Replacing One Cash-Flow Schedule With Another

What It Means to Replace a Cash-Flow Schedule

A payment schedule can often be replaced by a different schedule without changing its financial value at a chosen date. The payments need not occur on the same dates or have the same amounts. The condition is that, under the stated interest assumption, the two schedules have equal value at the same focal date.

This is useful when a payment is rescheduled, a set of obligations is consolidated, or a proposed payment plan is compared with an existing one. The calculation is a value equation: bring every payment in both schedules to one date using the same interest assumption, then set the two totals equal. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Two cash-flow schedules are equivalent at a focal date under a specified interest model if their values at that date are equal. The schedule value is found by valuing each cash flow at the focal date under that model and adding the resulting values. Equivalence at one focal date does not mean the schedules contain identical payments.

Before writing an equation, identify the payment dates and amounts in both schedules, the focal date, and the interest assumption. A payment before the focal date is accumulated forward. When the model and dates permit it, a payment after the focal date is discounted back. Keep the direction of each valuation clear, and do not mix interest assumptions between the schedules.

A General Value Equation

Suppose the original schedule has signed cash flows $C_1,\ldots,C_n$ at times $t_1,\ldots,t_n$, and the replacement schedule has signed cash flows $D_1,\ldots,D_m$ at times $s_1,\ldots,s_m$. A positive sign can represent an amount received, and a negative sign an amount paid, provided the same sign convention is used for both schedules. Let $V_T(C,t)$ denote the value at focal time $T$ of a cash flow $C$ at time $t$, using the stated interest model. The value equation is

\sum_{j=1}^{n} V_T(C_j,t_j)=\sum_{k=1}^{m} V_T(D_k,s_k).

For example, under compound interest at an effective annual rate $i$, a payment $C$ made $r$ whole years before the focal date accumulates to $C(1+i)^r$. Under simple interest at annual rate $i$, a payment $C$ made $r$ years before the focal date accumulates directly to $C(1+ir)$. These are different models; use the one specified for the problem. For simple interest, the direct accumulation factor applies over the interval from the payment date to the focal date, not by repeatedly applying a one-year accumulation factor.

Key result: To find a replacement amount, calculate the value of the original schedule at the focal date, calculate the value of the known replacement payments at that same date, and solve for the unknown amount. Apply the same interest model to every cash flow.

Why the Values Can Be Added

The value of a schedule is the sum of the values of its individual cash flows. Under compound interest, for example, suppose all payments occur before focal time $T$, and a payment $C_j$ occurs at time $t_j$. Its focal-date value is $C_j(1+i)^{T-t_j}$. The whole schedule therefore has focal-date value

\sum_{j=1}^{n} C_j(1+i)^{T-t_j}.

Each term is the value of one payment, so adding the terms accounts for all payments without changing their dates or amounts. If two schedules are equivalent at $T$, these sums must be equal. The same additivity applies under simple interest when each payment is valued directly at the focal date using its simple-interest accumulation factor. This is the justification for forming one equation from the two schedules.

Choose the focal date.
Use the date stated in the problem, or the date at which the schedules are to be compared.

Value each schedule there.
For every payment, use the applicable factor for its time relative to the focal date.

Set the values equal.
Put the original schedule on one side and the replacement schedule on the other.

Solve and check.
Substitute the unknown amount back into the replacement schedule and verify that both focal-date values agree.

Worked Examples

Worked Example: Replacing Three Payments With One

An original schedule pays $1,000 at time 0, $500 at time 1, and $800 at time 3. Replace it with one payment $X$ at time 2. Find $X$ if interest is 5% effective annually under compound interest, and the focal date is time 3. All amounts are positive payments to the recipient.

Method. Accumulate every payment to time 3 under the same compound-interest assumption. Since the original payments occur at or before time 3, each is accumulated forward by the appropriate whole-year factor.

The original schedule has focal-date value

$$ 1000(1.05)^3+500(1.05)^2+800 =1157.625+551.25+800 =2508.875. $$

The replacement payment occurs one year before the focal date, so its value at time 3 is $1.05X$. Equating the values gives

1.05X=2508.875, \qquad X=\frac{2508.875}{1.05}=2389.4047619\ldots.

Rounded to the nearest cent, the replacement payment is $2,389.40. To check, its unrounded focal-date value is $2389.4047619\ldots(1.05)=2508.875$, equal to the original schedule’s value. The common mistake avoided here is comparing the amounts of the payments directly instead of first valuing them at the same date.

Worked Example: Finding One Amount in a Replacement Schedule

An original schedule pays $1,000 at time 0, $500 at time 1, and $600 at time 2. A replacement schedule pays $1,000 at time 0 and an unknown amount $X$ at time 2. Find $X$ at a focal date of time 2 if the effective annual compound interest rate is 10%.

Method. Accumulate the time-0 and time-1 payments to time 2. The payment at time 2 already has its value at the focal date.

The original schedule’s value at time 2 is

$$ 1000(1.10)^2+500(1.10)+600 =1210+550+600 =2360. $$

The replacement schedule’s value at time 2 is $1000(1.10)^2+X=1210+X$. Therefore,

1210+X=2360, \qquad X=1150.

The replacement payment is $1,150. The check is $1210+1150=2360$, matching the original schedule’s focal-date value. The common mistake avoided here is accumulating the original schedule correctly but forgetting to accumulate the known time-0 replacement payment as well.

Worked Example: A Replacement Under Simple Interest

An original schedule pays $800 at time 0 and $400 at time 1. Replace it with a payment of $500 at time 1 and an unknown payment $X$ at time 2. Find $X$ at focal time 3 under simple interest at an annual rate of 8%. All dates are measured in years, and every payment is valued by direct simple-interest accumulation from its payment date to time 3.

Method. For each payment before time 3, multiply by $1+0.08r$, where $r$ is the number of years from its payment date to the focal date.

The original payment at time 0 accumulates for 3 years, and the payment at time 1 accumulates for 2 years. Thus the original value is

800(1+0.08\cdot3)+400(1+0.08\cdot2) =800(1.24)+400(1.16) =992+464 =1456.

The known replacement payment at time 1 accumulates for 2 years, and $X$ at time 2 accumulates for 1 year. Equivalence at time 3 requires

$$ 500(1.16)+X(1.08)=1456, $$

580+1.08X=1456, \qquad X=\frac{876}{1.08}=811.1111111\ldots.

Rounded to the nearest cent, $X$ is $811.11. The check using the unrounded amount is $580+1.08(811.1111111\ldots)=580+876=1456$. The common mistake avoided is treating simple interest as compound interest by applying a one-year factor repeatedly. Here each amount is accumulated directly over its own interval to the focal date.

Common Mistakes and Important Limits

A replacement calculation is only as reliable as its dates, signs, and interest assumptions. Check that every term in the equation represents a value at the same focal date. The following errors can produce an equation that looks plausible but does not compare the schedules correctly.

Comparing unadjusted amounts. Payments on different dates are not directly comparable just because they are both stated in dollars. Move each payment to the focal date first.
Using different rates for the two schedules. Equivalence is defined under a specified interest assumption. Do not value one schedule at one rate and the other at another.
Using the wrong time interval. The exponent under compound interest, or the time $r$ under simple interest, is the number of years between that payment date and the focal date.
Mixing simple and compound interest. If the problem specifies simple interest, use the simple-interest factor directly over the interval. Do not compound it year by year.
Forgetting the sign convention. If a payment is an outflow from the perspective being used, represent it consistently as negative. A schedule can include receipts and payments, not only positive amounts.

For instance, in the first worked example, a wrong calculation would set $X=1000+500+800=2300$, treating the sum of the original payments as the replacement amount. That ignores both the different payment dates and the focal date. The correct equation values the original schedule at time 3 and then solves $1.05X=2508.875$, giving $X=2389.4047619\ldots$.

The focal date is part of the statement of equivalence. This tutorial’s method establishes equality of values at the specified date under the specified model. Do not silently infer more than that statement provides; retain the given timing and interest assumptions when interpreting the result.

Key takeaway: A replacement schedule is found by valuing both schedules at one stated focal date with one consistent interest model. The unknown replacement amount is whatever makes those two focal-date values equal.

Check Your Understanding

For each problem, state the focal-date value equation before solving. Amounts are in dollars, and payments are positive unless specified otherwise.

At 4% effective annual compound interest, an original schedule pays $600 at time 0 and $300 at time 2. Replace it with one payment $X$ at time 1, using time 3 as the focal date. Find $X$, rounded to the nearest cent. (Answer: $912.46)
At 5% effective annual compound interest, an original schedule pays $1,000 at time 1 and $700 at time 3. A replacement schedule pays $900 at time 0 and $X$ at time 2. Find $X$ at focal time 4, rounded to the nearest cent. (Answer: $724.42)
At 6% simple interest, an original schedule pays $1,200 at time 1 and $500 at time 3. Replace it with $X$ at time 0 and $700 at time 2. Find $X$ at focal time 5. (Answer: $940)
At 8% effective annual compound interest, a single payment of $3,412.80 is due at time 2. It is replaced by payments $A$ at time 0 and $B$ at time 1, with the additional condition $A=2B$. Find both amounts using time 2 as the focal date. (Answer: $A=$2,000$, $B=$1,000$)
At 4% effective annual compound interest, an original schedule pays $500 at time 0, $800 at time 1, and $200 at time 3. Replace it with $600 at time 0 and $X$ at time 2. Find $X$ at focal time 3, rounded to the nearest cent. (Answer: $916.15)

What You'll Learn