Linking NPV to Terminal Wealth

Why Link NPV to Terminal Wealth?

Net present value (NPV) expresses a sequence of dated cash flows in dollars at a valuation date, usually time 0. Terminal wealth expresses the same sequence at a later date after cash flows have been accumulated. A direct relationship connects the two when the NPV rate is also the rate used to reinvest interim net cash flows. This connection lets you translate a value stated today into its corresponding value at the end of a project horizon.

The rate assumption matters. NPV discounts each cash flow at a stated rate; terminal wealth accumulates each cash flow. If the same effective rate is used for both operations, discounting and accumulation reverse one another across the horizon. The resulting identity is useful for interpreting an NPV, checking calculations, and finding a break-even terminal value.

Definition: Consider signed cash flows $C_0,C_1,\ldots,C_n$ at times $0,1,\ldots,n$, where a receipt is positive and an outflow is negative. At effective annual rate $i$, the net present value at time 0 is

\operatorname{NPV}(i)=\sum_{t=0}^{n} C_t v^t =\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}, \qquad v=\frac{1}{1+i}.

If every cash flow is accumulated to time $n$ at the same effective annual rate $i$, the algebraic terminal wealth is

W_n(i)=\sum_{t=0}^{n} C_t(1+i)^{n-t}.

These definitions use annual periods with cash flows occurring at integer times. The initial cash flow $C_0$ is included in both sums: it is not discounted at time 0, but it is accumulated for $n$ years in the terminal-wealth calculation. A cash flow at time $n$ is included in terminal wealth without further accumulation.

The word algebraic is important. A negative cash flow reduces the total, just as a positive cash flow increases it. If there is an interim outflow, this formula treats that amount as a cash flow funded at the same rate for the remaining period. It does not, by itself, describe every possible financing arrangement or guarantee that an investor can actually reinvest every receipt at rate $i$.

The NPV–Terminal Wealth Identity

Under the stated assumptions, the terminal value is the NPV multiplied by the accumulation factor from time 0 to time $n$:

Theorem: For cash flows at integer annual times $0$ through $n$, if NPV is evaluated at effective annual rate $i$ and each cash flow is accumulated at that same rate, then

W_n(i)=(1+i)^n\operatorname{NPV}(i).

The identity follows by factoring the common accumulation factor out of each terminal-value term. For a cash flow at time $t$, accumulating it to time $n$ multiplies it by $(1+i)^{n-t}$. This is the same as first discounting it to time 0 by $(1+i)^{-t}$, then accumulating that time-0 value for $n$ years by $(1+i)^n$.

Write the terminal value.
Each $C_t$ is accumulated for the $n-t$ periods remaining after its payment date, giving $C_t(1+i)^{n-t}$.

Factor out the horizon accumulation factor.
Since $(1+i)^{n-t}=(1+i)^n(1+i)^{-t}$, every term contains the same factor $(1+i)^n$.

Recognize NPV.
The remaining sum is $\sum_{t=0}^{n} C_t(1+i)^{-t}$, exactly the definition of $\operatorname{NPV}(i)$.

\begin{aligned} W_n(i) &=\sum_{t=0}^{n}C_t(1+i)^{n-t}\\ &=(1+i)^n\sum_{t=0}^{n}C_t(1+i)^{-t}\\ &=(1+i)^n\operatorname{NPV}(i). \end{aligned}

For a positive effective rate, $(1+i)^n$ is positive. Therefore NPV and terminal wealth have the same sign. A zero NPV at the stated rate corresponds to zero algebraic terminal wealth at the same rate and horizon. This does not mean that a project’s actual account balance must be zero: the relationship concerns the signed cash-flow stream relative to the chosen valuation rate and reinvestment assumption.

Worked Applications

Worked Example: Positive NPV and Accumulated Terminal Value

A project requires an initial outlay of $1,000 and pays $400 at the end of year 1, $500 at the end of year 2, and $600 at the end of year 3. Find its NPV at an effective annual rate of 10%, then find the terminal wealth at the end of year 3 if all interim net cash flows are accumulated at 10%.

Method. First discount every signed cash flow to time 0. Then accumulate each signed cash flow to year 3. The rate is 10% effective per year, and the horizon is $n=3$.

The time-0 cash flow is $C_0=-1000$. The NPV is

\begin{aligned} \operatorname{NPV}(0.10) &=-1000+\frac{400}{1.10} +\frac{500}{1.10^2} +\frac{600}{1.10^3}\\ &=-1000+\frac{400}{1.10} +\frac{500}{1.21} +\frac{600}{1.331}\\ &=\frac{303}{1.331}\\ &\approx \$227.65. \end{aligned}

To calculate terminal wealth directly, 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 further accumulation:

\begin{aligned} W_3(0.10) &=-1000(1.10)^3+400(1.10)^2+500(1.10)+600\\ &=-1331+484+550+600\\ &=\$303.00. \end{aligned}

The identity gives the same result: $\operatorname{NPV}(0.10)(1.10)^3=(303/1.331)(1.331)=\$303.00$. The positive NPV corresponds to positive terminal wealth. All money results are rounded to the nearest cent.

Check. The direct accumulation and the NPV conversion both give $303.00. A common mistake is to multiply the NPV by $1.10^2$, using the number of interim receipts rather than the three-year horizon. The correct exponent is the time from the valuation date to the terminal date.

Worked Example: An Interim Outflow and Negative Terminal Wealth

A three-year plan has cash flows of $-\$2{,}000$ at time 0, $+\$300$ at the end of year 1, $-\$100$ at the end of year 2, and $+\$1{,}200$ at the end of year 3. Calculate terminal wealth at an effective annual rate of 5%, and then find NPV at that rate.

Method. Keep the year-2 cost negative. Accumulate each signed cash flow to year 3 at 5%, then divide by the three-year accumulation factor to recover NPV.

\begin{aligned} W_3(0.05) &=-2000(1.05)^3+300(1.05)^2-100(1.05)+1200\\ &=-2000(1.157625)+300(1.1025)-105+1200\\ &=-2315.25+330.75-105+1200\\ &=-\$889.50. \end{aligned}

The terminal value is negative, so the corresponding NPV is also negative. Since $(1.05)^3=1.157625$,

\begin{aligned} \operatorname{NPV}(0.05) &=\frac{W_3(0.05)}{(1.05)^3}\\ &=\frac{-889.50}{1.157625}\\ &\approx -\$768.38. \end{aligned}

The check is $(-\$768.38)(1.157625)\approx-\$889.50$, allowing for cent rounding. Money results are rounded to the nearest cent. The common mistake is to omit the interim outflow or enter it as a positive amount. Either error overstates both terminal wealth and NPV.

Worked Example: Finding the Break-Even Final Receipt

A project costs $1,000 initially. It pays $300 at the end of year 1 and $400 at the end of year 2. What payment $X$ at the end of year 3 makes NPV zero at an effective annual rate of 10%? Verify the result using terminal wealth.

Method. Set the NPV of all signed cash flows equal to zero and solve for the final receipt. By the identity, a zero NPV must also give zero terminal wealth at year 3.

\begin{aligned} 0&=-1000+\frac{300}{1.10}+\frac{400}{1.10^2}+\frac{X}{1.10^3}\\ X&=1000(1.10)^3-300(1.10)^2-400(1.10)\\ &=1331-363-440\\ &=\$528.00. \end{aligned}

To check, accumulate all cash flows to year 3:

\begin{aligned} W_3(0.10) &=-1000(1.10)^3+300(1.10)^2+400(1.10)+528\\ &=-1331+363+440+528\\ &=\$0.00. \end{aligned}

Thus the break-even receipt is $528.00. The result is exact to the cent, so no further rounding is needed. A common mistake is to accumulate the year-1 and year-2 payments for the wrong number of years: they earn two and one years of interest, respectively, before year 3.

Conditions, Interpretation, and Common Traps

The identity is an algebraic result, but its economic interpretation relies on matching assumptions. The NPV rate and the accumulation rate must be the same effective rate over the same periods. With annual cash flows and a constant effective annual rate $i$, this means that a cash flow received at time $t$ earns that rate for each of the $n-t$ remaining years.

If interim cash flows are actually reinvested at rates different from $i$, their actual terminal value is calculated using those realized rates, not by multiplying the NPV at $i$ by $(1+i)^n$. The identity still describes a hypothetical terminal value under accumulation at $i$, but it does not predict the realized value under different reinvestment conditions. Similarly, for irregular dates or fractional periods, use the applicable accumulation factors for those periods rather than applying annual powers without justification.

The relationship also helps separate a project’s NPV from its total cash receipts. NPV is a value relative to the stated rate: a positive NPV means the signed cash-flow stream has positive value at time 0 under that rate. Its corresponding terminal wealth is the accumulated value of that same stream. Neither measure alone describes an account balance that includes unrelated deposits, withdrawals, or financing.

NPV at rate $i$	Terminal wealth at the same rate	Interpretation
Positive	Positive	The signed cash-flow stream has positive value under the stated accumulation assumption.
Zero	Zero	The cash flows break even at rate $i$ over the specified horizon.
Negative	Negative	The signed cash-flow stream has negative value under the stated accumulation assumption.

Wrong horizon: Use $n$, the number of periods from valuation to terminal time, not the number of payments.
Wrong sign: Enter outflows as negative cash flows in both NPV and terminal wealth.
Mismatched rates: Do not use an NPV rate for discounting and a different reinvestment rate while applying the identity.
Wrong payment timing: A time-$t$ cash flow accumulates for $n-t$ periods. In particular, the time-$n$ cash flow accumulates for zero periods.

Key takeaway: When signed cash flows are discounted and accumulated using the same effective rate over the same horizon, terminal wealth equals NPV multiplied by the horizon accumulation factor. The sign is preserved, and a zero NPV corresponds to zero terminal wealth.

Check Your Understanding

For each problem, use the stated annual rate for both valuation and accumulation where applicable. Amounts are in dollars.

A two-year cash-flow stream is $-\$500$ at time 0, $+\$200$ at year 1, and $+\$400$ at year 2. Find its NPV at 4% effective annually and its terminal wealth at year 2. (Answer: NPV $\approx \$62.13$; terminal wealth $=\$67.20$.)
Cash flows are $-\$1{,}200$ at time 0, $+\$500$ at year 1, $-\$100$ at year 2, and $+\$900$ at year 3. Find terminal wealth at 6% effective annually. (Answer: $-\$73.42$.)
A cash-flow stream has NPV of $250 at 8% effective annually. What is its corresponding terminal wealth four years after time 0, assuming accumulation at 8%? (Answer: $\$340.12$.)
A project costs $2,000 initially and pays $600 at year 1, $700 at year 2, and $X$ at year 3. Find $X$ for zero NPV at 5% effective annually, and verify the implied terminal wealth. (Answer: $X=\$918.75$; terminal wealth $=\$0.00$.)
At 5% effective annually, Project A has NPV $80 and a three-year horizon. Project B has NPV $75 and a three-year horizon. If both are accumulated at 5%, which has the greater terminal wealth, and by how much? (Answer: Project A; its terminal wealth is $5.79 greater.)

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