Finding An Unknown Cash-Flow Date

Why a Cash-Flow Date Can Be Unknown

A value equation can determine when a payment occurs if its amount, the interest model, and the other transaction terms are known. The key is to express every cash flow at the same focal date, isolate the value attributable to the payment with the unknown date, and then solve for elapsed time. A date should always be stated relative to a clear origin: for example, “4 years after time 0.”

The method depends on the interest model. Under compound interest, the time between the payment and focal date appears as an exponent, so logarithms are used to solve for it. Under simple interest, the relevant accumulation factors are linear in time. In either model, a calculated time must be checked against the timeline and the assumptions used to form the value equation.

Definition: An unknown cash-flow date is the time $t$, measured from a stated reference date, at which a payment of known amount $C$ occurs. A value equation for solving $t$ equates the value of all cash flows at one focal date. The amount $C$, interest model, rate, focal date, and any other cash flows are treated as known.

Solving for a Date Under Compound Interest

Assume a constant effective annual interest rate $i$, with $i>-1$, and compound interest over the time interval being considered. If a payment $C$ occurs at time $t$, its value at focal time $T$ is $C(1+i)^{T-t}$. The exponent is the signed time from the payment to the focal date: it is positive when $T>t$, and negative when $T<t$.

V_T=C(1+i)^{T-t} \qquad\Longrightarrow\qquad t=T-\frac{\ln(V_T/C)}{\ln(1+i)}, \quad i\ne 0

Here $V_T$ is the value at time $T$ attributable to the payment $C$. This formula requires $C$ and $V_T$ to have the same sign so that $V_T/C>0$, and it requires $i\ne0$. If $i=0$, the value is $C$ at every date, so the value equation cannot identify the payment’s date by itself.

Fix the time origin and focal date.
Measure all dates from a stated reference point, and select a focal date at which the value equation will be written.

Find the unknown payment’s value at the focal date.
If other payments are present, first value those known payments at the same focal date and subtract their total from the stated value.

Solve the value equation for time.
Under compound interest, take logarithms after isolating the unknown payment’s accumulation or discount factor.

Check the answer.
Substitute the date into the original value equation and confirm that the result fits the stated time interval and payment direction.

Why the Compound-Interest Formula Works

Under compound interest, one unit invested at time $t$ accumulates to $(1+i)^{T-t}$ units at time $T$. Multiplying by $C$ gives the payment’s focal-date value. Dividing the value equation by $C$ isolates the accumulation factor; taking logarithms turns its exponent into a product:

\frac{V_T}{C}=(1+i)^{T-t} \quad\Longrightarrow\quad \ln(V_T/C)=(T-t)\ln(1+i) \quad\Longrightarrow\quad t=T-\frac{\ln(V_T/C)}{\ln(1+i)}.

For $i>0$, a payment made earlier has more time to accumulate to a later focal date. Thus, when $T$ is after the payment, a focal-date value greater than $C$ is consistent with a positive accumulation period. A result in the wrong direction is a reason to recheck the value ratio, signs, or dates.

Worked Example: Finding a Payment Date Under Compound Interest

A fund earns an effective annual rate of 5%. A payment of $4,000 is made at an unknown time $t$, where $0\leq t\leq6$. Its value at time 6 is $4,410. Find the payment date as elapsed time from time 0.

Method. Use compound accumulation from the payment date to the focal date, time 6. The payment value equation is

4{,}410=4{,}000(1.05)^{6-t}.

Divide by $4,000 to isolate the accumulation factor:

\frac{4{,}410}{4{,}000}=1.1025=(1.05)^{6-t}.

Since $1.1025=(1.05)^2$, it follows that $6-t=2$, so $t=4$. The payment occurs 4 years after time 0. This is within the stated interval.

Check. Accumulating the payment for 2 years gives $4{,}000(1.05)^2=4{,}000(1.1025)=\$4{,}410$, as required. The check avoids reporting a date merely because a logarithm calculation produced a number.

Solving for a Date Under Simple Interest

For simple interest at annual rate $i$, measured from time 0, the accumulation factor to time $s$ is $1+is$. This is not the compound-interest model: the factor is linear in elapsed time rather than a power of $1+i$. Assume the simple-interest factors are positive throughout the dates being valued.

A payment $C$ at time $t$ has value $C/(1+it)$ at time 0. Equating that value to a known amount $V_0$ gives a direct equation for the date. More generally, its value at focal time $T$, using the same time-0 simple-interest basis, is $C(1+iT)/(1+it)$. This ratio follows by moving the payment’s time-0 value to the focal date using the factor $1+iT$.

V_0=\frac{C}{1+it} \qquad\Longrightarrow\qquad t=\frac{C/V_0-1}{i}, \quad i\ne0

The formula is tied to the stated simple-interest model and time origin. It should not be replaced with a compound-interest exponent, nor should the simple-interest factor be treated as if interest were repeatedly earned on accumulated interest.

Worked Example: Finding a Payment Date Under Simple Interest

At a simple annual interest rate of 10%, a payment of $3,000 is due at an unknown time $t$ after time 0. Its value at time 0 is $2,500. Find $t$, assuming the simple-interest model applies and the accumulation factors are positive.

Method. Discount the payment to time 0 using the simple-interest factor $1+0.10t$:

2{,}500=\frac{3{,}000}{1+0.10t}.

Multiply both sides by the denominator:

2{,}500(1+0.10t)=3{,}000, \qquad 1+0.10t=\frac{3{,}000}{2{,}500}=1.2.

Therefore $0.10t=0.2$, so $t=2$. The payment is due 2 years after time 0.

Check. The simple-interest accumulation factor for 2 years is $1+0.10(2)=1.2$, and $3{,}000/1.2=\$2{,}500$. The equation is satisfied exactly. The common mistake avoided here is applying a compound discount factor $(1.10)^{-t}$ even though the problem specifies simple interest.

When Other Cash Flows Are Present

The date of one payment can still be found when the value equation includes other transactions, provided their dates and amounts are known. First move every known payment to the focal date. Then subtract the known payments’ total value from the total stated value. The remainder is the focal-date value attributable to the payment with the unknown date.

Worked Example: Isolating an Unknown-Date Payment in a Schedule

An account earns an effective annual rate of 4%. At time 5, the total value of two deposits is $5,494.528. One deposit is $2,000 at time 2; the other is $3,000 at unknown time $t$, with $0\leq t\leq5$. Find the unknown deposit date.

Method. Write the value equation at time 5. The $2,000 deposit accumulates for 3 years, and the $3,000 deposit accumulates for $5-t$ years:

5{,}494.528=2{,}000(1.04)^3+3{,}000(1.04)^{5-t}.

Calculate the known deposit’s value:

(1.04)^3=1.124864, \qquad 2{,}000(1.124864)=2{,}249.728.

Subtract this known value from the total:

5{,}494.528-2{,}249.728=3{,}244.800 =3{,}000(1.04)^{5-t}.

Divide by $3,000:

(1.04)^{5-t}=\frac{3{,}244.800}{3{,}000}=1.0816=(1.04)^2.

Thus $5-t=2$, so $t=3$. The $3,000 deposit was made 3 years after time 0.

Check. The unknown deposit accumulates for 2 years to $3{,}000(1.04)^2=3{,}244.800$. Adding the known deposit’s value gives $3{,}244.800+2{,}249.728=\$5{,}494.528$, exactly the stated total. The calculation avoids treating the total focal-date value as if it belonged to the unknown deposit alone.

Common Mistakes and Checks

A date calculation is particularly sensitive to which value ratio is used and how the time interval is defined. In the first worked example, the correct equation is $4{,}410/4{,}000=(1.05)^{6-t}$. Reversing that ratio gives the wrong calculation:

t=6-\frac{\ln(4{,}000/4{,}410)}{\ln(1.05)}=8.

This result lies outside $0\leq t\leq6$, and it would place the payment after the focal date even though its value at time 6 exceeds the payment amount under a positive interest rate. The correction is to use $V_T/C$, not $C/V_T$, in the compound formula. Substitution into the original value equation is the final check.

Other useful checks are:

State the origin. A result such as $t=3$ is incomplete unless the reference date and time unit are clear.
Use the correct interval. For a payment at $t$ valued at $T$, the compound-interest exponent is $T-t$, not automatically $t$ or $T$.
Respect the model. Simple interest uses linear accumulation factors; compound interest uses powers of $1+i$.
Isolate other payments first. In a multi-payment equation, only the residual focal-date value belongs to the unknown-date payment.
Check feasibility. Confirm that the date is consistent with any stated time bounds and with whether the payment occurs before or after the focal date.

Key takeaway: Fix a time origin, write a value equation at one focal date, isolate the known payment’s focal-date value, and solve using the specified interest model. Verify the date by substituting it back into the original equation.

Check Your Understanding

For each problem, state the unknown date as elapsed time from time 0. Use the stated interest model and check that the answer satisfies the value equation.

At an effective annual rate of 3%, a payment of $5,000 at unknown time $t$ has value $5,304.50 at time 4. Find $t$, given $0\leq t\leq4$. (Answer: 2 years after time 0)
Under simple interest at 10% per year, a payment of $2,420 at unknown time $t$ has value $2,200 at time 0. Find $t$. (Answer: 1 year after time 0)
At an effective annual rate of 6%, a payment of $1,000 at unknown time $t$ has value $1,191.016 at time 5. Find $t$, given $0\leq t\leq5$. (Answer: 2 years after time 0)
At an effective annual rate of 5%, the value at time 4 of a $4,000 payment at unknown time $t$ and a $1,000 payment at time 1 is $5,567.625. Find $t$, given $0\leq t\leq4$. (Answer: 2 years after time 0)
At an effective annual rate of 25%, a payment of $4,000 at unknown time $t$ has value $2,560 at time 2. Find $t$, given $t\geq2$. (Answer: 4 years after time 0)

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