Business Case Development and ROI Analysis Questions

Direct answer

A draft that counts only vendor fees is missing most of the cost of change: the migration, integration, training and change management, internal staff time, parallel running and decommissioning of the legacy system, contract exit and escalation terms, and probability-weighted risk. These can be as large as the fees; in the illustrative example below they roughly double the three-year total. I estimate each from a source other than the sponsor's optimism, state a confidence level, and show both views: fees-only and all-in.

Likely missing costs and how I estimate each credibly

Worked example (illustrative figures, 3 years)

The sponsor's draft: vendor fees of $240,000 a year, so $720,000. Contract allows a 5% annual price rise: $240,000 + $252,000 + $264,600 = $756,600.

All-in is about 1.98 times the fees-only draft of $720,000. This is a gross cost; for a fair decision I would put it beside the legacy system's avoided run cost, otherwise the comparison is one-sided.

Regulated customer data moving to the cloud: the expected-cost buckets are audit and certification work, legal and data-processing agreements, regional performance or residency fixes, and breach exposure. Example: a 40% chance the audit requires $50,000 of rework gives an expected $20,000 (0.40 x $50,000), the figure above. Show it separately, labelled expected value (the probability-weighted average cost), and put the unweighted worst case in the risk section.

Validate each estimate

Walk the table with the people who own each line (IT, security, the business units, finance, procurement, the buying and contracts team), record who gave the number and a confidence level, and ask the vendor to confirm the migration assumptions in writing. Estimates from the sponsor alone count as low confidence.

Presenting differently

• To procurement: negotiating levers: price-escalation cap, exit fees, SLA credits, migration support in the contract.
• To the executive sponsor: the all-in total, the range, the three largest line items, and the decision gate.

A related use, if you work on the vendor side: the same table shows what a customer faces in switching away from a product (implementation, training, integration, downtime and the opportunity cost of delayed projects). A seller can cite those costs as a retention argument, but they must be shown honestly rather than hidden, because a buyer who finds them later stops trusting the whole case.

Pitfalls

Treating "we already employ those people" as zero cost, counting SLA credits as savings, and presenting one total with no confidence levels.

Direct answer

Show one picture and three sentences, not a statistics lesson. The picture is a cumulative cash curve (money out first, then money back in) with a shaded band around it for the range of likely outcomes and a vertical marker where the central line (the single most likely case, as opposed to the median, the middle of many simulated outcomes) crosses zero. That crossing is the break-even point (also called payback: the month when cumulative benefit overtakes cumulative cost). The three sentences carry the numbers an executive can act on: (1) the break-even month as a range, not a single date; (2) the odds of paying back within the window they care about, stated as plain frequencies ("about one future in two pays back within three years"); (3) the downside, meaning how much money is at risk and what we would see early if things are going badly. Then name the one assumption that moves the answer most.

Which numbers go in front of executives, and which stay in the appendix

Keep out of the main page: IRR (internal rate of return, the discount rate at which NPV, net present value, is zero), the shape of the probability distributions (the exact way likelihood is spread across a range of values), and any p-values (statistical test results). They invite a debate about method instead of a decision.

Where the $40k checkpoint comes from: the project costs $1.2M up front and $20k a month to run. To pay back inside five years (60 months) it must net at least $1.2M / 60 = $20k a month, so the benefit must be at least $20k + $20k = $40k a month. That is the lowest steady benefit that can pay back in five years even before allowing for the ramp; with the 6-month ramp, a $40k benefit actually pays back around month 65 (run of the model below). So a benefit running under $40k at month 6 means the five-year case has already failed, which is why that is the stop or rescope line.

Which chart does which job

• Break-even curve (cumulative cash): the hero chart. Annotate the break-even month, the assumptions in a footnote, and the band.
• Fan chart: the same curve with a band for the 10th to 90th percentile. A percentile marks how many simulated outcomes have happened by a given point: P10 is the month by which only the luckiest 1 in 10 futures have paid back, P90 the month by which 9 in 10 have. Label the band "middle 80% of outcomes". In the run below P90 is month 60, the end of the 5-year window; the 9.7% of futures that never pay back inside the window sit beyond it, which is why "8 in 10 land between month 26 and month 60" and "about 1 in 10 does not pay back in five years" are consistent.
• Tornado chart: horizontal bars, one per assumption, ordered by how far the answer swings when that assumption goes to its low and high value. It answers "what should we worry about". With the swings printed below, the finished chart has two bars: a long benefit bar running from "never inside 60 months" to month 21, and a short ramp bar from month 36 to month 43, so benefit sits at the top.
• Waterfall: shows how a gross monthly benefit becomes net (for example $55,000 benefit minus $20,000 running cost equals $35,000 net per month). Use it only if someone asks where the money comes from.

One chart per question; do not put all four on one slide.

Worked example with a real run

Illustrative project: $1,200,000 up front, $20,000 a month to run, benefit per month between $30,000 and $90,000 with $55,000 most likely, and a ramp (months until the full benefit arrives) between 3 and 12 months, most likely 6. A Monte Carlo run repeats the calculation 10,000 times, each time drawing the uncertain inputs at random from those ranges, and counts how often each outcome occurs.

import random, statistics

random.seed(42)
UPFRONT = 1_200_000            # $ cost at month 0
RUN = 20_000                   # $ per month running cost

def draw():
    benefit = random.triangular(30_000, 90_000, 55_000)    # steady $/month benefit (low, high, most likely)
    ramp = random.triangular(3, 12, 6)                      # months to reach steady benefit
    return benefit, ramp

def payback_month(benefit, ramp, horizon=60):
    cum = -UPFRONT
    for m in range(1, horizon + 1):
        cum += benefit * min(m / ramp, 1) - RUN
        if cum >= 0:
            return m
    return None

paybacks = []
for _ in range(10_000):
    b, r = draw()
    paybacks.append(payback_month(b, r))
ranked = sorted(float("inf") if p is None else p for p in paybacks)
def pct(q):
    v = ranked[int(q * (len(ranked) - 1))]
    return "after month 60" if v == float("inf") else int(v)
within24 = sum(1 for p in paybacks if p is not None and p <= 24) / len(paybacks)
within36 = sum(1 for p in paybacks if p is not None and p <= 36) / len(paybacks)
never = sum(1 for p in paybacks if p is None) / len(paybacks)
print("P10 / P50 / P90 payback month:", pct(0.10), pct(0.50), pct(0.90))
print(f"Paid back within 24 months: {within24:.0%}; within 36: {within36:.0%}; not within 60: {never:.1%}")

# One-at-a-time swing for the tornado: payback month when one input is at its low or high, the other at most likely
for name, lo, hi in (("benefit $/month", (30_000, 6), (90_000, 6)), ("ramp months", (55_000, 12), (55_000, 3))):
    print(name, "-> payback month at unfavourable value:", payback_month(*lo), "| at favourable value:", payback_month(*hi))
print("Central case (55k/month, 6-month ramp) payback month:", payback_month(55_000, 6))
print("Checkpoint case (40k/month, 6-month ramp) payback month, horizon 120:", payback_month(40_000, 6, horizon=120))

How the code works, step by step: random.seed(42) fixes the random sequence so anyone re-running gets the same numbers. random.triangular(low, high, most_likely) draws a random value between a low and a high bound, favouring the most likely value (a triangle-shaped spread, which suits expert estimates of a low, a high and a best guess). payback_month walks forward month by month, adding that month's benefit (growing linearly during the ramp) minus running cost to a cumulative total that starts at minus the upfront cost, and returns the first month the total reaches zero, or None if it never does inside 60 months. The 10,000 results are sorted, with never-paid-back treated as infinity so it sorts to the end. P10, P50 and P90 are then read by position: the value 10%, 50% and 90% of the way along the sorted list (P10 = 26 means 10% of futures pay back by month 26).

Output:

P10 / P50 / P90 payback month: 26 37 60
Paid back within 24 months: 6%; within 36: 48%; not within 60: 9.7%
benefit $/month -> payback month at unfavourable value: None | at favourable value: 21
ramp months -> payback month at unfavourable value: 43 | at favourable value: 36
Central case (55k/month, 6-month ramp) payback month: 39
Checkpoint case (40k/month, 6-month ramp) payback month, horizon 120: 65

Reading it: the central case (most likely values) breaks even at month 39. The middle simulated outcome is month 37, slightly earlier, because the benefit range leans upward (average $58,333 against a most likely $55,000). The cautious end (P90) is month 60, and about 10% of simulated futures do not pay back inside five years. The tornado is read straight from the last lines: benefit swings from "never inside 60 months" to month 21, while the ramp only swings from month 36 to month 43. So benefit goes at the top of the tornado and gets the checkpoint.

What I would actually say: "Our best estimate is that this pays for itself in about three years. Across 10,000 simulated futures, roughly half pay back within three years and about one in ten does not pay back in five. What decides it is the monthly benefit, so we propose a review at month 6: if the benefit is running well below plan, we stop before most of the cost is spent."

Trade-offs and pitfalls

• False precision. "Break-even in 38.4 months" invites an argument about the decimal. Round to months and give the range.
• Averages hide the tail. An executive who only sees the average will be surprised by the one-in-ten bad case. Always show the bad case beside it.
• Ranges invented for effect. The ranges must come from data (similar past projects, pilot results) or be labelled as judgement. If the only evidence is a guess, say so on the page.
• Too many views. Lead with the decision they must make, then the picture. The method belongs in the appendix.
• Confidence language. Say "about one in two" rather than "50% confidence interval". Frequencies are read correctly by non-statisticians; interval jargon is not.
• If the executive says "give me one number", give the central case and the date you will know whether it is holding, not a probability.

Direct answer

I build the case in five moves: frame the decision, gather and validate data, model costs and benefits with scenarios, stage the risk, then present differently to the CFO (chief financial officer) and CEO (chief executive officer). The CFO needs defensible numbers and a clear owner for each; the CEO needs the decision, the options and the strategic trade-off. Throughout, the aim is a case that survives challenge, not one that wins on a single optimistic figure.

1. Frame the decision

One sentence: "Should we invest $2M over 12 months in X, versus the status quo and one cheaper alternative?" Name the sponsor, the functions that bear costs (engineering, data, operations, marketing) and the ones that capture benefits. Agree the horizon (3 to 5 years) and the hurdle rate (the minimum return an investment must earn to be worth doing; it is used as the discount rate, the percentage by which future money is shrunk to express it in today's value) with finance before modelling.

2. Data to gather

• Baseline: current revenue, volumes, conversion, cost-to-serve (what it costs to deliver the product to each customer) and failure rates, from the system of record, signed off by finance.
• Costs: quotes, engineering estimates with contingency, run costs, shared overhead, change management, and the opportunity cost of the people.
• Benefit drivers: the specific lever (uplift in conversion, hours saved, churn avoided), with benchmarks, pilots, or expert interviews where no internal data exists.
• Constraints: capacity, dependencies, regulatory points.

3. Model and scenarios (worked example)

A personalisation engine costing $2,000,000 to build and $250,000 a year to run. Online revenue is $400,000,000, contribution margin 30% (the share of each revenue dollar left after the variable costs of making that sale, so extra revenue is worth 30 cents on the dollar to profit). The uncertain driver is revenue uplift. Benefits begin after build; cash at year end, 5-year horizon, 10% discount rate. For simplicity the model below treats the $2,000,000 as spent today and the five years of benefit and run cost as starting in year 1. Sensitivity to that simplification: the proposal says the build takes 12 months, and if benefits and run costs only start in year 2 (four years of benefit), the discount-factor sum is 3.7908 - 0.9091 = 2.8817, so Base NPV falls from +$1,601,247 to +$737,611, Low to -$991,406, and break-even uplift rises from 0.65% to 0.79% ($2,000,000 / 2.8817 + $250,000 = $944,036 a year, divided by $400,000,000 x 0.30 = $120,000,000 of contribution per 100% uplift). The phase-1 pass threshold of 0.8% proposed in section 4 would then barely clear break-even, so I would set the gate higher or confirm the build timeline before fixing it.

How NPV works. Net present value (NPV) is the total of all future cash flows, each shrunk back to today's money, minus the cost paid today. A dollar arriving at the end of year t is worth 1 / 1.10^t dollars today at 10%. The five yearly discount factors are 0.9091, 0.8264, 0.7513, 0.6830 and 0.6209, and they add up to 3.7908. So a steady yearly net benefit is worth (yearly net) x 3.7908 today. Traced for the Base row: $950,000 x 3.7908 = $3,601,247 today, minus the $2,000,000 build = +$1,601,247. Low: $350,000 x 3.7908 - $2,000,000 = -$673,225. High: $1,550,000 x 3.7908 - $2,000,000 = +$3,875,719.

Break-even uplift is the benefit that makes NPV zero. Set (annual contribution - $250,000) x 3.7908 = $2,000,000 and solve: annual contribution = $2,000,000 / 3.7908 + $250,000 = $527,595 + $250,000 = $777,595 a year, which is 0.65% uplift ($777,595 / ($400,000,000 x 0.30)). So the project pays if the uplift exceeds about 0.65%, and the sensitivity is steep: the whole case hinges on one number we have not yet measured.

4. Stage the risk

Instead of approving $2M outright, propose $400,000 for phase 1: a controlled experiment (A/B test, with a holdout group that does not get the feature), with a pre-agreed pass threshold (for example, measured uplift above 0.8%, which clears the 0.65% break-even if benefits start in year 1 but only just clears the 0.79% break-even if they start in year 2, so I would confirm the build timeline before fixing the threshold) before releasing the remaining $1.6M. This converts a $2M bet into a $400,000 option on a $2M bet: like a financial option, you pay a small price now for the right, not the obligation, to make the larger investment later, and that right is worth something because you only exercise it if the evidence is good, so most of the downside never happens. Include KPIs (key performance indicators: measurable targets) with owners and dates, and a measurement plan that finance co-owns so benefits are not self-reported.

5. Presenting the trade-offs

• To the CFO: assumptions page, base/low/high NPV, payback, break-even, the contingency, who owns each benefit line, how realised value will be tracked and reported against the case.
• To the CEO: the decision, three options (do nothing, staged, full commit), the strategic reason (competitors, customer expectation), the capital at risk at each gate, and my recommendation (staged). One page, no model.

Same skeleton for two other shapes

Pitfalls

Presenting only the base case, burying the assumption that drives value, leaving benefits unowned, and asking for the full budget before any evidence exists.

Direct answer

I would show the numbers exactly as they are, including that they are marginal, put a price on whatever can honestly be priced (with a confidence label), score what cannot be priced on a rubric agreed before anyone looked at this project, and put the two side by side in one brief. My recommendation would be to fund a bounded first phase with explicit gates, not the whole amount up front, because a marginal case plus strategic value is exactly the situation where buying an option is better than committing in full. Here an option means paying a small amount now for the right, not the obligation, to commit the rest later once early evidence arrives.

Step 1: present the quantitative case honestly

Illustrative project: invest $1,200,000 now, benefits of $450,000 a year for 3 years, required return (discount rate) 8%. Discounting converts future dollars to today's dollars because money received later is worth less.

$\text{NPV} = -1{,}200{,}000 + 450{,}000 \times (0.9259 + 0.8573 + 0.7938) = -1{,}200{,}000 + 450{,}000 \times 2.5771 = -\$40{,}300$

Using exact discount factors, the NPV is -$40.3k (about -3% of the investment) and the IRR (the discount rate at which NPV is exactly zero) is 6.1%, below the 8% requirement. I would not hide that or inflate benefits to turn it positive. The three discount factors in brackets add to 2.5771, the annuity factor: the present value of $1 received at the end of each of the next 3 years at 8%. Multiplying a level yearly benefit by it is a shortcut for discounting each year separately.

Step 2: convert the "faster decisions, time saved, lower error rates" benefits with stated conversion factors

The $1,000 a day is an assumption, not a measurement, and it is worth being clear about why. The pure time value of a deal's margin is tiny (a $90,000 margin at 10% a year is about $25 a day), so $1,000 a day only holds if slower decisions also lose some deals or tie up sales capacity. That is exactly why this line is marked low confidence.

That is the $450,000. The "faster decisions" line is the softest, so I show it separately and rerun the NPV without it: it becomes -$427k, which tells the reader how much of the case depends on one low-confidence conversion factor.

Step 3: price the unpriced value as a break-even, not a guess

Instead of arguing about what strategic value is worth, ask the reverse: how much unpriced value per year would make the project break even? The NPV gap is $40.3k and the 3-year annuity factor is 2.5771, so the unpriced benefit has to be worth only $40.3k / 2.5771 = about $15.6k a year (roughly 3.5% of the priced benefit). A sponsor can judge whether strategic fit, reduced vendor dependence or capability is worth $15.6k a year far more easily than they can judge "strategic value".

Step 4: score the rest on a weighted rubric, shown beside the money

Agree the criteria and weights first, then score. The weights add to 100%.

Rule agreed in advance for this brief: a project whose NPV is within about 5% of the investment either side of zero may proceed on strategic grounds only if the weighted score is at least 3.5 and the break-even unpriced value is plausible. This project scores 3.65 and its break-even is $15.6k a year, so it clears both tests. Weights need a stress test. Swapping the risk and optionality weights (20% and 25%) gives 3.70, still a pass. Swapping the strategy and customer-impact weights (15% and 30%) gives 3.20 and fails, so this project's pass depends on the strategic-fit weight being legitimate. That is the point to put to the committee, and it is why the weights are fixed before scoring.

Step 5: recommendation and the decision brief

One page, in this order: (1) the recommendation and the exact ask; (2) the quantitative table (NPV -$40.3k, IRR 6.1%, undiscounted payback 1,200 / 450 = 2.7 years); (3) the break-even unpriced value, $15.6k a year; (4) the weighted score and who scored it; (5) what is not priced; (6) the gates.

Recommendation: approve Phase 1 only, about one third of the budget ($400,000, 4 months), with these gates before releasing the rest: measured hours saved annualise to at least 2,000, rework cases fall at least 25%, and the strategic sponsor confirms in writing the specific capability it unlocks. This turns a marginal bet into a cheap option on a strategic one: the downside is capped at $400,000, and the upside is the full programme if the early measurements land.

Trade-offs and pitfalls

• A scoring rubric can launder a bad project (make a weak one look respectable by dressing it in scores). It ranks projects against one another; it does not make a negative NPV positive. Hence the pre-agreed pass mark and the break-even.
• Never present the weighted score as a dollar figure or add it to the NPV. Two different units on one line is how a brief loses Finance's trust.
• What would change my call: if the break-even unpriced value were large (say above $200,000 a year), if the score fell below 3.5, or if capital is rationed (a fixed budget that cannot fund every worthwhile project) and other projects have clearly positive NPV, I would decline or defer.

Direct answer

Treat the benefit as a curve, not a number. Estimate the full run-rate benefit (what the improvement is worth per month once everyone is using it), multiply it each month by an adoption share that climbs from 0% toward its ceiling, and charge the costs on their real dates, which usually start before any benefit does. Then justify the curve with evidence (pilot data, rollout schedule, comparable launches), show how NPV and payback move when the curve is slower, and track the leading indicators so the ramp can be checked after approval.

Terms used below. Run-rate means the steady monthly amount once everything is fully in use. Loaded hourly cost is what an hour of an employee really costs the company (pay plus benefits, taxes and overhead share). Leading indicators are early measures (for example weekly active users) that move before the money does. NPV (net present value) is the sum of each future cash flow divided by (1 + rate) raised to the number of periods until it arrives, so later money counts for less. The discount rate is that rate (the return you could earn elsewhere, set by finance). IRR (internal rate of return) is the discount rate at which NPV is exactly zero. Payback is the first period in which cumulative cash turns non-negative. A monthly series needs a monthly rate: use (1.10)^(1/12) - 1, not 10% / 12, so twelve months of compounding gives exactly the annual 10%.

How to build the ramp

1. Full run-rate benefit. Hours saved per person x people x loaded hourly cost x the share of saved time that becomes real cash or real extra output. Freed time is only a cash saving if the capacity is redeployed to other paid work, backfilled less (a departing person is replaced by fewer new hires), or sold; say which.
2. Adoption curve = rollout x usage x proficiency. Rollout is the share of teams that have the tool (from the deployment schedule). Usage is the share of those people who use it in a given month. Proficiency is how much of the full gain a typical user gets while still learning (a new user may get half the time saving that an expert gets). For example, 80% rollout x 70% usage x 60% proficiency is 33.6% of the full benefit. Multiply them; each has its own evidence.
3. Anchor points, then interpolate. Pick a few anchors you can defend (end of year 1: 30%, year 2: 65%, year 3: 90%) and interpolate monthly (fill the months in between with evenly spaced steps from one anchor to the next). Anchors are easier for a reviewer to challenge than a formula. An S-shaped curve (slow start, fast middle, flattening) is usually more realistic than a straight line, but a straight line between honest anchors is fine.
4. Costs on their own dates. Upfront spend lands in month 0 and run costs begin immediately; neither waits for adoption.
5. Justify the shape. In order of strength: your own pilot cohort (monthly usage of the first teams), earlier rollouts at your company, then outside benchmarks. Cap the ceiling below 100% (here 90%) because not every role gets full value. State what would make it faster or slower (training, executive mandate, integration with existing tools).

Worked example (illustrative inputs)

There are two examples. Part A is the main one: a monthly 36-month model of a tool whose benefit ramps up by anchors. Part B is a second, simpler yearly example that tests how sensitive a savings ramp is to slower adoption; it uses year-end cash flows, not monthly ones. Hand check for Part A: month 1 adoption is 30% x 1/12 = 2.5%, so benefit is $80,000 x 2.5% = $2,000, less $10,000 to run = -$8,000. Each month is then discounted by (1 + 0.0080)^month, where 0.0080 is the monthly rate (1.10^(1/12) - 1 = 0.797%); the headline NPV is the sum of all 36 of these plus the -$600,000 at month 0.

A tool is worth $80,000 per month at full run-rate, costs $600,000 upfront and $10,000 per month to run, at a 10% annual rate over 36 months.

# Part A: 36 monthly cash flows for a productivity gain that ramps up
full_monthly = 80_000.0   # $/month once every team has adopted
upfront = 600_000.0       # $ spent in month 0
run_cost = 10_000.0       # $/month licences and support
annual_rate = 0.10
r_m = (1 + annual_rate) ** (1 / 12) - 1     # equivalent monthly rate
anchors = {0: 0.0, 12: 0.30, 24: 0.65, 36: 0.90}   # share of full benefit reached

def adoption(m):
    ks = sorted(anchors)
    for a, b in zip(ks, ks[1:]):
        if a <= m <= b:
            return anchors[a] + (anchors[b] - anchors[a]) * (m - a) / (b - a)

def npv(flows, rate):           # flows[0] happens now (t = 0)
    return sum(cf / (1 + rate) ** t for t, cf in enumerate(flows))

def payback_index(flows):       # first period where cumulative cash >= 0
    cum = 0.0
    for t, cf in enumerate(flows):
        cum += cf
        if cum >= 0:
            return t
    return None

ramp = [-upfront] + [full_monthly * adoption(m) - run_cost for m in range(1, 37)]
flat = [-upfront] + [full_monthly * 0.90 - run_cost for m in range(1, 37)]  # day-one 90%
for m in (1, 6, 12, 18, 24, 30, 36):
    print(f"month {m:2d}: adoption {adoption(m):5.1%}  net cash ${ramp[m]:>9,.0f}")
print(f"ramped:   NPV ${npv(ramp, r_m):,.0f}, payback month {payback_index(ramp)}")
print(f"day-one:  NPV ${npv(flat, r_m):,.0f}, payback month {payback_index(flat)}")

# Part B (second example, yearly): five-year linear savings ramp, 10% to 40% of an addressable cost base
# (addressable cost base = the annual cost the tool can actually reduce)
def irr(flows):
    lo, hi = -0.9, 10.0
    for _ in range(200):
        mid = (lo + hi) / 2
        if npv(flows, mid) > 0:
            lo = mid
        else:
            hi = mid
    return mid

base_cost, invest, run_yr, rate = 1_000_000.0, 450_000.0, 50_000.0, 0.10
def case(stretch):                 # stretch 1.0 = plan, 0.8 = 20% slower adoption
    flows = [-invest]
    for t in range(1, 6):
        # savings share rises 7.5 points a year from a 2.5% start: 10%, 17.5%, 25%, ... 40% at stretch 1
        pct = 0.025 + 0.075 * (stretch * t)     # 10% at t=1, 40% at t=5 when stretch=1
        flows.append(base_cost * pct - run_yr)
    return flows
for name, s in (("plan", 1.0), ("20% slower", 0.8)):
    f = case(s)
    # break-even = first year cumulative cash >= 0; discounted break-even = same, but each year's cash
    # is first discounted to today's money
    cum, disc, be, dbe = 0.0, 0.0, None, None
    for t, cf in enumerate(f):
        cum += cf
        disc += cf / (1 + rate) ** t
        if be is None and cum >= 0: be = t
        if dbe is None and disc >= 0: dbe = t
    print(name, [round(x) for x in f])
    print(f"  NPV ${npv(f, rate):,.0f}  IRR {irr(f):.1%}  break-even year {be}  discounted break-even year {dbe}")

Output:

month  1: adoption  2.5%  net cash $   -8,000
month  6: adoption 15.0%  net cash $    2,000
month 12: adoption 30.0%  net cash $   14,000
month 18: adoption 47.5%  net cash $   28,000
month 24: adoption 65.0%  net cash $   42,000
month 30: adoption 77.5%  net cash $   52,000
month 36: adoption 90.0%  net cash $   62,000
ramped:   NPV $228,345, payback month 29
day-one:  NPV $1,333,560, payback month 10
plan [-450000, 50000, 125000, 200000, 275000, 350000]
  NPV $254,174  IRR 25.0%  break-even year 4  discounted break-even year 4
20% slower [-450000, 35000, 95000, 155000, 215000, 275000]
  NPV $94,386  IRR 16.0%  break-even year 4  discounted break-even year 5

Reading it: adoption is only 30% at month 12, so net cash in month 1 is negative and turns positive around month 6. The ramped case gives NPV $228,345 and payback in month 29. A model that assumed 90% of the benefit from month 1 would show NPV $1,333,560 and payback in month 10, overstating NPV by about $1.1M and understating the payback time by 19 months. The ramp, not the headline benefit, is what the approval decision turns on.

Five-year linear ramp. For a savings rate that climbs linearly from 10% of an addressable cost base in year 1 to 40% in year 5 (base $1,000,000 a year, $450,000 upfront, $50,000 a year to run, 10% rate, cash at year end), the savings share is 10%, 17.5%, 25%, 32.5% and 40% of $1,000,000 (the formula 0.025 + 0.075 x t, which is 10% at t = 1 and 40% at t = 5), less $50,000 a year to run, so the cash flows are -$450,000, then $50,000, $125,000, $200,000, $275,000, $350,000. Discounted at 10% these are -$450,000, $45,455, $103,306, $150,263, $187,829 and $217,322, which add to NPV $254,174, IRR is 25.0%, and cumulative cash turns positive in year 4 (-$75,000 at the end of year 3, +$200,000 at the end of year 4). If adoption is 20% slower (the ramp reaches in 5 years only what the plan reaches in 4; the savings share is 8.5%, 14.5%, 20.5%, 26.5% and 32.5%), the savings rates are lower in every year, NPV falls to $94,386 (about 63% lower), IRR to 16.0%, and the discounted break-even moves from year 4 to year 5. The case still clears a 10% rate but with much less cushion, and adoption pace is the assumption it is most sensitive to because the costs are fixed.

Trade-offs and pitfalls

• Ramp the benefit but not the costs: costs start on their real dates, so the early months are negative. Also model the learning dip (productivity can fall slightly during onboarding); a benefit curve that rises smoothly from month 1 and ignores that dip flatters the first year.
• An S-shaped curve is one that starts slowly, speeds up in the middle and flattens near the ceiling. Do not let the ceiling creep: the curve's end point matters as much as its slope.
• Present base, slower and faster ramps together, and name the leading indicator (monthly active users, share of work done in the tool) that tells you in month 6 which one you are on.
• Dividing the annual rate by 12 gives a slightly different (wrong) monthly rate; use the equivalent monthly rate.