your principal to zero and your amortization period is six years.
In most relatively small purchases, the amortization term and amortization period are identical. For example, a vehicle loan has an agreedupon interest rate and payments for a fixed term. At the end of the term, the loan is fully repaid. However, larger purchases such as real estate transactions typically involve too much money to be repaid under short time frames. Financial institutions hesitate to agree to amortization terms of much more than five to seven years because of the volatility and fluctuations of interest rates. As a result, a term of five years may be established with an amortization period of 25 years. When the five years elapse, a new term is established as agreed upon between the borrower and lender. The conditions of the new term reflect prevailing interest rates and a payment plan that continues to extinguish the debt within the original amortization period.
The figure shows the timeline for a 25year mortgage in which the borrower establishes five sequential fiveyear terms throughout the 25year amortization period to extinguish the debt. With regard to amortization, all loans take the structure of either simple ordinary annuities or general ordinary annuities unless otherwise stated. Therefore, this textbook will primarily focus on ordinary annuities.
Calculating Interest and Principal Components for a Single Payment
At any point during amortization you can precisely calculate how much any single payment contributes toward principal and interest. Businesses must separate the principal and interest components for two reasons:
 Interest Expense. Any interest paid on a debt is an accounting expense that must be reported in financial statements. In addition, interest expenses have tax deduction implications for a business.
 Interest Income. Any interest that a company receives is a source of income. This must be reported as revenue in its financial statements and is subject to taxation rules.
The Formula
To calculate the interest and principal components of any annuity payment, follow this sequence of two formulas.
 Calculate the interest portion of the payment (Formula 13.1).
 Calculate the principal portion of the payment (Formula 13.2).
How It Works
Follow these steps to calculate the interest and principal components for a single annuity payment:
Step 1: Draw a timeline (seen below). Identify the known time value of money variables, including \(IY, CY, PY\), Years, and one of \(PV_{ORD}\) or \(FV_{ORD}\). 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, solve for it 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 the payment to two decimals.
Step 3: Calculate the future value of the original principal immediately prior to the payment being made. Use Formulas 9.1 (Periodic Interest Rate), 9.2 (Number of Compounding Periods for Single Payments), and 9.3 (Compound Interest for Single Payments). For example, when you calculate the interest and principal portions for the 22nd payment, you need to know the balance immediately after the 21st payment.
Step 4: Calculate the future value of all annuity payments already made. Use Formulas 11.1 (Number of Annuity Payments) and 11.2 (Ordinary Annuity Future Value). For example, if you need to calculate the interest and principal portions for the 22nd payment, you need to know the future value of the first 21 payments.
Step 5: Calculate the balance (\(BAL\)) prior to the payment by subtracting step 4 (the future value of the payments) from step 3 (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.
Step 6: Calculate the interest portion of the current annuity payment using Formula 13.1.
Step 7: Calculate the principal portion of the current annuity payment using
Formula 13.2.Important Notes
Investment Annuities
The formulas and techniques being discussed in this section also apply to any type of investment annuity from which annuity payments are received. For example, most people receive annuity payments from their accumulated RRSP savings when in retirement. In these cases, view the investment as a loan to the financial institution at an agreedupon interest rate. The financial institution then makes annuity payments to the retiree to extinguish its debt at some future point; these payments consist of the principal and interest being earned.
Your BAII Plus Calculator.
The function that calculates the interest and principal components of any single payment on your BAII Plus calculator is called AMORT. It is located on the 2nd shelf above the PV button.
The Amortization window has five variables (use ↓ or ↑ to scroll through them). The first two, P1 and P2, are data entry variables. The last three, BAL, PRN, and INT, are output variables.
 P1 is the starting payment number. The calculator works with a single payment or a series of payments.
 P2 is the ending payment number. This number is the same as P1 when you work with a single payment. When you work with a series of payments later in this section, you set it to a number higher than P1.
 BAL is the principal balance remaining after the P2 payment number. The cash flow sign is correct as indicated on the calculator display.
 PRN is the principal portion of the payments from P1 to P2 inclusive. Ignore the cash flow sign.
 INT is the interest portion of the payments from P1 to P2 inclusive. Ignore the cash flow sign.
To use the Amortization function, the commands are as follows:
 You must enter all seven time value of money variables accurately (\(N, I/Y, PV, PMT, FV, P/Y\), and \(C/Y\)). If \(PMT\) was computed, you must reenter it with only two decimals and the correct cash flow sign.
 Press 2nd AMORT.
 Enter a value for P1 and press Enter followed by ↓.
 Enter a value for P2 and press Enter followed by ↓. Note that the higher the numbers entered in P1 or P2, the longer it takes the calculator to compute the outputs. It is possible that your calculator will go blank for a few moments before displaying the outputs.
 Using the ↓ and ↑, scroll through BAL, PRN, and INT to read the output.
Things To Watch Out For
A common misunderstanding when using the AMORT function on the calculator occurs when inputting the values for P1 and P2. Many people think that when they solve for a single payment they need to set these values one apart. For example, if they are looking for the 22nd payment they imagine that P1 = 21 and P2 = 22. This is incorrect, as the calculator would then compute the total values for both payments 21 and 22.
If you are interested in a single payment, you must set P1 and P2 to the exact same value. In the example, if you want the 22nd payment then both P1 = 22 and P2 = 22.
 For any loan, if you calculated the interest portions of the second payment and the tenth payment, which payment has a smaller interest portion?
 For any loan, if you calculated the principal portions of the fifth payment and the twelfth payment, which payment has a smaller principal portion?
 Holding all other variables constant, if Loan A had an interest rate of 4% while Loan B had an interest rate of 6%, which loan has the higher interest portion on any payment?
 Answer

 The tenth payment, since the principal is much smaller by then so less interest is charged.
 The fifth payment, since the balance is higher so more interest is charged.
 Loan B, since the interest rate is higher.
The accountant at the accounting firm of Nichols and Burnt needs to separate the interest and principal on the tenth loan payment. The company borrowed $10,000 at 8% compounded quarterly with monthend payments for two years.
Solution
Note that this is an ordinary general annuity. Calculate the principal portion (\(PRN\)) and the interest portion (\(INT\)) of the tenth payment on the twoyear loan.
What You Already Know
Step 1:
The information about the accounting firm's loan are in the timeline. \(PV_{ORD}\) = $10,000, \(IY\) = 8%, \(CY\) = 4, \(PY\) = 12, Years = 2, \(FV\) = $0
How You Will Get There
Step 2:
PMT is unknown. Apply Formulas 9.1, 11.1, and 11.4.
Step 3:
Calculate the future value of the loan principal using Formulas 9.2 and 9.3.
Step 4:
Calculate the future value of the first nine payments using Formulas 11.1 and 11.2.
Step 5:
Calculate the principal balance after nine payments through \(BAL = FV − FV_{ORD}\).
Step 6:
Calculate the interest portion by using Formula 13.1.
Step 7:
Calculate the principal portion by using Formula 13.2.
Perform
Step 2:
\(i=8 \% / 4=2 \% ; N=12 \times 2=24 \) payments
\[\$ 10,000=PMT\left[\dfrac{1\left[\dfrac{1}{(1+0.02)^{\frac{4}{12}}}\right]^{24}}{(1+0.02)^{\frac{4}{12}}1}\right] \nonumber \]
\[PMT=\dfrac{\$ 10,000}{\left[\dfrac{1\left[\dfrac{1}{(1+0.02)}^{\frac{4}{12}}\right]^{24}}{(1+0.02)^{\frac{4}{12}}1}\right]}=\dfrac{\$ 10,000}{\left[\dfrac{0.146509}{0.006622}\right]}=\$ 452.03 \nonumber \]
Step 3:
\[N=4 \times \dfrac{9}{12}=3 \text { compounds; } FV=\$ 10,000(1+0.02)^{3}=\$ 10,612.08 \nonumber \]
Step 4:
\(N=12 \times \dfrac{9}{12}=9\) payments
\[FV_{ORD}=\$ 452.03\left[\dfrac{\left[(1+0.02)^{\frac{4}{12}}\right]^{9}1}{(1+0.02)^{\frac{4}{12}}1}\right]=\$ 4,177.723934 \nonumber \]
Step 5:
\[BAL=\$ 10,612.08\$ 4,177.723934=\$ 6,434.356066 \nonumber \]
Step 6:
\[INT=\$ 6,434.356066 \times\left((1+0.02)^{\frac{4}{12}}1\right)=\$ 42.612871 \nonumber \]
Step 7:
\[PRN=\$ 452.03\$ 42.612871=\$ 409.42 \nonumber \]
Calculator Instructions
N  I/Y  PV  PMT  FV  P/Y  C/Y 

24  8  10000 
Answer: 452.032375 Rekeyed as: 452.03 
0  122  4 
P1  P2  BAL (output)  PRN (output)  INT (output) 

10  10  6,024.938937  409.417128  42.612871 
The accountant for Nichols and Burnt records a principal reduction of $409.42 and an interest expense of $42.61 for the tenth payment.
Baxter has $50,000 invested into a fiveyear annuity that earns 5% compounded quarterly and makes regular endofquarter payments to him. For his fifth payment, he needs to know how much of his payment came from his principal and how much interest was earned on the investment.
Solution
Note that this is an ordinary simple annuity. Calculate the principal portion (\(PRN\)) and the interest portion (\(INT\)) of the fifth payment on the fiveyear investment annuity.
What You Already Know
Step 1:
You know the following about the investment annuity, as illustrated in the timeline. \(PV_{ORD}\) = $50,000, \(IY\) = 5%, \(CY\) = 4, \(PY\) = 4, Years = 5, \(FV\) = $0
How You Will Get There
Step 2:
\(PMT\) is unknown. Apply Formulas 9.1, 11.1, and 11.4.
Step 3:
Calculate the future value of the loan principal using Formulas 9.2 and 9.3.
Step 4:
Calculate the future value of the first four payments using Formulas 11.1 and 11.2.
Step 5:
Calculate the principal balance after four payments through \(BAL = FV − FV_{ORD}\).
Step 6:
Calculate the interest portion by using Formula 13.1.
Step 7:
Calculate the principal portion by using Formula 13.2.
Perform
Step 2:
\(i=5 \% / 4=1.25 \% ; N=4 \times 5=20 \) payments
\[\$ 50,000=PMT\left[\dfrac{1\left[\dfrac{1}{(1+0.0125)^{\frac{4}{4}}}\right]^{20}}{(1+0.0125)^{\frac{4}{4}}1}\right] \nonumber \]
\[PMT=\dfrac{\$ 50,000}{\left[\dfrac{1\left[\dfrac{1}{(1+0.0125)^{\frac{4}{4}}}\right]^{20}}{[(1+0.0125)^{\frac{4}{4}}]1}\right]}=\dfrac{\$ 50,000}{\left[\dfrac{0.219991}{0.0125}\right]}=\$ 2,841.02 \nonumber \]
Step 3:
\[N=4 \times 1=4 \text { compounds; } FV=\$ 50,000(1+0.0125)^{4}=\$ 52,547.26685 \nonumber \]
Step 4:
\(N=4 \times 1=4 \) payments
\[FV_{ORD}=\$ 2,841.02\left[\dfrac{\left[(1+0.0125)^{\frac{4}{4}}\right]^{4}1}{(1+0.0125)^{\frac{4}{4}}1}\right]=\$ 11,578.93769 \nonumber \]
Step 5:
\[BAL=\$ 52,547.26685\$ 11,578.93769=\$ 40,968.32916 \nonumber \]
Step 6:
\[INT =\$ 40,968.32916 \times\left((1+0.0125)^{\frac{4}{4}}1\right)=\$ 512.104114 \nonumber \]
Step 7:
\[PRN=\$2,841.02\$512.104114=\$ 2,328.92 \nonumber \]
Calculator Instructions
N  I/Y  PV  PMT  FV  P/Y  C/Y 

20  5  50000 
Answer: 2,841.019482 Rekeyed as: 2,841.01 
0  4  4 
P1  P2  BAL (output)  PRN (output)  INT (output) 

5  5  38,639.41327  2,328.915886  512.104114 
On Baxter's fifth payment of $2,841.02, he has $2,328.92 deducted from his principal and the remaining $512.10 comes from the interest earned on his investment.
Calculating Interest and Principal Components for a Series of Payments
Many times in business, you need to know the principal and interest portions for a series of annuity payments. For example, when completing tax forms a company needs the total loan interest paid annually. If the loan payments are monthly, using Formula 13.1 and Formula 13.2 requires you to perform the calculations 12 times (once for each payment) to arrive at the total interest paid. Clearly, that is time consuming and tedious. In this section, you learn new formulas and a process for calculating the principal and interest portions involving a series of payments.
The Formula
Formulas 13.3 and 13.4 are used to determine the interest and principal components for a series of annuity payments.
How It Works
Follow these steps to calculate the interest and principal components for a series of annuity payments:
Step 1: Draw a timeline. Identify the known time value of money variables, including \(IY, CY, PY\), Years, and one of \(PV_{ORD}\) or \(FV_{ORD}\). 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, solve for it 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 the payment to two decimals.
Step 3: Calculate the future value of the original principal immediately prior to the series of payments being made. Use Formulas 9.1 (Periodic Interest Rate), 9.2 (Number of Compounding Periods for Single Payments), and 9.3 (Compound Interest for Single Payments). For example, when calculating the interest and principal portions for the 22nd through 25th payments, you need the balance immediately after the 21st payment.
Step 4: Calculate the future value of all annuity payments already made prior to the first payment in the series. Use Formulas 11.1 (Number of Annuity Payments) and 11.2 (Ordinary Annuity Future Value). For example, when calculating the interest and principal portions for the 22nd through 25th payments, you need the future value of the first 21 payments.
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 fundamental concept of time value of money allows you to combine these two numbers on the same focal date. Do not round this number.
Steps 6 to 8: Repeat steps 3 to 5 to calculate the future value of the original principal immediately after the last payment in the series is made. For example, when calculating the interest and principal portions for the 22nd through 25th payments, you need the balance immediately after the 25th payment.
Step 9: Calculate the principal portion of the series of payments using Formula 13.3.
Step 10: Calculate the interest portion of the series of payments using Formula 13.4.
Important Notes
Working with a series of payments on the BAII Plus calculator requires you to enter the first payment number into the P1 and the last payment number into the P2. Thus, if you are looking to calculate the interest and principal portions of payments four through seven, set P1 = 4 and P2 = 7. In the outputs, the \(BAL\) window displays the balance remaining after the last payment entered (P2 = 7), and the \(PRN\) and \(INT\) windows display the total principal interest portions for the series of payments.
Things To Watch Out For
A common mistake occurs in translating years into payment numbers. For example, assume payments are monthly and you want to know the total interest paid in the fourth year. In error, you might calculate that the fourth year begins with payment 36 and ends with payment 48, thus looking for payments 36 to 48. The mistake is to fail to realize that the 36th payment is actually the last payment of the third year. The starting payment in the fourth year is the 37th payment. Hence, if you are concerned only with the fourth year, then you must look for the 37th to 48th payments.
There are two methods to calculate the correct payment numbers:
 Calculate the payment at the end of the year in question, then subtract the payment frequency less one (\(PY − 1\)) to arrive at the first payment of the year. In the example, the last payment of the fourth year is 48. With monthly payments, or \(PY\) = 12, then 48 − (12 − 1) = 37, which is the first payment of the fourth year.
 You could determine the last payment of the year prior to the year of interest and add one payment to it. Thus, the end of the third year is payment #36, so the first payment of the fourth year is 36 + 1 = 37. The last payment of the fourth year remains at payment 48.
Revisit Example \(\PageIndex{1}\). The accountant at the accounting firm of Nichols and Burnt is completing the tax returns for the company and needs to know the total interest expense paid during the tax year that encompassed payments 7 through 18 inclusively. Remember, the company borrowed $10,000 at 8% compounded quarterly with monthend payments for two years.
Solution
Note that this is an ordinary general annuity. Calculate the total principal portion (\(PRN\)) and the total interest portion (\(INT\)) of the 7th to the 18th payments on the twoyear loan.
What You Already Know
Step 1:
The following information about the accounting firm's loan is known, as illustrated in the timeline.
\(PV_{ORD}\) = $10,000, \(IY\) = 8%, \(CY\) = 4, \(PMT\) = $452.03, \(PY\) = 12, Years = 2, \(FV\) = $0
How You Will Get There
Step 2:
\(PMT\) is known. Skip this step.
Step 3:
Calculate the future value of the loan principal prior to the first payment in the series using Formulas 9.2 and 9.3.
Step 4:
Calculate the future value of the first six payments using Formulas 11.1 and 11.2.
Step 5:
Calculate the principal balance prior to the 7th payment through \(BAL_{P1} = FV − FV_{ORD}\).
Steps 6 to 8:
Repeat steps 3 to 5 for the 18 payment to calculate \(BAL_{P2}\).
Step 9:
Calculate the principal portion by using Formula 13.3.
Step 10:
Calculate the interest portion by using Formula 13.4.
Perform
Step 3:
Recall \(i=2 \% ; N=4 \times \dfrac{6}{12}=2\) compounds; \(FV=\$ 10,000(1+0.02)^{2}=\$ 10,404.00\)
Step 4:
\(N=12 \times \dfrac{6}{12}=6\) payments
\[FV_{ORD}=\$ 452.03\left[\dfrac{\left[(1+0.02)^{\frac{4}{12}}\right]^{6}1}{(1+0.02)^{\frac{4}{12}1}}\right]=\$ 2,757.483449 \nonumber \]
Step 5:
\[BAL_{P1}=\$ 10,404.00\$ 2,757.483449=\$ 7,646.516551 \nonumber \]
Step 6:
\(N=4 \times \dfrac{18}{12}=6 \) compounds; \(FV=\$ 10,000(1+0.02)^{6}=\$ 11,261.62419 \)
Step 7:
\(N=12 \times \dfrac{18}{12}=18 \) payments
\[FV_{ORD}=\$ 452.03\left[\dfrac{\left[(1+0.02)^{\frac{4}{12}}\right]^{18}1}{(1+0.02)^{\frac{4}{12}}1}\right]=\$ 8,611.157995 \nonumber \]
Step 8:
\[BAL_{P2}=\$ 11,261.62419\$ 8,611.157995=\$ 2,650.466195 \nonumber \]
Step 9:
\[PRN=\$ 7,646.516551\$ 2,650.466195=\$ 4,996.05 \nonumber \]
Step 10:
N = 7th through 18th payment inclusive = 12 payments;
\[INT=12 \times \$ 452.03\$ 4,996.06=\$ 5,424.36\$ 5,064.96=\$ 428.30 \nonumber \]
Calculator Instructions
N  I/Y  PV  PMT  FV  P/Y  C/Y 

24  8  10000  452.03  0  12  4 
P1  P2  BAL (output)  PRN (output)  INT (output) 

7  18  2,650.466197  4,996.050354  428.309646 
For the tax year covering payments 7 through 18, total payments of $5,424.36 are made, of which $4,996.05 was deducted from principal while $428.31 went to the interest charged.
Revisit Example \(\PageIndex{2}\), in which Baxter has $50,000 invested into a fiveyear annuity that earns 5% compounded quarterly and makes regular endofquarter payments to him. For his third year, he needs to know how much of his payments came from his principal and how much was interest earned on the investment.
Solution
Note that this is an ordinary simple annuity. Calculate the principal portion (\(PRN\)) and the interest portion (\(INT\)) of the thirdyear payments for the fiveyear investment annuity. This is the 9th through the 12th payments inclusive.
What You Already Know
Step 1:
The following information about the investment annuity is known, as illustrated in the timeline.
\(PV_{ORD}\) = $50,000, \(IY\) = 5%, \(CY\) = 4, \(PMT\) = $2,841.02, \(PY\) = 4, Years = 5, \(FV\) = $0
How You Will Get There
Step 2:
\(PMT\) is known. Skip this step.
Step 3:
Calculate the future value of the loan principal prior to the first payment in the series using Formulas 9.2 and 9.3.
Step 4:
Calculate the future value of the first eight payments using Formulas 11.1 and 11.2.
Step 5:
Calculate the principal balance prior to the ninth payment through \(BAL_{P1} = FV − FV_{ORD}\).
Steps 6 to 8:
Repeat steps 3 to 5 for the 12th payment to calculate \(BAL_{P2}\).
Step 9:
Calculate the principal portion by using Formula 13.3.
Step 10:
Calculate the interest portion by using Formula 13.4.
Perform
Step 3:
Recall \(i=1.25 \% ; N=4 \times 2=8\) compounds; \(FV=\$ 50,000(1+0.0125)^{8}=\$ 55,224.30506\)
Step 4:
\(N=4 \times 2=8 \) payments
\[FV_{ORD}=\$ 2,841.02\left[\dfrac{\left[(1+0.0125)^{\frac{4}{4}}\right]^{8}1}{(1+0.0125)^{\frac{4}{4}}1}\right]=\$ 23,747.76825 \nonumber \]
Step 5:
\[BAL_{Pl}=\$ 55,224.30506\$ 23,747.76825=\$ 31,476.53681 \nonumber \]
Step 6:
\(N=4 \times 3=12\) compounds; \(FV=\$ 50,000(1+0.0125)^{12}=\$ 58,037.72589 \)
Step 7:
\(N=4 \times 3=12\) payments
\[FV_{ORD}=\$ 2,841.02\left[\dfrac{\left[(1+0.0125)^{\frac{4}{4}}\right]^{12}1}{(1+0.0125)^{\frac{4}{4}}1}\right]=\$ 36,536.544 \nonumber \]
Step 8:
\[BAL_{P2}=\$ 58,037.72589\$ 36,536.544=\$ 21,501.18189 \nonumber \]
Step 9:
\[PRN=\$ 31,476.53681\$ 21,501.18189=\$ 9,975.35 \nonumber \]
Step 10:
N = 9th through 12th payment inclusive = 4 payments;
\[INT=4 \times \$ 2,841.02\$ 9,975.35=\$ 11,364.08\$ 9,975.35=\$ 1,388.73 \nonumber \]
Calculator Instructions
N  I/Y  PV  PMT  FV  P/Y  C/Y 

20  5  50000  2,841.02  0  4  4 
P1  P2  BAL (output)  PRN (output)  INT (output) 

9  12  21,501.18189  9,975.354914  1,388.725086 
In the third year, Baxter receives a total of $11,364.08 in payments, of which $9,975.35 is deducted from the principal and $1,388.73 represents the interest earned on the investment.