Introduction
A value equation can show that two cash-flow arrangements have the same value at a chosen date under stated interest assumptions. That conclusion is useful, but it answers a particular question: are the arrangements financially equivalent at that focal date under this model? It does not, by itself, say how interest is credited, accrued, or paid between dates.
Keeping those questions separate matters in exam problems and in practice. A calculated value is a result of a valuation model. The actual timing of interest depends on the financial arrangement being described. One cannot infer the latter from the former unless the model or contract supplies that information.
What a Value Equation Says
For a single amount \(C\) at time \(s\), under a constant effective annual compound rate \(i\), its value at time \(t\) is \(C(1+i)^{t-s}\) if \(t\) is later than \(s\), and \(C/(1+i)^{s-t}\) if \(t\) is earlier than \(s\). This description assumes the stated compound-interest model applies throughout the interval; it does not describe an account whose contractual terms use a different rule.
A value equation places all amounts on one common date and compares their values there. For a set of payments, each payment is moved to that date according to the assumed interest model, with inflows and outflows assigned consistent signs. Equality of the resulting values establishes model-based equivalence. It does not prove that the arrangements have matching intermediate account balances, interest postings, or access to funds.
The focal date is a comparison point, not necessarily a payment date or an interest-credit date. In the examples below, the stated rate is used to value amounts. When an example describes a hypothetical product, its contractual payment or crediting terms are stated separately from its model value.
Why Equivalence Can Be Carried to Another Date
Under constant compound interest, a fixed amount carried from one date to another is multiplied by the accumulation factor for the interval. This gives a useful result: if two amounts are equal in value at one date, carrying both forward or backward over the same interval preserves equality, provided the same model applies and there are no intervening cash flows.
The equality is simple, but its assumptions matter. It concerns two equal values \(V\) at time \(t_1\), each carried to time \(t_2\) under the same constant effective annual rate. If payments occur between those dates, they must be included; simply multiplying the original values would omit them.
For example, take two amounts \(C_1\) and \(C_2\) due at the same date \(s\). If they are equivalent at \(t_1\), then
Thus the equality remains true at \(t_2\). This reasoning is a consequence of the common accumulation rule; it is not evidence that interest was actually posted at either focal date.
Worked Example: One Amount, Two Focal Dates
A payment of $700 is due at time 0. Under a constant 5% effective annual compound rate, find the equivalent single payment at time 4 and verify the common value at time 2. Report money to the nearest cent.
Method. Accumulate the time-0 payment to time 4. Then value each alternative at time 2 using the same 5% model.
The time-4 payment is
Rounded to the nearest cent, the equivalent payment at time 4 is $850.85.
At time 2, the original payment has value
The time-4 payment has value at time 2 of
Both arrangements therefore have a time-2 value of $771.75 under the stated model. To the nearest tenth, that value is $771.8. The check is direct: $771.75 accumulated for two years at 5% becomes $850.854375, the unrounded time-4 amount. This establishes equivalence at the stated dates under the model; it does not describe actual interest postings.
Common mistake avoided: Replacing compound interest with simple interest would give \(700(1+4(0.05))=\$840\). That is not the stated 5% effective annual compound model.
Testing Equivalence Rather Than Assuming It
A value equation can also show that two amounts are not equivalent. Both must be valued at the same focal date using the same rate assumption. A visually plausible later amount is not enough: its value must match the earlier amount.
Worked Example: Checking a Proposed Equivalent Payment
A $500 payment at time 0 is compared with a payment of $629.856 at time 3. The assumed rate is a constant 7% effective annual compound rate. Determine whether the payments are equivalent and calculate the time-0 value of the later payment. Report money to the nearest cent.
Method. Discount the time-3 payment to time 0, then compare it with $500. First calculate the three-year accumulation factor:
The time-0 value of the later payment is
Rounded to the nearest cent, this is $514.15, not $500. The payments are not equivalent under the stated assumptions.
A second check is to accumulate $500 from time 0 to time 3:
That is the time-3 payment that would be equivalent to $500 at time 0. The proposed $629.856 is larger. Discounting it gave $514.15, consistent with this comparison.
Common mistake avoided: The later amount might be treated as equivalent because it is “about what $500 grows to.” The calculation shows otherwise: at 7%, $500 grows to $612.52, not $629.856. Equivalence requires equality under the specified model, not an informal estimate.
Equal Values Do Not Reveal Interest Timing
An endpoint value can be consistent with more than one description of what happens between dates. Consider two hypothetical arrangements, each starting with $1,000 at time 0 and providing $1,210 at time 2.
Worked Example: Same Maturity Value, Different Interest Information
Arrangement A credits interest annually at a 10% effective annual compound rate. Arrangement B is a hypothetical two-year term arrangement that promises a single $1,210 maturity payment at time 2; no interim credit or withdrawal value is specified. Compare their time-2 amounts and explain what can be concluded about the interest timing.
Step 1: Calculate Arrangement A’s balances. At time 1, the balance is
The first year’s interest credit is \(1100-1000=\$100\). At time 2, the balance is
The second year’s interest credit is \(1210-1100=\$110\). The total increase over two years is $210.
Step 2: Compare at maturity. Arrangement B also pays $1,210 at time 2. Thus the time-2 amounts are equal: both are $1,210. They are equivalent at time 2 when compared as maturity amounts.
Step 3: Separate valuation from contractual timing. If the 10% compound model is used to calculate a time-1 value for Arrangement B’s promised maturity payment, the result is
That $1,100 is a model value at time 1. The problem does not state that Arrangement B credits $100 of interest at time 1, or that $1,100 is available for withdrawal then. Its interim balance or surrender value cannot be inferred from the time-2 promise alone.
Common mistake avoided: It would be incorrect to divide the $210 total increase by two and assert that Arrangement B earns or credits $105 each year. The endpoint gives the total difference between the initial amount and maturity payment, not the dates at which interest is credited or available. Arrangement A’s stated credits are $100 and $110; Arrangement B’s interim terms are unspecified.
| Question | What the stated information can establish | What it cannot establish by itself |
|---|---|---|
| Are two payments equivalent at a focal date? | Yes, if their values at that date match under the stated assumptions | Whether the actual arrangements have identical terms |
| What is a future payment worth at an earlier date? | A model value, using the selected rate and valuation rule | The account’s actual interim balance or withdrawal value |
| How much interest is credited in each period? | The credits, if the crediting schedule is specified | The schedule, if only a terminal payment is given |
A Reliable Way to State the Conclusion
When solving a value-equation problem, make the scope of the conclusion explicit. The following sequence keeps the mathematical comparison separate from claims about the underlying arrangement.
Identify the interest rate, compounding convention, dates, and any assumption about how the rule applies between dates.
Move every relevant cash flow to that date using the stated model and consistent signs.
Equality establishes equivalence at that focal date under those assumptions; unequal values do not.
Do not infer interim balances, credit dates, or interest payments unless the problem supplies the relevant terms.
This distinction is useful whenever a problem compares alternative payment schedules, asks for a value on a date between payments, or gives only a terminal amount. The equation answers the valuation question. A statement about actual interest timing requires additional information.
Common Exam Trap
A frequent error is to treat a value at one date as a complete description of the financial arrangement. For instance, from a $1,000 initial amount and a $1,210 payment two years later, one can calculate a two-year growth factor of \(1.21\). Under a constant compound model, the corresponding annual effective rate is 10%, since \((1.10)^2=1.21\). But this calculation does not show that the arrangement credited 10% at the end of each year. The annual rate is a model-consistent summary; the actual crediting schedule is a separate contractual fact.
Likewise, if two schedules are equal at a focal date, that is not a guarantee that they have equal values at all dates under every possible valuation rule. The equivalence conclusion must retain its conditions: the chosen focal date, the specified interest assumptions, and the cash flows included.
Check Your Understanding
For each problem, use the stated interest model and keep the conclusion limited to what the data establish.
- At a constant 4% effective annual compound rate, compare $900 at time 0 with $1,052.872704 at time 4. Are they equivalent? Give their common time-2 value to the nearest cent. (Answer: Yes; $973.44 at time 2.)
- At a constant 5% effective annual compound rate, compare $600 at time 1 with $700 at time 3. Are they equivalent? Give the time-1 value of the later payment to the nearest cent. (Answer: No; $634.92.)
- At a constant 6% effective annual compound rate, find the time-3 payment equivalent to $1,200 at time 0, then find their common time-2 value. Round both amounts to the nearest cent. (Answer: Time-3 payment $1,429.22; common time-2 value $1,348.32.)
- A hypothetical arrangement starts with $2,000 at time 0. One description credits 5% effective annual compound interest in each of the first two years and 0% in the third year. Another promises $2,205 only at time 3 and gives no interim value. Compare their time-3 amounts and state whether their interim interest schedules are shown to be the same. (Answer: Both provide $2,205 at time 3; no, the second arrangement’s interim interest schedule is not specified.)
- At a constant 5% effective annual compound rate, a schedule pays $300 at time 0 and $500 at time 2. What single payment at time 1 is equivalent to that schedule? Give the amount to the nearest cent. (Answer: $791.19 at time 1.)