Arithmetic-Increasing Sinking-Fund Deposits

Introduction

A sinking fund may be designed so that contributions grow over time rather than remain level. For example, a borrower might deposit a fixed amount at the end of the first year and increase each later deposit by the same number of dollars. To value the fund correctly, separate the original deposit level from the increases and accumulate each payment from its actual deposit date.

This tutorial models annual deposits made at the end of each year into a fund earning a constant effective annual interest rate. It develops the maturity-value formula, shows how to solve for a required first deposit, and explains why the first payment is not one increment larger than the stated starting deposit. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: An arithmetic-increasing sinking-fund schedule has deposits $R,\ R+G,\ R+2G,\ldots,R+(n-1)G$ at the ends of years $1,2,\ldots,n$. Here $R$ is the first deposit, $G$ is the fixed dollar increase per year, and $n$ is the number of deposits. The fund earns an effective annual rate $i$, credited once per year. The model assumes deposits are made as scheduled, the rate is constant, and no other withdrawals or deposits occur.

Accumulating the Deposits

A deposit made at the end of year $k$ earns interest for $n-k$ years before the end of year $n$. Its accumulated value at maturity is therefore its amount multiplied by $(1+i)^{n-k}$. The total maturity value is the sum of all deposits accumulated to that date.

F_n=\sum_{k=1}^{n}\bigl(R+(k-1)G\bigr)(1+i)^{n-k}

Split each deposit into a level part $R$ and an increasing part $(k-1)G$. The level deposits accumulate to $R s_{\overline{n}|}$, where $s_{\overline{n}|}$ is the accumulated value of an annuity-immediate of 1 per year at rate $i$. The increases have accumulated value $G C_n$, where

C_n=\sum_{k=1}^{n}(k-1)(1+i)^{n-k}.

Theorem: For end-of-year deposits $R,\ R+G,\ldots,R+(n-1)G$, the maturity value is $$ F_n=R s_{\overline{n}|}+G C_n, \qquad C_n=\frac{(1+i)^n-1-ni}{i^2}\quad (i\ne 0). $$ If $i=0$, then $s_{\overline{n}|}=n$ and $C_n=\frac{n(n-1)}{2}$, so $F_n=nR+G\frac{n(n-1)}{2}$.

The formula for $C_n$ assumes the increase begins at zero: the first deposit is $R$, not $R+G$. This timing detail is built into the factors $k-1$ in the sum. If instead the first deposit were $R+G$, the schedule would be a different one.

Why the Increasing-Deposit Factor Works

Write $q=1+i$ and index the increasing deposits by how many years they earn interest. A deposit of $rG$, for $r=0,1,\ldots,n-1$, is made $n-r$ years before maturity. Thus $C_n=\sum_{r=0}^{n-1}(n-1-r)q^r$. To simplify this sum, use the finite geometric series:

\begin{aligned} iC_n &=\sum_{r=0}^{n-1}(n-1-r)(q-1)q^r\\ &=\sum_{r=0}^{n-1}(n-1-r)(q^{r+1}-q^r)\\ &=\sum_{r=0}^{n-1}q^r-n\\ &=\frac{q^n-1}{i}-n. \end{aligned}

In the second-to-last line, like powers cancel after the two sums are expanded, leaving the geometric sum and the subtraction of $n$. Dividing by $i$ gives $C_n=((1+i)^n-1-ni)/i^2$. When $i=0$, division by $i$ is not valid; use the original sum instead. It contains $0+1+\cdots+(n-1)=n(n-1)/2$, which gives the stated zero-rate result.

Worked Examples

Worked Example: Accumulating a Fund With a Fixed Annual Increase

A hypothetical fund receives four end-of-year deposits. The first is $1,000, and each subsequent deposit is $200 larger than the previous one. The fund earns 5% effective annual interest. Find the maturity value at the end of year 4. Round the final answer to the nearest cent.

Method. Apply the split between the level deposit and the arithmetic increases. The deposits are $1,000, $1,200, $1,400, and $1,600. Here $R=1{,}000$, $G=200$, $n=4$, and $i=0.05$.

First calculate the annuity accumulation factor:

s_{\overline{4}|}=\frac{(1.05)^4-1}{0.05} =\frac{1.21550625-1}{0.05} =4.310125.

Next calculate the factor for the increases:

C_4=\frac{(1.05)^4-1-4(0.05)}{(0.05)^2} =\frac{1.21550625-1-0.20}{0.0025} =6.2025.

The maturity value is

F_4=1{,}000(4.310125)+200(6.2025) =4{,}310.125+1{,}240.50 =5{,}550.625.

Rounded to the nearest cent, the maturity value is $5,550.63. As a check, accumulating the deposits individually gives $1{,}000(1.05)^3=1{,}157.625$, $1{,}200(1.05)^2=1{,}323$, $1{,}400(1.05)=1{,}470$, and $1{,}600$. Their total is $5,550.625, as required. The schedule starts at $1,000; treating the first deposit as $1,200 would avoid neither the timing error nor the resulting overstatement.

Worked Example: Solving for the Required First Deposit

A hypothetical reserve must have $5,895 immediately after the third annual deposit. The deposits are made at the ends of years 1, 2, and 3, and each is $300 larger than the previous deposit. The fund earns 10% effective annual interest. Find the first deposit.

Method. Use the maturity-value equation and isolate $R$. The increase is $G=300$, the term is $n=3$, and $i=0.10$.

The level-payment accumulation factor is

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

The factor for the increases is

C_3=\frac{1.331-1-3(0.10)}{(0.10)^2} =\frac{1.331-1-0.30}{0.01} =3.1.

Substitute into $F_n=R s_{\overline{n}|}+G C_n$ and solve:

5{,}895=3.31R+300(3.1) =3.31R+930, $$ $$ R=\frac{5{,}895-930}{3.31} =\frac{4{,}965}{3.31} =1{,}500.

The first deposit must be $1,500, giving deposits of $1,500, $1,800, and $2,100. Check their values at the end of year 3: $1{,}500(1.10)^2=1{,}815$, $1{,}800(1.10)=1{,}980$, and $2,100. The total is $5,895. The calculation avoids the mistake of treating the specified $300 increase as part of the first deposit.

Worked Example: Solving for the Annual Increase

A hypothetical account will receive three end-of-year deposits. The first deposit is $800, the fund earns 10% effective annual interest, and the required value immediately after the third deposit is $3,000. Find the fixed dollar increase between consecutive deposits.

Method. Calculate the value of the level deposits, then assign the remaining required value to the increasing part. From the preceding factor calculation, $s_{\overline{3}|}=3.31$ and $C_3=3.1$.

The level part accumulates to $800(3.31)=2{,}648$. Therefore, the increases must accumulate to $3{,}000-2{,}648=352$. Since their accumulation is $G C_3$,

352=3.1G, \qquad G=\frac{352}{3.1}=113.548387\ldots.

To the nearest cent, the fixed increase is $113.55 per year. The unrounded increase makes the formula give exactly $3,000. The deposits are approximately $800, $913.55, and $1,027.10 when each is displayed to cents; checking with those rounded deposits can produce a small difference from the target. The calculation itself uses the unrounded increase, so it does not treat intermediate rounding as exact.

Solving and Checking the Model

Once $n$, $i$, and $G$ are specified, the required first deposit follows directly from the maturity-value equation:

R=\frac{F_n-GC_n}{s_{\overline{n}|}}.

This expression is useful when a target fund value is known. For a nonnegative first deposit, the target must satisfy $F_n\geq G C_n$, assuming $i\geq 0$, $G\geq 0$, and $n\geq 1$. If the target is smaller than the accumulated value of the increases alone, the equation produces a negative $R$; that is not a valid nonnegative starting deposit under this schedule.

If $R$ is known instead, solve for the increase:

G=\frac{F_n-Rs_{\overline{n}|}}{C_n}, \qquad C_n>0.

For this second calculation, $C_n>0$ when $n\geq2$. With only one deposit, no increase is ever made, so the maturity value is independent of $G$ and the increase cannot be inferred. Likewise, if a required increase calculated from the equation is negative, the target calls for decreasing rather than increasing deposits.

Fix the dates.
Write down when each deposit is made and the date at which the fund is valued. For the formula here, deposits occur at the ends of years 1 through $n$, and value is measured at the end of year $n$.

State the schedule.
Identify the first deposit $R$ and the fixed increase $G$. The deposit at the end of year $k$ is $R+(k-1)G$.

Accumulate the two parts.
Use $R s_{\overline{n}|}$ for the level portion and $G C_n$ for the increases, with the rate applying over each annual deposit interval.

Check the result.
Confirm that the deposits rise by exactly $G$, that each is accumulated for the correct number of years, and that the calculated maturity value matches the target before rounding.

Common Timing Mistake

A frequent error is to interpret “the deposits increase by $G each year” as meaning the first deposit is $R+G$. Under the definition used here, $R$ is the first deposit, so the increase has been applied zero times at the end of year 1, once at the end of year 2, and $n-1$ times at the end of year $n$.

For the first worked example, incorrectly starting at $1,200 would shift every deposit upward by $200. The incorrect maturity value would be

5{,}550.625+200s_{\overline{4}|} =5{,}550.625+200(4.310125) =6{,}412.65.

That is not a rounding discrepancy: it is the value of a different deposit schedule. Always write the first few deposits before using a formula. This also helps identify a timing mismatch: deposits made at the beginning rather than the end of each year would earn one additional year of interest and would require a different accumulation.

Key takeaway: For end-of-year deposits $R,\ R+G,\ldots,R+(n-1)G$, accumulate the level part and the increases separately: $F_n=R s_{\overline{n}|}+G C_n$. The first deposit is $R$, and the increase is applied only to later deposits.

Check Your Understanding

For each problem, assume deposits are made at the stated annual dates and the fund earns a constant effective annual rate. Round dollar answers to the nearest cent.

A fund receives four end-of-year deposits. The first is $700, each later deposit is $150 larger than the preceding one, and the fund earns 4% annually. Find the value immediately after the fourth deposit. (Answer: $3,896.76)
A fund earns no interest and receives five end-of-year deposits increasing by $100 each year. The target value immediately after the fifth deposit is $6,000. Find the first deposit. (Answer: $1,000)
A fund earns 5% annually and receives four end-of-year deposits, starting at $900 and increasing by $120 each year. Find the value immediately after the fourth deposit. (Answer: $4,623.41)
A fund earns 10% annually and receives three end-of-year deposits, starting at $800. What fixed increase is needed for the value immediately after the third deposit to equal $3,000? (Answer: $113.55)
A hypothetical fund earns 6% annually and receives deposits of $500, $600, and $700 at the beginnings of years 1, 2, and 3, respectively. Find its value at the end of year 3. (Answer: $2,011.67)

What You'll Learn