The Deposit Is Not a Loan Principal Repayment

Why the Deposit Is Not a Principal Repayment

A borrower may make two payments during each period of a sinking-fund loan: interest to the lender and a deposit into a separate fund. Although both payments affect the borrower’s finances, they do not have the same effect on the loan. The interest payment satisfies the loan’s interest obligation; the deposit builds an asset that may later be used to repay principal. Unless the lender receives a principal payment, the contractual loan balance does not fall.

The distinction matters whenever someone is asked for the outstanding loan balance, the interest due next period, or the amount still needed to redeem the loan. A fund deposit is not automatically a payment to the lender, and the fund’s current value is not a substitute for the contractual balance.

Definition: The contractual loan balance is the amount of principal still owed under the loan agreement. A sinking fund is a separate account into which deposits accumulate to help provide money for repayment at a specified future date. A fund deposit reduces the contractual loan balance only if it is actually paid to the lender and applied to principal.

Two Separate Ledgers

To keep the accounting clear, track the loan and fund separately. In the model used here, the loan has an initial principal $P$, a constant effective annual loan rate $i_L$, and annual periods. Interest is paid at each year-end. The borrower deposits $R$ into a separate fund at each year-end, and that fund earns a constant effective annual rate $i_F$. Assume there are no fees, taxes, missed payments, withdrawals, or transfers between the loan and the fund before maturity.

Let $B_k$ be the contractual loan balance just after the payment at time $k$, and $F_k$ the sinking-fund value at that same time. If each year’s loan payment covers only the interest due, the loan ledger follows one rule; the fund ledger follows another:

B_k=B_{k-1}(1+i_L)-i_LP,\qquad F_k=F_{k-1}(1+i_F)+R.

The first expression assumes the payment to the lender is $i_LP$, the annual interest on the original principal. The second assumes a deposit $R$ is made at the end of the year, after the existing fund balance has earned interest. Neither expression transfers the fund deposit to the lender.

Key result: If the borrower pays only the interest due on a loan of principal $P$, then $B_k=P$ for every period before redemption. Meanwhile, the fund value changes according to its own interest rate and deposit schedule. The difference $P-F_k$ can measure how much of the principal is not yet covered by the fund, but it is not the contractual loan balance.

Why the Contractual Balance Stays at Principal

The balance recurrence gives a direct justification. Suppose the balance at the start of a year is $P$. It accrues interest of $i_LP$, so the amount due before payment is $P+i_LP$. If the borrower pays $i_LP$ to the lender, the remaining balance is $P$. Repeating the same calculation in the next year leaves the balance unchanged again.

$$ B_k=P(1+i_L)-i_LP=P. $$

By contrast, the fund earns interest on its own balance and receives the deposit. Starting from $F_0$, the recurrence gives $F_n=F_0(1+i_F)^n+R\,s_{\overline{n}|}$, where $s_{\overline{n}|}$ is the accumulated value of an annuity-immediate of unit annual deposits at rate $i_F$. This expression describes the fund asset; it does not describe the loan balance.

The difference follows from the destination of the payments. Interest goes to the lender to satisfy the interest obligation. A sinking-fund deposit goes to the fund. Only a payment applied to principal changes the contractual balance.

Update the loan ledger.
Accrue loan interest, then subtract only the amount paid to the lender and applied to the loan.

Update the fund ledger.
Accrue fund interest, then add the deposit made into the fund.

Compare at the repayment date.
Compare the fund value with the contractual principal to find the amount still required or the surplus available.

Worked Examples

Worked Example: A Fund That Falls Short at Maturity

A borrower takes a $10,000 loan for three years at an effective annual loan rate of 8%. The borrower pays annual interest to the lender and deposits $3,000 into a separate sinking fund at each year-end. The fund earns an effective annual rate of 5%. Find the loan balance after two years, the fund value at maturity, and the additional amount needed to repay principal at maturity.

Method. Use separate recurrences for the loan and fund. The common mistake to avoid is subtracting the deposits from the loan principal.

Loan ledger. Annual interest is $0.08(10{,}000)=\$800$. Each $800 payment covers the interest due, so the contractual balance remains $10,000 after year 1 and after year 2. In particular, after two deposits of $3,000, the balance is still $10,000, not $4,000.

Fund ledger. After year 1, $F_1=\$3{,}000$. After year 2, $F_2=3{,}000(1.05)+3{,}000=\$6{,}150$. After year 3, $F_3=6{,}150(1.05)+3{,}000=\$9{,}457.50$.

The accumulated-value check gives the same result: $s_{\overline{3}|}=(1.05^3-1)/0.05=3.1525$, so $3{,}000(3.1525)=\$9{,}457.50$.

Maturity comparison. The principal due is $10,000, so the additional amount needed from outside the fund is $\$10{,}000-\$9{,}457.50=\$542.50$. The check is consistent: the fund is $542.50 short of the unchanged contractual principal.

Worked Example: An Existing Fund and Equal Annual Deposits

A $20,000 loan has a four-year term and an effective annual loan rate of 6%. Interest is paid annually, while an existing $2,000 fund balance grows at an effective annual rate of 10%. The borrower also deposits $5,000 at each year-end. Find the contractual loan balance after three years and the fund surplus or shortfall at maturity.

Method. Carry the initial fund balance forward separately from the annual deposits. The trap is to treat deposits made toward the future redemption as principal payments already received by the lender.

Loan ledger. Annual interest is $0.06(20{,}000)=\$1{,}200$. Since only interest is paid, the balance remains $20,000 after each year, including year 3.

Fund ledger. Start with $F_0=\$2{,}000$. The year-end recurrence gives:

After year 1: $F_1=2{,}000(1.10)+5{,}000=\$7{,}200$.
After year 2: $F_2=7{,}200(1.10)+5{,}000=\$12{,}920$.
After year 3: $F_3=12{,}920(1.10)+5{,}000=\$19{,}212$.
After year 4: $F_4=19{,}212(1.10)+5{,}000=\$26{,}133.20$.

The fund has accumulated for four years, including the initial balance. As a check, $2{,}000(1.10^4)=\$2{,}928.20$, and the four deposits accumulate to $5{,}000s_{\overline{4}|}$, where $s_{\overline{4}|}=(1.10^4-1)/0.10=4.641$. Thus the deposits accumulate to $23,205, and the total is $\$2{,}928.20+\$23{,}205=\$26{,}133.20$.

At maturity the fund exceeds the $20,000 principal by $\$26{,}133.20-\$20{,}000=\$6{,}133.20$. This is a fund surplus, not a reduction in the contractual balance during the four-year term. The balance remains $20,000 until the principal is actually redeemed.

Worked Example: The Wrong Balance Produces the Wrong Interest

A borrower owes $12,000 for three years at an effective annual loan rate of 5%. The borrower pays annual interest to the lender and deposits $3,600 at each year-end into a fund earning an effective annual rate of 4%. Find the balance after two years, the correct interest due in year 3, and the amount still needed to redeem principal after the third deposit.

Method. First calculate the contractual loan balance, then calculate the fund independently. The common mistake is to subtract the accumulated deposits from the loan and use that smaller number to calculate loan interest.

Loan ledger. The annual interest payment is $0.05(12{,}000)=\$600$. As these payments cover interest only, the contractual balance is $12,000 after year 1 and after year 2. The correct year-3 interest due is therefore also $600.

Fund ledger. After year 1, $F_1=\$3{,}600$. After year 2, $F_2=3{,}600(1.04)+3{,}600=\$7{,}344$. After year 3, $F_3=7{,}344(1.04)+3{,}600=\$11{,}237.76$.

The fund check is $3{,}600s_{\overline{3}|}$, with $s_{\overline{3}|}=(1.04^3-1)/0.04=3.1216$. Hence $3{,}600(3.1216)=\$11{,}237.76$. The amount still needed for principal redemption is $\$12{,}000-\$11{,}237.76=\$762.24$.

A mistaken calculation would subtract the two $3,600 deposits from principal and report a $4,800 balance after year 2, then calculate year-3 interest as $0.05(4{,}800)=\$240$. Both results are wrong: deposits were made to the fund, not to the lender. The correct balance is $12,000, the correct interest is $600, and the fund is $762.24 short of principal at maturity.

Common Mistakes and Practical Checks

A fund may be intended to make a loan’s eventual redemption possible, but intention does not change the legal or mathematical treatment of payments. Keep the ledgers separate until an actual payment is made to the lender. At the repayment date, the fund may be withdrawn and used to meet the principal obligation; before that date, it remains a separate asset.

Subtracting deposits from principal. Do not reduce the contractual balance unless a payment is applied to principal by the lender.
Using the fund rate for loan interest. The loan’s interest is calculated using the loan rate and contractual balance. The fund rate applies only to the fund asset.
Calling the difference the loan balance. The difference between principal and fund value is a funding shortfall or net exposure measure, not the contractual loan balance.
Forgetting the payment date. Under the model here, each deposit is made at year-end. A deposit earns fund interest only in later periods, not in the period ending on its deposit date.

A useful check is to ask where each payment goes. If the payment goes to the lender as interest, it satisfies interest due but leaves principal unchanged in an interest-only arrangement. If it goes into a separate fund, it changes the fund value but not the lender’s claim. Only at redemption do the fund proceeds become available to pay the loan.

This separation of asset and liability is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In a problem, label each value by what it represents: contractual balance, interest payment, fund deposit, accumulated fund value, or maturity shortfall. Clear labels prevent an accumulated asset from being mistaken for a principal repayment.

Check Your Understanding

For each problem, keep the loan balance and fund value separate. Assume annual periods, end-of-year deposits, and no fees or transfers before maturity.

A $8,000 interest-only loan has an effective annual loan rate of 6% and a two-year term. The borrower deposits $2,500 at each year-end into a fund earning 4%. What is the contractual balance after year 1, the fund value at maturity, and the maturity shortfall? (Answer: Balance after year 1 = $8,000; fund = $5,100; shortfall = $2,900.)
A $11,000 interest-only loan earns interest at 7% annually. An existing fund of $1,500 and three year-end deposits of $3,000 accumulate at 5%. Find the maturity fund value and surplus or shortfall. (Answer: Fund = $11,193.94; surplus = $193.94; contractual balance remains $11,000.)
A $15,000 interest-only loan has an effective annual loan rate of 9% and a four-year term. The borrower deposits $3,500 at each year-end into a fund earning 6%. Find the annual loan interest and the maturity fund surplus or shortfall. (Answer: Annual interest = $1,350; fund = $15,311.16; surplus = $311.16; contractual balance remains $15,000.)
A $25,000 interest-only loan has an effective annual loan rate of 8% and a four-year term. A separate fund starts at $4,000 and receives $4,500 at each year-end; it earns 7%. Find the year-4 contractual balance, the year-4 fund value, and the amount left after the fund is used to redeem principal. (Answer: Contractual balance = $25,000; fund = $25,222.93; surplus after redemption = $222.93.)

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