Why Equivalent Values Do Not Depend on the Focal Date

Introduction

An equation of value compares cash-flow streams at a chosen date, called the focal date. The date is a calculation choice: if two streams are equivalent under a consistent interest model, they remain equivalent when both are moved to another focal date. Their common value will generally change, however. It is the equivalence, not the numerical value, that is independent of the focal date.

This distinction matters in multi-stage problems. A cash flow may need to be accumulated through one rate regime and then another, or discounted backward through the same regimes. The key is to apply the same accumulation schedule to every cash flow. Changing the focal date cannot repair an incorrect equation of value, and it cannot make two genuinely unequal streams equivalent.

Definition: A focal date is the date to which all cash flows in an equation of value are accumulated or discounted. Two signed cash-flow streams are equivalent under a specified interest model if their values are equal at a focal date. Under a consistent accumulation schedule, equality at one focal date implies equality at every other focal date.

Accumulation Across Rate Regimes

Assume deterministic effective annual rates that are constant within each stated time interval, with the rate schedule known in advance. For dates $s<t$, let $A(s,t)$ be the accumulation factor from $s$ to $t$: the product of the effective accumulation factors for all periods between those dates. Thus a cash flow $C$ at $s$ is worth $C A(s,t)$ at $t$. Discounting from $t$ to $s$ divides by the same factor.

For example, if the effective annual rate is $4\%$ from time 0 to 1 and $8\%$ from time 1 to 3, then $A(0,3)=(1.04)(1.08)^2$. The rate applying to each interval must be used only for the time spent in that interval. In particular, it would be incorrect to apply the $8\%$ rate to the entire three-year span.

Valuation rule: For signed cash flows $C_k$ at dates $t_k$, the value at focal date $f$ is the sum of the values of the individual cash flows:

If $t_k\leq f$, the contribution is $C_k A(t_k,f)$. If $t_k>f$, the contribution is $C_k/A(f,t_k)$. A cash flow at $f$ contributes $C_k$ itself. The cash-flow signs and the accumulation schedule must be handled consistently.

A positive cash flow is typically a receipt and a negative one a payment, although either sign convention is acceptable if applied to every stream. It is often helpful to value the streams separately first, then compare them. The difference between their values is the value of a single signed stream formed by subtracting one from the other.

Why the Focal Date Can Change

The essential property is that accumulation factors compose. For $s<u<t$, accumulating from $s$ to $u$ and then from $u$ to $t$ gives the same result as accumulating directly from $s$ to $t$:

$$ A(s,t)=A(s,u)A(u,t). $$

This holds because the factors for the intervals from $s$ to $u$ and from $u$ to $t$ together contain exactly the same rate periods as the full interval from $s$ to $t$. It remains true when $u$ is a rate boundary.

Move one dated cash flow.
Let a cash flow $C$ occur at date $t$, and take two focal dates $f<g$. Its value at $f$, multiplied by $A(f,g)$, equals its value at $g$.

Check all possible positions of the payment date.
If $t\leq f$, then $C A(t,f)A(f,g)=C A(t,g)$. If $f<t\leq g$, then $\bigl(C/A(f,t)\bigr)A(f,g)=C A(t,g)$. If $g<t$, then $\bigl(C/A(f,t)\bigr)A(f,g)=C/A(g,t)$, because $A(f,t)=A(f,g)A(g,t)$.

Add the cash flows.
Each dated cash flow has the same transformation, so summing them gives $V_g=V_f A(f,g)$, where $V_f$ and $V_g$ are the stream values at the two focal dates.

Apply the result to two streams.
For streams $L$ and $R$, $V_g(L)-V_g(R)=\bigl(V_f(L)-V_f(R)\bigr)A(f,g)$. Since $A(f,g)>0$, a zero difference at $f$ remains zero at $g$, and a nonzero difference remains nonzero.

This derivation explains both parts of the result. Equivalent streams have equal values at every focal date because their difference is zero and remains zero after multiplication by an accumulation factor. But the common value generally changes: moving the focal date from $f$ to $g$ multiplies it by $A(f,g)$. The result assumes that the same deterministic accumulation schedule applies to both streams and that no extra costs or cash flows arise merely from changing the valuation date.

Worked Examples Across Changing Rates

Worked Example: Detecting a False Equivalence

The effective annual rate is $10\%$ from time 0 to 1 and $5\%$ from time 1 to 2. Stream $L$ pays $\$200$ at time 0 and $\$100$ at time 1. Stream $R$ pays $\$346.50$ at time 2. Determine whether the streams are equivalent, and compare their values at times 0 and 2.

Method. Value both streams at time 0 by discounting, then value them at time 2 by accumulating. The full factor from time 0 to 2 is $(1.10)(1.05)=1.155$.

At time 0: Stream $L$'s time-0 payment is already at the focal date. Its time-1 payment is discounted one year at $10\%$:

V_0(L)=200+\frac{100}{1.10}=290.909090\ldots

Stream $R$'s payment is discounted across both rate periods:

V_0(R)=\frac{346.50}{(1.10)(1.05)} =\frac{346.50}{1.155}=300.00.

The values differ, so the streams are not equivalent. At time 2, accumulate each payment of $L$ only across the periods after its payment date:

V_2(L)=200(1.10)(1.05)+100(1.05) =231+105=336.00, \qquad V_2(R)=346.50.

The difference at time 2 is $\$10.50$, while the difference at time 0 is $300-290.909090\ldots=\$9.090909\ldots$. The time-0 difference multiplied by $1.155$ is $\$10.50$, as the focal-date relationship predicts.

Common mistake avoided. A tempting but incorrect calculation is to call both streams worth $\$300$ at time 0 and $\$346.50$ at time 2. Those figures are the values of $R$, not $L$. The calculation for $L$ must include its actual $\$200$ at time 0 and discount or accumulate its $\$100$ at time 1 according to the relevant interval.

Worked Example: Verifying Equivalence at Three Focal Dates

The effective annual rate is $4\%$ from time 0 to 1, $8\%$ from time 1 to 3, and $3\%$ from time 3 to 4. Stream $X$ pays $\$100$ at time 0 and $\$50$ at time 2. Stream $Y$ pays $\$180.564768$ at time 4. Verify whether the streams are equivalent by valuing them at times 2, 4, and 0.

Method. First accumulate the payments of $X$ to time 4 to check the terminal amount in $Y$. Then compare the streams at the other focal dates using the same rate factors. Rates are effective annual rates and each interval spans whole years.

From time 0 to 4, the $\$100$ accumulates across all three regimes. From time 2 to 4, the $\$50$ accumulates at $8\%$ for one year and $3\%$ for one year:

$$ V_4(X)=100(1.04)(1.08)^2(1.03)+50(1.08)(1.03) =124.944768+55.62=180.564768. $$

This equals $V_4(Y)$, since $Y$ pays exactly $\$180.564768$ at time 4. At time 2, the $\$100$ has accumulated through the first two rate periods, while the $\$50$ is already at the focal date:

$$ V_2(X)=100(1.04)(1.08)+50=112.32+50=162.32. $$

Discounting $Y$ from time 4 to time 2 uses the $8\%$ and $3\%$ periods:

V_2(Y)=\frac{180.564768}{(1.08)(1.03)} =\frac{180.564768}{1.1124}=162.32.

At time 0, discount $X$'s time-2 payment across the $4\%$ and $8\%$ periods. Discount $Y$ across the entire schedule:

V_0(X)=100+\frac{50}{(1.04)(1.08)} =100+44.5156695\ldots =144.5156695\ldots $$ $$ V_0(Y)=\frac{180.564768}{(1.04)(1.08)^2(1.03)} =144.5156695\ldots

Thus the streams are equivalent. The common values are $\$144.515670$ at time 0, $\$162.32$ at time 2, and $\$180.564768$ at time 4. Values are shown to six decimal places where needed; no rounding is used in the intermediate factors.

Common mistake avoided. The time-2 payment of $\$50$ is not accumulated at $4\%$, because that rate ended at time 1. Likewise, $Y$'s time-4 payment must be discounted through every intervening regime, not by applying a single rate to four years.

Worked Example: Comparing Streams with Payments on Both Sides of a Focal Date

The effective annual rate is $6\%$ from time 0 to 1, $4\%$ from time 1 to 2, and $10\%$ from time 2 to 3. Stream $A$ pays $\$300$ at time 0 and $\$200$ at time 2. Stream $B$ pays $\$583.792$ at time 3. Determine whether they are equivalent by comparing their values at times 1 and 3.

Method. At time 3, accumulate each earlier payment of $A$ forward across the rate intervals it actually spans. At time 1, accumulate the time-0 payment and discount the time-2 payment; discount $B$'s payment back across the last two regimes.

At time 3, the $\$300$ at time 0 accumulates at $6\%$, then $4\%$, then $10\%$. The $\$200$ at time 2 accumulates for one year at $10\%$:

$$ V_3(A)=300(1.06)(1.04)(1.10)+200(1.10) =363.792+220=583.792. $$

Since $V_3(B)=\$583.792$, the streams have equal values at time 3. Now move the focal date to time 1. Stream $A$'s time-0 payment accumulates at $6\%$; its time-2 payment is discounted one year at $4\%$:

V_1(A)=300(1.06)+\frac{200}{1.04} =318+192.307692\ldots =510.307692\ldots

Stream $B$ is discounted from time 3 to time 1 using $4\%$ from time 1 to 2 and $10\%$ from time 2 to 3:

V_1(B)=\frac{583.792}{(1.04)(1.10)} =\frac{583.792}{1.144} =510.307692\ldots

The values agree at time 1 as well as time 3, so the streams are equivalent. The common value changes by the factor $(1.04)(1.10)=1.144$ when the focal date moves from time 1 to time 3: $510.307692\ldots(1.144)=583.792$.

Common mistake avoided. A payment before the focal date is accumulated, while a payment after it is discounted. Treating the time-2 payment as if it were already at time 1, or discounting it with the $6\%$ rate from time 0 to 1, would use the wrong dates and rate interval.

How to Use Focal-Date Invariance

In an exam problem, the focal date can be chosen for convenience, but it cannot be chosen to make the arithmetic inconsistent. A good choice often places the date at a rate boundary or at a major payment date, reducing the number of distinct factors. Regardless of the choice, each cash flow must be carried from its own payment date to the focal date through the correct sequence of regimes.

A reliable procedure is:

Write the time line, marking every payment date and every rate boundary.
Choose a focal date and identify which payments are accumulated and which are discounted.
Write the factor for each payment separately, using only the periods between its date and the focal date.
Sum the signed values for each stream and compare them.
If changing focal date, multiply the value difference by the accumulation factor between the two focal dates; do not recompute with a different or inconsistent rate schedule.

Focal-date invariance is useful for checking work. If two streams are equal at one date, their values at another date must be equal under the same model. If they are not, at least one calculation has used the wrong time interval, omitted a cash flow, or applied inconsistent signs. Conversely, if the streams differ at one date, the difference cannot disappear at another date under a positive accumulation factor.

Key takeaway: An equation of value is independent of focal date in the sense that equality or inequality is preserved. The value itself is not fixed: moving the focal date scales each stream's value, and their difference, by the accumulation factor between the dates.

Check Your Understanding

For each problem, use the stated effective annual rates and assume no fees or other cash flows. Keep the rate schedule consistent when changing focal dates.

The effective annual rate is $5\%$ from time 0 to 2. Stream $P$ pays $\$500$ at time 0 and $\$100$ at time 1. Stream $Q$ pays $\$656.25$ at time 2. Are the streams equivalent, and what is their common value at time 1? (Answer: Yes; $\$625.00$ at time 1.)
The effective annual rate is $3\%$ from time 0 to 1 and $7\%$ from time 1 to 3. Stream $M$ pays $\$800$ at time 0 and $\$100$ at time 2. What single payment at time 3 is equivalent to $M$? (Answer: $\$1,050.3976$.)
The effective annual rate is $4\%$ from time 0 to 2 and $9\%$ from time 2 to 4. Stream $R$ pays $\$250$ at time 1 and $\$400$ at time 3. What is its value at time 2, and what single payment at time 4 is equivalent to it? (Answer: $\$626.9725$ at time 2; $\$744.906$ at time 4.)
Two streams have values $\$1,250$ and $\$1,200$ at time 2. The effective annual rate is $6\%$ from time 2 to 3 and $4\%$ from time 3 to 4. What is the difference in their values at time 4, and can they become equivalent at time 4? (Answer: The difference is $\$55.12$; no, they cannot become equivalent.)
State why equal values at one focal date imply equal values at another focal date under a deterministic rate schedule, and explain why the common value may nevertheless change. (Answer: Each stream value, and hence its difference from the other stream's value, is multiplied by the positive accumulation factor between focal dates. A zero difference remains zero, while the common value is scaled by that factor.)

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