Interest and principal repayment in a payment

Exam FM – Financial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the Breakdown: Interest vs. Principal

Hi there! If you’ve ever looked at a car loan or a mortgage statement, you might have noticed that your monthly payment stays the same, but the amount that actually "pays down" the debt changes every month. In this chapter, we are going to pull back the curtain and see exactly how each payment is split between the bank’s profit (Interest) and your debt reduction (Principal).

Don't worry if this seems tricky at first—once you see the pattern, it’s as predictable as a heartbeat! By the end of these notes, you'll be able to calculate exactly how much of the 42nd payment of a loan goes toward the balance versus the interest. Let’s dive in!

1. The Golden Rule of Loan Payments

Every time you make a level payment (let's call it $ R $) on a loan, that payment is doing two jobs at once:

1. It pays the Interest ($ I_t $) that has piled up since your last payment.
2. It pays a chunk of the Principal ($ P_t $) to reduce the total amount you owe.

This gives us our most important starting formula:
$ R = I_t + P_t $

Think of it like this: Imagine you owe a friend $100 and you’ve agreed to pay 5% interest. If you give them $15, they first take the $5 you owe them for interest. The remaining $10 is used to lower your debt to $90. In this case, $ R = 15 $, $ I = 5 $, and $ P = 10 $.

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Quick Review: Key Variables

$ R $: The total periodic payment (the installment).
$ I_t $: The interest portion of the $ t $-th payment.
$ P_t $: The principal portion (amount of capital repaid) of the $ t $-th payment.
$ OB_t $: The Outstanding Balance immediately after the $ t $-th payment.
$ i $: The effective interest rate per period.

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2. Calculating the Interest Portion ($ I_t $)

The bank only charges you interest on what you currently owe. Therefore, the interest part of your payment is always based on the balance you had before you made that payment.

The Formula:
\n$ I_t = i \times OB_{t-1} $

Step-by-Step Logic:
\n1. Look at the balance from the previous period ($ OB_{t-1} $).
\n2. Multiply it by the interest rate ($ i $).
\n3. That result is the interest portion ($ I_t $) of your current payment.

Summary Takeaway: As the loan balance ($ OB $) goes down over time, the interest portion ($ I_t $) of each payment will also go down. This is why, at the start of a long mortgage, most of your money goes to interest!

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3. Calculating the Principal Portion ($ P_t $)

If you know your total payment ($ R $) and you’ve calculated the interest ($ I_t $), finding the principal is simple subtraction.

The Formula:
\n$ P_t = R - I_t $

Substituting the interest formula from above, we get:
\n$ P_t = R - (i \times OB_{t-1}) $

Summary Takeaway: Since the interest portion decreases with every payment, the principal portion must increase to keep the total payment ($ R $) the same. You are paying off the loan faster and faster as time goes on!

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4. The Geometric Growth of Principal

Here is a secret "shortcut" that is very popular on Exam FM. The principal portions of consecutive payments actually follow a geometric progression.

The Relationship:
\n$ P_{t+1} = P_t(1+i) $

This means if you know the first principal payment ($ P_1 $), you can find any other principal payment ($ P_t $) using this formula:
\n$ P_t = P_1(1+i)^{t-1} $

Memory Aid: The principal "grows" at the interest rate. Just like money in a savings account grows by $ (1+i) $, the amount of debt you pay off each month also grows by $ (1+i) $.

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Did you know?

This geometric relationship only works for level payments. If the payment amount $ R $ changes, the pattern changes too!

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5. Direct Formulas (Using $ n $ and $ t $)

Sometimes the exam won't give you the balance; they’ll just give you the total number of payments ($ n $) and the interest rate ($ i $). In these cases, you can use these "direct" formulas for a loan of 1 (where $ R = 1 $):

$ P_t = v^{n-t+1} $
$ I_t = 1 - v^{n-t+1} $

Wait, what if the payment is $ R $ instead of 1? Just multiply the result by $ R $!
\n$ P_t = R \cdot v^{n-t+1} $
\n$ I_t = R(1 - v^{n-t+1}) \approx R \cdot a_{\overline{n-t+1}|} \cdot i $

Common Mistake: Be careful with the exponent! It is $ n-t+1 $. If you are looking for the 5th payment of a 10-year loan, the exponent is $ 10 - 5 + 1 = 6 $.

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6. Summary Table for Amortization

If you were to look at an "Amortization Schedule," here is how the values behave over time:

\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

Component	Over Time...	Reason
Payment ($ R $)	Constant	Fixed by contract.
Interest ($ I_t $)	Decreases	Balance is shrinking.
Principal ($ P_t $)	Increases	More of the level payment is "left over."
Outstanding Balance ($ OB_t $)	Decreases	Principal is being paid off.

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7. Example Walkthrough

Scenario: A loan of $10,000 is being repaid with annual payments over 5 years at an effective annual interest rate of 6%. Calculate the principal and interest portions of the 3rd payment.

Step 1: Calculate the level payment $ R $.
$ 10,000 = R \cdot a_{\overline{5}|0.06} $
Using your calculator: $ R = 10,000 / 4.21236 = 2,373.96 $.

Step 2: Calculate the Principal portion $ P_3 $.
Using our direct formula: $ P_t = R \cdot v^{n-t+1} $
$ P_3 = 2,373.96 \cdot (1.06)^{-(5-3+1)} = 2,373.96 \cdot (1.06)^{-3} $
$ P_3 = 2,373.96 \cdot 0.83962 = 1,993.22 $.

Step 3: Calculate the Interest portion $ I_3 $.
$ I_3 = R - P_3 $
$ I_3 = 2,373.96 - 1,993.22 = 380.74 $.

Verification: You could also find $ OB_2 $ and multiply by 0.06. Try it! You’ll get the same answer.

8. Pro-Tips for the Exam

1. The $ P_t $ Ratio: If you are given two principal repayments, say $ P_5 = 100 $ and $ P_{10} = 150 $, you can find the interest rate quickly: $ 150 = 100(1+i)^5 $. Solve for $ i $!
2. The "Sum of Principals": The sum of all $ P_t $ from $ t=1 $ to $ n $ must equal the original loan amount ($ L $). If it doesn't, you've made a calculation error.
3. Round Carefully: In the real world, we round to cents. On the exam, keep as many decimals as possible until the very end to avoid "rounding drift."

Key Takeaway Summary

Interest is what you pay for the privilege of borrowing (based on previous balance).
Principal is what you pay to actually get rid of the debt.
As time goes on, the Interest portion falls and the Principal portion rises geometrically at rate $ i $.
Total Payment = Interest + Principal.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the basic relationship between the total payment, interest portion, and principal portion in a loan?

The total payment $ R $ is the sum of the interest portion $ I_t $ and the principal portion $ P_t $: $ R = I_t + P_t $.

How is the interest portion of a specific payment ($ I_t $) calculated?

$ I_t = i \times OB_{t-1} $, where $ i $ is the interest rate and $ OB_{t-1} $ is the balance from the previous period.

How is the principal portion ($ P_t $) of a payment determined from the total payment and interest?

The principal portion is the remainder of the payment after interest is paid: $ P_t = R - I_t $.

What is the relationship between consecutive principal payments ($ P_t $ and $ P_{t+1} $)?

Principal payments grow geometrically at the interest rate: $ P_{t+1} = P_t(1+i) $.

How can you find any principal portion $ P_t $ using only the first principal portion $ P_1 $?

$ P_t = P_1(1+i)^{t-1} $.

What is the direct formula for the principal portion $ P_t $ given total periods $ n $?

$ P_t = R \cdot v^{n-t+1} $, where $ v = (1+i)^{-1} $.

What is the direct formula for the interest portion $ I_t $ in terms of the discount factor $ v $?

$ I_t = R(1 - v^{n-t+1}) $.

In a standard amortization schedule, how do Interest, Principal, and Balance behave over time?

The payment remains constant, Interest decreases, Principal increases, and the Balance decreases.

What is the relationship between the sum of all principal portions and the original loan amount?

The sum of all principal portions ($ P_1 $ through $ P_n $) must equal the original loan amount $ L $.

How can you find the interest rate $ i $ if you are given two principal repayments, $ P_k $ and $ P_m $?

Use the ratio: $ P_m = P_k(1+i)^{m-k} $, where $ m > k $.

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Interest and principal repayment in a payment

Welcome to the Breakdown: Interest vs. Principal

1. The Golden Rule of Loan Payments

Quick Review: Key Variables

2. Calculating the Interest Portion (\( I_t \))

3. Calculating the Principal Portion (\( P_t \))

4. The Geometric Growth of Principal

Did you know?

5. Direct Formulas (Using \( n \) and \( t \))

6. Summary Table for Amortization

7. Example Walkthrough

8. Pro-Tips for the Exam

Key Takeaway Summary

Can you answer these now?

Ready to test yourself?

More Exam FM – Financial Mathematics chapters

Put These Notes into Practice