Why the Final Deposit May Need Adjustment
A sinking-fund schedule is often calculated using an exact periodic deposit, but actual deposits must usually be stated in currency units. Rounding each regular deposit can change the amount accumulated by the redemption date. If the rounded schedule falls short, the final deposit can be adjusted to close that shortfall.
The key is to value every deposit at the same date. A deposit made earlier earns interest for more periods; the final deposit, made on the target date, earns none before that date. The exact final deposit is therefore the target amount minus the accumulated value of all earlier, rounded deposits.
If the exact final deposit is positive and the fund must meet or exceed the target, a cent-denominated deposit should be rounded up to the next cent when ordinary rounding to nearest cent would leave the fund below the target. If the exact amount is already a whole number of cents, no upward adjustment is needed. This is different from simply rounding every deposit to the nearest cent.
Why the Accumulation Formula Works
A deposit made at the end of period \(k\) is in the fund for the remaining \(n-k\) periods. With an effective rate \(j\) per period, one dollar grows to \((1+j)^{n-k}\) dollars over that time. Thus \(D_k\) grows to \(D_k(1+j)^{n-k}\) by time \(n\). Adding the accumulated values of the first \(n-1\) deposits gives the total value available before the final deposit. Since the final deposit is made at time \(n\), it contributes exactly \(D_n\), with no additional accumulation. Equating the total to \(F\) and rearranging gives the formula above.
For equal deposits \(R\) at the ends of all \(n\) periods, the target is \(R s_{\overline{n}|}\), where \(s_{\overline{n}|}=\big((1+j)^n-1\big)/j\) for \(j\ne 0\). This can determine an exact planned deposit. But once earlier deposits are rounded, the final deposit should be calculated from their actual accumulated values—not by assuming the exact schedule still holds.
Worked Example: Equal Deposits Rounded to Cents
A fund must accumulate to $2,000 at the end of five annual periods. It earns an effective annual rate of 4%, and deposits are made at each year-end. The planned deposits are equal, but each of the first four deposits is made to the nearest cent. Find the final deposit, in cents, that ensures the target is met. Ignore fees and taxes.
Method. First calculate the planned equal deposit, then accumulate the actual rounded first four deposits to time 5 and find the remaining amount.
The accumulation factor for five equal end-of-year deposits is \(s_{\overline{5}|}=(1.04^5-1)/0.04=5.41632256\). The exact planned deposit is therefore $$ R=\frac{2{,}000}{5.41632256}\approx 369.2542. $$ Rounded to the nearest cent, each of the first four deposits is $369.25, not $369.26.
At time 5, those four deposits have accumulated for 4, 3, 2, and 1 years, respectively. Their accumulated value is $$ \begin{aligned} 369.25(1.04^4+1.04^3+1.04^2+1.04) &=369.25(1.16985856+1.124864+1.0816+1.04)\\ &=369.25(4.41632256)\\ &=1{,}630.72710528. \end{aligned} $$
The exact final deposit is $$ 2{,}000-1{,}630.72710528=369.27289472. $$ A deposit of $369.27 would leave the fund short by \(369.27289472-369.27=\$0.00289472\). To ensure the target is met, round upward to $369.28.
Check. The actual accumulated fund is \(1{,}630.72710528+369.28=\$2{,}000.00710528\), which is $0.00710528 above the target. The cents deposit meets the target. The trap is to round the exact final amount to the nearest cent without checking whether that rounding leaves a shortfall.
Rounding a Regular Deposit to Whole Dollars
The same calculation applies if regular deposits are rounded to whole dollars rather than cents. The final deposit is still based on the target less the value of the deposits actually made. A larger rounding unit can create a more noticeable adjustment, so it is especially important not to use the original unrounded schedule in the final calculation.
Worked Example: Regular Deposits Rounded to Dollars
A fund is intended to reach $10,000 at the end of eight annual periods. It earns an effective annual rate of 6%. An equal-deposit calculation is made, but the first seven deposits are rounded to the nearest whole dollar. Find the final deposit in cents that ensures the target is reached. Assume deposits occur at each year-end and there are no fees or taxes.
Method. Find the rounded regular deposit, accumulate the seven actual deposits to time 8, and use the target-date shortfall for the final deposit.
The accumulation factor for eight equal deposits is $$ s_{\overline{8}|}=\frac{1.06^8-1}{0.06} =\frac{1.5938480743-1}{0.06} \approx 9.897467905. $$ The planned equal deposit is \(10{,}000/9.897467905\approx \$1{,}010.36\), so the first seven deposits, rounded to the nearest dollar, are each $1,010.
At time 8, the first seven deposits have accumulated for seven down to one years. Their accumulation factor is \(1.06+1.06^2+\cdots+1.06^7=s_{\overline{8}|}-1\), or approximately \(8.897467905\). Thus their accumulated value is $$ 1{,}010(8.897467905)=8{,}986.44258405. $$ The exact final deposit is $$ 10{,}000-8{,}986.44258405=1{,}013.55741595. $$ Rounding upward to cents gives a final deposit of $1,013.56.
Check. The accumulated total is \(8{,}986.44258405+1{,}013.56=\$10{,}000.00258405\), slightly above the target. The method avoids the mistake of making the eighth deposit $1,010 simply because the first seven were rounded regular deposits. The final deposit must compensate for the accumulated shortfall.
Adjusting a Varying Deposit Schedule
A final adjustment is not limited to equal deposits. If the intended regular deposits vary—for example, as in an arithmetic-increasing schedule—accumulate each actual rounded deposit separately. The amount deposited at period \(k\) has its own accumulation factor, \((1+j)^{n-k}\). It is not generally correct to treat varying deposits as though they were one equal payment or to apply the same adjustment to every period.
Worked Example: Final Deposit After Rounding a Varying Schedule
A fund must contain $1,913.30 at the end of period 4. It earns an effective rate of 5% per period. The intended arithmetic-increasing deposits are $300.004, $400.004, $500.004, and $600.004 at the ends of periods 1 through 4. The first three deposits are rounded to the nearest cent. Find the final cent-denominated deposit that ensures the target is met. Assume no fees or withdrawals.
Method. Accumulate the first three deposits after rounding them as actually made, then subtract that amount from the target.
The first three rounded deposits are $300.00, $400.00, and $500.00. At time 4, the first has earned interest for three periods, the second for two, and the third for one. Their accumulated value is $$ \begin{aligned} 300(1.05^3)+400(1.05^2)+500(1.05) &=300(1.157625)+400(1.1025)+500(1.05)\\ &=347.2875+441+525\\ &=1{,}313.2875. \end{aligned} $$
The exact final deposit needed at time 4 is \(1{,}913.30-1{,}313.2875=\$600.0125\). A $600.01 deposit would produce a total of \(1{,}313.2875+600.01=\$1{,}913.2975\), which is $0.0025 short. Therefore the final deposit must be rounded upward to $600.02.
Check. The fund then contains \(1{,}313.2875+600.02=\$1{,}913.3075\), exceeding the target by $0.0075. The final deposit differs from the intended $600.004 because the earlier deposits were rounded and the target must be met using the amounts actually deposited.
Common Mistakes and Exam Traps
- Using the wrong number of interest periods. A deposit at the end of period \(k\) earns interest for \(n-k\) periods by the end of period \(n\). The last regular deposit earns one period; the final deposit at time \(n\) earns none before the target date.
- Rounding the exact final amount without checking the target. If the problem requires the fund to meet or exceed a stated amount, ordinary rounding to nearest cent can leave a shortfall. Compare the rounded deposit with the exact amount and check the resulting accumulated total.
- Rounding the planned deposit upward automatically. The exact equal deposit may be rounded up or down to the nearest currency unit. The appropriate final adjustment depends on the deposits actually made, not on an assumption about the direction of rounding.
- Accumulating rounded amounts as if they were exact. When earlier deposits differ from the planned amounts, use those actual rounded amounts in the target-date accumulation.
- Using a final deposit as though it earns a full period of interest. A deposit made at the target date has no time to earn interest before that date. Adding a factor of \(1+j\) to it understates the required deposit.
Check Your Understanding
For each problem, assume deposits are made at the ends of the stated periods, the fund earns the stated effective rate per period, and there are no fees or withdrawals. When asked for a deposit that ensures the target, round the final deposit upward to cents if needed.
- A fund target is $1,200 at the end of period 4. The rate is 4% per period, and the first three deposits are each $250. Find the final deposit in cents that ensures the target. (Answer: $388.39.)
- A fund target is $3,000 at the end of period 5. The rate is 2% per period, and the first four deposits are each $600. Find the final deposit in cents that ensures the target. (Answer: $477.58.)
- A fund target is $5,000 at the end of period 6. The rate is 5% per period, and the first five deposits are each $800. Find the final deposit in cents that ensures the target. (Answer: $358.47.)
- A fund target is $2,500 at the end of period 4. The rate is 3% per period. Deposits at the ends of periods 1, 2, and 3 are $400, $500, and $600, respectively. Find the final deposit in cents that ensures the target. (Answer: $914.46.)
- A fund target is $10,000 at the end of period 10. The rate is 4% per period, and the first nine deposits are each $800. Find the final deposit in cents that ensures the target. (Answer: $1,195.12.)