Why a Fixed Planning Horizon Changes the Comparison
Projects with different useful lives can still be compared when the decision-maker has a specified planning horizon. The horizon might be set by a contract, a strategic review date, or a forecast period. The comparison must then include the cash flows that occur during that period—not an assumed infinite sequence of replacements—and must treat the position of each asset at the horizon consistently.
The key task is to put every alternative on the same dated timeline. A project may end before the horizon, requiring an explicit decision about what happens next. It may also remain useful beyond the horizon, in which case the analysis needs an end-of-horizon value if the remaining asset has value. Replacement costs and proceeds from selling an old asset belong on their actual dates, even when several cash flows occur at the same date.
Model and Timeline Method
Assume a single effective annual discount rate \(i\), constant over the horizon; cash flows occur at year-ends unless stated otherwise; and each cash flow is known with certainty for the calculation. The alternatives are evaluated from the same time 0 and over the same horizon \(H\). If cash flows are uncertain, a risk-consistent rate or expected-cash-flow approach is needed; that issue is not resolved merely by aligning timelines.
For annual cash flows, the NPV over \(H\) years is the sum of each dated cash flow discounted to time 0. A cash flow at time 0 is not discounted. A terminal value at time \(H\) is included once, as a cash flow at \(H\). If the stated operating cash flow already includes sale proceeds, do not add those proceeds again.
This formula does not assume that a project lasts exactly \(H\) years. It says only that cash flows relevant to the decision are recorded through \(H\). A project that ends earlier may have a replacement, closure, or other consequence before \(H\). A project that could continue after \(H\) contributes no post-horizon operating cash flows to this calculation; instead, any value it retains at \(H\) is represented by a horizon-end value when appropriate.
Write down time 0, the specified horizon \(H\), and the annual rate \(i\) used to value all alternatives.
Record initial outlays, operating cash flows, replacement purchases, and disposal proceeds at their actual dates. Combine amounts that occur at the same date.
For an asset still in service at \(H\), include its stated end-of-horizon value if the analysis calls for one. For an asset that expires before \(H\), specify whether and when a replacement is made.
Calculate each alternative’s NPV from its complete timeline. For mutually exclusive alternatives, choose the higher NPV when the objective is to maximize value and the cash-flow estimates and discount rate are comparable.
Why Discounting Each Date Works
The formula follows from the time value of money and the additivity of present values. At effective annual rate \(i\), one dollar received at time \(t\) has present value \(1/(1+i)^t\): investing that amount at time 0 grows it to one dollar at \(t\). Therefore, a cash flow \(CF_t\) has present value \(CF_t/(1+i)^t\). Since each dated cash flow can be valued separately and the values can be added, the project’s total present value is their sum. A cost is a negative cash flow, so the same reasoning discounts it with a negative sign.
The time index is essential: a dollar at time 0 and a dollar at time \(H\) are not interchangeable. An end-of-horizon value is received at \(H\), so it must be discounted by \((1+i)^H\) when calculating its present value.
Worked Example: Comparing Two Alternatives Over Four Years
Method and assumptions. Compare two mutually exclusive alternatives over a fixed four-year horizon at an effective annual discount rate of 10%. All amounts are dollars, flows occur at year-end, and the stated year-4 horizon values are additional to the annual operating cash flows. Alternative A costs $90 at time 0, generates $40 at each year-end for four years, and has a $20 value at year 4. Alternative B costs $110 at time 0, generates $35 at each year-end for four years, and has a $45 value at year 4. Which has the higher NPV?
First, the discount factors are \(v=1/1.10=0.9090909\), \(v^2=0.8264463\), \(v^3=0.7513148\), and \(v^4=0.6830135\). The four-year annuity discount-factor sum is \(0.9090909+0.8264463+0.7513148+0.6830135=3.1698654\).
For A, the time-0 outlay is \(-$90\), the operating cash flows have present value \(40(3.1698654)=$126.7946\), and the year-4 value has present value \(20(0.6830135)=$13.6603\). Thus, before final rounding, \(NPV_A=-90+126.7946+13.6603=$50.4549\), or $50.45 to the nearest cent.
For B, the operating cash flows have present value \(35(3.1698654)=$110.9453\), and its year-4 value has present value \(45(0.6830135)=$30.7356\). Thus, \(NPV_B=-110+110.9453+30.7356=$31.6809\), or $31.68 to the nearest cent.
The check is that both alternatives include four annual operating cash flows and exactly one year-4 value; each year-4 amount is discounted by \(v^4\). Alternative A has the higher NPV by \($50.45-$31.68=$18.77\). The common mistake this method avoids is treating the horizon value as if it arrived at time 0: doing so would overstate A’s NPV by \(20-20v^4=$6.34\).
Replacement Timing and Partial Project Lives
A project whose equipment life is shorter than the planning horizon requires an explicit replacement decision. If the asset is replaced, the old asset’s sale or salvage proceeds and the new asset’s purchase cost are recorded at the replacement date. The new asset’s subsequent operating cash flows are then recorded only for dates through the horizon. A replacement chain is not automatically extended beyond the specified horizon.
Conversely, if equipment is expected to remain useful after the horizon, the analysis ends at \(H\). The remaining value is represented by a horizon-end value if one is supplied or can be estimated under the stated model. Do not include operating cash flows after \(H\) as well as a horizon value that already reflects those future benefits: that would count value beyond the horizon twice.
Worked Example: Replacing Equipment Before the Horizon
Method and assumptions. A firm plans over five years and uses an effective annual discount rate of 8%. At time 0 it buys a machine for $100. It produces net operating cash flow of $50 at the end of each of years 1, 2, and 3. At the end of year 3, the old machine can be sold for $20 and a replacement costs $110. The replacement produces $55 at the end of years 4 and 5, and has a $40 value at the end of year 5. All amounts are dollars, year-end flows are certain, and the year-5 value is additional to year-5 operations. Find the NPV over the stated horizon.
At year 3, three amounts occur: the old machine’s $50 operating cash flow, $20 sale proceeds, and the \(-$110\) replacement purchase. The combined year-3 cash flow is \(50+20-110=-$40\). The complete timeline is therefore \(-$100\) at time 0; \(+$50\) at years 1 and 2; \(-$40\) at year 3; \(+$55\) at year 4; and \(+$95\) at year 5, where the last amount is \(55+40\).
The discount factors at 8% are \(v=0.9259259\), \(v^2=0.8573388\), \(v^3=0.7938322\), \(v^4=0.7350299\), and \(v^5=0.6805832\). Discount each timeline amount:
\(NPV_5=-100+50(0.9259259)+50(0.8573388)-40(0.7938322)+55(0.7350299)+95(0.6805832)\).
The present-value contributions are \(-$100.0000\), \(+$46.2963\), \(+$42.8669\), \(-$31.7533\), \(+$40.4266\), and \(+$64.6554\). Their sum is $62.4920 before rounding, so the NPV is $62.49 to the nearest cent.
The timeline check confirms that the replacement cost is included at year 3, the old machine’s sale proceeds are also included at year 3, and the year-5 value is discounted for five years. Omitting the replacement purchase would overstate NPV by \(110v^3=$87.32\). Treating the replacement as if it occurred at time 0 would also put its cost on the wrong date.
Worked Example: Stopping Before an Asset’s Useful Life Ends
Method and assumptions. A company evaluates alternatives over a three-year planning horizon at an effective annual rate of 10%. Each alternative’s equipment would remain useful for five years, but the company plans to sell it at the end of year 3. Alternative A costs $120 at time 0, generates $55 at each year-end through year 3, and can be sold for $45 at year 3. Alternative B costs $100, generates $45 at each year-end through year 3, and can be sold for $35 at year 3. Treat the stated sale proceeds as additional to that year’s operating cash flow. Find and compare the horizon NPVs.
The relevant factors are \(v=0.9090909\), \(v^2=0.8264463\), and \(v^3=0.7513148\). Their sum is \(2.4868520\). No year-4 or year-5 operating cash flows are included because the decision horizon ends after year 3.
For A, the operating cash flows have present value \(55(2.4868520)=$136.7769\). The year-3 sale proceeds have present value \(45(0.7513148)=$33.8092\). Therefore, \(NPV_A=-120+136.7769+33.8092=$50.5860\), or $50.59 to the nearest cent.
For B, the operating cash flows have present value \(45(2.4868520)=$111.9083\). The sale proceeds have present value \(35(0.7513148)=$26.2960\). Therefore, \(NPV_B=-100+111.9083+26.2960=$38.2044\), or $38.20 to the nearest cent.
A has the higher horizon NPV by $12.38. The check is that both alternatives use the same three operating dates and discount their sale proceeds at \(v^3\). Extending either project’s operating cash flows to year 5 would answer a different planning question; omitting the sale proceeds would fail to recognize the value realized at the specified horizon.
Common Traps in Fixed-Horizon Comparisons
| Trap | Why it is wrong | Correction |
|---|---|---|
| Discounting the horizon value for the wrong number of years | A value received at time \(H\) is not received at time 0. | Discount it by \(1/(1+i)^H\). |
| Leaving out a replacement cost | The project’s later operating cash flows require the replacement investment that enables them. | Record the purchase cost at the actual replacement date. |
| Comparing alternatives over different periods | The NPVs then cover different sets of dated cash flows. | Use the specified common horizon and consistently treat each alternative at its endpoint. |
| Counting terminal proceeds twice | Sale proceeds may already be included in a stated terminal cash flow or value. | Check exactly what each cash-flow figure includes before adding a separate value. |
| Adding post-horizon operations to a horizon value | A horizon value may already reflect the asset’s remaining economic value. | Include only cash flows through \(H\), with the horizon value defined consistently. |
A fixed horizon does not, by itself, determine the correct end-of-horizon value. The analyst must use the value specified by the problem or a clearly stated valuation assumption. A book value is not automatically a sale price, and a value estimated at the horizon should not be confused with the present value of that amount. These are distinct quantities with different dates and meanings.
Check Your Understanding
For each problem, use year-end cash flows and round dollar NPVs to the nearest cent.
- Over a three-year horizon at an effective annual rate of 8%, Project X costs $75 initially, pays $30 at each year-end, and has a $10 value at year 3. Project Y costs $80 initially and pays $32 at each year-end, with no year-3 value. Which has the higher NPV? (Answer: X; \(NPV_X=$10.25\) and \(NPV_Y=$2.47\).)
- A project costs $90 now and pays $60 at the end of each of the next two years. Its additional year-2 horizon value is $20. At 5% effective annual interest, calculate its two-year NPV. (Answer: $39.71.)
- A four-year plan uses a 10% effective annual discount rate. Equipment costs $100 now and generates $45 at the end of years 1 and 2. At the end of year 2 it is sold for $25 and replaced for $70. The replacement generates $40 at the end of years 3 and 4 and has an additional year-4 value of $15. Find the NPV. (Answer: $8.53.)
- A project costs $50 now and generates $22 at the end of each of years 1, 2, and 3. Its additional year-3 horizon value is $8. At 12% effective annual interest, calculate its three-year NPV and state which date is used to discount the $8. (Answer: NPV is $8.53; discount the $8 from year 3.)
- An asset will remain useful for two years beyond a firm’s four-year planning horizon. The firm’s forecast gives operating cash flows only through year 4 and an estimated sale value at the end of year 4. Explain which amounts enter the horizon NPV and whether year-5 and year-6 operating cash flows should also be added. (Answer: Include cash flows dated 0 through 4 and discount the year-4 sale value by \(1/(1+i)^4\); do not add year-5 or year-6 operations to that horizon comparison.)