Cumulative Principal Repaid

What Cumulative Principal Repaid Measures

A loan payment contains both interest and principal. Consequently, adding the payments made so far does not tell you how much of the original debt has been repaid: part of each payment compensates the lender for interest. To measure principal repaid through a particular payment, compare the original loan amount with the loan balance remaining immediately after that payment.

This calculation relies on a consistent definition of the balance. Let $L$ be the original principal, $i$ the effective interest rate per payment period, $R$ the level payment made at the end of each period, and $B_k$ the scheduled balance immediately after payment $k$. Let $B_0=L$. The rate $i$ must match the payment period: for monthly payments, use an effective monthly rate.

Definition: The cumulative principal repaid through payment $k$ is the original principal minus the scheduled balance immediately after payment $k$: $$ C_k=L-B_k. $$ This definition applies to the scheduled loan balance, under the stated loan terms. It assumes the payments and balance are measured on the same schedule and that no unscheduled extra payments, fees, or balance adjustments are being included.

The phrase through payment $k$ matters. The balance immediately before payment $k$ still includes the amount due at that payment. The balance immediately after payment $k$ does not. Therefore, $L-B_k$ counts principal repaid by the first $k$ payments, not by the first $k-1$.

Why Subtracting the Balance Gives Principal Repaid

Suppose each payment is made at the end of its period, the interest rate per period is fixed at $i$, and the payment amount is $R$. During period $k$, the balance $B_{k-1}$ earns interest $iB_{k-1}$. The payment first covers that interest; the remainder reduces principal. Thus the principal portion of payment $k$ is $R-iB_{k-1}$, and the balance recurrence is $B_k=B_{k-1}(1+i)-R$.

Rearranging the recurrence shows that the reduction in balance during period $k$ equals that payment's principal portion. Adding those reductions from payment 1 through payment $k$ makes the intermediate balances cancel:

\begin{aligned} B_{j-1}-B_j &= R-iB_{j-1},\\ \sum_{j=1}^{k}(B_{j-1}-B_j) &= (B_0-B_1)+(B_1-B_2)+\cdots+(B_{k-1}-B_k)\\ &= B_0-B_k\\ &= L-B_k. \end{aligned}

The left side is the sum of the principal portions of the first $k$ payments; the right side is the original principal less the remaining balance. This telescoping argument explains why the subtraction method works even when the individual principal portions vary from payment to payment.

Result: Under a fixed-rate, level-payment schedule with payments at period-end, the scheduled balance after $k$ payments can be found either from the recurrence $B_k=B_{k-1}(1+i)-R$ or as the value at time $k$ of the remaining payments. If the loan has $n$ payments in total, then $$ B_k=R\,a_{\overline{n-k}|} \qquad (0\leq k\leq n), $$ where $a_{\overline{m}|}$ is valued using the effective rate $i$ per payment period. Hence $C_k=L-R\,a_{\overline{n-k}|}$. The remaining-payment formula assumes the scheduled payments are level and the loan is valued at the same rate used in its schedule.

Worked Examples

Worked Example: Cumulative Principal After the First Payment

A hypothetical loan is repaid by two equal annual payments of $600, made at the end of each year. The effective annual interest rate is 20%. Find the cumulative principal repaid immediately after the first payment. Assume no fees or extra payments.

Method. First find the original principal as the present value of the two scheduled payments. Then find the balance after payment 1 and subtract it from the original principal. This avoids treating the entire first payment as principal.

The original principal is the value at time 0 of the two payments:

\begin{aligned} L &=600a_{\overline{2}|}\\ &=600\left(\frac{1}{1.20}+\frac{1}{1.20^2}\right)\\ &=600\left(\frac{5}{6}+\frac{25}{36}\right)\\ &=600\left(\frac{55}{36}\right)\\ &=\frac{2750}{3} =916.666\ldots \end{aligned}

The interest during the first year is $0.20(2750/3)=550/3=183.333\ldots$ dollars. Therefore, the principal portion of the first $600 payment is $600-550/3=1250/3=416.666\ldots$ dollars. Equivalently, the balance after the payment is:

B_1=L(1.20)-600 =\frac{2750}{3}(1.20)-600 =1100-600 =500.

Thus cumulative principal repaid is $C_1=L-B_1=2750/3-500=1250/3$. To the nearest cent, $416.67 of principal has been repaid.

Check. The first payment splits into $183.33 of interest and $416.67 of principal, totaling $600. The post-payment balance is $500; at 20%, it grows to $600 by the next payment, so the second payment clears the loan.

Worked Example: Using the Balance of the Remaining Payments

A hypothetical loan is repaid by three annual end-of-year payments of $110 at an effective annual interest rate of 10%. Find the cumulative principal repaid immediately after payment 2. Assume the schedule uses exact, unrounded payments and has no fees.

Method. After the second payment, exactly one scheduled payment remains. Its value one year before it is due is the balance at that time. Compare that balance with the original principal.

The original principal is the present value of all three payments. The annuity factor is:

\begin{aligned} a_{\overline{3}|} &=\frac{1}{1.10}+\frac{1}{1.10^2}+\frac{1}{1.10^3}\\ &=0.9090909+0.8264463+0.7513148\\ &=2.4868520. \end{aligned}

Therefore, $L=110(2.4868520)=273.55372$ dollars, using the displayed rounded factor for presentation. The scheduled balance after payment 2 is the present value of the one remaining $110 payment:

B_2=110a_{\overline{1}|} =\frac{110}{1.10} =100.

Subtracting gives $C_2=273.55372-100=173.55372$. To the nearest cent, cumulative principal repaid is $173.55.

Check. The recurrence gives $B_1=273.55372(1.10)-110$, approximately $190.91, and then $B_2\approx190.91(1.10)-110=$100.00$. Small differences in the intermediate check are from displaying the original principal to five decimal places; the unrounded annuity value gives a balance of exactly $100.

Worked Example: Monthly Payments and a Monthly Rate

A hypothetical loan is repaid by six equal monthly payments of $900 at the end of each month. The nominal annual interest rate is 12% convertible monthly, so the effective rate per payment period is $i=0.12/12=0.01$, or 1% per month. Find the cumulative principal repaid immediately after payment 3. Assume no fees, no extra payments, and exact scheduled payments.

Method. Value the six payments at time 0 to obtain the original principal. Then value the three remaining payments immediately after payment 3 to obtain $B_3$. The difference is cumulative principal repaid.

For the original principal, the six-payment annuity factor is:

\begin{aligned} a_{\overline{6}|} &=\frac{1-(1.01)^{-6}}{0.01}\\ &\approx 5.795476. \end{aligned}

Thus $L=900(5.795476)\approx$5,215.93$. The remaining three payments have a value, immediately after payment 3, of:

\begin{aligned} B_3 &=900a_{\overline{3}|}\\ &=900\left(\frac{1-(1.01)^{-3}}{0.01}\right)\\ &\approx900(2.940985)\\ &\approx \$2,646.89. \end{aligned}

Therefore, $C_3=L-B_3\approx5,215.93-2,646.89=$ $2,569.04, rounding only the final dollar result to the nearest cent.

Check. The original principal less the balance after payment 3 is approximately $2,569.04. This is less than the $2,700 total of the first three payments because those payments also cover interest. Using the annual 12% rate directly as a monthly rate would not match the stated monthly conversion.

Reading a Cumulative Principal Result

A cumulative principal amount is a balance reduction, not a payment total. If a loan began at $10,000 and the scheduled balance after payment 5 is $7,200, then $10,000-$7,200=$2,800 of principal has been repaid through payment 5. The five payments may total much more than $2,800 because they include interest.

The result also depends on the valuation instant. If a question gives a balance immediately before payment $k$, that amount includes the payment due at time $k$. To use $C_k=L-B_k$, first obtain the balance immediately after payment $k$, or carefully identify whether the payment itself is included in the requested cumulative amount. For an end-of-period payment, the balance immediately before it is $B_{k-1}(1+i)$; subtracting the payment $R$ gives the post-payment balance $B_k$.

The remaining-payment formula is especially useful when $k$ is near the end of the loan. It may require valuing only a few payments rather than building the entire schedule. The recurrence is often more convenient for a small $k$, and a period-by-period schedule is useful when the problem asks for individual interest and principal portions.

Common Mistake: Counting Payments as Principal

A frequent error is to calculate cumulative principal repaid as $kR$, the total of the first $k$ payments. That incorrectly assumes every dollar paid reduces the loan balance. For the first example, that wrong calculation would report $600 of principal after payment 1. But the first year's interest is $183.33, so only $416.67 of the payment is principal. The balance check confirms the correction: $916.67-500=$416.67$, not $600.

A second trap is mixing rate periods. If payments are monthly, the balance recurrence must use the effective monthly interest rate, not the annual nominal rate as though it applied each month. Convert the quoted rate according to its stated compounding convention before valuing payments or rolling the balance forward.

Key takeaway: To find principal repaid through payment $k$, use the original principal and the balance immediately after payment $k$: $C_k=L-B_k$. Confirm the balance is at the correct time and that the interest rate is expressed per payment period.

Check Your Understanding

For each problem, use the scheduled balance immediately after the specified payment. Round dollar answers to the nearest cent.

A loan has original principal $5,000 and a scheduled payment of $1,000 at each year-end. The effective annual rate is 4%. Find the cumulative principal repaid immediately after the third payment. (Answer: $2,497.28)
A loan has principal $1,200, a scheduled annual payment of $400, and an effective annual rate of 8%. Find the cumulative principal repaid through payment 2. (Answer: $632.32)
A loan has principal $3,000 and scheduled quarterly payments of $550. The effective rate is 2% per quarter. Find cumulative principal repaid after the fourth payment. Assume the stated schedule continues for at least four payments. (Answer: $2,019.59)
A loan has principal $10,000 and scheduled semiannual payments of $2,800. The nominal annual rate is 8% convertible semiannually. Find cumulative principal repaid after the second payment. (Answer: $4,896.00)
A loan has principal $5,000, a scheduled annual payment of $700, and an effective annual rate of 3%. Find cumulative principal repaid immediately after payment 5. Assume the schedule continues through that payment. (Answer: $2,920.02)

What You'll Learn