Solving for an Unknown Cash Flow

Why Solve for an Unknown Cash Flow?

A dated cash-flow stream may include several known receipts and payments but one amount that must be determined. The unknown could be a deposit needed to reach a savings target, a payment that makes two sides of a transaction equivalent, or a cash flow that reconciles a stated value with the rest of a schedule. The calculation is not a new valuation method: it uses present value or future value, but treats one amount as an unknown.

The essential discipline is to value every cash flow at the same date. Once the known amounts and the target have been expressed at a common date, the unknown amount appears as one term in an equation. Solving that equation gives the required cash flow. This method is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: An unknown cash flow is an amount $X$ at a specified date in a stream whose other cash flows, interest assumption, and target value are known. A solution is found by writing a present-value or future-value equation at a common valuation date and isolating $X$.

Set Up the Equation at One Date

Use a consistent sign convention. For example, record receipts as positive cash flows and payments or withdrawals as negative cash flows. A target value must use the same convention. The equation then values the net stream, rather than adding receipts and payments as though they all had the same direction.

For the examples below, assume a constant effective annual interest rate $i$, annual compounding, and cash flows at integer year-ends unless a cash flow occurs at time 0. Let $C_k$ be the signed cash flow at time $k$, and let $X$ occur at time $s$. The discount factor is $v=1/(1+i)$. When the target is present value $V_0$ at time 0, each cash flow is discounted to time 0:

V_0=\sum_k C_kv^k

When $X$ is one of the cash flows, separate it from the known amounts. The result is:

V_0=Xv^s+\sum_{k\ne s}C_kv^k \qquad\Longrightarrow\qquad X=\frac{V_0-\sum_{k\ne s}C_kv^k}{v^s}

For a target $V_T$ at time $T$, accumulate each cash flow to time $T$. If the cash flow at time $k$ occurs before $T$, it accumulates for $T-k$ years:

V_T=\sum_k C_k(1+i)^{T-k}

In particular, when $X$ occurs at time $s$, its contribution at time $T$ is $X(1+i)^{T-s}$. Isolating it gives:

X=\frac{V_T-\sum_{k\ne s}C_k(1+i)^{T-k}}{(1+i)^{T-s}}

These expressions assume that the stated rate applies over every period being valued. If the rate changes by period, each cash flow must instead be valued using the applicable rates for its own dates. The formulas also require a target date and cash-flow signs that are interpreted consistently.

Why the Isolation Formula Works

The present-value equation follows from the definition of discounting: one unit payable at time $s$ has present value $v^s$. Therefore, a cash flow $X$ at time $s$ contributes $Xv^s$ to the total present value. All other cash flows contribute the sum of their own discounted values. Subtracting that known sum from the target leaves the present value attributable to $X$; dividing by $v^s$ converts that present value back to the amount at time $s$:

V_0=Xv^s+\text{PV of known cash flows}

V_0-\text{PV of known cash flows}=Xv^s \quad\Longrightarrow\quad X=\frac{V_0-\text{PV of known cash flows}}{v^s}

The future-value formula has the same logic. A cash flow at time $s$ grows by the accumulation factor $(1+i)^{T-s}$ to time $T$. Subtract the accumulated known cash flows from the target, then divide by this factor to recover the amount at time $s$. These steps justify both formulas; they are not separate rules to memorize without regard to the dates.

Worked Examples

Worked Example: Finding a Cash Flow From a Future-Value Target

An account earns 10% effective annually. It receives $100 at time 0, an unknown amount $X$ at the end of year 1, and $200 at the end of year 2. The target balance at the end of year 3 is $400. Find $X$, treating each receipt as positive.

Method. Accumulate every receipt to year 3, then isolate the year-1 amount. The $100 receipt accumulates for 3 years, $X$ for 2 years, and the $200 receipt for 1 year. The equation is:

$$ 100(1.10)^3+X(1.10)^2+200(1.10)=400 $$

Evaluate the known amounts. The first receipt contributes $100(1.331)=133.10$ at year 3. The last receipt contributes $200(1.10)=220.00$. The unknown receipt has accumulation factor $(1.10)^2=1.21$. Thus:

$$ 133.10+1.21X+220.00=400 $$

The known accumulated receipts total $133.10+220.00=353.10$, so the amount attributable to $X$ at year 3 is $400-353.10=46.90$. Therefore:

X=\frac{46.90}{1.21}=38.760330\ldots

Rounded to the nearest cent, $X=\$38.76$. Check. Substitution gives $133.10+1.21(38.76)+220.00=399.9996$, which rounds to the target balance of $400.00. The small difference is due solely to rounding $X$ to cents. The method avoids the common mistake of accumulating $X$ for only one year: it is paid at year 1 and valued at year 3, so it earns interest for two years.

Worked Example: Finding a Cash Flow From a Present-Value Target

A fund earns 10% effective annually. It is to have a present value of $2,000 at time 0 from receipts of $1,000 at the end of year 1, $X$ at the end of year 2, and $1,000 at the end of year 3. Find $X$, treating all three receipts as positive.

Method. Discount the receipts to time 0. Here $v=1/1.10$, so the present-value equation is:

\frac{1000}{1.10}+\frac{X}{(1.10)^2}+\frac{1000}{(1.10)^3}=2000

The known present values are $1000/1.10=909.090909\ldots$ and $1000/(1.10)^3=751.314801\ldots$. Their sum is $1660.405710\ldots$. Subtracting this from the target leaves the present value that must come from $X$:

\frac{X}{1.21}=2000-1660.405710\ldots=339.594290\ldots

The year-2 cash flow must be converted from its present value back to an amount at year 2 by multiplying by $(1.10)^2=1.21$:

X=339.594290\ldots(1.21)=410.908092\ldots

To the nearest cent, $X=\$410.91$. Check. The present value of the rounded amount is $410.91/1.21=339.595041\ldots$. Adding the two known present values gives approximately $2000.000751$, which rounds to $2,000.00. The method avoids treating the year-2 amount as though it were already a time-0 value; it must be discounted in the original equation and multiplied by 1.21 when solving for its dated amount.

Worked Example: Solving With Both Receipts and a Withdrawal

An account earns 10% effective annually. It receives $100 at the end of year 1 and $200 at the end of year 4, and pays out $50 at the end of year 3. An additional receipt $X$ arrives at the end of year 2. The balance at the end of year 5 is $971.41. Find $X$.

Method. Use receipts as positive and the withdrawal as negative, and accumulate every cash flow to year 5. The periods of accumulation are four years for the year-1 receipt, three for $X$, two for the year-3 withdrawal, and one for the year-4 receipt:

$$ 100(1.10)^4+X(1.10)^3-50(1.10)^2+200(1.10)=971.41 $$

The known terms at year 5 are $100(1.4641)=146.41$, $-50(1.21)=-60.50$, and $200(1.10)=220.00$. Their net value is $146.41-60.50+220.00=305.91$. Hence:

$$ 1.331X=971.41-305.91=665.50 $$

Solving gives $X=665.50/1.331=500$, so $X=\$500.00$. Check. Its value at year 5 is $500(1.331)=665.50$; adding the known net value gives $665.50+305.91=971.41$, exactly the target. The signed calculation avoids the common error of adding the withdrawal as a positive amount.

Common Errors and a Reliable Check

An incorrect number of accumulation or discount periods changes the coefficient of $X$, and therefore changes the answer even if the rest of the arithmetic is sound. Count periods from the date of the unknown cash flow to the valuation date. In the first example, year 1 to year 3 is two years, so the coefficient is $(1.10)^2$, not $1.10$.

A second frequent error is to subtract known amounts without first valuing them at the target date. A $200 receipt at year 2 is not $200 at year 3 when interest is positive; at 10%, its year-3 value is $200(1.10)$. Similarly, at a present-value target, a future receipt must be discounted before it is combined with cash flows at other dates.

After solving, substitute the amount into the original equation, with its original dates and signs. The resulting value should match the stated target, allowing only for the effect of rounding the final cash flow. Also consider whether the sign and size of the answer make sense: a negative $X$ can be mathematically correct if the target is below the value of the known receipts, but it represents a net payment under the stated convention.

Choose the target date.
Use the date at which the problem states the present-value or future-value target.

Value every known cash flow there.
Discount or accumulate each amount for the number of periods between its date and the target date, retaining its sign.

Isolate the unknown.
Subtract the known values from the target, then divide by the unknown cash flow's discount or accumulation factor.

Check the result.
Substitute the unrounded solution into the original equation before rounding the final amount.

Check Your Understanding

For each problem, state the valuation date you use and keep receipts and payments signed consistently.

An account earns 5% effective annually. It receives $300 at time 0, $X$ at the end of year 2, and $500 at the end of year 4. The year-4 balance is $1,200. Find $X$ to the nearest cent. (Answer: $304.17)
At 10% effective annually, a stream consists of $100 at time 0, $200 at the end of year 1, $X$ at the end of year 2, and $300 at the end of year 3. Its present value is $700. Find $X$ to the nearest cent. (Answer: $233.27)
An account earns 4% effective annually. It receives $1,000 at time 0 and $X$ at the end of year 2, then pays $300 at the end of year 3. Its year-4 balance is $1,500. Find the positive receipt $X$ to the nearest cent. (Answer: $593.70)
At 10% effective annually, a stream has a $600 receipt at the end of year 1, an unknown cash flow $X$ at the end of year 2, and a $200 payment at the end of year 3. Its net present value at time 0 is $400. Use positive signs for receipts and a negative sign for payments; report $X$ to the nearest cent. (Answer: $5.82)

What You'll Learn