How Shifting a Payment Date Changes Present Value

Why the Payment Date Matters

Two promises to pay the same dollar amount are not necessarily worth the same amount today. If one payment arrives sooner, it can be invested for longer; if it arrives later, fewer periods of interest remain before payment. This timing effect is the reason present-value comparisons must put cash flows on a common valuation date.

The result in this tutorial concerns a fixed payment: its amount and all other assumptions stay the same while its date changes. The discount rate is positive and unchanged. The direction of the result is straightforward, but a sign or exponent error can easily reverse a calculation.

Definition: For a positive payment $C$ due $n$ whole years from a valuation date, assuming a constant effective annual interest rate $i$, its present value at that date is $Cv^n$, where $v=1/(1+i)$. A payment moved earlier has a smaller $n$; one moved later has a larger $n$. All dates in a comparison must be measured from the same valuation date.

The assumptions matter. The payment amount $C$ is fixed and positive, the valuation date is fixed, and the same effective annual rate applies over the periods being compared. The formulas below use whole years. For fractional-year dates, a fractional-period accumulation convention must also be specified before applying a power with a fractional exponent.

Why Earlier Timing Raises Present Value

Suppose a payment of $C$ is due in $n$ years. Under annual compounding at rate $i$, an amount $C v^n$ invested at the valuation date accumulates to $C v^n(1+i)^n=C$. Thus $C v^n$ is the amount at the valuation date that is equivalent to receiving $C$ at year $n$.

Now compare two dates $n_1$ and $n_2$, with $n_1<n_2$. Their present values are $Cv^{n_1}$ and $Cv^{n_2}$. Since $i>0$, $1+i>1$, so $v=1/(1+i)$ is between zero and one. Increasing the exponent of a number between zero and one reduces its value. Therefore $v^{n_1}>v^{n_2}$, and multiplying by the positive amount $C$ preserves the inequality.

n_1<n_2,\quad i>0,\quad C>0 \quad\Longrightarrow\quad Cv^{n_1}>Cv^{n_2}.

The ratio makes the size and direction of the change especially clear:

\frac{Cv^{n_1}}{Cv^{n_2}} = v^{n_1-n_2} = (1+i)^{n_2-n_1} > 1.

So moving the payment from $n_2$ years to $n_1$ years multiplies its present value by $(1+i)^{n_2-n_1}$. Equivalently, the increase is $Cv^{n_1}-Cv^{n_2}$. Moving it in the opposite direction reverses the comparison: present value falls. The ratio is useful for checking direction, while the difference gives the dollar change.

Theorem: At a positive effective interest rate, moving a fixed positive payment to an earlier date increases its present value at a common valuation date. Moving it later decreases present value. If $i=0$, the two dates have equal present value. The positive-rate conclusion does not apply unchanged at a negative rate.

Worked Applications

Worked Example: Moving a Receipt Two Years Earlier

A hypothetical agreement promises a receipt of $12,000. The valuation date is today, the effective annual interest rate is 5%, and annual compounding applies. Compare payment at the end of year 4 with payment at the end of year 2. Find the increase in present value if the payment is moved earlier.

Method. Discount each version of the same receipt to today, then subtract the later-date value from the earlier-date value. This avoids comparing dollar amounts located at different dates.

For payment at year 2, $v^2=1/(1.05)^2=1/1.1025$, so its present value is $12{,}000/1.1025=\$10{,}884.35$, rounded to the nearest cent. For payment at year 4, $(1.05)^4=1.21550625$, so its present value is $12{,}000/1.21550625=\$9{,}872.43$, rounded to the nearest cent.

The increase is $\$10{,}884.35-\$9{,}872.43=\$1{,}011.92$. As a check, moving two years earlier multiplies the year-4 present value by $(1.05)^2=1.1025$; $\$9{,}872.43\times1.1025$ is approximately $\$10{,}884.35$, subject to rounding.

Common mistake avoided: The earlier date has exponent 2, not 4, because it is two years from the valuation date. A larger discount exponent belongs to the later payment.

Worked Example: Moving a Receipt One Year Later

A hypothetical fund is due to receive $7,500 at the end of year 3. The effective annual interest rate is 8%, compounded annually. The payer asks to move the payment to the end of year 4. Calculate the decrease in present value at today’s valuation date.

Method. Compute each present value using its number of years from today. Since the payment is delayed by one year, the ratio of the new value to the old value should be $v$, which is less than 1.

At year 3, $(1.08)^3=1.259712$, giving a present value of $7{,}500/1.259712=\$5{,}953.74$. At year 4, $(1.08)^4=1.36048896$, giving a present value of $7{,}500/1.36048896=\$5{,}512.72$. Amounts are rounded to the nearest cent.

The decrease is $\$5{,}953.74-\$5{,}512.72=\$441.02$. For a check on direction and scale, the year-4 value equals the year-3 value divided by 1.08, before rounding. It must therefore be smaller.

Common mistake avoided: Do not multiply the year-3 present value by 1.08 to value the delayed payment at today’s date. That would move a value forward in time. Discounting one more year requires division by 1.08.

Worked Example: Shifting One Payment in a Cash-Flow Stream

A hypothetical project has receipts of $4,000 at the end of year 1, $9,000 at the end of year 3, and $5,000 at the end of year 5. The effective annual rate is 6%, with annual compounding. The $9,000 receipt is moved from year 3 to year 2; the other receipts are unchanged. Find the change in the present value of the whole stream.

Method. Value all receipts at today’s date. The unchanged receipts cancel when comparing the two totals, but calculating both totals is a useful check.

The discount factors needed are $v=1/1.06=0.9433962264$, $v^2=1/1.1236=0.889996439$, $v^3=1/1.191016=0.839619283$, and $v^5=1/1.3382255776=0.747258173$, shown to ten decimal places where applicable.

Before the change, the present value is $4{,}000v+9{,}000v^3+5{,}000v^5$. The three contributions are $3,773.58, $7,556.57, and $3,736.29, respectively, giving a total of $15,066.45 when rounded to cents.

After the change, the present value is $4{,}000v+9{,}000v^2+5{,}000v^5$. The contributions are $3,773.58, $8,009.97, and $3,736.29, giving a total of $15,519.84. The increase is therefore $\$15{,}519.84-\$15{,}066.45=\$453.39$.

As a separate check, only the shifted payment changes the comparison. Its increase is $9{,}000(v^2-v^3)=\$453.39$ when rounded to cents. This agrees with the difference between the two stream values.

Common mistake avoided: Revaluing the entire stream as if every payment moved would incorrectly change the two unchanged receipts. Isolate the changed term, or include all terms in both totals and verify that the unchanged terms cancel.

Receipts, Costs, and the Meaning of “Value”

The theorem is stated for a positive receipt, so “higher present value” means a larger positive number. A cost is often written as a negative cash flow. Moving a fixed negative payment earlier makes its signed present value more negative, even though the present value of the cost’s positive dollar magnitude becomes larger. For example, the present value of a $1,000 cost due in year 1 is greater in magnitude than the present value of the same cost due in year 2 when the rate is positive.

That distinction matters when comparing alternatives. State whether the amount is a receipt, a cost, or a signed cash flow. For a positive receipt, earlier means a larger present value; for a cost, earlier means a larger present-value cost and a lower signed value. The timing result does not by itself say which alternative is preferable: other cash flows and the decision context also matter.

Common Timing and Discounting Traps

Using the wrong exponent. Count periods from the valuation date, not from the date of another payment. A receipt at year 4 is discounted by $v^4$ when the valuation date is year 0.
Reversing the direction of the factor. Moving a value to an earlier date reverses some discounting and multiplies its present value by a factor greater than 1. Applying $v$ again would reduce it instead.
Changing more than the date. The comparison assumes the payment amount and rate remain fixed. If either changes, the simple timing comparison is not enough.
Mixing valuation dates. Cash flows at separate dates cannot be compared directly as if their dollar amounts were at the same time. Discount both to the same date before subtracting.
Ignoring the sign of a cost. An earlier cost has a higher present-value magnitude, but its signed present value is lower because it is more negative.

Key takeaway: With a positive effective rate, $v<1$. For a fixed positive payment, reducing the number of periods to the payment date reduces the discount exponent and raises present value. Check the date count, the cash-flow sign, and the common valuation date before interpreting the result.

Check Your Understanding

For each question, use the stated effective annual rate and annual compounding. Round monetary answers to the nearest cent.

A $2,500 receipt due at the end of year 4 is moved to the end of year 3. The effective annual rate is 4%. By how much does its present value at today’s date increase? (Answer: $85.48)
A $6,400 receipt due at the end of year 5 is moved to the end of year 2. At an effective annual rate of 5%, calculate the increase in present value at today’s date. (Answer: $790.42)
A hypothetical $18,000 cost due at the end of year 1 is delayed until the end of year 3. At an effective annual rate of 7%, by how much does the present-value magnitude of the cost decrease? (Answer: $2,129.07)
A cash-flow stream contains receipts of $3,000 at year 1, $5,000 at year 3, and $2,000 at year 4. At an effective annual rate of 6%, only the $5,000 receipt moves to year 2. How much does the stream’s present value increase? (Answer: $251.89)
A stream has a $10,000 receipt at year 2 and a $12,000 receipt at year 6. At an effective annual rate of 5%, the first receipt moves to year 1 and the second to year 4. Find the increase in the stream’s present value at today’s date. (Answer: $1,371.36)

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