Present Value Sums and Net Present Value

Why a Sum of Present Values Is Needed

A project or financial arrangement may produce several cash flows at different times. Adding the amounts directly does not account for the fact that money received sooner can earn interest before a later date. To assess the whole set of cash flows at one date, first convert each amount to a value at that common date, then sum those values.

This tutorial uses time 0 as the valuation date and assumes cash flows occur at the ends of annual periods. The discount rate is a specified, constant effective annual rate. The aim is to calculate the present value sum and net present value (NPV), not to find a rate that makes NPV equal to zero.

Definition: A cash flow $C_t$ is the signed amount paid or received at time $t$. Use a positive sign for a receipt and a negative sign for a payment. At annual effective interest rate $i$, the present value at time 0 of a cash flow $C_t$ at the end of year $t$ is $C_t v^t$, where $v=1/(1+i)$. For cash flows $C_0,C_1,\ldots,C_n$, the net present value at time 0 is $$ \operatorname{NPV}_0=\sum_{t=0}^{n} C_t v^t =\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}. $$

The model requires the stated rate to be effective for the same annual periods used to measure the cash-flow times. It also assumes that the amounts and their dates are known and that one constant rate is to be used for every cash flow. If the rate is quoted for a different period, it must first be put on a compatible basis. A time-0 cash flow is already at the valuation date, so its discount factor is $v^0=1$; it is not discounted.

Why Each Cash Flow Is Discounted

The discounting formula follows directly from compound interest. If an amount $P$ at time 0 accumulates for $t$ years at effective annual rate $i$, it becomes $P(1+i)^t$. To find the time-0 amount equivalent to a known amount $C_t$ at year $t$, set those amounts equal at year $t$ and solve for $P$.

P(1+i)^t=C_t \quad\Longrightarrow\quad P=\frac{C_t}{(1+i)^t}=C_t v^t.

Thus, each term in the NPV formula is the value at time 0 of one cash flow. Since all the terms are expressed at the same date and in the same currency, they can be summed to give the net value at that date. A receipt contributes a positive present value; a payment contributes a negative present value.

Interpretation: A positive NPV means the present value of the signed cash flows is positive at the specified rate. A negative NPV means it is negative at that rate. An NPV of zero means the signed present values exactly offset. These conclusions are about the stated cash-flow model and discount rate; they do not, by themselves, account for uncertainty or other constraints.

A Reliable Calculation Process

For a finite set of annual cash flows, organize the calculation by time. The exponent is the number of years between the valuation date and the cash flow, not the cash flow's position in a list. Keep payments negative throughout the calculation. A calculator or spreadsheet can help with powers, but the meaning of each term should remain visible.

Fix the valuation date and rate.
Identify the date at which NPV is required and confirm that the discount rate is effective for the cash-flow period.

Record signed cash flows and dates.
Write each receipt as positive and each payment as negative, with its actual time index.

Discount each amount.
For a cash flow at year $t$, multiply by $v^t=1/(1+i)^t$. Leave a time-0 amount unchanged.

Sum and report.
Add the discounted signed amounts. Keep full precision during the calculation and round the final monetary answer as requested.

Worked Examples

Worked Example: A Project with Three Annual Receipts

A hypothetical equipment project costs $1,600 immediately and produces receipts of $500 at the end of year 1, $1,000 at the end of year 2, and $800 at the end of year 3. Calculate its NPV at an effective annual discount rate of 25%.

Method. Discount each receipt to time 0 using the given annual rate, then include the immediate cost with a negative sign. This avoids comparing amounts from different dates without adjusting for interest.

The discount factor is $v=1/(1.25)=0.8$. The signed cash flows are $C_0=-1{,}600$, $C_1=500$, $C_2=1{,}000$, and $C_3=800$. Their present values are:

\begin{aligned} C_0v^0&=-1{,}600,\\ C_1v^1&=500(0.8)=400,\\ C_2v^2&=1{,}000(0.8)^2=1{,}000(0.64)=640,\\ C_3v^3&=800(0.8)^3=800(0.512)=409.60. \end{aligned}

Therefore,

\operatorname{NPV}_0=-1{,}600+400+640+409.60=-150.40.

The NPV is −$150.40. As a check, the receipts have an undiscounted total of $2,300, but their time-0 value is only $1,449.60; subtracting the $1,600 immediate cost gives the same negative result.

Worked Example: A Cash Flow Occurring in Year 4

A hypothetical investment requires $2,500 at time 0. It pays $1,000 at the end of year 1, $1,250 at the end of year 2, and $1,500 at the end of year 4. Find its NPV at an effective annual discount rate of 25%.

Method. Use the stated time of each payment as the exponent. In particular, the last receipt is discounted for four years even though it is the third receipt.

Here $v=0.8$, and the time-0 cost is $-$2,500$. The present values of the receipts are:

\begin{aligned} 1{,}000v^1&=1{,}000(0.8)=800,\\ 1{,}250v^2&=1{,}250(0.64)=800,\\ 1{,}500v^4&=1{,}500(0.8)^4 =1{,}500(0.4096)=614.40. \end{aligned}

The present value of all receipts is $800+800+614.40=$2{,}214.40$. Including the initial cost gives

\operatorname{NPV}_0=-2{,}500+2{,}214.40=-285.60.

The NPV is −$285.60. The year-4 factor is $v^4$, not $v^3$: the exponent is determined by when the cash flow occurs, not by its place in the list.

Worked Example: Including a Later Payment

A hypothetical machine has a time-0 purchase cost of $2,800. It generates a receipt of $1,440 at the end of year 1, requires a $720 payment at the end of year 2, and generates a receipt of $3,456 at the end of year 3. Calculate NPV at an effective annual rate of 20%.

Method. Discount every amount, including the year-2 payment, with its sign intact. This avoids treating a later cost as though it were a receipt.

The discount factor is $v=1/(1.20)=5/6$. The present values are:

\begin{aligned} C_0v^0&=-2{,}800,\\ C_1v^1&=1{,}440\left(\frac{5}{6}\right)=1{,}200,\\ C_2v^2&=-720\left(\frac{5}{6}\right)^2 =-720\left(\frac{25}{36}\right)=-500,\\ C_3v^3&=3{,}456\left(\frac{5}{6}\right)^3 =3{,}456\left(\frac{125}{216}\right)=2{,}000. \end{aligned}

Summing the signed present values,

\operatorname{NPV}_0=-2{,}800+1{,}200-500+2{,}000=-100.

The NPV is −$100.00. As a check, the year-2 payment has present value $-$500$, so it reduces the NPV rather than increasing it. The result is exactly determined by the displayed values, with no intermediate rounding.

Common Mistakes and Their Corrections

The most frequent errors are small changes to the formula that have a substantial effect on the result. The following checks help catch them:

Using the wrong discount factor. The present value of a year-$t$ amount is $C_t/(1+i)^t$, not $C_t(1+i)^t$. Multiplication by $(1+i)^t$ accumulates a value forward in time.
Using the count of receipts as the exponent. In the year-4 example, the final receipt is discounted by $v^4$, even though it is the third receipt listed.
Discounting a time-0 amount. The exponent is zero at the valuation date, so $C_0v^0=C_0$.
Changing a payment into a positive amount. A payment is negative under the stated convention. Discounting changes its magnitude, not its sign.
Rounding each term too early. Retain full calculator precision for discount factors and present values, then round the final NPV. If intermediate values are displayed in rounded form, their displayed sum may differ slightly from the final answer computed at full precision.

Key takeaway: NPV is the sum of signed cash flows after each has been discounted to the same valuation date using the specified rate. Check the rate basis, the time exponent, and the sign of every cash flow before interpreting the result.

Check Your Understanding

For each problem, use annual effective discounting at the stated rate and report NPV to the nearest cent.

A project costs $1,000 at time 0 and pays $550 at the end of year 1 and $605 at the end of year 2. Find its NPV at 10%. (Answer: $0.00)
An investment costs $1,800 immediately and returns $1,200 at the end of year 1 and $960 at the end of year 2. Find its NPV at 20%. (Answer: −$133.33)
A project costs $2,500 now and pays $900, $1,200, and $800 at the ends of years 1, 2, and 3, respectively. Find its NPV at 5%. (Answer: $136.65)
A project requires $3,200 now, pays $1,000 at the end of year 1, requires a $400 payment at the end of year 2, and pays $3,000 at the end of year 4. Find its NPV at 25%. (Answer: −$1,427.20)

What You'll Learn