Finding the Required Level Fund Deposit

Why the Required Deposit Matters

A sinking fund is built by depositing money into a separate fund so that it will accumulate to a specified amount at a future date. In this tutorial, the target is the principal amount to be repaid at maturity, and the question is how large each equal deposit must be to reach that target. The calculation depends on both the fund’s interest rate and how often deposits are made.

The model below assumes deposits are made at the end of each deposit period. It treats the fund as earning a constant rate, with interest credited at the end of each period. Beginning-of-period deposits use a different accumulation factor; that timing distinction is not part of this calculation.

Definition: A level sinking-fund deposit is a constant amount deposited into a fund at regular intervals. The required deposit is the amount that makes the fund’s accumulated value at the target date equal to the stated target, under the specified fund-rate and deposit-timing assumptions.

Match the Rate to the Deposit Period

Let $F$ be the target fund amount, $R$ the level deposit, $n$ the number of deposits, and $j$ the effective interest rate per deposit period. The deposit frequency and the rate conversion must describe the same period. For example, if deposits are quarterly, $j$ must be the effective rate earned over one quarter.

If the fund earns a nominal annual rate $i^{(m)}$, convertible $m$ times per year, and deposits are made every conversion period, then the rate per deposit period is $j=i^{(m)}/m$. If instead the fund rate is an effective annual rate $i$, and there are $q$ equal deposit periods per year, the equivalent rate per deposit period is $j=(1+i)^{1/q}-1$. These conversions assume a constant rate and compound accumulation.

When the quoted rate’s conversion frequency differs from the deposit frequency, do not simply divide the quoted rate by the number of deposits per year. First determine the effective rate over one deposit period. For instance, an effective annual rate must be converted by taking the appropriate fractional power, not by treating one-quarter of the annual rate as the exact quarterly rate.

The Required-Deposit Formula

After the first end-of-period deposit earns interest for $n-1$ periods, the second earns interest for $n-2$ periods, and the last earns no interest before the target date. The accumulated value is therefore the sum of a geometric sequence.

F=R\left(1+(1+j)+(1+j)^2+\cdots +(1+j)^{n-1}\right) =R\frac{(1+j)^n-1}{j},\qquad j\ne 0.

The factor $\frac{(1+j)^n-1}{j}$ is the accumulated value of $n$ unit end-of-period payments, commonly written $s_{\overline{n}|}$ at rate $j$. Solving the accumulated-value equation for the unknown deposit gives the required level deposit.

R=\frac{F}{s_{\overline{n}|}} =\frac{Fj}{(1+j)^n-1},\qquad j\ne 0.

If the fund earns zero interest, the positive-rate expression involves division by zero and does not apply. In that case the deposits simply add, so $F=nR$ and $R=F/n$. This is also the limiting result as $j$ approaches zero.

Key result: For $n$ equal end-of-period deposits earning a constant effective rate $j$ per deposit period, divide the target fund amount by the accumulation factor $s_{\overline{n}|}$. Use $R=F/n$ if $j=0$. The formula does not apply unchanged to beginning-of-period deposits, unequal deposits, or a fund whose rate changes over time.

A Reliable Calculation Process

Identify the target and deposit count.
Write down the amount required at maturity and count the deposits that occur before that date. For end-of-period deposits, the final deposit is made at the end of the final period.

Find the rate per deposit period.
Use the stated nominal conversion rate directly only when its conversion periods match the deposit periods. Otherwise convert the effective annual rate to the deposit-period rate.

Calculate the accumulation factor and solve.
Evaluate $s_{\overline{n}|}$, then calculate $R=F/s_{\overline{n}|}$, or use $F/n$ for a zero-rate fund.

Check the rounded deposit.
Multiply the rounded deposit by the same accumulation factor. If the fund must meet or exceed the target, a deposit rounded to the nearest cent may need to be rounded upward instead.

Worked Applications

Each example treats the target as the required fund balance on the maturity date. Loan interest payments are not included in these calculations: the deposit is determined by the fund accumulation requirement alone.

Worked Example: Monthly Deposits at a Nominal Annual Rate

A hypothetical sinking fund must accumulate $5,000 after 24 month-end deposits. The fund earns a nominal annual rate of 6% convertible monthly. Find the required monthly deposit, then check the amount accumulated when the deposit is rounded to the nearest cent.

Method. The nominal rate is convertible monthly, and deposits are monthly, so the effective rate per deposit period is $j=0.06/12=0.005$. There are $n=24$ deposits.

s_{\overline{24}|} =\frac{(1.005)^{24}-1}{0.005} =\frac{1.12715978-1}{0.005} \approx 25.431955.

The required unrounded monthly deposit is

R=\frac{5000}{25.431955}\approx 196.6031.

Rounded to the nearest cent, the deposit is $196.60. Check the accumulated value using the displayed accumulation factor:

196.60(25.431955)\approx 4999.92.

The rounded deposit accumulates to about $4,999.92, a shortfall of about $0.08. If the fund is required to meet or exceed $5,000, use $196.61 instead; its accumulated value is approximately $4999.92+0.01(25.431955)=5000.18$. The small difference from the target after rounding is expected, and checking its direction avoids mistaking a shortfall for a surplus.

Worked Example: Quarterly Deposits for a Maturity Target

A hypothetical fund must accumulate $18,000 in three years through 12 quarter-end deposits. It earns a nominal annual rate of 8% convertible quarterly. Find the required quarterly deposit and check the rounded amount.

Method. The deposit and conversion frequencies both are quarterly. Thus $j=0.08/4=0.02$, and the number of deposits is $n=3(4)=12$.

s_{\overline{12}|} =\frac{(1.02)^{12}-1}{0.02} =\frac{1.26824179-1}{0.02} \approx 13.4120897.

The unrounded deposit is

R=\frac{18000}{13.4120897}\approx 1342.0727.

Therefore the required deposit to the nearest cent is $1,342.07 per quarter. Multiplying back gives

1342.07(13.4120897)\approx 17999.96.

This is about $0.04 below the target. If the requirement is to ensure that the fund has at least $18,000, a deposit of $1,342.08 gives approximately $17999.96+0.01(13.4120897)=18000.10$. The check uses the accumulation factor, not just the rounded deposit amount in isolation.

Worked Example: Semiannual Deposits at a Nominal Rate

A hypothetical sinking fund is to contain $50,000 after 10 years. The fund earns a nominal annual rate of 4% convertible semiannually, and deposits are made at the end of each half-year. Find the level deposit and verify the nearest-cent result.

Method. The nominal rate is convertible at the same frequency as the deposits, so the semiannual rate is $j=0.04/2=0.02$. There are $n=10(2)=20$ deposits.

s_{\overline{20}|} =\frac{(1.02)^{20}-1}{0.02} =\frac{1.48594740-1}{0.02} \approx 24.2973698.

The unrounded amount is

R=\frac{50000}{24.2973698}\approx 2057.8359.

Rounded to the nearest cent, the deposit is $2,057.84. The check is

2057.84(24.2973698)\approx 50000.10.

The rounded deposits produce a surplus of about $0.10. The result is consistent: the exact deposit lies just below $2,057.84, and rounding it upward by a fraction of a cent creates a small positive difference at maturity.

Common Mistakes and Why the Formula Matters

A frequent error is to apply the annual rate directly to every deposit period, or to confuse the number of years with the number of deposits. Both errors change the accumulation factor and therefore the deposit. Always state $j$ and $n$ explicitly before substituting values.

Another mistake is to reverse the annuity accumulation factor. The factor $s_{\overline{n}|}$ answers, “How much will one unit deposited each period accumulate to?” It is a multiplier on the deposit, so the target is divided by the factor. Multiplying the target by the factor would produce an amount far larger than the target, not the required contribution.

Common exam trap: With 24 monthly deposits at a nominal annual rate of 6% convertible monthly, using $j=0.06$ incorrectly treats 6% as the monthly rate. The correct rate is $j=0.06/12=0.005$, because the quoted nominal rate is convertible 12 times per year. Also check the order of operations: the required deposit is $F$ divided by the entire accumulation factor $\big((1+j)^n-1\big)/j$, not $F$ divided by its numerator and then divided by $j$.

The calculation is useful whenever a stated maturity amount must be funded through regular contributions, including a sinking fund set aside to repay principal. It isolates the funding question: given the assumed fund yield, payment frequency, and time available, what equal contribution reaches the target? It does not, by itself, calculate the borrower’s separate loan-interest payments or compare the fund yield with the loan rate.

This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In an exam calculation, write the target date, number of deposits, rate per deposit period, and deposit timing before using the formula. That short setup makes the assumptions visible and helps expose a frequency or timing mismatch.

Check Your Understanding

For each problem, assume equal end-of-period deposits and a constant fund rate. Use the stated rate convention and round the deposit to the nearest cent.

A fund earns a nominal annual rate of 4% convertible quarterly. Find the quarterly deposit needed to accumulate $9,600 after 16 quarter-end deposits. (Answer: $556.27)
A fund earns a nominal annual rate of 12% convertible monthly. Find the monthly deposit needed to accumulate $10,000 after 12 month-end deposits. (Answer: $788.49)
A fund earns a nominal annual rate of 4% convertible semiannually. Find the semiannual deposit needed to accumulate $15,000 after eight half-year-end deposits. (Answer: $1,747.65)
A fund earns no interest. Find the monthly deposit needed to accumulate $8,750 after 14 month-end deposits. (Answer: $625.00)

What You'll Learn