Comparing Settlement Installment Schedules

Why Settlement Schedules Need a Common Basis

A settlement may offer a series of installments rather than one payment today. The schedule with the larger total of stated dollars is not necessarily more valuable: a dollar received sooner can be invested or used sooner. To compare alternatives, first choose a common valuation date, then express every payment as a value at that date using a consistent interest-rate assumption.

The comparison is conditional on the model. A present-value calculation does not establish that a schedule is legally fair or that its payments are certain. It answers a narrower question: given the stated payment amounts, dates, and rate, what is the value of each schedule at the chosen date?

Definition: The present value of a settlement schedule is the sum of the present values of its individual payments, all measured at one valuation date. If payment $C_j$ is due $t_j$ years after that date and the effective annual interest rate $i$ is constant, the present value at time 0 is

PV_0=\sum_{j=1}^{n} C_j(1+i)^{-t_j}

Here $C_j$ is the amount of the $j$th payment, $t_j$ is its time in years from the valuation date, and $i>-1$. A payment at time 0 has $t_j=0$, so its present value is its full amount.

This formula assumes the payments are measured consistently, are payable as stated, and are discounted at the same effective annual rate. It also assumes that rate is appropriate for the comparison and that there are no additional fees, taxes, payment contingencies, or default adjustments. If those factors matter, they must be reflected separately in the cash flows or in an appropriate valuation model. Do not mix nominal and inflation-adjusted cash flows with an inconsistent rate.

For a recipient, the schedule with the greater present value is more valuable under the assumptions. For a payer, the lower present value is less costly. State whose perspective is being used; otherwise, “better” can be ambiguous.

A Reliable Comparison Method

Write each schedule as dated cash flows before calculating. The valuation date is the common reference point, and every time $t_j$ is measured from it. With a constant effective annual rate $i$, a payment at time $t_j$ is multiplied by $v^{t_j}$, where $v=1/(1+i)$. The exponent is the elapsed time in years, not the payment number in a list.

Fix the valuation date.
For example, set today as time 0 and measure every due date from today.

List amounts and dates.
Keep each installment paired with its own time. Do not combine payments with different due dates before discounting.

Apply the stated rate consistently.
Discount each amount using its own elapsed time and the stated compounding convention.

Sum and compare.
Compare the present values at the common date, report the difference, and state which perspective makes that result favorable.

Why the Present-Value Sum Works

The formula follows from moving each payment separately to the same date. Under a constant effective annual rate $i$, one dollar at time 0 accumulates to $1+i$ dollars after one year and to $(1+i)^t$ dollars after $t$ years under the model. Therefore, the amount at time 0 equivalent to receiving $C_j$ at time $t_j$ is the amount which accumulates to $C_j$:

PV_j(1+i)^{t_j}=C_j \qquad\Longrightarrow\qquad PV_j=\frac{C_j}{(1+i)^{t_j}}

The schedule consists of all its individual payments. Since values at the same date can be added, the schedule’s total present value is the sum of these individual present values. This reasoning relies on using the same rate convention and valuation date for every installment; discounting different payments with incompatible assumptions would not produce a coherent comparison.

Worked Example: Two Schedules with Equal Present Values

A hypothetical settlement recipient can choose between Schedule A, which pays $11,000 at the end of year 1, $12,100 at the end of year 2, and $14,641 at the end of year 4, and Schedule B, which pays $16,500 at the end of year 1 and $19,965 at the end of year 3. Compare them at time 0 using an effective annual rate of 10%. Assume all payments are certain and there are no fees or taxes.

Method. Discount every installment to time 0, then add the discounted amounts within each schedule. This avoids judging the offers only by their undiscounted totals.

For Schedule A, $1.10^1=1.10$, $1.10^2=1.21$, and $1.10^4=1.4641$. Thus:

PV_A=\frac{\$11{,}000}{1.10} +\frac{\$12{,}100}{1.21} +\frac{\$14{,}641}{1.4641} =\$10{,}000+\$10{,}000+\$10{,}000 =\$30{,}000

For Schedule B, $1.10^3=1.331$. The two discounted payments are:

PV_B=\frac{\$16{,}500}{1.10} +\frac{\$19{,}965}{1.331} =\$15{,}000+\$15{,}000 =\$30{,}000

The present values are equal, so neither schedule has a present-value advantage at 10% under these assumptions. Schedule A’s nominal total is $\$11{,}000+\$12{,}100+\$14{,}641=\$37{,}741$. Schedule B’s nominal total is $\$16{,}500+\$19{,}965=\$36{,}465$. The larger nominal total for A does not make it more valuable at the stated rate. Check: each installment in A has a present value of $10,000, and each installment in B has a present value of $15,000, confirming both totals of $30,000.

Comparing Unequal Present Values

When present values differ, report the difference in dollars at the valuation date. Keep the rate and perspective visible in the conclusion: a schedule’s value is not independent of the assumptions used to calculate it. The next example also illustrates why a payment due immediately must not be discounted.

Worked Example: Selecting Between Two Uneven Installment Plans

A hypothetical recipient is offered Schedule A: $11,000 at the end of year 1 and $18,150 at the end of year 2. Schedule B pays $8,000 immediately, $12,100 at the end of year 1, and $8,470 at the end of year 2. Compare the schedules at time 0 using an effective annual rate of 10%. Assume payments are certain and there are no fees or taxes.

Method. Use $1.10$ for year 1 and $1.10^2=1.21$ for year 2. A payment immediately at time 0 is already at the valuation date.

For Schedule A, the year-1 payment has present value $\$11{,}000/1.10=\$10{,}000$, and the year-2 payment has present value $\$18{,}150/1.21=\$15{,}000$. Therefore:

PV_A=\$10{,}000+\$15{,}000=\$25{,}000

For Schedule B, the immediate $8,000 payment contributes $8,000. The year-1 payment contributes $\$12{,}100/1.10=\$11{,}000$, and the year-2 payment contributes $\$8{,}470/1.21=\$7{,}000$. Therefore:

PV_B=\$8{,}000+\$11{,}000+\$7{,}000=\$26{,}000

From the recipient’s perspective, Schedule B has a present value higher by $\$26{,}000-\$25{,}000=\$1{,}000$. Common mistake: discounting the immediate $8,000 as though it were due in year 1 gives an incorrect calculation:

\frac{\$8{,}000}{1.10}+\frac{\$12{,}100}{1.10} +\frac{\$8{,}470}{1.21} =\$7{,}272.73+\$11{,}000+\$7{,}000 =\$25{,}272.73

The correction is to use $t=0$ for the immediate payment, so its present value is $8,000, not $7,272.73. Check: the correctly valued installments in Schedule B sum to $26,000, which is $1,000 more than Schedule A for the recipient.

The Rate Assumption Can Change the Ranking

A schedule with earlier payments often benefits more from a higher discount rate than one with later payments, but the ranking of two schedules depends on their entire pattern of cash flows. A rate should therefore be stated as part of the comparison, not presented as a hidden calculation choice.

The following example uses a deliberately simple pair of schedules. Both include the same year-2 payment, so their difference depends only on their year-1 and time-0 payments. This makes the rate’s role transparent.

Worked Example: Finding the Rate at Which the Ranking Changes

Schedule A pays $11,000 at the end of year 1 and $12,100 at the end of year 2. Schedule B pays $10,000 immediately and $12,100 at the end of year 2. Find the rate at which their present values are equal, then compare them at effective annual rates of 8% and 12%. Assume certain payments and no fees or taxes.

Method. First equate the two present-value expressions. Because both schedules have the same year-2 payment, that term cancels when the values are compared.

PV_A-PV_B =\left(\frac{\$11{,}000}{1+i} +\frac{\$12{,}100}{(1+i)^2}\right) -\left(\$10{,}000 +\frac{\$12{,}100}{(1+i)^2}\right) =\frac{\$11{,}000}{1+i}-\$10{,}000

The schedules are equal when $\$11{,}000/(1+i)=\$10{,}000$. Multiplying by $1+i$ gives $1+i=1.10$, so the equality rate is $i=0.10$, or 10%. Below 10%, Schedule A is higher; above 10%, Schedule B is higher.

At 8%, Schedule A’s present value is $\$11{,}000/1.08+\$12{,}100/1.1664=\$10{,}185.19+\$10{,}373.80=\$20{,}558.98$. Schedule B’s present value is $\$10{,}000+\$10{,}373.80=\$20{,}373.80$. A is higher by $185.19, using unrounded values before rounding the difference to cents.

At 12%, Schedule A’s present value is $\$11{,}000/1.12+\$12{,}100/1.2544=\$9{,}821.43+\$9{,}646.56=\$19{,}467.98$. Schedule B’s present value is $\$10{,}000+\$9{,}646.56=\$19{,}646.56$. B is higher by $178.57. Values are rounded to the nearest cent only at the final reporting step. Check: at 10%, both schedules have present value $20,000; the two comparisons on either side of that rate agree with the derived ranking.

Assumptions and Common Traps

A sound comparison makes its assumptions explicit. In the examples, the rate is an effective annual rate and each date is measured in years from time 0. If a problem instead gives a nominal rate convertible $m$ times per year, use its stated compounding convention and align the period count with the payment dates. Do not substitute a nominal rate directly for an effective annual rate.

Other frequent errors are avoidable by writing the dates beside the amounts before calculating:

Comparing nominal totals only. Totals do not account for when installments are paid. Calculate each schedule’s present value before deciding which is more valuable.
Discounting by installment number. The second listed payment is not automatically a year-2 payment. Use its actual date measured from the valuation date.
Discounting a time-0 amount. Its discount factor is $(1+i)^0=1$, so it is unchanged.
Mixing assumptions. Use one valuation date and consistent rate conventions for both schedules. If the rate changes, recalculate both alternatives.
Leaving out the perspective. A higher present value favors the recipient but costs more from the payer’s perspective.

The calculation is useful only to the extent that its inputs represent the decision being made. If payments are contingent, delayed by a condition, or subject to fees, the stated cash flows may need adjustment. A present-value ranking should be reported together with the valuation date, rate, and key assumptions rather than as an unconditional claim about the settlement.

Key takeaway: To compare settlement installment schedules, discount each dated payment to a common date, sum within each schedule, and compare the results from a stated perspective. The schedule ranking can depend on the assumed rate as well as on the amounts and dates.

Check Your Understanding

For each problem, treat the payments as certain, use the stated rate convention, and compare values from the recipient’s perspective unless otherwise specified.

At an effective annual rate of 5%, Schedule A pays $10,000 immediately and $11,025 at the end of year 2. Schedule B pays $10,500 at the end of year 1 and $11,025 at the end of year 2. Find both present values and compare the nominal totals. (Answer: Both present values are $20,000.00; A’s nominal total is $21,025 and B’s is $21,525.)
At an effective annual rate of 10%, Schedule X pays $7,700 at the end of year 1 and $14,520 at the end of year 2. Schedule Y pays $5,000 immediately, $7,260 at the end of year 1, and $8,470 at the end of year 2. Which is worth more, and by how much? (Answer: $PV_X=\$19,000.00$, $PV_Y=\$18,600.00$; X is higher by $400.00.)
A schedule pays $4,000 at the end of year 1, $5,000 at the end of year 2, and $6,000 at the end of year 4. Calculate its present value at an effective annual rate of 6%, rounding to cents. (Answer: $12,976.13.)
A schedule pays $10,600 after 6 months and $11,910.16 after 18 months. The rate is nominal 12% per year convertible semiannually. Compare its present value with an immediate payment of $19,700. (Answer: The installment schedule has present value $20,000.00 and is higher by $300.00.)
Schedule C pays $11,000 at the end of year 1 and $12,100 at the end of year 2. Schedule D pays $10,000 immediately and $12,100 at the end of year 2. Find the equality rate and state which schedule has the greater present value at 8%. (Answer: Equality occurs at 10%; at 8%, C is higher by $185.19.)

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