Comparing Lump-Sum Offers at a Common Date

Why a Common Date Is Necessary

A lump-sum offer is not fully described by its dollar amount. The date on which it is paid matters because money available earlier can earn interest, while money received later must be discounted to compare it with money available sooner. Thus, an offer of $20,000 today cannot be compared directly with an offer of $21,000 three years from now by looking only at the two amounts.

The solution is to express every offer at the same valuation date. An earlier payment is accumulated forward to that date; a later payment is discounted back to it. Once the dates match, the resulting values can be compared directly. The calculation is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: A common-date value is the value, at a specified valuation date, of a payment made at another date. Under a fixed annual effective interest rate $i$, a payment of $C$ at time $t$, measured in years, has value at time $T$ $$ V_T=C(1+i)^{T-t}. $$ If $T>t$, the payment is accumulated forward. If $T<t$, the negative exponent discounts it back. This formula assumes the same effective annual rate applies over the interval and that the payment is made in full at the stated time.

The valuation date is a comparison tool, not a change to the contractual payment date. For example, valuing a year-3 payment at year 5 does not mean the recipient actually gets the money in year 5; it means we calculate the equivalent value at year 5 under the stated interest assumption.

Moving a Single Payment to a Common Date

For an effective annual interest rate $i$, one dollar at a date grows to $1+i$ dollars one year later. It grows to $(1+i)^n$ dollars over $n$ years when the same rate applies throughout. Reversing this movement discounts by the reciprocal factor $v=1/(1+i)$. This is why the same formula works whether the payment is moved forward or backward.

V_T=C(1+i)^{T-t}

In practical use, first list the payment amount and its date for each offer. Then select a common date, apply the appropriate growth or discount factor to each amount, and compare the results. A date at or after the latest payment often makes all calculations accumulations; a date at or before the earliest payment makes all calculations discounts. Either choice is valid if applied consistently.

State the terms.
Record each payment amount, its time, the effective annual rate, and any assumptions that affect the comparison.

Choose a common date.
Use the same valuation date for every offer; do not compare amounts that remain at different dates.

Move each payment.
Use $C(1+i)^{T-t}$. A positive exponent accumulates forward; a negative exponent discounts backward.

Compare and interpret.
Subtract the smaller common-date value from the larger and identify which offer has the greater value at that date.

Worked Comparisons

The examples treat payment amounts as deterministic contractual cash flows. No mortality assumption or probability distribution is involved. Unless stated otherwise, taxes, transaction costs, payment default, and changes in the interest rate are ignored. These assumptions matter: the common-date result is conditional on the rate and cash flows specified.

Worked Example: Comparing a Year-3 Settlement With a Year-5 Settlement

A claimant can accept a hypothetical settlement of $33,920 at the end of year 3 or $38,000 at the end of year 5. Assume a fixed annual effective interest rate of 6% for both years, with annual compounding. Compare the offers at the end of year 3. The amounts are certain, and there are no taxes, costs, or default risk.

Method. Year 3 is the common date. The first offer is already stated at that date. The second is paid two years later, so discount it by two years:

V_3^{(1)}=\$33{,}920

V_3^{(2)} =\frac{\$38{,}000}{(1.06)^2} =\frac{\$38{,}000}{1.1236} \approx \$33{,}819.86.

The difference is $$ \$33{,}920.00-\$33{,}819.86=\$100.14. $$ At the year-3 valuation date, the year-3 payment is greater by $100.14. Amounts are rounded to cents only at the final step. As a check on the direction, moving the later payment backward must reduce it because the assumed rate is positive.

Common mistake avoided. Multiplying the year-5 amount by $(1.06)^2$ would move it forward to year 7, not backward to year 3. The correct factor is division by $(1.06)^2$. This comparison does not say the offers differ by $100.14 on their actual payment dates; that difference is expressed at year 3.

Worked Example: Comparing Payment Now With Payment in Two Years

A customer is offered either $24,000 immediately or $27,000 at the end of year 2 to settle a hypothetical contract dispute. Assume a fixed annual effective rate of 5%, annual compounding, certain payments, and no taxes or costs. Compare the offers at the end of year 2.

Method. Accumulate the immediate payment for two years. The year-2 offer is already at the common date:

V_2^{(\text{now})} =\$24{,}000(1.05)^2 =\$24{,}000(1.1025) =\$26{,}460.00

V_2^{(\text{later})}=\$27{,}000.00.

Subtracting gives $$ \$27{,}000.00-\$26{,}460.00=\$540.00. $$ The delayed payment has the greater year-2 value by $540.00. The intermediate one-year values for the immediate payment are $24,000(1.05)=$25,200 after year 1 and $25,200(1.05)=$26,460 after year 2, confirming the two-year accumulation.

Check at another common date. At time 0, the immediate offer is $24,000. The year-2 offer has present value $$ \frac{\$27{,}000}{1.1025}=\$24{,}489.80 $$ to the nearest cent. It is larger by $489.80 at time 0. This agrees with the year-2 comparison: both use the same rate and differ only by a common positive accumulation factor before rounding.

Common mistake avoided. Comparing $24,000 directly with $27,000 ignores the two-year wait. The later amount is larger in dollars, but the meaningful comparison requires accounting for the value the immediate payment could earn.

Worked Example: A Close Comparison of Two Settlement Dates

Under a hypothetical settlement, one option pays $18,500 at the end of year 1 and another pays $20,000 at the end of year 3. Assume a fixed annual effective rate of 4%, annual compounding, certain payments, and no taxes or costs. Compare at the end of year 3, rounding money to cents.

Method. The first payment is received two years before the common date. Accumulate it for one year and then for a second year:

\$18{,}500(1.04)=\$19{,}240

V_3^{(1)} =\$19{,}240(1.04) =\$20{,}009.60.

The other offer is paid at the common date, so $V_3^{(2)}=\$20{,}000.00$. Therefore, $$ \$20{,}009.60-\$20{,}000.00=\$9.60. $$ The year-1 offer is greater by $9.60 at year 3. The calculation is checked by the two-year factor $(1.04)^2=1.0816$: $\$18{,}500(1.0816)=\$20{,}009.60$.

Common mistake avoided. Rounding the growth factor too early can obscure a small difference between offers. Keep the given rate and factor through the calculation, and round the final dollar values to cents.

Why the Common-Date Comparison Works

The comparison is based on a consistent model of how money moves through time. If a payment $C$ is made at time $t$, its value one year later is $C(1+i)$. Repeating that one-year growth $T-t$ times gives the value at a later date $T$:

C\underbrace{(1+i)(1+i)\cdots(1+i)}_{T-t\text{ annual periods}} =C(1+i)^{T-t}.

For a date earlier than the payment, reverse the same steps. Since one-year accumulation multiplies by $1+i$, reversing it divides by $1+i$. Applying that reversal for $t-T$ years gives $$ \frac{C}{(1+i)^{t-T}}=C(1+i)^{T-t}. $$ Thus both directions follow from the same accumulation relationship, rather than from separate unrelated rules.

There is also a useful consistency check. Suppose one offer is valued at time $T_1$, and both offers are then moved to a later common date $T_2$ using the same positive rate. Each value is multiplied by the same positive factor $(1+i)^{T_2-T_1}$. Multiplication by a positive number preserves which value is larger. Therefore, with the same fixed rate and assumptions, choosing a different common date should not reverse the ranking. Small discrepancies can arise if intermediate amounts are rounded.

Assumptions and Common Traps

A common-date value is only as meaningful as the assumptions used to calculate it. In particular, the rate must be appropriate for the period being valued. A quoted nominal rate is not automatically the effective annual rate $i$; if the problem specifies a nominal rate with a compounding frequency, first determine the effective rate or use the stated periodic rate consistently. This tutorial's examples specify effective annual rates directly.

Comparing amounts at different dates. First convert all payments to one common date. A larger nominal amount may still be less valuable if it arrives much later.
Using accumulation when discounting is required. Moving a payment to an earlier date uses a negative exponent, or equivalently division by the accumulation factor.
Using different assumptions for different offers. Apply the same stated rate and valuation convention to all offers unless the problem explicitly supplies different terms.
Forgetting what the result means. A year-3 value is a model-based equivalent at year 3; it does not change when the cash is contractually paid.
Rounding before comparison. Carry full precision in intermediate calculations and round money at the final reporting step.

In a real settlement decision, a calculated value is not necessarily the only consideration. Payment certainty, taxes, legal costs, liquidity needs, and risk may matter. If those factors are excluded from the stated model, the result is a financial comparison under the stated assumptions—not a complete personal or legal recommendation.

Key takeaway: To compare lump sums at different times, select one valuation date and move every payment to it using the same effective rate and consistent timing assumptions. Accumulate payments that occur earlier than the valuation date; discount payments that occur later.

Check Your Understanding

For each problem, use the stated annual effective rate and compare all offers at the requested common date. Treat payments as certain, with no taxes or costs, and round final money amounts to cents.

An organization can receive $12,000 now or $13,000 at the end of year 2. At an annual effective rate of 3%, compare at year 2. Which offer is larger, and by how much? (Answer: The year-2 offer is larger by $269.20; the accumulated value of $12,000 is $12,730.80.)
A contract offers $21,000 at the end of year 1 or $25,000 at the end of year 4. At an annual effective rate of 5%, compare at year 4. (Answer: The year-4 offer is larger by $689.88; the year-1 payment accumulates to $24,310.13.)
A hypothetical settlement pays $32,000 at the end of year 2 or $36,000 at the end of year 5. At an annual effective rate of 4%, compare at year 5. (Answer: The year-5 offer is larger by $4.35; the year-2 payment accumulates to $35,995.65.)
Choose between $40,000 at the end of year 1 and $44,500 at the end of year 3. The annual effective rate is 6%. Compare at year 3 and state which offer is larger. (Answer: The year-1 offer is larger by $444.00; its year-3 value is $44,944.00.)
A payer offers $50,000 now or $55,000 at the end of year 3. At an annual effective rate of 3.5%, compare at year 3. (Answer: The immediate offer is larger by $435.89; its year-3 value is $55,435.89.)

What You'll Learn