Moving a Cash-Flow Value Between Dates

Why Move a Value Between Dates?

A cash-flow value is always tied to a date. A value of $10,000 today and a value of $10,000 three years from today are not economically equivalent when interest is earned. To compare or check amounts, first place them at a common date using a stated rate convention. This is the basic operation of moving a value between dates.

The operation is useful both for a single known value and for the value of a complete cash-flow stream. The key is consistency: use the same interest model and rate convention in each step. In particular, do not treat a nominal rate as though it were an effective annual rate, or count periods in years when the stated rate compounds more frequently.

Definition: Under a fixed effective annual interest rate $i$, a value $V_s$ at date $s$ moved to date $t$ is $V_t=V_s(1+i)^{t-s}$. If $t>s$, this accumulates the value forward. If $t<s$, the negative exponent discounts the value backward. Dates are measured in years, and this form assumes the effective annual rate applies over the elapsed time.

For annual dates, let $n$ be the number of years between the two dates. Forward movement uses the accumulation factor $(1+i)^n$; backward movement uses its reciprocal, $v^n$, where $v=1/(1+i)$. The same rate is used in both directions. A value moved forward and then backward over the same interval must return to its starting amount, apart from any rounding made for display.

V_{\text{later}}=V_{\text{earlier}}(1+i)^n, \qquad V_{\text{earlier}}=V_{\text{later}}v^n =\frac{V_{\text{later}}}{(1+i)^n}

Moving a Known Value

Before calculating, identify the date of the known value, the target date, the number of periods between them, and the rate convention. Use an exponent that counts the periods between the dates—not the date number of the target by itself. If a value is known at date 2 and is needed at date 5, the interval is three years.

Worked Example: Accumulating and Checking a Known Value

A project appraisal assigns a value of $9,000 at date 1. Assume a fixed effective annual rate of 7%, with annual dates and annual compounding. Find the equivalent value at date 4, then check by discounting it back to date 1.

Method. The target is three years after the known value, so accumulate by $(1.07)^3$. The common mistake to avoid is using date 4 as the exponent instead of the three-year interval.

Forward calculation. The accumulation factor is $1.07^3=1.225043$. Thus $V_4=\$9{,}000(1.225043)=\$11{,}025.387$, which is $11,025.39 to the nearest cent.

Check. Discount the unrounded result using the same three-year interval and rate: $\$11{,}025.387/1.225043=\$9{,}000$. The original value is recovered. Rounding to cents before the check can create a difference of a fraction of a cent, so the check uses the unrounded amount.

The reverse direction is not a different economic rule. It is the same accumulation relationship solved for the earlier value. Dividing by $(1+i)^n$ is equivalent to multiplying by $v^n$. A common sign check is helpful: at a positive interest rate, moving a positive value forward makes it larger, while moving it backward makes it smaller.

Moving the Value of a Whole Stream

A stream may contain several receipts and payments on different dates. Assign each cash flow a sign, using positive values for receipts and negative values for payments. The value of the complete stream at a chosen date is the sum of each cash flow moved to that date. A cash flow before the chosen date is accumulated forward; one after the chosen date is discounted backward.

Definition: For signed cash flows $C_j$ occurring at dates $j$, the value of the complete stream at date $t$, under one fixed effective annual rate $i$, is $V_t=\sum_j C_j(1+i)^{t-j}$. This expression assumes the same rate convention applies to every period and every cash flow in the stream.

This definition lets us express a stream at any selected date, including a date that is neither its first nor its last cash-flow date. It is important to distinguish this equivalent value from an account balance that is actually held over time: if cash flows are deposited into or withdrawn from an account between two dates, those intervening transactions must be included in the balance calculation.

Worked Example: Re-Expressing an Uneven Stream at Two Dates

A hypothetical contract has receipts of $5,000 at date 0, $4,000 at date 1, and $6,000 at date 3. Assume an effective annual rate of 10%, annual dates, and no other cash flows. Find the stream’s value at dates 1 and 3, then verify that the values are related by accumulation.

Method. Move each receipt to the requested date and add. The common mistake to avoid is accumulating the receipt at date 3 when finding the value at date 1; that receipt must instead be discounted two years.

Value at date 1. The date-0 receipt is accumulated one year: $\$5{,}000(1.10)=\$5{,}500$. The date-1 receipt is already at the valuation date: $\$4{,}000$. The date-3 receipt is discounted two years: $\$6{,}000/(1.10)^2=\$6{,}000/1.21=\$4{,}958.67768595\ldots$. Therefore, $V_1=\$5{,}500+\$4{,}000+\$4{,}958.67768595\ldots=\$14{,}458.67768595\ldots$, or $14,458.68 to the nearest cent.

Value at date 3. Accumulate the date-0 receipt three years, the date-1 receipt two years, and leave the date-3 receipt unchanged: $\$5{,}000(1.10)^3=\$6{,}655$; $\$4{,}000(1.10)^2=\$4{,}840$; and $\$6{,}000$ at date 3. The sum is $V_3=\$6{,}655+\$4{,}840+\$6{,}000=$ $17,495.00.

Check. The dates 1 and 3 are two years apart. Accumulating the unrounded date-1 value gives $\$14{,}458.67768595\ldots(1.10)^2=\$17{,}495$, agreeing with the direct date-3 calculation. The stream has one consistent value at either date, expressed in date-specific dollars.

Why the Stream’s Value Changes by the Accumulation Factor

The relationship between the values at two dates follows directly from the stream formula. This is more than a check on an example: it explains why the entire stream can be moved as one value once its value at one date has been calculated.

Write the stream value at date $s$.
Using the same fixed rate for every cash flow, $V_s=\sum_j C_j(1+i)^{s-j}$.

Accumulate that value from $s$ to $t$.
Multiply by the factor for the interval: $V_s(1+i)^{t-s}=\left[\sum_j C_j(1+i)^{s-j}\right](1+i)^{t-s}$.

Combine the exponents.
For each cash flow, $(1+i)^{s-j}(1+i)^{t-s}=(1+i)^{t-j}$. Therefore $V_s(1+i)^{t-s}=\sum_j C_j(1+i)^{t-j}=V_t$.

The derivation works whether a cash flow is before, between, or after the two valuation dates: negative exponents represent discounting, and positive exponents represent accumulation. Its essential conditions are that each cash flow is included once with the correct sign and that the same accumulation rule is used throughout. If the rate convention changes across periods, the one-factor relationship may no longer apply in this form.

Rate Conventions Must Match the Periods

An effective annual rate $i$ measures the actual accumulation over one year. A nominal annual rate $i^{(m)}$, convertible $m$ times per year, instead specifies a rate of $i^{(m)}/m$ for each conversion period. For $n$ complete years under that nominal convention, there are $mn$ periods, so the accumulation factor is $(1+i^{(m)}/m)^{mn}$. The equivalent effective annual rate is $(1+i^{(m)}/m)^m-1$.

These are alternative ways to describe the same accumulation only when they are converted consistently. Do not use a nominal rate as the annual effective rate in $(1+i)^n$. Also, do not use the annual number of years as the exponent on a periodic factor when the exponent should count the shorter conversion periods.

Worked Example: Converting a Nominal Rate Before Moving a Value

A reserve value of $6,000 is stated at date 1. The applicable rate is a nominal annual rate of 8% convertible semiannually. Find the equivalent value at date 3, and check the result using the equivalent effective annual rate.

Method. The nominal rate gives a semiannual rate of $0.08/2=0.04$. There are two years, or four semiannual periods, from date 1 to date 3. The common mistake to avoid is using $1.08^2$, which treats 8% as an effective annual rate instead of a nominal rate convertible twice per year.

Periodic calculation. The accumulation factor is $1.04^4=1.16985856$. Hence the date-3 value is $\$6{,}000(1.16985856)=\$7{,}019.15136$, or $7,019.15 to the nearest cent.

Check using an effective annual rate. The effective annual rate is $1.04^2-1=0.0816$, or 8.16%. Over two years its factor is $1.0816^2=1.16985856$, the same as the four semiannual periods. Thus this method also gives $\$6{,}000(1.16985856)=\$7{,}019.15$. Using $1.08^2$ would instead give $6,998.40, which is inconsistent with the stated nominal convertible rate.

Common Mistakes and Practical Checks

A short set of checks catches many date-conversion errors before they affect a larger calculation:

Count the interval, not the endpoint. Moving from date 2 to date 5 uses three years, not five.
Use the direction of movement to choose the operation. At a positive rate, forward movement multiplies by an accumulation factor; backward movement divides by it.
Keep the rate convention aligned with the period count. An annual effective rate and a nominal rate convertible monthly or quarterly are not interchangeable without conversion.
Check the date of every cash flow. For a stream value, accumulate flows before the valuation date and discount flows after it.
Keep full precision until the end. Round the final reported dollar amount to cents; rounding intermediate factors or components can create a small discrepancy in the check.

If a stream is being valued at two dates, a useful independent check is to calculate it directly at each date and then verify $V_t=V_s(1+i)^{t-s}$. A disagreement usually points to a missed cash flow, a wrong interval, an inconsistent rate, or premature rounding.

Key takeaway: Under one consistent compound-interest convention, moving a value between dates means multiplying by the accumulation factor for the interval or dividing by that factor to move backward. A complete stream obeys the same rule because every cash flow’s exponent shifts by the same amount.

Check Your Understanding

For each problem, use the stated rate convention and round dollar answers to the nearest cent.

A fund value of $7,500 is measured at date 2. Assume an effective annual rate of 4% and annual dates. What is its value at date 5? (Answer: $8,436.48)
A value at date 4 is $11,576.25. Under an effective annual rate of 5%, what is its equivalent value at date 1? (Answer: $10,000.00)
A stream pays $2,000 at date 0, $5,000 at date 2, and $3,000 at date 4. At an effective annual rate of 8%, find the stream’s value at date 2 and verify it by moving the value to date 4. (Answer: Date 2 value $9,904.82; date 4 value $11,552.98)
A value of $6,000 at date 1 accumulates to date 3 at a nominal annual rate of 8% convertible semiannually. Find the date-3 value and state the equivalent effective annual rate. (Answer: $7,019.15; effective annual rate 8.16%)
A valuation of $13,310 at date 3 is based on a 10% effective annual rate. Find its equivalent values at dates 1 and 0, and check the movement from date 0 to date 3. (Answer: Date 1 value $11,000.00; date 0 value $10,000.00; $10,000.00(1.10)^3=$13,310.00)

What You'll Learn