Terminal and Salvage Cash Flows

Why the Final Cash Flow Matters

A project can produce operating cash flows throughout its life and still have an important final cash flow when it ends. Equipment may be sold, a facility may have salvage value, or an asset may require a final removal payment. These amounts belong in the project’s net present value (NPV) on the dates they are expected to occur. Omitting a valuable sale receipt understates NPV; counting an asset’s value both as a sale receipt and as a separate salvage receipt overstates it.

The key is to treat each final amount like any other project cash flow: establish its sign, date, and value, then discount it to the valuation date. The examples use deterministic cash flows and an annual effective interest rate. No mortality or probability distribution is assumed.

Definition: A terminal cash flow is a cash flow associated with a project’s end or with the final disposal or realization of a project asset. A sale receipt or salvage value is an inflow; a dismantling or disposal payment is an outflow. If a receipt and a cost both occur at the end, their signed sum is the net terminal cash flow.

Putting a Terminal Cash Flow into NPV

Let $CF_t$ be the project’s net cash flow at time $t$, measured in years from the valuation date $t=0$. Positive values are inflows and negative values are outflows. Assume a constant annual effective discount rate $i$, with $i>-1$, and that each cash flow is paid at its stated date. The discount factor for $t$ years is $v^t=(1+i)^{-t}$. Then the project’s NPV at time 0 is the sum of the discounted signed cash flows:

\operatorname{NPV}_0=\sum_t CF_t v^t.

If the project has operating cash flows $C_1,\ldots,C_n$, an initial outlay $I$, and a terminal cash flow $T$ at time $n$, this can be written as

\operatorname{NPV}_0=-I+\sum_{t=1}^{n} C_t v^t+T v^n.

Here $T$ is the net amount at time $n$ if more than one terminal receipt or cost occurs then. If disposal takes place at a different time $s$, use $T v^s$, not $T v^n$. For irregular dates, the exponent is the elapsed time in years from the valuation date, consistent with the discounting method being used.

Key result: Under a constant annual effective rate $i$, a terminal amount $T$ paid at time $s$ contributes $T(1+i)^{-s}$ to NPV at time 0. A positive receipt adds value; a cost, represented by a negative amount, reduces value.

Why the Discounting Formula Applies

The terminal amount is not a special kind of money for valuation purposes. It is a cash flow at a particular date. To express a payment $T$ at time $s$ as an equivalent amount at time 0, accumulate an amount $P$ at the annual effective rate $i$ for $s$ years. Its value at time $s$ is $P(1+i)^s$. Setting that equal to the payment gives

P(1+i)^s=T \quad\Longrightarrow\quad P=T(1+i)^{-s}=Tv^s.

Thus, the terminal contribution to NPV is simply its time-0 equivalent. Since NPV adds the time-0 equivalents of all project cash flows, adding the terminal contribution follows directly from the definition of NPV. This reasoning assumes the same effective rate is appropriate for discounting the cash flows and that the amount and payment date are as specified.

Worked Examples

Worked Example: Operating Receipts Plus End-of-Project Salvage

A machine costs $100,000 at time 0. It is expected to generate $30,000 at the end of each of years 1 through 4 and to be sold for $20,000 at the end of year 4. Find the project NPV at an annual effective rate of 10%. Treat all amounts as certain and use year-end payment dates.

Method. Discount the four operating receipts and the sale receipt to time 0, then include the initial outlay with a negative sign. The sale receipt is an additional cash flow at year 4, not a replacement for the year-4 operating receipt.

The discount factors needed are $v^4=(1.10)^{-4}=0.683013455$ and $a_{\overline{4}|}=(1-v^4)/0.10=3.16986545$. The present value of the operating receipts is

30{,}000a_{\overline{4}|} =30{,}000(3.16986545) =\$95{,}095.96.

The present value of the sale receipt is

20{,}000v^4 =20{,}000(0.683013455) =\$13{,}660.27.

Therefore,

\operatorname{NPV}_0 =-100{,}000+95{,}095.96+13{,}660.27 =\$8{,}756.23.

Check. The discounted inflows total $108,756.23, which exceeds the $100,000 outlay by $8,756.23. The NPV is positive. The common mistake avoided here is omitting the sale proceeds merely because they occur on the same date as the final operating receipt.

Worked Example: A Larger Receipt Paid Later

A project requires $60,000 at time 0 and produces $20,000 at the end of each of years 1, 2, and 3. Under Plan A, the asset is sold for $18,000 at the end of year 3. Under Plan B, it is retained and sold for $21,000 at the end of year 4; assume there are no other cash flows in year 4. Compare the NPVs at an annual effective rate of 8%. All cash flows are deterministic.

Method. The operating cash flows and initial outlay are identical in the two plans. Discount them once, then compare the sale receipts at their respective dates. The different receipt dates must not be ignored.

Since $v^3=(1.08)^{-3}=0.793832241$, $v^4=(1.08)^{-4}=0.735029853$, and $a_{\overline{3}|}=(1-v^3)/0.08=2.57709699$, the present value of the three operating receipts is

20{,}000a_{\overline{3}|} =20{,}000(2.57709699) =\$51{,}541.94.

For Plan A, the present value of the year-3 sale receipt is $18{,}000(0.793832241)=\$14{,}288.98$. Thus,

\operatorname{NPV}_A =-60{,}000+51{,}541.94+14{,}288.98 =\$5{,}830.92.

For Plan B, the present value of the year-4 sale receipt is $21{,}000(0.735029853)=\$15{,}435.63$. Thus,

\operatorname{NPV}_B =-60{,}000+51{,}541.94+15{,}435.63 =\$6{,}977.57.

Plan B’s NPV is higher by $\$6{,}977.57-\$5{,}830.92=\$1{,}146.65$. Equivalently, the present value of its sale receipt exceeds Plan A’s by $\$15{,}435.63-\$14{,}288.98=\$1{,}146.65$. Check. Both comparisons give the same difference because all other cash flows are identical. A larger amount paid later is not automatically more valuable: compare its discounted value, not its face amount.

Worked Example: Netting a Sale Receipt Against Closure Costs

A project requires $50,000 at time 0 and generates $13,000 at each year-end for five years. At the end of year 5, the equipment can be sold for $12,000, but closing the site will cost $4,000 at that same time. Find NPV at an annual effective rate of 6%. Assume the sale and closure amounts are certain and paid at the end of year 5.

Method. Include both final amounts with their correct signs. The net terminal cash flow is $T=12{,}000-4{,}000=\$8{,}000$.

The required factors are $v^5=(1.06)^{-5}=0.747258173$ and $a_{\overline{5}|}=(1-v^5)/0.06=4.21236378$. The present value of the operating receipts is

13{,}000a_{\overline{5}|} =13{,}000(4.21236378) =\$54{,}760.73.

The net terminal contribution is

(12{,}000-4{,}000)v^5 =8{,}000(0.747258173) =\$5{,}978.07.

Therefore,

\operatorname{NPV}_0 =-50{,}000+54{,}760.73+5{,}978.07 =\$10{,}738.80.

Check. Discounting the sale receipt and closure cost separately gives $8,967.10 and $-\$2,989.03$, respectively, for a net terminal contribution of $5,978.07. The common mistake avoided is counting only the gross sale proceeds while overlooking the final cost.

How Changes in the Terminal Cash Flow Affect Value

When two scenarios differ only in the terminal amount or its date, the change in NPV is the difference between the terminal amounts’ present values. If the first scenario pays $T_1$ at time $s_1$ and the second pays $T_2$ at time $s_2$, then

\Delta\operatorname{NPV}_0=T_2v^{s_2}-T_1v^{s_1}.

This comparison is valid when the other cash flows and the discount rate are unchanged. If a receipt is moved later without changing its amount, its present value falls when $i>0$. If the amount also changes, neither the date nor the face amount alone determines which scenario has the greater present value.

A terminal amount can also be solved for directly. For an initial outlay $I$, operating cash flows $C_t$, and terminal receipt $T$ at time $n$, the value that makes NPV zero satisfies

0=-I+\sum_{t=1}^{n}C_tv^t+Tv^n \quad\Longrightarrow\quad T=\frac{I-\sum_{t=1}^{n}C_tv^t}{v^n}.

This is an algebraic break-even calculation under the stated cash-flow and interest assumptions; it does not establish that the terminal amount will actually be realized.

Common Mistakes and an Exam Trap

Suppose a project has a $10,000 terminal receipt at the end of year 3, and the annual effective rate is 10%. The correct present value is

10{,}000(1.10)^{-3} =10{,}000(0.751314801) =\$7{,}513.15.

A wrong calculation adds the undiscounted $10,000 directly to the time-0 NPV. That mixes a year-3 amount with time-0 values. The correction is to discount it to time 0 before adding it.

Using the wrong date. A receipt at the end of year $n$ is discounted by $v^n$, even if the operating cash flows stop earlier or later. Use the actual payment date.
Double-counting the asset’s final value. If the sale proceeds already represent the asset’s salvage value, do not add a second receipt for the same asset.
Ignoring a terminal cost. Give closure, removal, or disposal payments a negative sign and include them on their payment date.
Comparing face amounts instead of present values. When dates differ, discount each scenario’s terminal amount before judging its contribution to NPV.

For exam calculations, a timeline is a useful check: mark the initial outlay, each operating cash flow, and each final receipt or cost at its date. Then check that every amount is included once, has the correct sign, and is discounted to the same valuation date.

Check Your Understanding

For each problem, use the stated annual effective rate and assume deterministic cash flows paid on the stated dates. Round dollar answers to the nearest cent.

A project costs $25,000 at time 0, returns $10,000 at the end of each of years 1 and 2, and has a $5,000 sale receipt at the end of year 2. Find NPV at 5%. (Answer: $-\$1,870.75$)
A project costs $40,000 and pays $15,000 at the end of each of years 1 through 3, plus a $7,000 salvage receipt at the end of year 3. Find NPV at 10%. (Answer: $2,561.98)
At 6%, compare a $30,000 terminal receipt at the end of year 4 with a $33,000 receipt at the end of year 5. Which has the greater present value, and by how much? (Answer: The year-5 receipt, by $896.71)
A project costs $20,000 at time 0, returns $9,000 at the end of each of years 1 through 3, receives $4,000 from a sale at the end of year 3, and pays a $1,000 closure cost at that same time. Find NPV at 8%. (Answer: $5,575.37)
A project costs $50,000 at time 0 and returns $12,000 at the end of each of years 1 through 4. What terminal receipt at the end of year 4 makes NPV zero at 8%? (Answer: $13,951.10)

What You'll Learn