What Makes a Loan a Sinking-Fund Loan?
A borrower commonly repays a loan by making payments that reduce the amount owed over time. A sinking-fund arrangement separates that process into two accounts: the borrower pays interest on the loan, while also setting money aside in a fund intended to accumulate toward repayment of the principal at maturity. The fund is a separate asset; its deposits do not, by themselves, reduce the contractual loan balance.
This separation is important in calculations and in interpreting a payment schedule. A dollar paid to the lender as loan interest, a dollar deposited into the fund, and a dollar earned by the fund have different roles. At maturity, the borrower owes the loan principal under the loan contract. The fund can provide money toward that obligation, but its balance and the loan amount are not automatically the same.
For clarity, suppose there are \(n\) annual periods, the loan charges an effective annual rate \(i\), and the fund earns an effective annual rate \(j\). Assume the loan principal is not reduced before maturity, and ignore fees, taxes, and changes in rates. Under these assumptions, the loan interest for each year is \(Li\). A fund deposit is a separate cash flow, and interest earned by the fund belongs to the fund balance.
The rates \(i\) and \(j\) need not be equal. They apply to different balances: \(i\) applies to the outstanding loan principal, while \(j\) applies to money already in the fund. Confusing the rates or balances can lead to an incorrect interest amount or an incorrect maturity settlement.
The Two-Account View
The simplest way to read a sinking-fund arrangement is to keep a loan account and a fund account separately. The loan account records what the borrower owes the lender. The fund account records the borrower’s accumulated asset. A deposit moves cash into the fund; it does not change the amount recorded as loan principal unless the lender actually receives it as a principal repayment.
| Item | Where it belongs | Effect in the standard arrangement |
|---|---|---|
| Loan principal \(L\) | Loan account | Remains owed until repaid under the contract |
| Loan interest | Payment to lender | Compensates the lender for the loan; does not accumulate in the fund |
| Fund deposit | Fund account | Adds to the borrower’s separate asset; does not reduce loan principal |
| Fund earnings | Fund account | Increase the fund balance; do not pay loan interest automatically |
| Maturity repayment | Loan account settlement | Principal is due; fund assets may be used to provide the repayment cash |
A useful bookkeeping identity is that, at a given time, the borrower has a loan liability and may also have a fund asset. If both are valued in dollars at that time, subtracting the fund asset from the loan liability gives a simple net position. That net position is not the same as the contractual loan balance: unless the contract permits offset, the borrower still owes the lender the stated loan amount.
Why the Loan Balance Does Not Fall When the Fund Grows
The distinction follows directly from the contracts and cash flows. The loan contract determines the lender’s claim. The fund arrangement determines how a separate pool of assets changes. A fund deposit is made to the fund rather than to the lender, so it cannot be treated as a principal repayment merely because its purpose is to provide future repayment money.
These expressions apply when the principal remains unchanged for the full year and the annual rate is effective. If a borrower actually repays part of the principal to the lender, the loan balance can fall and the interest calculation may change. That is a different cash flow from making a deposit into a separate fund.
The fund balance changes according to fund activity. For example, if the fund balance at the start of a year is \(B\), the fund earns effective annual rate \(j\), and a deposit \(D\) is made at the end of the year, then the balance immediately after the deposit is \(B(1+j)+D\). This describes the fund asset only. It does not replace the loan-interest calculation \(Li\).
The timing convention matters: the displayed fund expression assumes that the existing balance earns interest for the full year and that the deposit is made at year-end, so that the deposit earns no interest during that year. A beginning-of-year deposit would have a different accumulation period. In either case, the loan balance is governed by the loan contract, not by the fund’s growth.
Worked Examples: Separate the Cash Flows
Worked Example: Identifying Each Part of a Two-Year Arrangement
A borrower receives a loan of \(L=\$12{,}000\) for two years. The loan charges an effective annual rate of \(i=5\%\), with interest paid at each year-end. The borrower also deposits \(\$5{,}000\) into a separate fund at the end of each year. The fund earns an effective annual rate of \(j=4\%\). Assume no fees, taxes, or other cash flows. Identify the annual loan interest, the fund balance at maturity, and the amount of principal not covered by the fund.
Method. Calculate interest from the loan principal and loan rate. Separately accumulate the fund deposits using the stated timing. Do not subtract the deposits from the loan principal.
Loan interest. The loan principal remains \(\$12{,}000\) during both years. Thus the interest due at each year-end is
The borrower pays \(\$600\) of loan interest at the end of year 1 and another \(\$600\) at the end of year 2.
Fund at the end of year 1. The first \(\$5{,}000\) deposit is made at year-end, so it has not earned interest by that date. The fund balance is \(\$5{,}000\).
Fund at the end of year 2. The existing \(\$5{,}000\) earns \(4\%\), giving \(\$5{,}000(1.04)=\$5{,}200\). Add the second year-end deposit:
At maturity, the contractual principal is \(\$12{,}000\). The fund supplies \(\$10{,}200\), leaving a principal shortfall of \(\$12{,}000-\$10{,}200=\$1{,}800\). The borrower has also paid \(\$1{,}200\) of loan interest over the two years. The check is that the fund has two deposits totaling \(\$10{,}000\) and \(\$200\) of earnings, for a total of \(\$10{,}200\).
Common mistake avoided: The \(\$5{,}000\) deposits are not principal payments to the lender. The loan remains \(\$12{,}000\) until maturity, even though the borrower has accumulated a fund asset.
Worked Example: A Deposit Does Not Reduce Next Year’s Interest
A company borrows \(\$18{,}000\) for three years at an effective annual loan rate of \(i=7\%\). It pays loan interest at each year-end and deposits \(\$4{,}000\) into a separate fund at each year-end. The fund earns \(j=5\%\) effective annually. Assume the loan principal is due at the end of year 3 and there are no other payments. Find the year-2 loan interest, the fund balance at the end of year 2, and the amount of loan principal still owed then.
Method. Keep the loan calculation and fund calculation in separate accounts. In particular, calculate the year-2 loan interest on the loan principal specified by the contract, not on the principal less the year-1 fund deposit.
Year-2 loan interest. No principal has been paid to the lender, so the loan principal is still \(\$18{,}000\). Therefore,
Fund balance at the end of year 2. The end-of-year-1 deposit of \(\$4{,}000\) earns one year of fund interest:
The fund balance immediately before the year-2 deposit is \(\$4{,}000+\$200=\$4{,}200\). After adding the second \(\$4{,}000\) deposit, the balance is \(\$8{,}200\).
Loan principal still owed. The loan principal remains \(\$18{,}000\). The year-2 deposit is in the fund and is not a payment to the lender.
The result is consistent with the account separation: the year-2 loan interest is based on \(\$18{,}000\), while the fund has a separate \(\$8{,}200\) asset. A tempting but incorrect calculation would use \(\$18{,}000-\$4{,}000=\$14{,}000\) as the year-2 loan balance and obtain \(\$14{,}000(0.07)=\$980\). That treats the first fund deposit as if it had been paid to the lender; the correct year-2 interest is \(\$1{,}260\).
Worked Example: Fund Surplus at Maturity
A borrower takes a two-year loan of \(\$10{,}000\) at an effective annual rate of \(i=4\%\). Loan interest is paid at each year-end, and the principal is due at the end of year 2. The borrower deposits \(\$4{,}800\) into a separate fund at the end of year 1 and \(\$5{,}400\) at the end of year 2. The fund earns \(j=5\%\) effective annually. Find the fund value at maturity, the loan principal repayment, and any fund surplus after repayment.
Method. Accumulate the first fund deposit for one year, then add the second deposit made at maturity. Separately identify the principal due under the loan contract.
The first deposit accumulates to \(\$4{,}800(1.05)=\$5{,}040\) by the end of year 2. The second deposit is made at that date, so it adds \(\$5{,}400\) without earning interest during the term:
The loan principal due is still \(\$10{,}000\). If the fund is used to repay it, the remaining fund surplus is \(\$10{,}440-\$10{,}000=\$440\). The loan interest paid is \(\$10{,}000(0.04)=\$400\) at each year-end, for \(\$800\) total over the two years. The fund surplus does not erase those interest payments; it is what remains after the principal repayment.
As a check, the fund’s \(\$10{,}440\) consists of total deposits of \(\$10{,}200\) plus \(\$240\) of earnings on the first deposit. The principal settlement uses \(\$10{,}000\), leaving exactly \(\$440\).
Reading the Maturity Settlement Correctly
At maturity, compare the fund available for repayment with the principal due under the loan contract. If the two amounts are equal, the fund exactly covers the principal. If the fund is smaller, the difference is a shortfall that must be met from another source. If the fund is larger, the difference is a surplus after principal is repaid. These comparisons concern principal funding; they do not change the interest already due under the loan terms.
This comparison also has a limitation. It describes a standard arrangement in which the loan and fund are separate and the fund is available to the borrower at maturity. A contract could impose other terms, and fees, taxes, withdrawals, or a fund shortfall could alter the cash required. Do not assume that a fund is legally available to offset the lender’s claim unless the problem states that it is.
Identify the principal, loan rate, interest-payment timing, and date principal is due.
Use the loan balance and loan rate. In an interest-only sinking-fund arrangement with unchanged principal \(L\), each full-period interest payment is \(Li\).
Apply the fund’s own rate to the fund balance and record deposits according to their timing.
Compare the fund available with the contractual principal due, and state any surplus or shortfall.
Common Exam Traps
- Subtracting deposits from loan principal. A deposit reduces loan principal only if it is paid to the lender and recognized as principal repayment.
- Applying the loan rate to the fund. The loan rate applies to the loan balance; the fund rate applies to the fund balance. Use each rate only on its corresponding account.
- Counting fund earnings as a loan payment. Fund earnings increase the fund asset. They do not automatically pay interest owed to the lender.
- Forgetting deposit timing. A deposit made at year-end does not earn interest during the year that has just ended. A different timing assumption changes fund accumulation.
- Assuming the fund exactly matches the principal. The fund may fall short or exceed the amount due. Compare the two values rather than presuming they are equal.
Check Your Understanding
For each problem, keep the contractual loan balance separate from the fund balance.
- A two-year loan of \(\$9{,}000\) charges \(6\%\) effective annual interest, payable annually, with no principal repayment before maturity. What is the loan interest due at the end of each year? (Answer: \(\$540\) each year.)
- A three-year loan of \(\$20{,}000\) charges \(5\%\) effective annual interest. The borrower deposits \(\$2{,}500\) into a separate fund at each year-end. What is the annual loan interest, and do those deposits reduce the contractual principal? (Answer: \(\$1{,}000\) each year; no, the principal remains \(\$20{,}000\) until repaid to the lender.)
- A two-year loan of \(\$14{,}000\) charges \(4\%\) effective annual interest. A fund earns \(3\%\), and the borrower deposits \(\$3{,}000\) at the end of each year. What is the fund balance at the end of year 2, the year-2 loan interest, and the principal shortfall if the loan matures then? (Answer: fund balance \(\$6{,}090\); year-2 loan interest \(\$560\); principal shortfall \(\$7{,}910\).)
- A two-year loan of \(\$11{,}000\) charges \(8\%\) effective annual interest. The fund earns \(4\%\), and \(\$5{,}000\) is deposited at each year-end. Find the fund at maturity, the principal shortfall, and the loan interest paid over the term. (Answer: fund \(\$10{,}200\); shortfall \(\$800\); total loan interest \(\$1{,}760\).)
- At maturity, a borrower owes \(\$16{,}000\) of loan principal and has \(\$16{,}640\) in the separate fund. The loan rate was \(6\%\) effective annually and principal was unchanged during the final year. What principal amount is due, what fund surplus remains after repayment, and what was the final year’s loan interest? (Answer: principal due \(\$16{,}000\); fund surplus \(\$640\); final-year loan interest \(\$960\).)