Why a Sinking-Fund Schedule Needs Two Balances
A sinking-fund loan has two distinct financial tracks. The borrower pays interest on the loan, while separate deposits accumulate in a fund intended to repay the loan principal at maturity. A schedule makes both tracks visible and helps prevent a common error: treating the fund deposit as if it were principal paid directly to the lender.
The schedule methods here are aligned with the SOA Exam FM (Financial Mathematics) syllabus. Unless stated otherwise, assume a level original loan principal \(P\), no principal repayment to the lender before maturity, equal periods, and a fund earning a constant effective rate per period. Ignore fees, taxes, and default risk. Loan interest is paid at each period-end; deposits are made at each period-end unless the example states otherwise.
Let \(i_L\) be the effective loan rate per period and \(i_F\) the effective fund rate per period. If the loan principal remains \(P\), loan interest each period is \(Pi_L\). If the fund’s beginning balance in period \(k\) is \(F_{k-1}\), and the deposit \(D\) is made at the end of the period, the fund earns interest only on the beginning balance during that period.
The schedule has two different interest calculations. Fund interest depends on the changing fund balance. Loan interest stays level in this model because the loan principal stays at \(P\). The schedule does not apply when principal is being repaid to the lender during the term; in that case, the loan interest must be calculated on the changing loan balance.
Why the Fund Recurrence Accumulates the Deposits
The recurrence gives a direct way to check the schedule’s final fund balance. With no initial fund balance, \(F_0=0\). After the first end-of-period deposit, \(F_1=D\). After the second, the first deposit has earned one period of interest, so \(F_2=D(1+i_F)+D\). Repeating this reasoning gives the general result:
This is an accumulation annuity-immediate: each deposit is made at a period-end and accumulates only in later periods. If the target fund balance is \(P\), the required level deposit is therefore \(D=P/s_{\overline{n}|i_F}\). That target condition assumes the fund is intended to repay exactly \(P\); a different contractual redemption amount would require a different target.
Identify the number of periods and whether each deposit occurs at the beginning or end of a period.
Use effective rates per period. If a rate is nominal, convert it using its stated compounding frequency.
Use the appropriate accumulation factor, then calculate each period’s fund interest and ending balance.
Calculate loan interest from the loan balance, not the fund balance, and add the deposit to find total outlay.
Confirm that the final fund balance equals the intended redemption amount, subject to any stated rounding convention.
Worked Schedules
Worked Example: Annual End-of-Year Deposits
A borrower receives a loan of $33,100. The loan requires interest-only payments for three years, with principal due at the end of year 3. The effective annual loan rate is 8%, and the fund earns an effective annual rate of 10%. Equal deposits are made at each year-end. Find the deposit and construct the schedule.
Method. First match the deposits’ accumulation to the $33,100 maturity principal. For three end-of-year deposits, the accumulation factor is \(s_{\overline{3}|0.10}=1+1.10+1.10^2=1+1.10+1.21=3.31\). Thus \(D=33{,}100/3.31=10{,}000\).
Loan interest each year is \(33{,}100(0.08)=2{,}648\). In year 1 the beginning fund balance is zero, so fund interest is zero. In year 2 it is \(10{,}000(0.10)=1{,}000\). In year 3 it is \(21{,}000(0.10)=2{,}100\). Each year’s total borrower outlay is \(2{,}648+10{,}000=12{,}648\).
| Year | Beginning fund | Deposit | Fund interest | Ending fund | Loan interest | Total outlay |
|---|---|---|---|---|---|---|
| 1 | $0 | $10,000 | $0 | $10,000 | $2,648 | $12,648 |
| 2 | $10,000 | $10,000 | $1,000 | $21,000 | $2,648 | $12,648 |
| 3 | $21,000 | $10,000 | $2,100 | $33,100 | $2,648 | $12,648 |
Check. The last fund balance is \(21{,}000+10{,}000+2{,}100=33{,}100\), exactly the principal due. The borrower’s three periodic outlays total \(3(12{,}648)=37{,}944\), consisting of $30,000 in deposits and $7,944 in loan interest. At maturity, the fund provides the $33,100 principal repayment. The common mistake avoided here is adding fund interest to the loan-interest column: the $2,100 is earned by the fund, not charged by the lender.
Worked Example: Semiannual Periods and Nominal Rates
A $4,246.464 interest-only loan is outstanding for two years, with four semiannual periods. The loan rate is 6% nominal, convertible semiannually, and the fund rate is 8% nominal, convertible semiannually. A level deposit is made at each period-end to accumulate to the principal. Construct the schedule and report dollar amounts to the nearest cent.
Method. Convert each nominal rate to a rate per half-year. The loan rate per period is \(0.06/2=0.03\); the fund rate per period is \(0.08/2=0.04\). The accumulation factor for four deposits is \(s_{\overline{4}|0.04}=1+1.04+1.04^2+1.04^3=4.246464\). Since \(4.246464(1{,}000)=4{,}246.464\), the deposit is $1,000 per half-year.
Loan interest each period is \(4{,}246.464(0.03)=127.39392\). The fund balances follow the recurrence: \(F_1=0(1.04)+1{,}000=1{,}000\); \(F_2=1{,}000(1.04)+1{,}000=2{,}040\); \(F_3=2{,}040(1.04)+1{,}000=3{,}121.60\); and \(F_4=3{,}121.60(1.04)+1{,}000=4{,}246.464\). Each period’s outlay before rounding is \(1{,}000+127.39392=1{,}127.39392\).
| Period | Beginning fund | Deposit | Fund interest | Ending fund | Loan interest | Total outlay |
|---|---|---|---|---|---|---|
| 1 | $0.00 | $1,000.00 | $0.00 | $1,000.00 | $127.39 | $1,127.39 |
| 2 | $1,000.00 | $1,000.00 | $40.00 | $2,040.00 | $127.39 | $1,127.39 |
| 3 | $2,040.00 | $1,000.00 | $81.60 | $3,121.60 | $127.39 | $1,127.39 |
| 4 | $3,121.60 | $1,000.00 | $124.86 | $4,246.46 | $127.39 | $1,127.39 |
Check. The exact, unrounded maturity fund is $4,246.464, equal to principal; the table rounds money to cents only for display. Using unrounded amounts, total periodic outlay over four periods is \(4(1{,}127.39392)=4{,}509.57568\), or $4,509.58. The trap avoided is treating 6% and 8% as rates per half-year; they are nominal annual rates that must first be divided by two.
Worked Example: Beginning-of-Period Deposits
A $36,410 interest-only loan is due in three years. Its effective annual rate is 5%. A fund earns 10% effective annually, and the borrower makes three equal deposits at the beginnings of years 1, 2, and 3. Find the deposit and prepare the schedule from each deposit date to the next year-end.
Method. Each deposit earns interest during the period in which it is made. The accumulation factor for three beginning-of-period deposits is \(1.10+1.10^2+1.10^3=1.10(1+1.10+1.21)=3.641\). Therefore \(D=36{,}410/3.641=10{,}000\).
In each row, the beginning fund balance is measured immediately before that period’s deposit. Fund interest is applied to the balance after the deposit: \(i_F(F_{k-1}+D)\). Loan interest each year is \(36{,}410(0.05)=1{,}820.50\), so total outlay per year, counting the deposit and interest paid during that year, is \(10{,}000+1{,}820.50=11{,}820.50\).
| Year | Beginning fund before deposit | Beginning deposit | Fund interest | Ending fund | Loan interest | Total outlay |
|---|---|---|---|---|---|---|
| 1 | $0 | $10,000 | $1,000 | $11,000 | $1,820.50 | $11,820.50 |
| 2 | $11,000 | $10,000 | $2,100 | $23,100 | $1,820.50 | $11,820.50 |
| 3 | $23,100 | $10,000 | $3,310 | $36,410 | $1,820.50 | $11,820.50 |
Check. The ending balance is \(23{,}100+10{,}000+3{,}310=36{,}410\), which equals the principal due. This schedule differs from an end-of-period deposit schedule: the deposit is included in the amount earning fund interest during that period. The timing convention must be visible in the schedule heading and in the interest calculation.
Common Timing and Accounting Traps
For end-of-period deposits, the fund interest in period \(k\) is \(i_FF_{k-1}\), not \(i_F(F_{k-1}+D)\). For beginning-of-period deposits, it is \(i_F(F_{k-1}+D)\). Applying the wrong timing convention changes every later fund balance.
For example, suppose a fund starts at zero, earns 10% per year, and receives a $1,000 deposit at each year-end for two years. The correct first-year interest is \(0.10(0)=0\), so the first ending balance is $1,000. The second-year interest is $100, making the final balance $2,100. A mistaken calculation that gives the first deposit a full year of interest produces $1,100 after year 1 and \(1{,}100(1.10)+1{,}000=2{,}210\) after year 2. It overstates the fund by $110 because it treats an end-of-year deposit as if it had been invested earlier.
- Do not combine the two interest columns. Loan interest is charged on the loan balance; fund interest is earned on the fund balance.
- Do not count the deposit as a lender principal payment. The deposit contributes to borrower outlay, but remains in the fund until used for the maturity payment.
- Do not mix annual and periodic rates. A schedule with semiannual periods requires semiannual loan and fund rates.
- Check the last row. A mismatch between the ending fund and the required redemption amount signals an error in the deposit, rate, timing, or rounding.
Check Your Understanding
For each problem, use the stated deposit timing and report dollar amounts to the nearest cent where needed.
- A $12,000 interest-only loan lasts three years. The effective annual loan rate is 7%, the fund earns 5% effective annually, and equal deposits are made at each year-end. Find the level deposit and annual borrower outlay. (Answer: \(s_{\overline{3}|0.05}=3.1525\), deposit \(=$3,806.50\), annual outlay \(=$4,646.50\).)
- A $20,000 interest-only loan lasts two years. The loan rate is 6% effective annually, the fund earns 10% effective annually, and deposits are made at each year-end. Find the deposit, the fund balance after year 1, and annual outlay. (Answer: deposit \(=$9,523.81\); year-1 fund balance \(=$9,523.81\); annual outlay \(=$10,723.81\).)
- A fund receives $4,000 at each of three semiannual period-ends. Its rate is 12% nominal, convertible semiannually. The associated interest-only loan has principal equal to the fund’s maturity value and a rate of 8% nominal, convertible semiannually. Find the fund maturity value and the borrower’s outlay per period. (Answer: fund maturity value \(=$12,734.40\); loan interest \(=$509.38\) per period; outlay \(=$4,509.38\) per period.)
- A loan is due in three years and is funded by three equal deposits made at the beginnings of years 1, 2, and 3. The fund earns 10% effective annually, and each deposit is $5,000. The loan rate is 5% effective annually. Find the maturity fund, the loan principal if it equals that fund, and annual borrower outlay. (Answer: maturity fund and loan principal \(=$18,205\); loan interest \(=$910.25\); annual outlay \(=$5,910.25\).)
- An end-of-year fund starts at zero, earns 10% effective annually, and receives $1,000 at the end of each year for two years. A calculation incorrectly credits the first deposit with a full year of interest. Find the correct maturity balance and the amount of the error. (Answer: correct balance \(=$2,100\); incorrect balance \(=$2,210\); overstatement \(=$110\).)