The Fund Balance After Any Number of Deposits

Why the Valuation Date Matters

A sinking fund balance is the accumulated value of deposits made into a fund. To calculate it after a specified number of deposits, track two things: how many deposits have been made and how long each deposit has earned interest. The phrase “after the $k$th deposit” usually means immediately after that deposit, before it earns interest for another period.

Deposit timing changes the calendar date associated with that balance. With end-of-year deposits, the balance immediately after deposit $k$ is at time $k$. With beginning-of-year deposits, it is at time $k-1$. In either case, if the deposits are equal and one period apart, the immediate post-deposit balance has the same accumulation pattern: the most recent deposit has earned no interest, the preceding deposit has earned one period of interest, and so on.

Definition: Suppose a fund starts with a zero balance and receives a level deposit of $R$ at the end of each year. The balance $B_k$ immediately after the $k$th deposit is the accumulated value, at that date, of all $k$ deposits. If the fund earns an effective annual rate $i$, then each deposit is accumulated for the number of complete years between its deposit date and the valuation date.

The same definition applies to beginning-of-year deposits, provided the valuation date is specified. In this tutorial, $B_k$ denotes the value immediately after the $k$th deposit unless another date is explicitly stated. Assume a constant effective annual fund rate $i$, level deposits, and no withdrawals, fees, or other cash flows. These formulas do not directly apply when the fund rate varies or when deposits are unequal.

Accumulated-Value Formula

For end-of-year deposits, the first deposit earns interest for $k-1$ years by the time the $k$th deposit is made. The second earns interest for $k-2$ years, continuing down to the last deposit, which earns no interest before the valuation date. Thus

B_k=R\left((1+i)^{k-1}+(1+i)^{k-2}+\cdots +(1+i)+1\right).

The terms form a geometric series. For $i\ne 0$, its sum is $s_{\overline{k}|}$, the accumulated value of an annuity-immediate of $k$ annual payments of 1 at effective annual rate $i$. Therefore,

B_k=R s_{\overline{k}|} =R\frac{(1+i)^k-1}{i}.

The formula also gives the immediate post-deposit balance for beginning-of-year deposits. The deposits then occur at times $0,1,\ldots,k-1$, and the value immediately after deposit $k$, at time $k-1$, is again $R$ times the sum of powers from $(1+i)^{k-1}$ down to 1. The calendar dates differ, but the deposit ages at the respective immediate post-deposit dates are the same.

Important timing distinction: For beginning-of-year deposits, if the valuation date is one full year after the $k$th deposit, the balance is $B_k(1+i)$, not $B_k$. The deposit has earned an additional year of interest. State the valuation date before choosing the formula.

Why the Formula Holds: A Recurrence

A recurrence gives the same balance by building it one deposit at a time. Let $B_0=0$. Between two successive deposits, the existing balance earns one year of interest; then the next end-of-year deposit is added. Hence

B_k=(1+i)B_{k-1}+R,\qquad k\geq 1.

For example, the first balances are $B_1=R$, $B_2=R(1+i)+R$, and $B_3=R(1+i)^2+R(1+i)+R$. Continuing this pattern gives

B_k=R\left((1+i)^{k-1}+(1+i)^{k-2}+\cdots+1\right),

which is the accumulated-value formula. This explains why the recurrence and the annuity formula agree: each recurrence step grows every earlier deposit by one more period and then includes the new deposit. The recurrence is especially useful for checking an answer or tracking balances through intermediate deposit numbers.

For beginning-of-year deposits, the recurrence depends on the balance date. If $B_k$ still means immediately after deposit $k$, the prior post-deposit balance earns a year before the next deposit is made, so $B_k=(1+i)B_{k-1}+R$ as well. If instead the balance is measured at the end of each year after a beginning-of-year deposit has earned interest, the recurrence must include interest on that year’s deposit too. A clear time index prevents these two balances from being confused.

Worked Example: End-of-Year Deposits and Two Calculation Methods

A hypothetical reserve fund starts at zero and receives deposits of $250 at the end of each year. It earns an effective annual rate of 4%. Find the fund balance immediately after the fifth deposit.

Method: accumulated value. The five deposits have earned interest for 4, 3, 2, 1, and 0 years, respectively. The needed powers are $1.04^2=1.0816$, $1.04^3=1.124864$, and $1.04^4=1.16985856$. Therefore,

B_5=250(1.16985856+1.124864+1.0816+1.04+1) =250(5.41632256) =\$1{,}354.08064.

Method: recurrence. Start with $B_0=0$, and apply $B_k=1.04B_{k-1}+250$:

\begin{aligned} B_1&=1.04(0)+250=250,\\ B_2&=1.04(250)+250=510,\\ B_3&=1.04(510)+250=780.40,\\ B_4&=1.04(780.40)+250=1{,}061.216,\\ B_5&=1.04(1{,}061.216)+250=1{,}353.66464. \end{aligned}

The recurrence values reveal an arithmetic error: the fourth-step calculation should be $1.04(780.40)=811.616$, so $B_4=1{,}061.616$. Then $B_5=1.04(1{,}061.616)+250=1{,}354.08064$, agreeing with the accumulated-value method. Rounding only at the final step, the balance is $1,354.08. The independent formula check catches the deliberately visible multiplication slip before it becomes the reported answer.

Common mistake avoided: Using five years of interest on the first deposit would shift every deposit one period too far. At the stated date, the first deposit has earned four years, not five.

Worked Example: Beginning-of-Year Deposits and a Later Valuation Date

A hypothetical fund receives $180 at the beginning of each year and earns 5% effective annual interest. Find its balance (a) immediately after the fourth deposit and (b) one full year later, before the next deposit.

(a) Immediately after deposit four. The deposits occur at times 0, 1, 2, and 3. At time 3, their respective accumulation periods are 3, 2, 1, and 0 years. Thus

B_4=180(1.05^3+1.05^2+1.05+1) =180(1.157625+1.1025+1.05+1) =180(4.310125) =\$775.8225.

The recurrence gives the same result: $B_1=180$, $B_2=1.05(180)+180=369$, $B_3=1.05(369)+180=567.45$, and $B_4=1.05(567.45)+180=775.8225$.

(b) One year later. No additional deposit has yet been made, so the entire balance earns one year of interest:

775.8225(1.05)=\$814.613625.

Rounding only at the final step, the answers are $775.82 immediately after the fourth deposit and $814.61 one year later. The later amount is not the immediate post-deposit balance; its extra factor of 1.05 reflects the additional year.

Common mistake avoided: Treating a balance “one year later” as if it were still at the deposit date omits the interest earned after the last deposit.

Worked Example: Finding an Intermediate Balance in a Longer Schedule

A hypothetical fund receives $400 at each year-end and earns 3% effective annual interest. Calculate the balance immediately after deposit three and immediately after deposit six. Use the first balance to check the later one by recurrence.

After three deposits. The three accumulation periods are 2, 1, and 0 years:

B_3=400(1.03^2+1.03+1) =400(1.0609+1.03+1) =400(3.0909) =\$1{,}236.36.

Continue by recurrence. Apply $B_k=1.03B_{k-1}+400$ for each of the next three deposits:

\begin{aligned} B_4&=1.03(1{,}236.36)+400=1{,}673.4508,\\ B_5&=1.03(1{,}673.4508)+400=2{,}123.654324,\\ B_6&=1.03(2{,}123.654324)+400=2{,}587.36395372. \end{aligned}

As a formula check, $1.03^3=1.092727$, $1.03^4=1.12550881$, and $1.03^5=1.1592740743$. Therefore,

B_6=400(1+1.03+1.0609+1.092727+1.12550881+1.1592740743) =400(6.4684098843) =\$2{,}587.36395372.

The two methods agree. Rounded to the nearest cent, the balances are $1,236.36 after deposit three and $2,587.36 after deposit six. Keeping the unrounded balance between recurrence steps avoids accumulating rounding error.

Zero Interest, Checks, and Common Traps

When $i=0$, the expression $\big((1+i)^k-1\big)/i$ involves division by zero and should not be used. With no interest, each deposit remains unchanged, so $B_k=kR$. The recurrence still works: $B_k=B_{k-1}+R$. For a positive fund rate, $B_k$ should be at least $kR$, since earlier deposits earn interest. If a computed balance is smaller than total deposits under a nonnegative rate, check the rate, exponent, and timing.

Counting periods instead of deposits: After $k$ deposits, there are $k$ payment amounts in the sum, but the first end-of-year deposit has earned only $k-1$ years of interest.
Moving the valuation date: An immediate post-deposit balance and a balance one period later differ by a factor of $1+i$ if no cash flow occurs between the dates.
Mixing a due value with an immediate post-deposit value: Beginning-of-year deposits do not mean the balance immediately after the last deposit has earned an extra period. Add that period’s interest only when the requested date is one period later.
Rounding too early: Carry full precision through the recurrence and round the final dollar amount to the nearest cent.

For an exam calculation, label the dates first, identify the age of each deposit at the requested valuation date, and then select either the geometric sum or recurrence. The two methods provide a useful cross-check, especially when the requested number of deposits is not the full length of a fund schedule.

Check Your Understanding

For each problem, assume a zero initial fund balance, a constant effective annual rate, and no withdrawals or fees. Round dollar answers to the nearest cent.

A fund receives $120 at each year-end and earns 5%. Find its balance immediately after the third deposit. (Answer: $378.30)
A fund receives $250 at the beginning of each year and earns 4%. Find its balance immediately after the fifth deposit. (Answer: $1,354.08)
A fund receives $600 at each year-end and earns 2%. Find its balance immediately after the fourth deposit. (Answer: $2,472.96)
A fund receives $300 at each year-end and earns 6%. Find its balance immediately after the seventh deposit. (Answer: $2,518.15)
A fund receives $175 at the beginning of each year and earns 3%. Find the balance immediately after deposit four, then find the balance one year later before another deposit. (Answer: $732.13 immediately after deposit four; $754.10 one year later)

What You'll Learn