What the Reinvestment Assumption Means
Net present value compares a project’s dated cash flows with an appropriate required return. A related interpretation compounds those cash flows to a common future date: when all interim cash flows are accumulated at the required return, their net terminal amount equals NPV accumulated to that date. This is a useful way to understand the arithmetic, but it does not mean the NPV accept-or-reject rule requires the company to follow a particular plan for reinvesting interim receipts.
The distinction matters because the compounding identity is a mathematical consequence of discounting at a consistent rate. An actual company may invest receipts elsewhere, hold them as cash, or use them to reduce borrowing. Those choices can change the eventual cash balance, but they do not change the project’s NPV when its cash flows and required return remain the same.
Here, \(C_0\) is usually the initial investment and is therefore negative; later cash flows may be receipts or additional outlays. The time unit must match the rate period. For example, if periods are years, \(i\) must be an effective annual required return. The setup assumes the project cash flows are measured at the stated dates and that the same required return applies in each period. With changing period-specific required returns, the matching discount and accumulation factors must instead use those rates period by period.
Why the Terminal-Value Identity Holds
The identity follows by accumulating each discounted cash flow to the same terminal date. For any individual cash flow at time \(t\), discounting to time 0 divides by \((1+i)^t\), while accumulating from time 0 to time \(n\) multiplies by \((1+i)^n\). The combined factor is therefore \((1+i)^{n-t}\), exactly the factor that accumulates the cash flow directly from time \(t\) to time \(n\).
This is a derivation, not an additional decision rule. It says the time-0 NPV and the terminal-date equivalent express the same net value at different dates, given the same rate and cash flows. Multiplying a time-0 value by \((1+i)^n\) changes its date, not its underlying project assessment.
Write every cash flow’s date and use a required return whose period matches the intervals being discounted.
Discount each dated cash flow to time 0 using the required return.
Multiply each cash flow by the required-return accumulation factor from its date to the chosen terminal date.
For a constant per-period rate and terminal date \(n\), the accumulated net amount must equal \(\mathrm{NPV}(1+i)^n\).
Worked Examples
Worked Example: Checking NPV Against the Terminal Equivalent
A hypothetical project requires an outlay of $100,000 now and pays $40,000 at the end of year 1, $50,000 at the end of year 2, and $30,000 at the end of year 3. The effective annual required return is 10%. Calculate NPV and the equivalent net amount at the end of year 3. Report money to the nearest cent.
Method. Calculate NPV using annual discount factors, then accumulate that value three years at the required return. Independently accumulate each project cash flow to year 3 as a check. This avoids confusing the terminal value with NPV itself.
NPV calculation. In thousands of dollars, the cash flows are \(-100, 40, 50, 30\), and the rate is \(i=0.10\) per year:
The displayed discounted amounts are shown to nine decimal places; the calculation retains unrounded values until the final result. Thus NPV is approximately \(0.225395192\) thousand dollars, or $225.40 to the nearest cent.
Terminal equivalent. Accumulate the initial outlay for three years, the year-1 receipt for two years, and the year-2 receipt for one year. The year-3 receipt needs no accumulation:
The check gives \(0.2253951916\ldots \times 1.331=0.300\) thousand dollars. The terminal equivalent is $300.00, not the NPV of $225.40; they are values at different dates. The common mistake is to report the accumulated amount as time-0 NPV.
Worked Example: Actual Reinvestment at a Different Rate
A hypothetical project costs $1,000 today and returns $600 at the end of each of the next two years. Its effective annual required return is 10%. Calculate NPV and its year-2 equivalent. Then suppose the year-1 receipt is actually invested for one year at 5%; compare the resulting terminal surplus with the initial cost accumulated at the required return.
Method. First apply the NPV formula using the 10% required return. Then separately calculate the specified actual terminal cash position. This distinguishes the valuation rate from the project’s assumed or actual handling of receipts.
NPV and required-return equivalent.
NPV is approximately $41.32. At year 2, its equivalent is \(41.322314050(1.10)^2=41.322314050(1.21)=\$50.00\). Direct accumulation confirms this: the outlay grows to \(-\$1,210\), the year-1 receipt to \(\$660\), and the year-2 receipt remains \(\$600\); the sum is \(-1{,}210+660+600=\$50\).
Actual 5% reinvestment. If the year-1 receipt earns 5% for one year, the two receipts together provide \(600(1.05)+600=\$1,230\) at year 2. The initial $1,000 outlay accumulated at the 10% required return is $1,210, so this comparison produces a terminal surplus of \(1,230-1,210=\$20\). That is different from the $50 terminal equivalent obtained by accumulating every project cash flow at 10%. The NPV remains $41.32: changing the actual reinvestment choice does not alter the project cash flows or the required-return calculation. The mistake to avoid is changing the discount rate in the NPV calculation to 5% merely because the interim receipt earns 5%.
Worked Example: Cash Flows at Unequal Annual Dates
A hypothetical project costs $12,000 now, pays $5,000 at the end of year 1, and pays $9,000 at the end of year 3. There is no year-2 cash flow. The effective annual required return is 6%. Find NPV and its equivalent at the end of year 3. Report money to the nearest cent.
Method. Discount each receipt according to its actual year, then compound the NPV for three years. Checking by accumulating each dated cash flow helps catch the error of treating the year-1 receipt as if it occurred at year 2 or year 3.
NPV calculation. The year-1 receipt is discounted one year, and the year-3 receipt is discounted three years:
The NPV rounds to $273.55 to the nearest cent. The year-3 equivalent, using unrounded NPV, is \(273.554679(1.06)^3=273.554679(1.191016)=\$325.808\), or $325.81 to the nearest cent.
Independent terminal check. The initial outlay accumulates for three years to \(-12{,}000(1.191016)=-\$14{,}292.192\). The year-1 receipt accumulates for two years to \(5{,}000(1.1236)=\$5{,}618\). The year-3 receipt remains $9,000. Their sum is \(-14{,}292.192+5{,}618+9{,}000=\$325.808\), matching the NPV equivalent. The common mistake is to give the year-1 cash flow only one year of accumulation to year 3; it has two years remaining after it is received.
What the Identity Does and Does Not Say
Compounding interim receipts at the required return is one way to translate the project’s NPV into a value at a later date. It is not a claim that the project’s receipts are guaranteed to earn that rate, nor that the firm must reinvest them in the project. The required return is used to evaluate cash flows at their dates; an actual investment decision about interim receipts is separate.
| Statement | Correct interpretation |
|---|---|
| “NPV equals a terminal amount when accumulated at the required return.” | True as an algebraic equivalence when the rate and periods are applied consistently. |
| “A firm must actually reinvest every interim receipt at the required return for NPV to be valid.” | False. NPV is calculated from dated cash flows and the required return; the actual use of receipts is not an input to that calculation. |
| “If the actual reinvestment return differs, the NPV must change.” | False, provided the project cash flows and required return used for evaluation are unchanged. The realized terminal cash position may differ. |
A positive NPV means the project’s discounted inflows exceed its discounted outflows at the required return. The terminal-value identity expresses that same net value at a later date. It does not replace the NPV decision rule or create a new condition for applying it. This distinction is aligned with the SOA VEE Accounting and Finance syllabus.
Common Mistake: Treating an Interpretation as a Requirement
Suppose an analyst calculates NPV at a required return of 10%, then learns that an interim receipt may earn only 5% in an outside account. The incorrect conclusion is that the original NPV is invalid because the receipt will not be reinvested at 10%. A related wrong calculation is to substitute 5% for the 10% required return in the project NPV formula solely because of that reinvestment choice.
The correction is to keep the two calculations distinct. Use the 10% required return to evaluate the project’s dated cash flows. If the question also asks for the actual terminal cash position under a 5% reinvestment plan, calculate that separately and identify the comparison being made. If the required return itself changes because the project’s risk or financing assumptions change, that is a different valuation input; it is not changed merely by selecting a different destination for interim receipts.
Check Your Understanding
For each problem, keep the required return used for NPV separate from any actual reinvestment rate stated. Money answers are rounded to the nearest cent unless otherwise specified.
- A project costs $2,000 at time 0 and returns $1,000 at year 1 and $1,400 at year 2. The effective annual required return is 10%. Find NPV and its year-2 equivalent. (Answer: NPV = $66.12; year-2 equivalent = $80.00.)
- A project costs $1,000 now and returns $500 at year 1 and $700 at year 2. At a 10% effective annual required return, find NPV and the terminal equivalent at year 2. (Answer: NPV = $33.06; year-2 equivalent = $40.00.)
- A project costs $2,000 now and returns $1,000 at year 1 and $1,300 at year 2. The required return is 10%, but the year-1 receipt is actually held without interest until year 2. Find NPV and the terminal surplus when the receipts are compared with the initial cost accumulated at the required return. (Answer: NPV = −$16.53; terminal surplus = −$120.00.)
- A project costs $5,000 now, returns $2,000 at year 1, and returns $3,000 at year 3. The effective annual required return is 10%. Find NPV and its year-3 equivalent. (Answer: NPV = −$927.87; year-3 equivalent = −$1,235.00.)
- True or false: If a firm will reinvest an interim receipt at a rate below the required return, it must replace the required return with that lower rate when calculating the project’s NPV. (Answer: False; calculate project NPV using the required return, and analyze the actual reinvestment choice separately.)