Industry: Construction and Infrastructure
Product: ModelRisk
Application: Urban Planning — Population, Infrastructure, Climate

A 40% population-growth forecast with a 50% range — the policy that fails for one fails for the other

A metropolitan planning authority projected 40% population growth over a 20-year horizon. The central forecast — 2.8M people growing to 3.9M — drove a $4.2B capital plan for water, transit and housing. The Monte Carlo on the same migration / fertility / mortality assumptions said the P10 was 3.4M and the P90 was 4.6M — a 1.2M range. Sized for the central case, the plan over-builds by 18% if the P10 realises and under-builds by 19% if the P90 realises. The cost of over-building is a stranded asset; the cost of under-building is a city in crisis. Neither is symmetric, and neither shows up in the deterministic line.

ModelRisk was used to build a probabilistic capacity model that funded infrastructure on a 90th-percentile envelope rather than the 50th-percentile point.

A population model with compounding uncertainty

Population at year t was modelled as the multiplicative product of independent annual growth rates:

\[ P_t = P_0 \cdot \prod_{i=1}^{t}(1 + r_i) \]

with r_i drawn from:

• Natural growth (births - deaths) — Normal(μ = 0.55%/yr, σ = 0.18 pp), correlated with itself year-on-year at ρ = 0.6 via an AR(1) on the rate (demographic momentum).
• Net migration rate — Lognormal-on-deviation; nominal +1.4%/yr, σ_log 0.45 on the migration component. Lognormal because migration shocks are asymmetric — a city can attract 2× its baseline migration in a boom, but cannot lose more than 100% (and net negative migration is a thin left tail).
• Economic-cycle modifier — discrete: 6% chance per year of recession year (net migration -1.2%, applied as multiplier), with a Markov "stickiness" so recessions cluster.

After 20 years of compounded multiplicative noise, the P10–P90 envelope on population was 3.4M–4.6M against a deterministic central of 3.9M — a 29% spread that is purely a function of compounding, not exotic assumptions.

The fan chart shows the forecast cone widening with time, as it should — the Year-5 P10/P90 spread is 13% of central; the Year-20 spread is 29%, with the median tracking the deterministic 3.9M plan line almost exactly while the bands fan out around it. Any policy framed against the central line is implicitly betting that 20 years of compounding error will cancel out, which they statistically will not.

Infrastructure capacity sized for which percentile?

Housing demand was modelled as P(t) / household-size, with household size itself a slowly-varying distribution (Beta on the 2.1–2.7 range, mean 2.4 declining 0.005/yr). Transit ridership was modelled as a function of population times trip-rate-per-capita (Triangular(3.1, 3.6, 4.2)) with a modal-share Beta on transit penetration. Water demand was modelled as gallons-per-capita-per-day (Triangular(110, 130, 165)) times population, with a climate-driven multiplier (warming summers, +0.4%/yr drift, σ 0.5%).

The aggregated capacity demands were:

• Year-20 housing units — P50 1.65M, P90 1.92M.
• Year-20 peak transit ridership — P50 4.8M trips/day, P90 5.7M trips/day.
• Peak water demand (over the 20-year horizon) — P50 673 MGD, P90 802 MGD.

Water demand is not uniform across the city. Projecting the central path by district and review year shows where the load concentrates and how the fast-growing greenfield and north-mesa districts pull the Year-20 total toward 660 MGD:

The deterministic plan sized water at 660 MGD. The Monte Carlo said there was a ~50% probability of exceeding it at some point in the 20-year horizon (a single high-growth year combined with a hot summer was enough). Reframing capacity as a confidence question, the sweep below plots P(no rationing over the horizon) against capacity investment above the $4.2B base plan: hitting 90% confidence required roughly $450M of additional capacity (sizing to the P90 of peak demand, ~802 MGD) — eliminating the rationing risk the deterministic plan was silently accepting.

Flood losses: where the deterministic mean is a dangerous number

Climate-driven flood loss was modelled as a compound Poisson-Lognormal:

• Annual flood-event count — Poisson, with mean rising from 0.18/yr today to 0.34/yr by Year 20 (climate-projection central) ± 30% σ on the rate path.
• Per-event loss — Lognormal in $M, median $42M, σ_log 1.1. The lognormal mean is exp(ln 42 + 1.1²/2) = $77M; the 99th percentile is $543M.

Over the 20-year horizon, the aggregate flood-loss distribution showed:

• Mean $420M.
• Median $346M.
• P90 $818M.
• P99 $1.59B.

The deterministic central — 5.2 events × $76M = $395M over 20 years — sat at the P57 of the simulation. A flood-defence investment of $310M was estimated to halve the per-event loss distribution; the Monte Carlo showed this cut P90 from $818M to $714M (the defended-plus-spend tail), a $104M tail reduction net of the $310M investment.

Three policy paths

The authority compared three capital programmes:

• Central — sized to deterministic P50 population, $4.2B capital.
• Robust — sized to P75 population, $5.0B capital (more units / capacity, less stranded-asset risk if low growth realises).
• Modular — initial build to P50 + pre-engineered expansion options that can be triggered at Year-10 if growth tracks above central, $4.5B fixed + $0.9B optional.

The Modular policy dominated on expected NPV-per-resident and had the narrowest distribution — option value is real, and Monte Carlo on the trigger condition (an explicit option-pricing exercise) gave the council the cost of buying that flexibility against the cost of committing now. The deterministic plan would have ranked Robust as wasteful; the probabilistic plan showed Modular as cheapest in expectation and lowest-variance — by a meaningful margin.

What the model changed

The authority adopted the Modular plan, sized the initial water and transit capacity to the P60 of the population fan, and committed $310M to flood defences ahead of the central climate projection. A Year-10 review trigger was contracted to re-run the simulation with the realised 10 years of data and re-decide the expansion options. Three years into the plan, observed population growth tracked the P55 — the option to expand is being held open, not yet exercised, at a sunk cost of ~$30M for the pre-engineered designs.

ModelRisk Functionality Used

• AR(1) on annual growth rates captured the demographic momentum that independent annual draws would have under-stated — turning a 31% Year-20 spread that is real into a 22% spread that is wrong.
• Lognormal migration component modelled the asymmetric upside risk of boom-driven inflows that a Normal would have truncated.
• Markov recession process linked economic-cycle years into clusters rather than scattering them, matching the historical pattern of multi-year downturns.
• Compound Poisson-Lognormal flood model with a 20-year rising-rate path showed the deterministic flood-cost central sat at the P57 — above the simulated median ($346M) yet still understating the simulated mean expected loss ($420M) by about 6%.
• Capacity-sizing percentile choice — making the P10 / P50 / P90 envelope explicit let the council debate "which percentile do we fund?" rather than treating the central forecast as truth.
• Modular vs committed comparison quantified $1.4B of NPV value in flexibility — option value that is invisible to deterministic NPV.

For long-horizon planning, the central forecast is the worst possible number to fund against. Monte Carlo turns "what will the population be?" into "what is the distribution of populations, and which percentile do we want to be ready for?" — which is the only honest question a planning authority can answer.