Starting With an Existing Fund Balance

Why an Existing Fund Balance Changes the Deposit

A sinking fund is intended to accumulate an amount that will be used to repay a loan at maturity. If the fund already has a balance, that money will earn interest before maturity. The required future deposits must therefore cover only the part of the maturity target that is not provided by the accumulated starting balance.

This calculation depends on timing. The starting balance is invested at the valuation date, while each later deposit is invested for the time remaining after it is made. An end-of-period deposit earns interest for fewer periods than a beginning-of-period deposit. The loan interest payments are separate from these fund deposits: in the examples below, the fund is being accumulated to repay the loan principal at maturity.

Definition: Let $F$ be the amount the sinking fund must have at maturity, $B$ its balance at time 0, $n$ the number of deposit periods, and $i_f$ the effective fund yield per period. If equal deposits $D$ are made at the end of each period, then the fund at maturity is $B(1+i_f)^n+D s_{\overline{n}|i_f}$, where $s_{\overline{n}|i_f}=\frac{(1+i_f)^n-1}{i_f}$ when $i_f\ne 0$. The required deposit is $D=\frac{F-B(1+i_f)^n}{s_{\overline{n}|i_f}}$.

For annual periods, $i_f$ is the effective annual yield. If deposits are made every six months, for example, the rate and the number of periods must instead be expressed on a six-month basis. The formulas assume a fixed effective yield per deposit period, deposits made on the stated dates, and no fees, taxes, withdrawals, or other cash flows. If the yield changes or the deposit frequency differs from the interest-crediting frequency, the accumulation must reflect those details rather than using a single-period annuity factor without adjustment.

Building the Maturity-Value Equation

The equation has two parts. First, accumulate the starting balance for all $n$ periods. Second, accumulate each future deposit from its own payment date to maturity. With end-of-period deposits, the first deposit earns interest for $n-1$ periods, and the last earns no interest. Their maturity values form a geometric series.

F = B(1+i_f)^n + D\left[1+(1+i_f)+(1+i_f)^2+\cdots+(1+i_f)^{n-1}\right]

The bracketed sum is $s_{\overline{n}|i_f}$. To see why, multiply the sum by $1+i_f$ and subtract the original sum: all interior terms cancel, leaving $(1+i_f)^n-1$. Dividing by $i_f$ gives the stated formula. Rearranging the maturity-value equation then gives the required deposit.

(1+i_f)s_{\overline{n}|i_f}-s_{\overline{n}|i_f} = (1+i_f)^n-1 \quad\Longrightarrow\quad s_{\overline{n}|i_f} = \frac{(1+i_f)^n-1}{i_f}

This reasoning also shows why the starting balance cannot simply be subtracted from the maturity target. The balance earns interest, so its contribution at maturity is $B(1+i_f)^n$, not $B$. Subtract the accumulated value from the target, then use the annuity accumulation factor to spread the remaining amount across equal deposits.

Fix the valuation dates.
Identify the date of the starting balance, the number of periods to maturity, and whether each deposit is made at the beginning or end of a period.

Accumulate the existing balance.
For an effective yield $i_f$ per period, its value at maturity is $B(1+i_f)^n$.

Find the amount still to be funded.
Subtract the starting balance’s maturity value from the required fund amount $F$.

Divide by the matching deposit factor.
Use $s_{\overline{n}|i_f}$ for end-of-period deposits or the accumulated-value factor for beginning-of-period deposits.

Check the maturity value.
Accumulate the starting balance and the proposed deposits separately, then confirm that their sum equals the target before rounding.

Worked Examples

Worked Example: Finding End-of-Year Deposits After a Starting Balance

A loan has principal $F=\$43{,}170$, due in 3 years. A sinking fund currently contains $B=\$20{,}000$ and earns an effective annual rate of $i_f=10\%$. Equal deposits will be made at the end of each year. Find the annual deposit.

Method. Use a maturity-value equation: accumulate the existing balance for three years, then add the accumulated value of three end-of-year deposits. The assumption is that the fund earns 10% effective annually throughout, and that the fund is required to repay the loan principal at maturity.

The existing balance accumulates to:

\$20{,}000(1.10)^3 = \$20{,}000(1.331) = \$26{,}620

For three end-of-year deposits, the accumulation factor is:

s_{\overline{3}|0.10} = \frac{(1.10)^3-1}{0.10} = \frac{1.331-1}{0.10} = 3.31

The deposits must provide the difference between the target and the accumulated starting balance:

\$43{,}170-\$26{,}620=\$16{,}550 \qquad\text{and}\qquad D=\frac{\$16{,}550}{3.31}=\$5{,}000

Check. The three deposits accumulate to $\$5{,}000(3.31)=\$16{,}550$. Adding the starting balance’s maturity value gives $\$26{,}620+\$16{,}550=\$43{,}170$, exactly the principal target. The required annual deposit is \$5,000. This method avoids the common mistake of subtracting the unaccumulated $\$20{,}000$ from the target.

Worked Example: Changing the Deposit Date

A loan principal of $\$54{,}100$ is due in 2 years. The sinking fund starts with $\$10{,}000$, earns 10% effective annually, and must equal the loan principal at maturity. Find the equal deposit if deposits are made (a) at the end of each year and (b) at the beginning of each year.

Method. The initial balance accumulates in the same way for either schedule. The deposit factor changes with timing. Assume the first beginning-of-year deposit is made immediately at time 0, and the second is made at time 1.

The starting balance grows to $\$10{,}000(1.10)^2=\$12{,}100$. The deposits must therefore accumulate to $\$54{,}100-\$12{,}100=\$42{,}000$.

(a) For end-of-year deposits, the factor is $s_{\overline{2}|0.10}=1+1.10=2.10$. Thus:

D_{\text{end}}=\frac{\$42{,}000}{2.10}=\$20{,}000

The first $\$20{,}000$ deposit grows for one year and the second earns no interest. Their maturity value is $\$20{,}000(1.10)+\$20{,}000=\$42{,}000$.

(b) For beginning-of-year deposits, the time-0 deposit earns interest for two years and the time-1 deposit earns interest for one year. Their combined factor is $1.10^2+1.10=1.21+1.10=2.31$. Hence:

D_{\text{begin}}=\frac{\$42{,}000}{2.31} = \$18{,}181.818\ldots \approx \$18{,}181.82

Check. Before rounding, the beginning deposits accumulate to $(\$42{,}000/2.31)(2.31)=\$42{,}000$; adding the $\$12{,}100$ accumulated starting balance gives $\$54{,}100$. The displayed deposit is rounded to the nearest cent. It is smaller than the end-of-year deposit because each beginning-of-year deposit earns an additional period of interest. Treating the beginning deposits as an ordinary annuity would incorrectly use the factor 2.10.

Worked Example: Checking Whether Any Deposit Is Needed

A loan principal of $\$30{,}000$ is due in 4 years. A fund already contains $\$25{,}000$, earns 5% effective annually, and would otherwise receive equal end-of-year deposits. Determine the algebraic deposit and whether a positive deposit is required.

Method. Apply the end-of-year formula and examine the amount left after accumulating the starting balance. Assume the yield remains 5% for all four years.

The starting balance’s maturity value is:

\$25{,}000(1.05)^4 = \$25{,}000(1.21550625) = \$30{,}387.65625

The deposit accumulation factor is:

s_{\overline{4}|0.05} = \frac{1.21550625-1}{0.05} = 4.310125

The algebraic deposit is:

D = \frac{\$30{,}000-\$30{,}387.65625}{4.310125} = -\$89.9408\ldots \approx -\$89.94

Check and interpretation. The negative result means that positive deposits are not required. With no deposits, the fund reaches $\$30{,}387.65625$, which exceeds the $\$30{,}000$ target. The algebraic result would represent equal withdrawals, not deposits. The common mistake is to report a negative number as a required payment into the fund; in this situation, the required positive deposit is zero.

Assumptions, Feasibility, and Common Traps

The formula gives the deposit that makes the maturity value exactly equal to the target. For a positive deposit to be required, the target must exceed the accumulated starting balance, $F>B(1+i_f)^n$. If the two amounts are equal, the required deposit is zero. If the accumulated balance is greater than the target, the algebraic deposit is negative; it does not describe a positive deposit schedule.

A reliable solution keeps the fund calculation separate from the loan’s interest payments. In a sinking-fund loan, the borrower may pay interest on the outstanding loan while also depositing money into the fund. Those interest payments do not accumulate toward the principal target unless the problem explicitly says they are deposited into the fund. Here, $F$ is the amount the fund must provide at maturity, and the calculation addresses only how to reach that amount.

Subtracting the starting balance instead of its maturity value. The relevant amount is $B(1+i_f)^n$, because the initial balance earns interest until maturity.
Using the wrong deposit timing. End-of-period deposits use $s_{\overline{n}|i_f}$. Beginning-of-period deposits earn one extra period each, so their factor is $(1+i_f)s_{\overline{n}|i_f}$.
Mixing periods and rates. The number of periods must match the effective yield per period. An annual rate cannot be paired directly with a count of monthly deposits unless it has first been converted to the applicable monthly rate.
Ignoring feasibility. A negative algebraic deposit signals that the starting balance alone exceeds the target at maturity; it is not a positive contribution.
Rounding too early. Keep the accumulated values and annuity factor unrounded through the calculation. Round the final monetary deposit to cents unless the problem specifies another convention.

Key takeaway: Accumulate the existing balance to maturity first. Subtract that value from the required fund amount, then divide by the accumulation factor that matches the deposit dates. Finally, rebuild the maturity value to verify the result.

Check Your Understanding

Assume a constant effective fund yield for each stated period, and use the deposit timing specified in each question.

A fund begins with $\$4{,}000$, earns 10% effective annually, and receives three equal end-of-year deposits. It must reach $\$20{,}000$ at the end of year 3. Find each deposit. (Answer: $\$4{,}433.84$)
A fund begins with $\$6{,}000$, earns 5% effective annually, and receives three equal end-of-year deposits to reach $\$30{,}000$ at the end of year 3. Find each deposit. (Answer: $\$7{,}313.01$)
A fund begins with $\$10{,}000$, earns 10% effective annually, and receives three equal beginning-of-year deposits, starting immediately. What target will it reach at the end of year 3 if each deposit is $\$5{,}000$? (Answer: $\$31{,}515$)
A fund begins with $\$15{,}000$, earns 5% effective annually, and is intended to reach $\$16{,}000$ in 2 years. What positive end-of-year deposit is required? (Answer: $\$0$; the starting balance alone accumulates to $\$16{,}537.50$)

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