Deriving the Present-Value Sum for Irregular Cash Flows

Why Each Cash Flow Needs Its Own Discounting Period

A stream of cash flows may include receipts and payments on different dates. To express that stream as one amount at time zero, each cash flow must be discounted from its own date. A receipt in year 1 is discounted for one period; a receipt in year 4 is discounted for four. Applying a single exponent to the whole stream would treat payments on different dates as though they occurred together.

This tutorial derives the present-value sum from the single-payment discount factor. The setup uses a constant effective annual interest rate and annual payment dates. It assumes cash-flow amounts and dates are known, and that the same rate applies to each period. The derivation does not cover changing rates or cash flows at nonannual intervals.

Definition: Let $C_k$ be the signed cash flow at the end of period $k$, with receipts positive and payments negative. Let $i$ be the effective interest rate per period and $v=1/(1+i)$ the discount factor. The present value at time zero of cash flows at times $1,\ldots,n$, with any time-zero amount $C_0$, is $$ PV_0=C_0+\sum_{k=1}^{n} C_kv^k =C_0+\sum_{k=1}^{n}\frac{C_k}{(1+i)^k}. $$

The exponent $k$ counts the number of periods from time zero to the date of $C_k$. If there is no cash flow at a particular period, its $C_k$ is zero and contributes nothing. A payment at time zero is already at the valuation date, so it is included as $C_0$, without discounting.

Deriving the Present-Value Sum

For a single amount, the discount factor follows from the accumulation rule. Under a constant effective rate $i$, one unit invested at time zero grows to $1+i$ after one period and to $(1+i)^k$ after $k$ periods. Therefore, an amount $C_k$ at time $k$ has a time-zero equivalent equal to $C_k/(1+i)^k$. For a stream, add the separate time-zero equivalents.

Start with one dated amount.
An amount $C_k$ at time $k$ has a time-zero equivalent $P_k$ satisfying $P_k(1+i)^k=C_k$.

Solve for its present value.
Dividing both sides by $(1+i)^k$ gives $P_k=C_k/(1+i)^k=C_kv^k$.

Combine the dated amounts.
Because all the individual equivalents are now stated at time zero, their signed values add: $PV_0=C_0+\sum_{k=1}^{n}P_k$.

Substitute each single-payment value.
Using $P_k=C_kv^k$ for every period gives $PV_0=C_0+\sum_{k=1}^{n}C_kv^k$.

The reason the sum works is that the value of a collection of amounts at one date is the sum of their values at that date. Discounting each amount first puts every component at the same valuation date; only then are they added. The derivation also explains the individual exponents: a cash flow at time $k$ is accumulated for $k$ periods when moving forward, so its equivalent at time zero is discounted for those same $k$ periods.

A cash flow that occurs after a gap still uses its actual time from the valuation date. For example, if there are no payments in years 2 and 3, a payment in year 4 is discounted by $v^4$, not by $v$ merely because it is the next listed payment. The exponent comes from the timeline, not the position of the payment in a written list.

Applying the Sum to a Dated Stream

A reliable calculation starts by writing the valuation date, rate per period, signed amounts, and dates. Then pair each amount with its own discount factor. Keep full precision in intermediate calculations and round the final monetary result to cents.

Worked Example: Discounting Payments With Gaps Between Dates

A hypothetical agreement produces receipts of $1,200 at the end of year 1, $800 at the end of year 3, and $1,500 at the end of year 4. Find their combined present value at time zero if the effective annual interest rate is 5%. There are no other cash flows.

Method. Apply $C_k/(1+i)^k$ separately at each dated payment. This avoids the common mistake of discounting every receipt by the same number of years or by the number of its position in the list.

The rate is $i=0.05$. The first receipt is at $k=1$, the second at $k=3$, and the third at $k=4$. Thus $$ PV_0=\frac{1200}{1.05^1}+\frac{800}{1.05^3}+\frac{1500}{1.05^4}. $$

The needed powers are $1.05^1=1.05$, $1.05^3=1.157625$, and $1.05^4=1.21550625$. The unrounded discounted amounts are $$ \frac{1200}{1.05}=1142.857142857,\qquad \frac{800}{1.157625}=691.070078825,\qquad \frac{1500}{1.21550625}=1234.053712188. $$

Adding the unrounded values gives $1142.857142857+691.070078825+1234.053712188=3067.980933870$. Rounded only at the final step to two decimal places, the present value is $3,067.98.

Check. Each discounted amount is less than its future receipt because $i$ is positive, and the three present values sum to the reported total.

Signed Cash Flows and Time-Zero Amounts

The formula handles payments and receipts in one sum by using signed amounts. A payment is negative, so its present value reduces the total. Do not discount a time-zero amount: its time exponent is zero, and $v^0=1$. These conventions avoid the need to calculate receipts and payments separately unless separate subtotals are useful for checking.

Worked Example: Including an Initial Payment

A hypothetical arrangement requires a $1,800 payment immediately. It then pays receipts of $605 at the end of year 1, $726 at the end of year 2, and $1,331 at the end of year 3. Calculate the present value of the signed stream at time zero using an effective annual rate of 10%.

Method. Put the immediate payment in $C_0$ and discount each later amount by its dated exponent. The common mistake is to discount the initial $1,800 by one period even though it occurs at time zero.

The signed cash flows are $C_0=-1800$, $C_1=605$, $C_2=726$, and $C_3=1331$, with $i=0.10$. Therefore, $$ PV_0=-1800+\frac{605}{1.10}+\frac{726}{1.10^2}+\frac{1331}{1.10^3}. $$

The period factors are $1.10^1=1.10$, $1.10^2=1.21$, and $1.10^3=1.331$. The discounted later amounts are $605/1.10=550$, $726/1.21=600$, and $1331/1.331=1000$. The time-zero amount remains $-1800$.

Combining the signed values gives $-1800+550+600+1000=350$. The present value of the stream is $350.00.

Check. The future receipts have a combined present value of $2,150, which exceeds the immediate $1,800 payment by $350.

Reading and Checking the Exponents

The exponent is determined by the time between the valuation date and the cash-flow date, measured in the periods used for $i$. In this tutorial, the valuation date is time zero, the rate is an effective annual rate, and dates are annual. If a stream has a payment in year 5, its discount factor is $v^5$, whether or not there are payments in years 1 through 4.

A useful check is to label each term with both its amount and its date before calculating. Then verify that every positive or negative amount has the correct sign, and that the exponent matches its date. Finally, compare the size of each discounted amount with its undiscounted amount: when $i>0$, discounting a positive future receipt reduces it, while discounting a negative future payment makes it closer to zero.

Worked Example: Combining Receipts and a Later Payment

A hypothetical project-related agreement has a receipt of $1,080 at the end of year 1, a receipt of $1,166.40 at the end of year 2, and a payment of $755.8272 at the end of year 3. There is also an immediate payment of $1,200. Find the present value of the signed stream at time zero using an effective annual interest rate of 8%.

Method. Discount each future amount to time zero, retain the payment signs, and then add the immediate payment. This avoids the mistake of either omitting the year-3 exponent or treating the later payment as a positive receipt.

The signed amounts are $C_0=-1200$, $C_1=1080$, $C_2=1166.40$, and $C_3=-755.8272$. With $i=0.08$, $$ PV_0=-1200+\frac{1080}{1.08}+\frac{1166.40}{1.08^2}-\frac{755.8272}{1.08^3}. $$

The rate powers are $1.08^1=1.08$, $1.08^2=1.1664$, and $1.08^3=1.259712$. The discounted amounts are $1080/1.08=1000$, $1166.40/1.1664=1000$, and $755.8272/1.259712=600$. The last amount remains negative because it is a payment.

Thus $PV_0=-1200+1000+1000-600=200$, or $200.00.

Check. The receipts have a combined present value of $2,000; the initial and later payments have a combined present value of $1,800. Their difference is $200.

Common Mistakes and When the Formula Applies

The present-value sum is a direct consequence of single-payment discounting and addition at a common date. It applies when the rate per period is constant and the dated cash flows are valued under that same rate. If rates differ by period, the single factor $v^k$ is not the appropriate factor for every date; the constant-rate assumption would have to be replaced with an appropriate period-by-period model. Likewise, here each time exponent counts whole annual periods. A time expressed in some other unit must be matched to a rate expressed per that unit before using a discount exponent.

Using one exponent for every amount. This incorrectly treats all cash flows as though they occurred on the same date. Use the period attached to each individual cash flow.
Counting listed payments instead of periods. A receipt in year 4 uses exponent 4 even if it is the third listed receipt.
Discounting time-zero cash flows. A time-zero amount is multiplied by $v^0=1$, so its value is unchanged.
Ignoring signs. A payment reduces the signed present value; it is not added as a positive amount.
Rounding every component too early. Retain unrounded values during the sum and round the final monetary amount to cents.

Key takeaway: For each cash flow, identify its signed amount and its date. Discount that amount by $v^k$, where $k$ is the number of periods from the valuation date, and then add all the resulting time-zero values.

Check Your Understanding

For each problem, use the stated effective annual rate and annual dates. Give monetary answers to the nearest cent.

A hypothetical contract pays $107 at the end of year 1 and $114.49 at the end of year 2, requires a payment of $57.245 at the end of year 2, and requires an immediate payment of $40. Find the signed present value at 7%. (Answer: $110.00)
A hypothetical plan has an immediate payment of $900, a receipt of $318 at the end of year 1, and a receipt of $561.80 at the end of year 2. Find the signed present value at 6%. (Answer: -$100.00)
A hypothetical arrangement pays $440 at the end of year 1 and $798.60 at the end of year 3, and requires a payment of $363 at the end of year 2. Find the present value at 10%. (Answer: $700.00)
A hypothetical stream pays $1,260 at the end of year 2 and $1,215.50625 at the end of year 4. Find its present value at 5%. (Answer: $2,142.86)
At an effective annual rate of 10%, a hypothetical stream has an immediate payment of $1,000, a receipt of $420 at the end of year 1, and a receipt of $484 at the end of year 2. What receipt at the end of year 3 makes the signed present value equal to zero? (Answer: $290.40)

What You'll Learn