Solving for an Unknown Cash Flow Using Spot Rates

Why an Unknown Cash Flow Requires a Spot-Rate Equation

A cash-flow schedule can have a known present value even when one of its future payments is not yet specified. If the spot rates for the payment dates are known, the present value of every known payment can be calculated. The remaining difference determines the present value, and then the amount, of the unknown cash flow.

The essential point is that each payment is discounted using the rate for its own maturity. A one-year cash flow and a two-year cash flow generally have different discount factors. The present-value equation must also preserve the signs of the cash flows: receipts are positive, while payments are negative.

Definition: A dated cash-flow schedule consists of amounts $C_t$ payable at specified times $t$, measured from the valuation date. Use $C_t>0$ for a receipt and $C_t<0$ for a payment. The signed present value is the sum of each cash flow multiplied by the discount factor for its maturity.

For this tutorial, assume that the valuation date is time 0, all amounts are in the same currency, and the supplied spot rates are annual effective rates. If the annual effective spot rate for maturity $t$ is $i_t$, its discount factor is $v_t=(1+i_t)^{-t}$, where $t$ is measured in years. An equivalent way to state the input is to provide the discount factors directly. Use the factor associated with each payment date; do not substitute one rate for the whole schedule.

V=\sum_t C_t v_t

Here $V$ is the schedule's signed present value. If a problem instead describes a liability's value as a positive magnitude, translate that into the signed value before writing the equation. For example, a liability with present-value magnitude \$250 has signed present value $-\$250$ when its future cash flows are payments.

Isolating One Unknown Amount

Suppose one cash flow $C_k$, due at time $k$, is unknown. Separate its present value from the contributions of all the known cash flows. This works whether the unknown is a receipt or a payment; the sign of the answer determines which it is.

Theorem: If the signed present value is $V$, and the other cash flows are known, then $$ C_k=\frac{V-\sum_{t\ne k}C_t v_t}{v_k}. $$ The unknown cash flow is found by subtracting the known cash flows' present values from the target value, then dividing the remainder by the unknown payment's discount factor.

The formula follows directly from the present-value equation. The known cash flows contribute $\sum_{t\ne k}C_t v_t$, while the unknown contributes $C_kv_k$. Therefore

V=\sum_{t\ne k}C_t v_t+C_kv_k \quad\Longrightarrow\quad C_kv_k=V-\sum_{t\ne k}C_t v_t \quad\Longrightarrow\quad C_k=\frac{V-\sum_{t\ne k}C_t v_t}{v_k}.

The final division is important: the remainder is the present value of the unknown cash flow, not its future amount. Since $v_k>0$ for an ordinary positive discount factor, the sign of the remainder is also the sign of $C_k$.

Set the time and sign conventions.
Identify the valuation date, express each payment time in years, and mark each receipt positive and each payment negative.

Find the maturity-specific factors.
For each dated cash flow, use its spot rate to calculate the corresponding discount factor, or use a provided factor directly.

Write and solve the equation.
Set the signed present value equal to the sum of discounted cash flows, then isolate the unknown amount.

Check the result.
Substitute the signed amount into the original equation and verify that the present value equals the stated target.

Worked Examples

Worked Example: Solving for a Future Receipt

An asset's future cash flows have a present value of \$1,000. It will produce a receipt of \$420 at the end of year 1 and an unknown receipt $X$ at the end of year 2. The one-year spot rate is 5% effective annually, and the two-year spot rate is 10% effective annually. Find $X$.

Method. Discount each receipt at its own maturity and solve the present-value equation. Both future cash flows are receipts, so both are positive.

The discount factors are

v_1=(1.05)^{-1}=\frac{1}{1.05}, \qquad v_2=(1.10)^{-2}=\frac{1}{1.21}.

The year-1 receipt has present value

420v_1=\frac{420}{1.05}=400.

The present value left for the year-2 receipt is $1{,}000-400=600$. Thus

\frac{X}{1.21}=600 \quad\Longrightarrow\quad X=600(1.21)=726.

Answer: The receipt at the end of year 2 is \$726.00. Check: $420/1.05+726/1.21=400+600=1{,}000$. The common mistake avoided here is discounting both receipts at the same rate; the year-2 receipt uses the two-year spot rate.

Worked Example: Solving for a Payment in a Liability Schedule

A liability has signed present value $-\$250$. It requires a payment of \$110 at the end of six months and a payment $X$ at the end of year 1. The six-month spot rate, quoted as an annual effective rate, is 21%; the one-year spot rate is 10% effective annually. Find the signed amount $X$.

Method. Keep the cash flows signed. The known payment is $-\$110$, the target signed present value is $-\$250$, and the unknown payment will also be negative if the schedule is to remain a liability.

For the six-month maturity, $t=0.5$ years. Its discount factor is

v_{0.5}=(1.21)^{-0.5}=\frac{1}{1.1}.

Therefore, the known payment's present value is

-110v_{0.5}=-\frac{110}{1.1}=-100.

The one-year factor is $v_1=1/1.10$. Substituting into the signed present-value equation gives

-250=-100+\frac{X}{1.10}.

The unknown payment's present value must be $-250-(-100)=-150$, so

\frac{X}{1.10}=-150 \quad\Longrightarrow\quad X=-150(1.10)=-165.

Answer: $X=-\$165.00$, meaning a payment of \$165.00. Check: $-110/1.1-165/1.1=-100-150=-250$. The common mistake avoided is equating a positive liability magnitude of \$250 to the signed present value of negative payments. Under the signed convention, the value is $-\$250$.

Worked Example: Finding an Unknown Receipt in a Mixed Schedule

A schedule has a signed present value of \$100. It includes a receipt of \$104 at the end of year 1, a payment of \$242 at the end of year 2, and an unknown receipt $X$ at the end of year 3. The annual effective spot rates are 4% for year 1, 10% for year 2, and 5% for year 3. Find $X$, to the nearest cent.

Method. Calculate the signed present value of each known cash flow, then solve for the present value and amount of the year-3 receipt.

At year 1, the discount factor is $v_1=1/1.04$, so the receipt's present value is

104v_1=\frac{104}{1.04}=100.

At year 2, $v_2=1/(1.10)^2=1/1.21$. The payment is negative, so its present value is

-242v_2=-\frac{242}{1.21}=-200.

The known cash flows therefore have combined present value $100-200=-100$. To reach the schedule's target of \$100, the year-3 receipt must have present value $100-(-100)=200$. Its discount factor is

v_3=(1.05)^{-3}=\frac{1}{1.157625}.

Thus

\frac{X}{1.157625}=200 \quad\Longrightarrow\quad X=200(1.157625)=231.525.

Answer: $X=\$231.53$, rounded to the nearest cent. Check: $104/1.04-242/1.21+231.525/1.157625=100-200+200=100$. The common mistake avoided is treating the known payment as positive; doing so would change the known present value from $-\$200$ to $+\$200$ and produce a different equation.

Interpreting the Answer and Avoiding Traps

A solved amount should be read together with its sign and due date. A positive answer is a receipt under the convention used here; a negative answer is a payment. If the question asks for a payment amount rather than a signed cash flow, report the positive magnitude and state that it is a payment. This distinction prevents a numerically correct magnitude from being attached to the wrong direction of cash flow.

A useful exam check is to compare the required residual present value with the known present values before calculating the unknown amount. For example, if a target present value is more negative than the known payments' combined present value, the unknown must contribute a further negative amount. A positive unknown would move the total in the wrong direction.

Do not use one rate for every date. A spot rate applies to a specified maturity. Calculate or select the discount factor separately for each date.
Do not confuse a present value with a future payment. First find the residual present value, then divide by the unknown cash flow's discount factor.
Do not discard cash-flow signs. Payments have negative signed values and receipts have positive signed values. Keep those signs throughout the equation.
Do not misread a rate's time basis. If a six-month spot rate is given as an annual effective rate, use $t=0.5$ in the annual-effective discount formula.

The method matters whenever a schedule is incomplete but its value is known, such as determining a balancing payment in a hypothetical contract or finding the missing receipt in an investment valuation. The equation itself is simple; accuracy depends on matching each cash flow to its maturity, preserving signs, and distinguishing the schedule's target value from the amount due at a future date. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: Write the signed present-value equation first. Subtract the known cash flows' present values from the target, then divide by the discount factor at the unknown cash flow's maturity. Substitute the answer back into the original equation to verify it.

Check Your Understanding

For each problem, use signed cash flows: receipts are positive and payments are negative. Rates are annual effective spot rates unless stated otherwise.

A schedule has present value \$500. It receives \$210 at year 1 and an unknown amount $X$ at year 2. The year-1 and year-2 spot rates are 5% and 10%, respectively. Find $X$. (Answer: \$363.00)
A liability has signed present value $-\$400$. It pays \$220 in six months and has an unknown payment $X$ at year 1. The six-month spot rate is 21% annual effective and the one-year spot rate is 10%. Find the signed amount $X$. (Answer: $-\$220.00$, a payment of \$220.00)
A schedule has present value \$75. It receives \$156 at year 1, pays \$242 at year 2, and receives $X$ at year 3. The spot rates for years 1, 2, and 3 are 4%, 10%, and 5%. Find $X$, to the nearest cent. (Answer: \$144.70)
A schedule has present value zero. It pays \$105 at year 1 and receives $X$ at year 2. The one-year spot rate is 5% and the two-year spot rate is 10%. Find $X$. (Answer: \$121.00)
A schedule has present value \$50. It receives \$220 in six months, pays \$242 at year 2, and receives $X$ at year 1. The six-month spot rate is 21% annual effective; the one-year and two-year rates are 5% and 10%. Find $X$. (Answer: \$52.50)

What You'll Learn