Rearranging Single-Sum Formulas to Find the Deposit

Finding a Deposit From a Target Amount

A single-sum problem often gives the amount needed at a future date and asks how much to set aside today. The key is to identify which amount is unknown and at which date it is measured. A deposit today must be smaller than a positive future target when interest is positive, because the deposit grows before the target date.

This tutorial uses annual compounding over a whole number of years. Assume a single deposit is made at time 0, remains invested for $n$ years, and earns a constant effective annual interest rate $i$. There are no additional deposits or withdrawals, and no fees or taxes. These assumptions are important: the formulas below describe one payment, not a stream of payments or an investment with changing terms.

Definition: Let $D$ be a deposit at time 0, $F$ the value of that deposit at time $n$, and $P$ the present value at time 0 of a single payment $F$ due at time $n$. Under annual compounding at effective annual rate $i$, the accumulation factor is $(1+i)^n$, and the discount factor is $v^n$, where $v=1/(1+i)$.

Two standard equations describe the same single-payment relationship from different directions. The future-value equation grows a deposit forward; the present-value equation discounts a future amount back. When the unknown is the deposit today, each equation leads to the same result.

F=D(1+i)^n \qquad\text{and}\qquad P=Fv^n=\frac{F}{(1+i)^n}.

If the problem gives the required future amount $F$, rearrange the future-value equation to isolate $D$. Equivalently, compute the present value of $F$. For this one payment, the deposit that will accumulate to $F$ is exactly its present value.

D=\frac{F}{(1+i)^n}=Fv^n=P.

Why the Rearrangement Works

The future-value formula follows from applying the same annual growth factor once for each year. After one year, a deposit $D$ is worth $D(1+i)$. After a second year, that balance earns interest again, giving $D(1+i)(1+i)=D(1+i)^2$. Repeating this multiplication for $n$ years gives $F=D(1+i)^n$. Dividing both sides by the positive factor $(1+i)^n$ isolates the original deposit.

F=D(1+i)^n \quad\Longrightarrow\quad \frac{F}{(1+i)^n}=D.

The present-value expression agrees because $v=1/(1+i)$. Raising both sides to the $n$th power gives $v^n=1/(1+i)^n$, so discounting $F$ by $v^n$ is precisely the same division used to solve for $D$. This equivalence assumes the same rate and the same number of annual periods are used in both calculations.

Key result: For one future payment $F$ due in $n$ years, the initial deposit required at time 0 is $D=F/(1+i)^n=Fv^n$. Use this relationship when the rate is a constant effective annual rate and $n$ is the number of whole annual periods.

Worked Examples

Worked Example: Accumulating a Reserve to a Target

A hypothetical organization wants a reserve of $13,310 exactly three years from today. It can make one deposit now into an account earning an effective annual rate of 10%, compounded annually. Find the deposit and check it by accumulating the result. Ignore fees and taxes.

Method. The target is a future value, so start with $F=D(1+i)^n$ and solve for $D$. The common mistake to avoid is multiplying the target by the accumulation factor: that would grow an amount already stated at the future date rather than find the amount needed today.

Step 1: Identify the inputs. The future target is $F=\$13{,}310$, the annual effective rate is $i=0.10$, and the term is $n=3$ years.

Step 2: Calculate the accumulation factor.

(1+i)^n=(1.10)^3=1.10\times1.10\times1.10=1.331.

Step 3: Isolate the deposit.

D=\frac{F}{(1+i)^n} =\frac{\$13{,}310}{1.331} =\$10{,}000.

Step 4: Check by accumulating. A deposit of $10,000 grows to $11,000 after one year, to $12,100 after two years, and to $13,310 after three years. Equivalently, $\$10{,}000(1.331)=\$13{,}310$, the required target. The deposit is $10,000, and the difference between the target and the deposit is $3,310.

Worked Example: Discounting a Single Payment Due Later

A hypothetical benefit fund must make a single payment of $14,641 four years from today. A separate account earns an effective annual rate of 10%, compounded annually, with no fees or taxes. How much must be deposited now to meet the payment, and how can the answer be checked?

Method. Find the present value of the payment by discounting four years. The common mistake to avoid is treating the amount due in four years as though it were already the deposit at time 0.

Step 1: State the inputs and factor. Here $F=\$14{,}641$, $i=0.10$, and $n=4$. The four-year accumulation factor is

(1.10)^4=1.10\times1.10\times1.10\times1.10=1.4641.

Step 2: Discount to time 0. The required deposit is the present value of the future payment:

D=P=Fv^n =\frac{\$14{,}641}{1.4641} =\$10{,}000.

Step 3: Check by accumulating. Growing the deposit forward gives $\$10{,}000(1.4641)=\$14{,}641$ after four years. The deposit therefore funds the stated single payment exactly under the assumptions. The discounted value and the deposit agree because both are measured at time 0.

Worked Example: Rounding a Discounted Deposit

A hypothetical project needs $30,000 in three years for a scheduled equipment replacement. A reserve account earns an effective annual rate of 4%, compounded annually. Find the deposit required today, report it to the nearest cent, and verify the rounded deposit. Assume no other cash flows, fees, or taxes.

Method. Discount the future target by the three-year accumulation factor. Retain precision in the calculation and round only the final deposit to cents. This avoids allowing an early rounding step to affect the result.

Step 1: Calculate the factor.

(1.04)^3=1.04\times1.04\times1.04=1.124864.

Step 2: Solve for the deposit.

D=\frac{\$30{,}000}{1.124864} =\$26{,}669.890760\ldots \approx \$26{,}669.89.

The final deposit is rounded to two decimal places. Do not round the accumulation factor to 1.12 before dividing; doing so would give $\$30{,}000/1.12=\$26{,}785.71$, which does not reflect the stated rate and term.

Step 3: Check the rounded result.

\$26{,}669.89(1.124864) =\$29{,}999.99914496 \approx \$30{,}000.00.

The small difference before rounding is caused by rounding the deposit to cents. The accumulated amount rounds to the required $30,000.00, so the result is consistent.

A Reliable Calculation Process

A short, consistent process helps keep the valuation date and the direction of calculation clear. The following steps apply to either a future-value equation or a present-value equation when the unknown is the initial deposit.

Mark the dates.
Identify time 0 as the deposit date and $n$ as the date of the single target payment. Confirm that $n$ counts annual periods.

Identify the known amount.
Decide whether the stated amount is the future target $F$ or a value already expressed at time 0. Do not assign the future target to the deposit variable.

Rearrange the matching equation.
For $F=D(1+i)^n$, divide by $(1+i)^n$. For $P=Fv^n$, calculate that discounted amount.

Check at the other date.
Accumulate the deposit forward and see whether it reaches the target, or discount the target back and see whether it equals the deposit.

Round only the reported result.
Keep sufficient precision in factors and intermediate values; report dollars to the nearest cent when the question requests a dollar amount.

Common Errors and When the Formula Applies

The most frequent error is using the right factor in the wrong direction. To find an unknown amount at a future date, multiply a present amount by $(1+i)^n$. To find an unknown amount at time 0 from a future target, divide by $(1+i)^n$, or multiply by $v^n$. These operations are inverses, not interchangeable choices.

For example, if $13,310 is needed in three years at 10%, the incorrect calculation $ \$13{,}310(1.331)=\$17{,}715.61 $ applies accumulation to a number that is already the future target. The correct deposit is $ \$13{,}310/1.331=\$10{,}000 $. Accumulating that answer checks the result: $ \$10{,}000(1.331)=\$13{,}310 $.

Before applying the formula, confirm that the problem has one deposit and one target payment, a constant effective annual rate, and a whole number of annual periods. If there are several payments, the present value must account for each one rather than treating their sum at different dates as a single payment. If the rate or compounding convention changes, the factor must match those terms; the constant annual-rate formula as written would not directly represent that arrangement.

Keep dates attached to amounts. The deposit is measured at time 0; the target $F$ is measured at time $n$.
Use the factor once in the proper direction. Discount a future target to find a deposit; accumulate a deposit to check a future target.
Preserve precision. Round the final money amount, not the factor or an intermediate quotient, unless the problem explicitly instructs otherwise.

Key takeaway: When one deposit today must become a specified amount $F$ after $n$ years at a constant effective annual rate $i$, solve $D=F/(1+i)^n=Fv^n$. A forward accumulation check should return the target, subject to any stated rounding.

Check Your Understanding

For each problem, assume annual compounding, no fees or taxes, and one deposit at time 0. Use the stated effective annual rate and round only as requested.

A hypothetical maintenance reserve needs $11,881 in two years. The account earns an effective annual rate of 9%. Find the deposit today and verify it by accumulating the deposit. (Answer: $10,000.00; $\$10{,}000(1.09)^2=\$11{,}881$.)
A single payment of $23,152.50 is due in three years. An account earns an effective annual rate of 5%. What deposit today has the same value as that payment? (Answer: $20,000.00.)
A hypothetical equipment fund requires $26,966.40 in two years, and the fund earns 6% effective annually. Find the initial deposit and state the factor used. (Answer: $24,000.00; factor $1.1236$.)
A single amount of $30,000 is required in three years. The effective annual rate is 4%. Calculate the deposit to the nearest cent, then state the accumulated value of that rounded deposit to the nearest cent. (Answer: $26,669.89; accumulated value $30,000.00.)
A hypothetical grant reserve must provide $14,693.28 in five years. The account earns 8% effective annually. A candidate multiplies the target by $(1.08)^5$. Identify the error and calculate the correct deposit. (Answer: The target must be discounted, not accumulated; $D=\$14{,}693.28/(1.08)^5=\$10{,}000.00$.)

What You'll Learn