Present Value of an Uneven Cash-Flow Stream

Why Timing Matters

An uneven cash-flow stream contains payments that may differ in amount and occur on different dates. To compare or summarize them at one date, each payment must be expressed in value at that same date. A dollar received sooner and a dollar received later do not have the same present value when interest is positive. The calculation therefore depends both on the amounts and on their timing.

This tutorial values a stream at time 0, the valuation date. The earlier material on building timelines for irregular cash flows provides a way to lay out amounts and dates before calculating. The key task here is to discount each dated amount to time 0 and add the resulting values.

Definition: The present value at time 0 of a cash-flow stream is the signed amount at time 0 equivalent, under a stated interest-rate model, to all cash flows in the stream. An inflow is positive and an outflow is negative. A payment made at time 0 is already at the valuation date and is not discounted.

The Discounting Model

Assume a constant effective annual interest rate $i$ applies over the period being valued, with annual compounding. Assume all cash flows are known, and their dates are integer numbers of years after time 0. These assumptions let us use the same annual discount rate for each payment. If the rate varies by date, or cash flows fall at non-integer times, this simple model needs adjustment; those cases are outside the calculations here.

Let $C_k$ be the signed cash flow at the end of year $k$, where $k$ is a nonnegative integer. The annual discount factor is $v=1/(1+i)$. The present value of the stream is:

\operatorname{PV}_0=\sum_{k=0}^{n} C_k v^k =\sum_{k=0}^{n}\frac{C_k}{(1+i)^k}.

Here $C_0$ is the amount at the valuation date, so its factor is $v^0=1$. A date with no payment can be left out of the sum. If more than one amount occurs on the same date, first add them using their signs, or discount them separately and add their present values; both methods give the same result.

Why the Present-Value Sum Works

The formula follows from the annual accumulation model and the fact that values at one date can be added. Consider one signed payment $C_k$ due at the end of year $k$. If its equivalent amount at time 0 is $P_k$, then accumulating $P_k$ for $k$ years at the effective annual rate gives $P_k(1+i)^k$. Equivalence requires that amount to equal the payment:

P_k(1+i)^k=C_k \quad\Longrightarrow\quad P_k=\frac{C_k}{(1+i)^k}=C_kv^k.

Each $P_k$ is the value at time 0 of one component of the stream. Since all the components are now at the same date, their signed values add. Thus the present value of the whole stream is $\sum_{k=0}^{n}C_kv^k$. This argument also explains why the exponent must match the payment’s date: it counts the number of annual periods between the valuation date and that payment.

Worked Example: A Project With Three Receipts

A hypothetical project requires an outflow of $12,000 at time 0. It is expected to produce receipts of $5,000 at the end of year 1, $6,000 at the end of year 2, and $4,000 at the end of year 3. Find the present value at time 0 using an effective annual interest rate of 8%.

Method. Assign signs, then discount each flow by its own number of years. The year-0 outflow is negative and is not discounted. The common mistake this method avoids is treating the receipts as if they all occurred at the same date.

The discount factor is $v=1/1.08=0.9259259\ldots$. The signed cash flows are $C_0=-12{,}000$, $C_1=5{,}000$, $C_2=6{,}000$, and $C_3=4{,}000$. Discount them one at a time:

\begin{aligned} C_0v^0&=-12{,}000,\\ C_1v^1&=\frac{5{,}000}{1.08}=4{,}629.6296\ldots,\\ C_2v^2&=\frac{6{,}000}{1.08^2} =\frac{6{,}000}{1.1664}=5{,}144.0329\ldots,\\ C_3v^3&=\frac{4{,}000}{1.08^3} =\frac{4{,}000}{1.259712}=3{,}175.3290\ldots. \end{aligned}

Now add the signed present values:

-12{,}000+4{,}629.6296+5{,}144.0329+3{,}175.3290 =948.9915\ldots

Rounded to the nearest cent, the present value is $948.99. As a check, the three receipts have a combined present value of $12,948.99; subtracting the $12,000 time-0 outflow leaves $948.99.

Payments on the Same Date

Different cash-flow descriptions can list several amounts due on one date. For valuation, each amount must be included, but their signs matter. For example, a receipt and an expense both due at the end of year 2 may be netted into one year-2 cash flow before discounting. This is an accounting convenience, not a change in value: discounting is multiplication by a common factor for that date, and multiplication distributes over addition.

Worked Example: Netting a Receipt and a Repair Cost

A hypothetical equipment arrangement has an outflow of $15,000 at time 0, receipts of $6,000 at the end of year 1 and $7,200 at the end of year 2, a repair cost of $1,200 also due at the end of year 2, and a receipt of $5,000 at the end of year 4. Calculate the present value at 6% effective annually.

Method. Net the two year-2 amounts, then discount each date-specific net flow. The common mistake avoided here is either omitting the repair cost or subtracting it at time 0 rather than at its actual date.

At year 2, the net flow is $7{,}200-1{,}200=6{,}000$. Thus the stream consists of $-15{,}000$ at time 0, $6{,}000$ at year 1, $6{,}000$ at year 2, and $5{,}000$ at year 4. With $i=0.06$, the discounted components are:

\begin{aligned} \text{Time 0:}&\quad -15{,}000,\\ \text{Year 1:}&\quad \frac{6{,}000}{1.06}=5{,}660.3774\ldots,\\ \text{Year 2:}&\quad \frac{6{,}000}{1.06^2} =\frac{6{,}000}{1.1236}=5{,}339.9786\ldots,\\ \text{Year 4:}&\quad \frac{5{,}000}{1.06^4} =\frac{5{,}000}{1.26247696}=3{,}960.4683\ldots. \end{aligned}

Adding the values gives

-15{,}000+5{,}660.3774+5{,}339.9786+3{,}960.4683 =-39.1757\ldots

The present value is -$39.18, rounded to the nearest cent. A direct check is to discount the year-2 receipt and repair cost separately: their combined value is $7{,}200/1.1236-1{,}200/1.1236=6{,}000/1.1236$, exactly the netted value used above.

Keep Each Date and Sign Attached to Its Flow

A reliable calculation preserves three pieces of information for every payment: its amount, its sign, and its date relative to the valuation date. The date determines the discount exponent, while the sign determines whether the discounted amount is added or subtracted. Losing any one of these can materially change the answer.

For a longer stream, a small table can make the calculation easier to audit. Write one row per date, net flows that occur together, calculate the applicable discount factor, and multiply. Retain extra digits during the arithmetic and round only the final total to cents. This avoids small rounding differences accumulating across several payments.

Worked Example: Operating Flows and a Later Cost

A hypothetical service contract requires an outflow of $24,000 at time 0. It produces receipts of $9,000 at the end of year 1 and $11,000 at the end of year 3. Maintenance costs of $2,000 are due at the end of year 2 and $3,000 at the end of year 4; a receipt of $7,000 is also due at the end of year 4. The final receipt is $8,000 at the end of year 5. Find the present value at 12% effective annually.

Method. First combine amounts with the same date, then discount all remaining flows. The year-4 net flow is $7{,}000-3{,}000=4{,}000$. The common mistake avoided is to discount the year-4 receipt and cost as if they were due in different years when they are actually simultaneous.

The date-0 flow is $-24{,}000$; the year-1, year-2, year-3, year-4, and year-5 flows are respectively $9{,}000$, $-2{,}000$, $11{,}000$, $4{,}000$, and $8{,}000$. Discounting each amount gives:

\begin{aligned} \text{Time 0:}&\quad -24{,}000,\\ \text{Year 1:}&\quad \frac{9{,}000}{1.12}=8{,}035.7143\ldots,\\ \text{Year 2:}&\quad \frac{-2{,}000}{1.12^2} =\frac{-2{,}000}{1.2544}=-1{,}594.3878\ldots,\\ \text{Year 3:}&\quad \frac{11{,}000}{1.12^3} =\frac{11{,}000}{1.404928}=7{,}829.5827\ldots,\\ \text{Year 4:}&\quad \frac{4{,}000}{1.12^4} =\frac{4{,}000}{1.57351936}=2{,}542.0723\ldots,\\ \text{Year 5:}&\quad \frac{8{,}000}{1.12^5} =\frac{8{,}000}{1.7623416832}=4{,}539.4148\ldots. \end{aligned}

Summing in dollars:

-24{,}000+8{,}035.7143-1{,}594.3878+7{,}829.5827 +2{,}542.0723+4{,}539.4148 =-2{,}647.6037\ldots

Rounded to the nearest cent, the present value is -$2,647.60. The check is that the time-0 outflow is included without discounting, the year-2 cost remains negative, and the two year-4 amounts have been combined into a positive $4,000 flow before discounting.

Common Error: One Discount for the Whole Stream

A frequent error is to add all cash flows first and divide the total by $1+i$, regardless of their dates. That operation gives every amount the same one-year discount factor, which is only appropriate if every amount is due at the end of year 1. For example, in the first worked example, this incorrect method would give:

\frac{-12{,}000+5{,}000+6{,}000+4{,}000}{1.08} =\frac{3{,}000}{1.08} =2{,}777.78.

This is not the present value of the stated stream: it discounts the time-0 outflow and treats year-2 and year-3 receipts as year-1 receipts. The correct calculation leaves the time-0 amount unchanged and uses powers $1.08^1$, $1.08^2$, and $1.08^3$ for the receipts, giving $948.99.

Before accepting a result, check that each exponent matches its date, that outflows have negative signs, that all amounts due on the same date are included, and that the final sum is reported at the requested valuation date. The present value is a value under the stated rate and timing assumptions; changing either can change the result.

Check Your Understanding

For each problem, calculate the present value at time 0 under the stated effective annual interest rate. Round money to the nearest cent.

A $6,000 outflow occurs at time 0, followed by receipts of $4,200 at the end of year 1 and $3,100 at the end of year 2. The rate is 5%. (Answer: $811.79)
A plan requires $9,000 now and pays $4,000 at the end of year 1 and $6,000 at the end of year 3. The rate is 7%. (Answer: -$363.89)
An outflow of $7,500 occurs now. At the end of year 2 there is a receipt of $5,000 and a cost of $800; at the end of year 4 there is a receipt of $4,200. The rate is 4%. (Answer: -$26.69)
A project has an initial cost of $10,000, a year-1 receipt of $2,500 and cost of $300, a year-2 receipt of $4,500, and a year-3 receipt of $5,000. The rate is 10%. (Answer: -$524.42)
A stream pays $1,200 at time 0, requires an outflow of $2,500 at the end of year 2, and pays $4,000 at the end of year 3 and $1,800 at the end of year 5. The rate is 8%. (Answer: $3,457.03)

What You'll Learn