EFN507 Chap.2 Net Present Value, Capital Rationing and Unequal Lives
Net Present Value, Capital Rationing and Unequal Lives
NPV as the benchmark
Week 2 makes NPV the yardstick for every other rule. An asset is worth its discounted future cash flows and present values add, so taking a project raises firm value by its NPV.
The cash flows are after tax with depreciation added back, discounted for now as if the project were all-equity financed, and the rule is to accept every positive-NPV project.
When money or lives differ
Two situations complicate the rule.
Under capital rationing the firm cannot fund every good project, so it ranks by NPV per dollar when one budget binds and projects are divisible, and otherwise solves a linear programme with Solver, reading shadow prices for each constraint; integer programming handles projects that must be taken whole.
When machines have different lives, the lecture replicates them to a common horizon or converts each NPV into an equivalent annual amount using the annuity factor on the formula sheet, preferably with real cash flows and a real rate.
What this chapter covers
- 01
Present value additivity and the NPV accept rule
- 02
After-tax cash flows with depreciation added back
- 03
Ranking by NPV per dollar under a single budget
- 04
Linear programming with Solver and shadow prices
- 05
Integer programming when projects cannot be split
- 06
Equivalent annual amounts and common-horizon replication
Worked example · free
Capital rationing with divisible and whole projects
- +1NPV per dollar: P 0.50, R 0.45, Q 0.40, S 0.30.
- +1Divisible: fund P and R ($10m) and 2/5 of Q for a total NPV of $5.6m.
- +1Whole projects: P + Q gives $5.0m, beating P + R at $4.8m and Q + R + S at $4.7m.
- +1The budget's shadow price in the divisible case is $0.40 of NPV per extra dollar, Q's ratio.
Key terms
- Capital rationing
- A limit on the funds a firm can invest, which stops it taking every positive-NPV project.
- Linear programming
- An optimisation that chooses project weights to maximise total NPV subject to linear funding constraints.
- Integer programming
- A version of the programme in which each project weight must be exactly zero or one.
- Value additivity
- The property that the present value of a combination of cash flows equals the sum of their separate present values.
Net Present Value, Capital Rationing and Unequal Lives FAQ
Why is depreciation added back when computing NPV cash flows?
NPV works from cash, not accounting profit. Depreciation reduces taxable income, which lowers the tax paid, but no cash leaves the firm when it is charged, so it is added back after tax is computed. Leaving it deducted understates the cash flows.
When does ranking by NPV per dollar give the right answer?
Only when one budget constraint binds and projects can be taken in part. Then filling the budget in order of NPV per dollar maximises total NPV. With whole projects or constraints in several periods, use linear or integer programming instead.
What does a shadow price tell a manager?
It is the increase in the maximum achievable NPV if a funding constraint is relaxed by one dollar. A high shadow price shows the period where extra finance would add most value; a zero shadow price means that constraint is not binding.
How is the equivalent annual amount calculated?
Divide the project's NPV by the annuity factor for its life at the cost of capital, one minus (1 + r) to the power minus n, all over r. The result is the level yearly amount with the same present value, so machines with different lives can be compared year by year.
Why should the equivalent annual cost use real figures?
The method assumes each replacement repeats the original exactly. In real terms that is true, but in nominal terms inflation makes each new machine dearer, so the lecture recommends real cash flows discounted at a real rate.
Can a zero-NPV project appear in an optimal rationing plan?
Yes. If it returns cash in a period where borrowing is capped, it relaxes that constraint and lets other positive-NPV projects be funded, so the programme may select it even though it adds no value on its own.
Exam move
Set up one rationing problem by hand, solve it in Solver as continuous and integer programmes, and read the shadow prices. Then compare two machines by equivalent annual amount and by a common-horizon chain.
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