Sinking Funds and Funding Schedules
Core Quiz 2 topic
The current syllabus expressly places Annuities and Sinking Funds in Week 8 Quiz 2, with a 1.5-hour assessment. A sinking fund accumulates money for a known future requirement by making regular deposits into an interest-earning account. The target might be equipment replacement or repayment of a debt principal. All cases in this unit are original study exercises, not predictions of the actual quiz.
Choose the right time-value problem
A loan-installment problem begins with money owed now and uses the present value of repayments. A sinking-fund problem begins with an amount needed later and uses the future value of deposits. The recurring cash amount may look similar, but its purpose determines the factor.
For n equal end-of-period deposits R at effective periodic fund yield i, target F at the instant of deposit n satisfies F = R s(i,n). Therefore R = Fi/[(1 + i)^n − 1]. If the fund earns zero interest, R = F/n.
For deposits at the beginning of each period and a target one period after the final deposit, R_due = F/[s(i,n)(1 + i)]. If the target is immediately after the last beginning-of-period deposit, locate that earlier date and use R s(i,n), not the standard due FV. Unit 4 explains the timing distinction.
Worked target case
A business needs PHP 900,000 three years from now and will deposit at each quarter end. Its fund earns 8% nominal compounded quarterly, so i = .02 and n = 12. Required deposit R = 900,000/s(.02,12) = PHP 67,103.64.
Total deposits = 12R = PHP 805,243.64; total fund interest = 900,000 − 12R = PHP 94,756.36. PHP 900,000/12 ignores earnings and overstates each deposit. Dividing by a PV factor would calculate a loan installment rather than funding the future reserve.
A required exact final target and cent-rounded scheduled deposits may need an adjusted last deposit. Unless the question explicitly rounds each actual payment, use unrounded R in theoretical totals and round only the final result.
Existing money changes the target still to be funded
Suppose PHP 120,000 is invested immediately in that same three-year fund. It earns all 12 quarters, unlike the periodic deposits. The equation is 900,000 = 120,000(1.02)^12 + R s(.02,12). Thus R = [900,000 − 120,000(1.02)^12]/s(.02,12) = PHP 55,756.49.
Subtracting PHP 120,000 directly from the future target ignores its earnings. The subtraction must happen at the target date. If the existing balance alone grows above the target, the algebra may give negative deposits; in an ordinary savings-only model, no positive deposit is required and the surplus must be interpreted separately.
Build a sinking-fund schedule from cash flows
For an ordinary fund beginning with balance B_0, each row follows:
- Interest during period k = i × opening balance B_(k−1).
- Add the end-of-period deposit R.
- Closing balance B_k = B_(k−1)(1 + i) + R.
- Total increase for the row = interest + deposit. It is not all deposit and not all interest.
An ordinary fund with B_0 = 0 earns zero interest in its first period because the first deposit arrives only at the end. In a due fund the deposit arrives first, so a row ending after its interest period instead follows B_k = (B_(k−1) + R)(1 + i).
Worked four-year schedule
Target PHP 100,000 after four end-of-year deposits at 5% annually. The unrounded deposit is 100,000/s(.05,4), displayed as PHP 23,201.18. The following values are displayed to cents, while calculations keep full precision.
| Year | Opening balance | Interest at 5% | Deposit | Closing balance |
|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 23,201.18 | 23,201.18 |
| 2 | 23,201.18 | 1,160.06 | 23,201.18 | 47,562.43 |
| 3 | 47,562.43 | 2,378.12 | 23,201.18 | 73,141.73 |
| 4 | 73,141.73 | 3,657.09 | 23,201.18 | 100,000.00 |
Cumulative balance also satisfies B_k = R s(i,k) when B_0 = 0. The recurrence and formula provide an independent check. As the fund grows, each period's interest grows; the deposit stays constant.
Recalculate after a shortfall or changed yield
At a review date immediately after a deposit, use the actual balance at that date, the remaining number of deposits, and the new future yield. Do not restart the original n-period plan or value the actual balance as if it had been invested at time 0.
Example: a PHP 600,000 target is due after 20 quarters. Immediately after quarter 8 the actual fund is PHP 200,000. The remaining 12 quarters will earn 2% each. New deposit R_new = [600,000 − 200,000(1.02)^12]/s(.02,12) = PHP 25,823.84. The current balance grows for 12 future quarters; twelve future end-of-quarter deposits supply the rest. Any previous deposits are already included in the actual balance and must not be counted again.
A missed deposit at period k creates a target-date shortfall R(1 + i)^(n−k). A replacement at a later period h must be R(1 + i)^(h−k) to recover the missed deposit's lost growth, assuming unchanged yield. A later payment of only R does not fully catch up.
Debt interest and a separate sinking fund
A PHP 500,000 debt principal remains legally outstanding for five years. It charges 6% annual interest-only payments; a separate reserve earns 4% annually and receives five year-end deposits. Annual debt interest is 500,000(.06) = PHP 30,000. Fund deposit = 500,000/s(.04,5) = PHP 92,313.56. Combined annual cash requirement = PHP 122,313.56.
Use the fund yield 4% in the sinking-fund factor, and the debt rate 6% only on the outstanding debt interest. Deposits into a separate fund do not reduce the contractual debt principal. At an interim date, face amount less accumulated fund balance may be called net unfunded debt; the face amount is still the contractual principal until the debt is actually repaid.
Unknown time and rate
With fixed end-of-period deposit R and target F, n = ln(1 + Fi/R)/ln(1 + i). For whole deposit periods, round up to reach at least F. Check the balance after the preceding deposit. If you seek a rate with fixed R,n,F, solve F = R s(i,n) numerically; with at least two positive ordinary deposits, FV increases with i. At zero rate F = nR, so a target above nR requires positive earnings if the deposit and count are fixed.
Final problem-solving checklist
- Mark the target date and first/last deposit; classify ordinary versus due.
- Convert the fund's rate to the deposit interval; count deposits in that same interval.
- Separate current funds from future deposits and move both to the target date.
- For schedule blanks use the stated row's opening balance, yield and deposit; do not invent intermediate values.
- For debt reserves distinguish borrowing interest, fund interest, deposits, contractual debt and net unfunded amount.
- Keep full precision unless each schedule row is explicitly rounded. With cent-rounded actual deposits, compute any final adjustment from the actual prior balance.
Source basis
Paraphrased from the course Chapter 20 textbook/slides, sinking-fund module and current syllabus. All numerical cases, table tasks and explanations are independently authored. Source files and answer keys were consulted locally and are not uploaded as public originals.