Exercise set

Engineering Investment NPV, Payback, IRR, and Capital Rationing Exercises

Solved engineering investment exercises for present value, NPV, payback, IRR, profitability index, capital rationing and approval gates.

Branch: Industrial and Management Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises practise engineering investment appraisal: present value, net present value, simple payback, discounted payback, IRR, profitability index, capital rationing, threshold savings, scenario NPV and approval gates.

The goal is to decide whether a proposed engineering investment earns its required return under stated technical assumptions. A project can pass payback but fail NPV, pass IRR but be weaker than a larger alternative, or pass finance metrics while technical evidence remains weak.

Assume the discount factors given in each exercise. Real investment approval should also check inflation basis, tax, commissioning risk, benefit evidence, operating constraints, downtime during implementation, safety gates and portfolio interactions.

Release Evidence Notes

Investment evidence should state initial cost, timing, benefits, residual value, discount rate, approval threshold and sign convention. Costs are negative and savings are positive unless stated otherwise.

NPV evidence should be preferred for value creation. Payback measures recovery speed, not total value, and IRR can be misleading for mutually exclusive or nonstandard cash flows.

Capital-rationing evidence should identify budget constraints, indivisible projects, dependencies and nonfinancial gates. A high profitability index does not override safety or reliability requirements.

Technical benefit evidence should be traceable to measured energy, throughput, quality, maintenance or downtime changes.

Engineering Boundary Notes

This page covers investment appraisal. Lifecycle cost and replacement economics belong in the lifecycle-cost exercise set. Tax, escalation and ramp-up cash-flow treatment belong in the cash-flow evidence exercise set.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Present value and NPV	1-5	discounted savings, residual value and NPV	Approve, reject or request evidence.
Payback and IRR	6-9	simple payback, discounted payback and IRR relative to MARR	Check recovery and return limits.
Capital rationing	10-14	profitability index, budget fit and threshold savings	Select portfolio under constraints.
Approval sensitivity	15-18	scenario NPV, cost overrun, benefit shortfall and hard gates	Stage, hold or approve investment.

Exercise 1: Present Value of Annual Savings

A retrofit saves:

S=32000

per year for five years. Use:

(P/A,8\%,5)=3.9938

Find present value of savings.

Solution

Present value:

PV_S=32000(3.9938)=127802

Engineering Comment

Savings should be traceable to measured or validated technical mechanisms.

Plausibility Check

Five undiscounted years are $160000$ , so present value below that is expected.

Exercise 2: Present Value of Residual Value

Residual value in year 5 is:

R=15000

Use:

(P/F,8\%,5)=0.6806

Find present value.

Solution

Present value:

PV_R=15000(0.6806)=10209

Engineering Comment

Residual value should be supported by resale, reuse or salvage evidence.

Plausibility Check

The future amount is discounted to about two thirds.

Exercise 3: Net Present Value

Initial cost is:

C_0=120000

Use:

PV_S=127802,\qquad PV_R=10209

Find NPV.

Solution

NPV:

NPV=-120000+127802+10209=18011

Engineering Comment

The project is positive under the stated assumptions. Approval should still check whether savings evidence is strong enough.

Plausibility Check

Savings plus residual exceed initial cost by about eighteen thousand.

Exercise 4: NPV Margin Ratio

NPV is:

18011

Initial cost is:

120000

Find NPV margin as percent of initial cost.

Solution

Margin:

M=\dfrac{18011}{120000}=0.150=15.0\%

Engineering Comment

A fifteen percent margin is useful, but not immune to benefit shortfall or cost overrun.

Plausibility Check

Eighteen thousand is about fifteen percent of one hundred twenty thousand.

Exercise 5: NPV with Higher Initial Cost

If initial cost rises to:

C_0=135000

while present benefits remain:

138011

find revised NPV.

Solution

Revised NPV:

NPV=-135000+138011=3011

Engineering Comment

The project remains positive but becomes marginal. Cost control becomes a release gate.

Plausibility Check

Increasing cost by $15000$ reduces the original $18011$ NPV to about $3000$ .

Exercise 6: Simple Payback

Project cost is:

C_0=95000

Annual saving:

S=28500

Find simple payback.

Solution

Simple payback:

P_s=\dfrac{95000}{28500}=3.33\ \text{years}

Engineering Comment

Simple payback ignores discounting and all benefits after payback.

Plausibility Check

Three years gives $85500$ recovered; a bit more than three years is needed.

Exercise 7: Discounted Payback

Discounted cumulative savings through year 3 are:

73445

Initial cost is:

95000

Year 4 discounted saving is:

20948

Estimate discounted payback.

Solution

Remaining after year 3:

95000-73445=21555

Interpolated payback:

P_d=3+\dfrac{21555}{20948}=4.03\ \text{years}

Engineering Comment

Discounted payback is longer than simple payback because future savings are worth less.

Plausibility Check

The remaining amount is slightly larger than year 4 discounted savings, so payback is just beyond year 4.

Exercise 8: NPV at MARR

A project costs:

100000

and returns:

28000

per year for five years. Use:

(P/A,10\%,5)=3.7908

Find NPV at MARR.

Solution

NPV:

NPV=-100000+28000(3.7908)=6142

Engineering Comment

Positive NPV at MARR means the project exceeds the required return under this cash-flow model.

Plausibility Check

Discounted benefits are slightly above one hundred thousand, so NPV is positive but modest.

Exercise 9: IRR Interpolation

At $12\%$ , NPV is:

934

At $13\%$ , NPV is:

-1518

Estimate IRR by linear interpolation.

Solution

IRR:

IRR\approx12\%+\dfrac{934}{934+1518}(1\%)=12.38\%

Engineering Comment

IRR is above a $10\%$ MARR, but NPV is still the better value measure.

Plausibility Check

The sign changes between $12\%$ and $13\%$ , so IRR must lie between them.

Exercise 10: Profitability Index

Project A costs:

100000

and has present benefits:

138000

Find profitability index.

Solution

Profitability index:

PI=\dfrac{138000}{100000}=1.38

Engineering Comment

PI helps under capital rationing but does not measure total dollars of value alone.

Plausibility Check

Benefits are thirty-eight percent above cost, so PI is $1.38$ .

Exercise 11: PI Ranking

Projects have:

Project	Cost	PV benefits
A	$100000$	$138000$
B	$75000$	$96000$
C	$60000$	$84000$

Rank by PI.

Solution

Compute:

PI_A=1.38,\quad PI_B=\dfrac{96000}{75000}=1.28,\quad PI_C=\dfrac{84000}{60000}=1.40

Ranking:

C,\ A,\ B

Engineering Comment

Ranking by PI can favor smaller projects. Check absolute NPV and strategic constraints.

Plausibility Check

C has the highest benefit per dollar.

Exercise 12: Capital Budget Selection

Budget is:

160000

Project costs and NPVs are:

Project	Cost	NPV
A	$100000$	$38000$
B	$75000$	$21000$
C	$60000$	$24000$

Choose the combination with highest NPV within budget.

Solution

Feasible combinations:

A+C=160000,\quad NPV=62000

B+C=135000,\quad NPV=45000

A+B=175000\quad \text{not feasible}

Choose A and C.

Engineering Comment

Capital rationing is a portfolio decision. Dependencies and resource limits should also be checked.

Plausibility Check

A plus C uses the full budget and has higher NPV than B plus C.

Exercise 13: Energy Price Threshold

Extra capital cost is:

35000

Energy saving is:

85\ \text{MWh/year}

Maintenance saving is:

1500\ \text{per year}

Find electricity price for four-year simple payback.

Solution

Required annual saving:

S=\dfrac{35000}{4}=8750

Energy saving needed:

S_E=8750-1500=7250

Price:

p=\dfrac{7250}{85}=85.29\ \text{per MWh}

Engineering Comment

The threshold should be compared with tariff structure and operating profile.

Plausibility Check

Maintenance covers part of the needed saving, leaving about $7250$ for energy.

Exercise 14: Minimum Annual Saving for NPV

A project costs:

180000

and lasts six years. Use:

(P/A,9\%,6)=4.4859

Find annual saving needed for zero NPV.

Solution

Set:

0=-180000+S(4.4859)

Solve:

S=\dfrac{180000}{4.4859}=40126

Engineering Comment

This is the break-even annual benefit. Approval should require margin above it.

Plausibility Check

Six-year undiscounted average would be $30000$ ; discounting raises the required annual saving.

Exercise 15: Scenario NPV

Base NPV is:

18011

Benefit shortfall reduces present benefits by:

22000

Find downside NPV.

Solution

Downside NPV:

NPV_d=18011-22000=-3989

Engineering Comment

The project is vulnerable to benefit shortfall. A pilot or staged approval may be appropriate.

Plausibility Check

The shortfall exceeds the base NPV, so downside NPV is negative.

Exercise 16: Cost Overrun Tolerance

Base NPV is:

18011

Find maximum initial cost overrun before NPV reaches zero.

Solution

Maximum overrun equals the NPV margin:

\Delta C_{max}=18011

Engineering Comment

If expected cost uncertainty is larger than this margin, approval should require contingency or redesign.

Plausibility Check

Every extra dollar of initial cost reduces NPV by one dollar.

Exercise 17: Benefit Realization Gate

Approved annual saving is:

32000

Measured first-year saving is:

27500

The gate requires at least $90\%$ of approved saving. Check.

Solution

Realization:

R=\dfrac{27500}{32000}=0.859=85.9\%

The gate fails:

85.9\%<90\%

Engineering Comment

Post-audit evidence matters. Under-realized benefits should feed future approval assumptions.

Plausibility Check

The measured saving is about $4500$ below target, so realization below ninety percent is plausible.

Exercise 18: Investment Approval Gate

An investment package has:

Gate	Requirement	Current result
NPV	positive	$18011$
discounted payback	at most $4$ years	$4.03$ years
downside scenario	nonnegative	$-3989$
benefit evidence	at least $90\%$ realized	$85.9\%$

Decide whether to release.

Solution

NPV passes, but payback, downside scenario and benefit evidence fail:

4.03>4,\quad -3989<0,\quad 85.9\%<90\%

The package should not receive unrestricted approval.

Engineering Comment

Positive base NPV is not enough when downside and measured benefit gates fail.

Plausibility Check

Multiple gates fail, so hold or stage approval is defensible.

Validation Package Checklist

A strong investment appraisal should check:

whether cash-flow signs and timing are explicit;
whether NPV, payback and IRR are interpreted for their proper purpose;
whether MARR and discount factors match the decision basis;
whether capital rationing uses feasible project combinations;
whether threshold savings and cost-overrun tolerance are visible;
whether scenario NPV checks downside exposure;
whether measured benefits support approval assumptions;
whether failed hard gates block unrestricted release.

Common Release Mistakes

Common mistakes include approving by simple payback alone, treating IRR as better than NPV for mutually exclusive alternatives, ignoring cost-overrun tolerance, using PI without budget feasibility, counting unvalidated benefits, omitting downside scenarios, and approving a project with positive base NPV while payback, downside or benefit-realization gates fail.

REF