Measuring a Sinking-Fund Surplus or Shortfall

Why Measure the Fund Against the Redemption Target?

A sinking fund is intended to accumulate money for a specified future payment, often a loan redemption. The projected fund balance at that date may be greater than, equal to, or less than the amount that must be paid. Measuring the difference tells the borrower whether the plan projects a surplus or a shortfall, and by how much.

The comparison is made at a single date: the redemption date. Deposits made earlier must be accumulated to that date using the fund’s assumed yield. The target is the amount actually required for redemption; it need not equal the original loan principal. The result is a projection under stated assumptions, not a guarantee of the fund’s realized value.

Definition: Let $F_n$ be the projected sinking-fund value at the redemption date and $R$ the required redemption amount at that same date. The surplus is $F_n-R$ when this difference is positive. The shortfall is $R-F_n$ when the difference is positive. If $F_n=R$, the fund exactly meets the target.

Projecting the Maturity Value

First specify the model. Suppose the fund earns a fixed effective rate $j$ per deposit period, deposits of $D$ are made at the end of each period, and there are $n$ periods to redemption. Assume there are no withdrawals, fees, taxes, or other cash flows unless they are explicitly included. The deposits are not loan repayments: they accumulate in the fund while the loan’s redemption obligation remains as stated in the loan terms.

Under these assumptions, the first deposit earns interest for $n-1$ periods, the second for $n-2$ periods, and the last earns no interest before redemption. Thus the projected value is the accumulated value of an annuity-immediate:

F_n=D\left((1+j)^{n-1}+(1+j)^{n-2}+\cdots +(1+j)+1\right) =D s_{\overline{n}|} =D\frac{(1+j)^n-1}{j}, \qquad j\ne 0.

If the fund earns zero interest, each deposit remains unchanged and $F_n=nD$. If there is an opening balance $B_0$ at the start of the first period, it also accumulates to redemption, so

F_n=B_0(1+j)^n+D s_{\overline{n}|}.

The rate $j$ must match the length of a deposit period. For example, annual deposits require an effective annual fund rate; quarterly deposits require a rate applicable to each quarter. If returns are not constant, the fixed-rate formula is not appropriate: project each balance using the returns that apply in each period, and compare the resulting value with the target.

Why the Annuity Formula Works

The accumulation formula follows by adding the value of each deposit at the redemption date. Equivalently, let $F_k$ be the fund immediately after the $k$th end-of-period deposit. The opening balance for period $k$ earns interest, and then the new deposit is added:

F_k=(1+j)F_{k-1}+D,\qquad F_0=0.

Applying this recurrence repeatedly gives $F_1=D$, $F_2=D(1+j)+D$, and $F_3=D(1+j)^2+D(1+j)+D$. Continuing to period $n$ gives the sum of $n$ powers in the annuity formula. For $j\ne0$, the geometric-sum identity yields $\big((1+j)^n-1\big)/j$. This derivation also explains why an end-of-period deposit earns no interest during the period in which it is made.

Once $F_n$ is projected, the comparison is direct:

\text{signed difference}=F_n-R.

A positive difference is a surplus; a negative difference indicates a shortfall whose amount is $R-F_n$. If a single additional payment is made exactly at redemption, that payment must equal the shortfall. If instead the plan is to add equal deposits at each period-end, and the same fixed-rate assumptions apply, the additional deposit $x$ satisfies $x s_{\overline{n}|}=\text{shortfall}$.

Worked Example: A Projected Surplus

A borrower must pay $R=\$50{,}000$ at the end of five years. The sinking fund receives $\$10{,}000$ at the end of each year and earns an effective annual rate of $5\%$. Find the projected surplus or shortfall at redemption.

Method. Use the accumulated value of five end-of-year deposits, then subtract the redemption target. The deposits are end-of-year payments, so the correct factor is $s_{\overline{5}|}$, not an annuity-due factor.

The annuity accumulation factor is

s_{\overline{5}|}=\frac{(1.05)^5-1}{0.05} =\frac{1.2762815625-1}{0.05} =5.52563125.

Therefore the projected fund value is

F_5=\$10{,}000(5.52563125)=\$55{,}256.3125.

Compare this with the required redemption amount:

F_5-R=\$55{,}256.3125-\$50{,}000=\$5{,}256.3125.

Rounded to the nearest cent, the projected surplus is \$5,256.31. As a check, the projected value exceeds the target, so the signed difference is positive and the result is a surplus.

Shortfalls and Corrective Payments

A shortfall does not change the amount owed under the loan contract. It measures the difference between the fund projection and the redemption requirement. A borrower may need to make an additional payment, increase future deposits, or use another source of funds. The appropriate action depends on the arrangement; the mathematical comparison alone does not establish that a surplus can automatically be withdrawn or that a shortfall will be covered.

Worked Example: A Shortfall at Redemption

A hypothetical loan requires a redemption payment of $\$80{,}000$ at the end of six years. The borrower makes six deposits of $\$11{,}000$, each at year-end. The fund earns an effective annual rate of $4\%$. Find the projected shortfall and the single extra payment required at the redemption date.

Method. Accumulate the six deposits using $s_{\overline{6}|}$, compare with the redemption amount, and treat any correction made at the redemption date as a payment with no further accumulation.

First calculate the annuity factor:

s_{\overline{6}|}=\frac{(1.04)^6-1}{0.04} =\frac{1.265319018496-1}{0.04} =6.6329754624.

The projected fund value is

F_6=\$11{,}000(6.6329754624)=\$72{,}962.7300864.

Since this is below the target, calculate the shortfall as target minus fund value:

R-F_6=\$80{,}000-\$72{,}962.7300864 =\$7{,}037.2699136.

Rounded to the nearest cent, the projected shortfall is \$7,037.27. A single additional payment of $\$7,037.27$ at the redemption date would bring the projected total to $\$80,000$, subject to cent-rounding in the actual transaction. The check is that the fund value plus the unrounded shortfall equals the target exactly.

Including an Existing Fund Balance

If money is already in the fund at the start of the projection, it must be accumulated for the entire remaining term. It does not receive the same accumulation factor as the later end-of-period deposits. This distinction matters because an opening balance is invested earlier than any of the future deposits.

Worked Example: Existing Balance and a Different Redemption Target

A hypothetical borrower has an opening fund balance of $\$8{,}000$ and will deposit $\$7{,}000$ at the end of each of the next four years. The fund earns an effective annual rate of $3\%$. The loan contract requires $\$40{,}000$ at redemption, even though the original principal was $\$42{,}000$. Find the projected surplus or shortfall.

Method. Accumulate the opening balance for all four years, accumulate the four future deposits as an annuity-immediate, add those values, and compare the total with the contractual redemption amount. The original principal is not the target in this calculation.

The opening balance accumulates to

\$8{,}000(1.03)^4 =\$8{,}000(1.12550881) =\$9{,}004.07048.

For the four future deposits,

s_{\overline{4}|}=\frac{(1.03)^4-1}{0.03} =\frac{1.12550881-1}{0.03} =4.183627,

so their projected value is

\$7{,}000(4.183627)=\$29{,}285.389.

The total projected fund value is

F_4=\$9{,}004.07048+\$29{,}285.389 =\$38{,}289.45948.

Compare with the stated redemption requirement:

R-F_4=\$40{,}000-\$38{,}289.45948 =\$1{,}710.54052.

The projected shortfall is \$1,710.54 to the nearest cent. The calculation uses $\$40{,}000$, not the original principal of $\$42{,}000$, because the problem states the actual redemption target.

Common Exam Trap: Applying the Wrong Timing or Rate

Two inputs deserve a deliberate check before calculating: when deposits are made and what rate the fund earns. An end-of-period deposit does not earn interest in the period in which it is deposited. Using an annuity-due factor would give every deposit one extra period of interest. In the first worked example, that incorrect calculation would be

\$10{,}000(1.05)s_{\overline{5}|} =\$58{,}019.128125,

which incorrectly reports a surplus of $\$8,019.13$. The correct end-of-year projection is $\$55,256.31$, a surplus of $\$5,256.31$. The timing correction is to use $s_{\overline{5}|}$, without the extra factor $1.05$.

Likewise, use the fund’s assumed yield to project the fund. The loan rate is relevant to the loan’s interest obligation, but it is not a substitute for the fund rate unless the problem explicitly states that the fund earns that rate. Keep the dates, target, deposit timing, and fund yield aligned before deciding whether there is a surplus or shortfall.

Key takeaway: Project every fund component to the redemption date under the stated fund assumptions, then compare that value with the required redemption amount at the same date. A positive difference is a surplus; a negative difference is a shortfall. Do not replace the redemption target with loan principal or confuse fund growth with principal repayment.

Check Your Understanding

For each problem, compare the projected fund value with the redemption target. Assume deposits are made at each year-end, and round dollar answers to the nearest cent.

A fund receives $\$3,000$ at the end of each year for four years and earns an effective annual rate of $5\%$. The redemption target is $\$13,000$. Find the projected surplus or shortfall. (Answer: $\$69.63$ shortfall)
An opening balance of $\$5,000$ earns an effective annual rate of $2\%$ for three years. In addition, $\$4,000$ is deposited at each of the next three year-ends. The target is $\$18,000$. Find the projected shortfall. (Answer: $\$452.36$ shortfall)
A fund receives five end-of-year deposits of $\$4,500$ and earns an effective annual rate of $4\%$. The required redemption amount is $\$25,000$. Find the projected shortfall. (Answer: $\$626.55$ shortfall)
An opening balance of $\$2,000$ earns $3\%$ effective annually for four years. Four deposits of $\$7,000$ are made at the end of each year. If the redemption target is $\$32,000$, find the projected surplus or shortfall. (Answer: $\$463.59$ shortfall)
A fund starts with $\$4,000$, earns an effective annual rate of $3\%$, and receives five end-of-year deposits of $\$6,000$. The target is $\$40,000$. How much additional money must be paid at the redemption date to meet the target? (Answer: $\$3,508.09$)

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