Why a Sinking-Fund Timeline Needs Two Accounts
A sinking-fund loan separates the payment of interest from the accumulation of principal. The borrower receives the loan proceeds, pays interest to the lender, and makes deposits into a separate fund. At maturity, the fund is used to redeem the principal. These are distinct cash flows, often occurring on the same date but serving different purposes.
A timeline makes the dates explicit. This matters because a deposit made earlier earns interest for longer, while a deposit made at maturity does not earn interest before redemption. The loan’s interest rate and the fund’s yield may also differ, so the interest paid to the lender cannot be inferred from the fund balance.
For a timeline, choose a perspective and state it. In the examples below, loan proceeds are inflows to the borrower; interest payments and fund deposits are outflows. Fund investment earnings are increases within the fund, not additional payments by the borrower. At maturity, the fund is liquidated and the principal is paid to the lender. When the fund exactly covers the principal, those two maturity-date transfers offset each other in the borrower’s overall cash position.
Set the Timeline Convention Before Calculating
The examples use annual periods. The loan rate \(i_L\) is the effective annual rate charged on the original principal, with interest paid at each year-end. The fund earns an effective annual rate \(i_F\). Deposits are made at year-end unless a problem specifies a different date. There are no taxes, fees, defaults, or other cash flows.
Let \(B_k\) be the fund balance immediately after the deposit at time \(k\), and let \(D_k\) be that end-of-year deposit. Over the following year, the existing fund balance earns interest. Thus the balance immediately before the next deposit is \(B_k(1+i_F)\). A deposit at the end of that year is added only after that accumulation.
This recurrence gives a precise way to read the timeline: carry the previous balance forward one period, then add the deposit dated at the end of the period. If the initial fund balance at time 0 is \(B_0\), repeated substitution gives the balance after \(n\) end-of-period deposits:
To justify the expression, the initial balance earns interest for \(n\) periods. A deposit \(D_k\), made at time \(k\), earns interest for the remaining \(n-k\) periods, so its value at time \(n\) is \(D_k(1+i_F)^{n-k}\). Adding the value of each dated amount gives the fund total. In particular, a deposit at time \(n\) has exponent zero and contributes \(D_n\), not \(D_n(1+i_F)\).
The loan interest is determined separately. If the original principal is \(L\), the loan rate is \(i_L\), and the principal remains outstanding throughout the term, the interest due at each year-end is \(Li_L\). At maturity, the principal \(L\) is also due. The fund balance must be checked against that amount.
Worked Timeline: Two-Year Loan With a Final Fund Deposit
Worked Example: Separating Interest From Principal Funding
A borrower receives a hypothetical \( \$20{,}000 \) loan for two years. The effective annual loan rate is 6%, with interest paid at each year-end and all principal due at the end of year 2. The sinking fund earns 4% effective annually. The borrower deposits \( \$8{,}000 \) at the end of year 1. Find the required year-2 deposit and the borrower’s outlay at each year-end. Assume no initial fund balance, fees, or taxes.
Method. First roll the fund forward on its own timeline. Then calculate the loan interest from the loan terms. Do not treat a fund deposit as a principal payment before maturity.
| Date | Loan event | Fund event and balance | Borrower cash flow on date |
|---|---|---|---|
| Time 0 | Receives \( \$20{,}000 \) | Fund starts at \( \$0 \) | Inflows \( \$20{,}000 \) |
| End of year 1 | Pays interest of \( \$1{,}200 \) | Deposits \( \$8{,}000 \); balance \( \$8{,}000 \) | Outflow \( \$9{,}200 \) |
| End of year 2 | Pays interest of \( \$1{,}200 \); principal \( \$20{,}000 \) is due | Fund grows to \( \$8{,}320 \), then receives the required deposit; it is liquidated for \( \$20{,}000 \) | Outflow from income for interest and deposit; fund redemption pays the principal |
Step 1: Find year-1 loan interest. The principal remains \( \$20{,}000 \), so the annual interest is
The year-1 fund deposit is \( \$8{,}000 \). The borrower’s outlay from other funds at the end of year 1 is therefore \( \$1{,}200+\$8{,}000=\$9{,}200 \).
Step 2: Accumulate the fund to year 2. The year-1 deposit earns one year of fund interest:
Step 3: Find the year-2 deposit. The fund must equal \( \$20{,}000 \) after the deposit at time 2. Therefore,
Step 4: Find the year-2 outlay. Interest is still calculated on the loan principal, not on the fund:
At maturity the fund contains \( \$8{,}320+\$11{,}680=\$20{,}000 \), exactly the principal due. The fund redemption pays the \( \$20{,}000 \) principal; the borrower’s year-2 outlay from income is the \( \$1{,}200 \) interest plus the \( \$11{,}680 \) deposit. The common mistake avoided here is adding another year of interest to the deposit made at maturity.
Worked Timeline: Several Deposits Before Maturity
Worked Example: Carrying Irregular Deposits Forward
A hypothetical \( \$30{,}000 \) loan has a three-year term. Interest is 5% effective annually, paid at each year-end, and the entire principal is due at the end of year 3. A separate fund earns 3% effective annually. The borrower deposits \( \$6{,}000 \) at time 1 and \( \$9{,}000 \) at time 2. Find the deposit needed at time 3 and the outlay from income at each year-end. Assume no other deposits or charges.
Method. Date each deposit, accumulate it only for the periods after its deposit date, and compare the total fund with the principal due at maturity.
| Date | Loan interest | Fund balance calculation | Outlay from income |
|---|---|---|---|
| Time 0 | — | \( \$0 \) | Loan proceeds received |
| End of year 1 | \( \$1{,}500 \) | \( \$6{,}000 \) deposited; balance \( \$6{,}000 \) | \( \$7{,}500 \) |
| End of year 2 | \( \$1{,}500 \) | \( \$6{,}000(1.03)+\$9{,}000=\$15{,}180 \) | \( \$10{,}500 \) |
| End of year 3 | \( \$1{,}500 \) | Prior balance grows to \( \$15{,}635.40 \), then the final deposit brings it to \( \$30{,}000 \) | Interest plus final deposit |
Step 1: Calculate the loan interest. The original principal remains outstanding until maturity, so the annual interest is
At the end of year 1, the borrower pays \( \$1{,}500 \) interest and deposits \( \$6{,}000 \), for an outlay of \( \$7{,}500 \). At the end of year 2, the borrower pays \( \$1{,}500 \) interest and deposits \( \$9{,}000 \), for an outlay of \( \$10{,}500 \).
Step 2: Find the fund at time 2. The first deposit earns one year of interest by time 2; the second deposit is made at time 2 and earns none yet:
Step 3: Carry the balance to time 3. Before the final deposit, the fund is
Step 4: Determine the final deposit and outlay. The final deposit must fill the gap to the \( \$30{,}000 \) principal:
The time-3 outlay from income is \( \$1{,}500+\$14{,}364.60=\$15{,}864.60 \). After that deposit, the fund is \( \$30{,}000 \), which is liquidated to pay the principal. The timeline check confirms that only the time-1 and time-2 deposits earn fund interest before maturity.
Worked Timeline: An Initial Fund Balance
Worked Example: Respecting a Deposit at Time 0
A borrower receives a hypothetical \( \$12{,}000 \), two-year loan at 8% effective annual interest. Interest is paid at each year-end, and the full principal is due at the end of year 2. The fund earns 5% effective annually. The borrower places \( \$1{,}000 \) in the fund at time 0 and deposits \( \$4{,}000 \) at the end of year 1. Find the required year-2 deposit and the borrower’s net cash position at time 0 and outlays at the two year-ends. Assume no other cash flows.
Method. Include the initial fund balance as a time-0 amount; it earns two full years of interest. Deposits made at years 1 and 2 earn one year and zero years, respectively, by maturity.
| Date | Loan event | Fund balance | Borrower cash flow |
|---|---|---|---|
| Time 0 | Receives \( \$12{,}000 \) | Deposits \( \$1{,}000 \) | Net inflow \( \$11{,}000 \) |
| End of year 1 | Pays \( \$960 \) interest | Balance grows to \( \$1{,}050 \), then receives \( \$4{,}000 \); balance \( \$5{,}050 \) | Outflow \( \$4{,}960 \) |
| End of year 2 | Pays \( \$960 \) interest; principal \( \$12{,}000 \) is due | Prior balance grows to \( \$5{,}302.50 \); final deposit completes the fund | Outflow is interest plus final deposit; fund pays principal |
Step 1: Calculate loan interest.
At time 0, the borrower’s net inflow is \( \$12{,}000-\$1{,}000=\$11{,}000 \). At the end of year 1, the initial fund balance grows to \( \$1{,}000(1.05)=\$1{,}050 \); after the \( \$4{,}000 \) deposit, the fund balance is \( \$5{,}050 \). The year-1 outlay is \( \$960+\$4{,}000=\$4{,}960 \).
Step 2: Accumulate the year-1 balance. By time 2, the fund balance before the last deposit is
Step 3: Complete the fund and find the year-2 outlay.
so the year-2 outlay from income is \( \$960+\$6{,}697.50=\$7{,}657.50 \). The fund then totals \( \$12{,}000 \), exactly the principal due. This also checks the timing: the initial \( \$1{,}000 \) earned two years, the year-1 deposit earned one year, and the time-2 deposit earned no interest before redemption.
Common Timeline Errors
A timeline is useful not only for finding amounts but for diagnosing a calculation that seems plausible yet uses the wrong date or account. The most frequent errors are:
- Giving a maturity-date deposit an extra period of growth. A deposit made at time \(n\) contributes \(D_n\) to the balance at time \(n\), because \((1+i_F)^0=1\).
- Using the fund yield to calculate loan interest. Loan interest is determined by the loan rate and the outstanding principal; fund earnings are determined by the fund yield and the fund balance.
- Calling a fund deposit principal repayment. A deposit increases an asset held for future repayment. Unless the contract says otherwise, the loan principal remains due until it is actually paid to the lender.
- Combining interest and principal into one unlabeled amount. At maturity, show the interest payment, fund deposit, fund redemption, and principal payment separately before calculating any net cash flow.
Check Your Understanding
For each problem, use the stated end-of-period deposit convention and distinguish the loan interest from fund contributions.
- A hypothetical two-year loan is \( \$10{,}000 \) at 7% effective annual interest, paid at each year-end, with principal due at time 2. The fund earns 4% and receives \( \$3{,}000 \) at time 1. Find the time-2 deposit and the time-2 outlay from income. (Answer: deposit \( \$6{,}880.00 \); time-2 outlay \( \$7{,}580.00 \).)
- A \( \$25{,}000 \), three-year hypothetical loan charges 6% effective annual interest at each year-end, with principal due at time 3. A fund earns 5% and receives \( \$7{,}000 \) at time 1 and \( \$8{,}000 \) at time 2. Find the final deposit and time-3 outlay. (Answer: final deposit \( \$8{,}882.50 \); time-3 outlay \( \$10{,}382.50 \).)
- A two-year \( \$16{,}000 \) hypothetical loan charges 5% effective annual interest, paid at each year-end. A fund earns 4%, starts with \( \$2{,}500 \) at time 0, and receives \( \$4{,}000 \) at time 1. Find the fund just before the final deposit, the required final deposit, and the time-2 outlay. (Answer: fund before deposit \( \$6{,}864.00 \); final deposit \( \$9{,}136.00 \); time-2 outlay \( \$9{,}936.00 \).)
- A hypothetical \( \$20{,}000 \), three-year loan charges 4% effective annual interest paid each year-end, with principal due at time 3. A fund earns 3% and receives \( \$6{,}000 \) at time 1 and \( \$5{,}000 \) at time 2. Find the fund at time 3 before any final deposit, the deposit needed to redeem the principal, and the total time-3 outlay from income. (Answer: fund before deposit \( \$11{,}515.40 \); final deposit \( \$8{,}484.60 \); time-3 outlay \( \$9{,}284.60 \).)