Separating Fund Deposits From Fund Earnings

Why Separate Deposits From Earnings?

A sinking fund grows from two sources: money deposited into it and interest earned on money already in the fund. At the valuation date, the fund balance combines both. Separating them shows how much of that balance came from the borrower’s contributions and how much arose through investment accumulation.

The distinction depends on deposit timing. A deposit made earlier has more time to earn interest than one made later. Therefore, subtracting total deposits from the maturity balance is meaningful only after the balance has been accumulated to the same date as those deposits. This tutorial assumes a hypothetical fund with no withdrawals or fees unless stated otherwise.

Definition: The total deposits through a valuation date are the sum of all contributions made to the fund by that date, without interest. The accumulated earnings at that date are the fund balance minus total deposits, provided there are no withdrawals, fees, or other cash flows that would affect that difference. The fund’s maturity balance is the accumulated value of the deposits and the interest earned on them.

Let $D_k$ be a deposit made at the end of year $k$, and let $B_k$ be the fund balance immediately after that deposit. Assume the fund earns an effective annual rate $i_f$, credited once each year, and that there are no other fund transactions. The subscript $f$ identifies the rate earned by the fund; it need not equal the interest rate charged on a loan.

B_k = B_{k-1}(1+i_f)+D_k

The balance entering year $k$ earns interest during that year; the end-of-year deposit $D_k$ does not earn interest until a later year. This timing convention is central to the decomposition.

Accumulating Each Deposit to the Valuation Date

For a valuation immediately after year $n$'s activity, a deposit made at the end of year $k$ accumulates for $n-k$ complete years. Its value at the valuation date is therefore $D_k(1+i_f)^{n-k}$. Adding the accumulated value of every deposit gives the fund balance. Adding the deposits without accumulation gives total contributions; the difference is accumulated earnings.

B_n=\sum_{k=1}^{n}D_k(1+i_f)^{n-k}, \qquad C_n=\sum_{k=1}^{n}D_k, \qquad E_n=B_n-C_n

Here $C_n$ denotes total deposits and $E_n$ accumulated earnings at time $n$. A deposit made at the valuation date has exponent zero, so its accumulated value is just the deposit itself. If there is no deposit in a given year, take $D_k=0$. These expressions assume a constant annual fund rate and no withdrawals, fees, taxes, or other cash flows.

Worked Example: Three Equal Deposits at Year-End

A hypothetical sinking fund earns an effective annual rate of 3%. The borrower deposits $700 at the end of each of years 1, 2, and 3. Find the fund balance immediately after the third deposit, the total deposits, and the accumulated earnings. Use the deposit-by-deposit method, and check the result against the recurrence.

Method. Accumulate each deposit to the end of year 3. The year-1 deposit earns for two years, the year-2 deposit for one year, and the year-3 deposit for zero years.

Year-1 deposit at year 3: $700(1.03)^2=700(1.0609)=\$742.63$.

Year-2 deposit at year 3: $700(1.03)=\$721.00$.

Year-3 deposit at year 3: $700(1.03)^0=\$700.00$.

The balance is $B_3=\$742.63+\$721.00+\$700.00=\$2,163.63$. Total deposits are $C_3=3(\$700)=\$2,100.00$, so accumulated earnings are $E_3=\$2,163.63-\$2,100.00=\$63.63$.

Check using the recurrence: $B_1=\$700.00$; $B_2=\$700(1.03)+\$700=\$1,421.00$; and $B_3=\$1,421.00(1.03)+\$700=\$2,163.63$. The balance agrees. The common mistake avoided is giving the first deposit only one year of interest: it earns for two complete years before the end-of-year-3 valuation.

Why the Difference Is the Accumulated Earnings

The decomposition follows directly from the fund’s annual growth. During year $k$, the interest credited is $i_fB_{k-1}$, because $B_{k-1}$ is the balance at the beginning of that year. The deposit $D_k$ is then added at year-end. Thus the balance increase is the year’s interest plus the new deposit.

B_k-B_{k-1}=i_fB_{k-1}+D_k

Summing this equation from year 1 through year $n$ makes the balance changes telescope: every intermediate balance appears once positively and once negatively. With an initial balance $B_0=0$, the result is $B_n=\sum_{k=1}^n i_fB_{k-1}+\sum_{k=1}^nD_k$. The first sum is all interest credited to the fund, while the second is all deposits. Therefore, final balance minus total deposits equals accumulated interest.

B_n-C_n=\sum_{k=1}^{n}i_fB_{k-1}

This derivation also clarifies the assumption behind the subtraction. If there are withdrawals, fees, or other cash flows, the difference between the balance and deposits need not equal gross interest earned. For example, a fee reduces the balance even though it is not a negative deposit. In that case, identify and account for each cash flow separately.

Worked Example: Unequal Deposits With a Year Remaining

A hypothetical fund earns an effective annual rate of 5%. The borrower deposits $500 at the end of year 1 and $900 at the end of year 2, with no deposit at the end of year 3. Find the balance, total deposits, and accumulated earnings immediately after year 3. State how long each deposit earns interest.

Method. Accumulate both deposits to the same valuation date, the end of year 3. The first deposit earns for two years; the second earns for one year.

The year-1 deposit accumulates to $ \$500(1.05)^2=\$500(1.1025)=\$551.25$. The year-2 deposit accumulates to $ \$900(1.05)=\$945.00$.

The balance is $B_3=\$551.25+\$945.00=\$1,496.25$. Total deposits are $C_3=\$500+\$900=\$1,400.00$. Accumulated earnings are $E_3=\$1,496.25-\$1,400.00=\$96.25$.

Check: Interest on the first deposit is $\$551.25-\$500=\$51.25$; interest on the second is $\$945-\$900=\$45.00$. Their sum is $\$51.25+\$45.00=\$96.25$, as found by subtraction. The common mistake avoided is treating the year-2 deposit as if it earned two years of interest; it was not in the fund during year 1.

Level Deposits and the Accumulated-Value Factor

When equal deposits $R$ are made at the end of each year for $n$ years, the same deposit-by-deposit principle gives $R$ accumulated for $n-1,n-2,\ldots,1,0$ years. In standard actuarial notation, the accumulated value factor for an annuity-immediate is $s_{\overline{n}|}$, evaluated at the fund rate. This provides a convenient shortcut, but the timing must still match end-of-year deposits.

B_n=R s_{\overline{n}|} =R\frac{(1+i_f)^n-1}{i_f}, \qquad C_n=nR, \qquad E_n=R\left(s_{\overline{n}|}-n\right)

The formula applies when there are $n$ equal deposits at the ends of years 1 through $n$, the rate is constant and effective annually, and the balance is valued immediately after the final deposit. It does not describe deposits at the beginnings of years without an adjustment: beginning-of-year deposits have an additional year of accumulation each.

Worked Example: Using the Annuity-Immediate Factor

A hypothetical fund earns 4% effective annually. A borrower deposits $600 at the end of each of years 1, 2, and 3. Find the year-3 balance, total deposits, and accumulated earnings using the annuity-immediate factor. Then verify the balance by accumulating the deposits individually.

Method. The three deposits are at year-end, so they form a three-payment annuity-immediate at rate $i_f=0.04$.

First calculate the accumulation factor: $s_{\overline{3}|}=\frac{(1.04)^3-1}{0.04}=\frac{1.124864-1}{0.04}=\frac{0.124864}{0.04}=3.1216$.

The balance is $B_3=\$600(3.1216)=\$1,872.96$. Total deposits are $C_3=3(\$600)=\$1,800.00$, and accumulated earnings are $E_3=\$1,872.96-\$1,800.00=\$72.96$.

Check by individual accumulation: the deposits earn for two, one, and zero years, respectively. Thus $B_3=\$600[(1.04)^2+1.04+1]=\$600[1.0816+1.04+1]=\$600(3.1216)=\$1,872.96$. The factor and deposit-by-deposit methods agree. The common mistake avoided is counting the final deposit as earning a full year; it is made at the valuation date and earns no interest by that date.

Irregular Timing and a Common Exam Trap

With unequal deposits, missing deposits, or a valuation date later than the last contribution, use a timeline before calculating. For each deposit, write its date, the valuation date, and the number of complete accumulation periods between them. Then calculate the balance, add the original deposits separately, and take the difference.

A frequent error is to use the number of deposits as the exponent or as the number of interest periods. These are different quantities. A fund can have three deposits, for example, while the earliest deposit earns for four years and the latest earns for one. Another error is to report the final fund balance as “earnings”; the balance includes contributed principal as well as growth.

Worked Example: Deposits in Nonconsecutive Years

A hypothetical fund earns an effective annual rate of 6%. Deposits of $1,000, $1,400, and $800 are made at the ends of years 1, 3, and 4, respectively. Find the balance at the end of year 5, total deposits, and accumulated earnings. No deposit is made at the end of year 2 or year 5.

Method. Measure each deposit’s accumulation period to the end of year 5. The deposits earn for four, two, and one years, respectively.

The year-1 deposit accumulates to $\$1,000(1.06)^4=\$1,000(1.26247696)=\$1,262.47696$. The year-3 deposit accumulates to $\$1,400(1.06)^2=\$1,400(1.1236)=\$1,573.04$. The year-4 deposit accumulates to $\$800(1.06)=\$848.00$.

Adding these values gives $B_5=\$1,262.47696+\$1,573.04+\$848.00=\$3,683.51696$, or $3,683.52 to the nearest cent. Total deposits are $C_5=\$1,000+\$1,400+\$800=\$3,200.00$. Accumulated earnings are $\$3,683.51696-\$3,200.00=\$483.51696$, or $483.52 to the nearest cent.

Check: the three deposits accumulate to a balance exceeding their $3,200.00 total by $483.51696. This matches the sum of the interest earned on each deposit: $\$262.47696+\$173.04+\$48.00=\$483.51696$. The common mistake avoided is assigning exponents based on deposit order rather than elapsed time to the end-of-year-5 valuation.

Interpretation and Practical Checks

The decomposition is useful whenever a problem asks how much of a sinking-fund balance reflects the borrower’s own contributions and how much reflects fund growth. The calculation concerns the fund account only. Its balance, deposits, and earnings should not be conflated: deposits are cash contributions, earnings are accumulated growth under the stated fund model, and the maturity balance is their combined value.

Before finalizing a calculation, verify the following:

One valuation date: accumulate every deposit to the same date before summing.
Correct periods: an end-of-year deposit earns only in complete periods after it is deposited.
Separate totals: sum the original deposits without accumulation to obtain total contributions.
Consistent assumptions: the simple balance-minus-deposits interpretation assumes no withdrawals, fees, or unrecorded cash flows.
Arithmetic check: confirm that balance equals deposits plus accumulated earnings.

Key takeaway: At a stated valuation date, accumulate each deposit according to its actual timing to find the fund balance. Add the unaccumulated deposits to find total contributions; when there are no other cash flows, the balance less total deposits is accumulated fund earnings.

Check Your Understanding

For each hypothetical fund, find the requested values and check that balance equals deposits plus earnings. All stated rates are effective annual rates, and deposits are made at the ends of the specified years.

A fund earns 5%. Deposits of $400 are made at the ends of years 1, 2, and 3. Find the balance immediately after the third deposit, total deposits, and accumulated earnings. (Answer: balance $1,261.00; deposits $1,200.00; earnings $61.00.)
A fund earns 4%. Deposits of $800 at the end of year 1 and $500 at the end of year 2 remain invested until the end of year 4. Find the balance, total deposits, and earnings. (Answer: balance $1,440.69; deposits $1,300.00; earnings $140.69.)
A fund earns 3%. Deposits of $300 and $700 are made at the ends of years 1 and 3, respectively. Find the balance at the end of year 4, total deposits, and earnings. (Answer: balance $1,048.82; deposits $1,000.00; earnings $48.82.)
A fund earns 2%. Deposits of $1,200 are made at the ends of years 1, 2, 3, and 4. Find the balance immediately after the fourth deposit, total deposits, and earnings. (Answer: balance $4,945.93; deposits $4,800.00; earnings $145.93.)

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