Scenario and Sensitivity Analysis Questions

Approach: run 10,000 Monte Carlo trials sampling cost and lift per trial, compute monthly incremental revenue from conversion lift, annualize (or choose horizon), compute ROI = (benefit - cost) / cost, collect ROI samples and report 90% credible interval (5th and 95th percentiles).

import numpy as np

N = 10_000
np.random.seed(0)

# Inputs / constants
cost_mu, cost_sigma = 50_000, 10_000
lift_mu, lift_sigma = 0.005, 0.002
baseline_traffic = 200_000
aov = 40.0
baseline_conv = 0.02
horizon_months = 12  # assume 12-month benefit horizon

# Sample
cost_samples = np.random.normal(cost_mu, cost_sigma, size=N)
lift_samples = np.random.normal(lift_mu, lift_sigma, size=N)

# If negatives are impossible, truncate
cost_samples = np.clip(cost_samples, 0, None)
lift_samples = np.clip(lift_samples, 0, None)

# Compute monthly incremental conversions and revenue
incremental_conversions_monthly = baseline_traffic * lift_samples
incremental_revenue_monthly = incremental_conversions_monthly * aov

# Total benefit over horizon
total_benefit = incremental_revenue_monthly * horizon_months

# ROI per trial
roi_samples = (total_benefit - cost_samples) / cost_samples

# Summary stats
roi_mean = roi_samples.mean()
ci_lower, ci_upper = np.percentile(roi_samples, [5, 95])

Key points:

• Assumes lift is absolute increase in conversion rate and benefits accrue monthly for chosen horizon; adjust horizon as business-appropriate.
• Truncate negatives (or sample from truncated normal) to avoid impossible values.
• Report distribution (histogram, percentiles) and credible interval [5th,95th].
• Consider correlations (e.g., higher cost correlating with higher lift) by sampling from a multivariate normal if justified.
• Sensitivity: run with different horizons, include churn, discounting (NPV) for longer horizons.

Approach: simulate 10,000 project paths for 5 years. Each year draw benefits ~ Normal(200k,50k), subtract fixed cost 80k to get net cashflow, discount at 10% per year, sum to get NPV. Aggregate NPVs and report descriptive and risk metrics.

import numpy as np

def simulate_npv(n_runs=10000, years=5, mu=200_000, sigma=50_000, cost=80_000, r=0.10, seed=42):
    rng = np.random.default_rng(seed)
    # simulate benefits: shape (n_runs, years)
    benefits = rng.normal(loc=mu, scale=sigma, size=(n_runs, years))
    # annual net cashflows = benefits - cost
    cashflows = benefits - cost
    # discount factors for years 1..years
    discounts = 1 / ((1 + r) ** np.arange(1, years + 1))
    # discounted NPVs per run
    npvs = (cashflows * discounts).sum(axis=1)
    return npvs

# run
npvs = simulate_npv()

# sample output metrics
mean_npv = np.mean(npvs)
median_npv = np.median(npvs)
std_npv = np.std(npvs, ddof=1)
p10, p90 = np.percentile(npvs, [10, 90])
prob_pos = np.mean(npvs > 0)

print(f"Mean NPV: ${mean_npv:,.0f}")
print(f"Median NPV: ${median_npv:,.0f}")
print(f"Std: ${std_npv:,.0f}")
print(f"10th/90th percentiles: ${p10:,.0f} / ${p90:,.0f}")
print(f"P(NPV>0): {prob_pos:.2%}")

Key concepts:

• Use vectorized simulation for speed; complexity O(n_runs * years).
• Ensure random seed for reproducibility.

Edge cases:

• Negative sampled benefits (allowed if realistic); if benefits can't be negative, use truncated Normal or lognormal.
• Validate n_runs>0 and reasonable sigma.

Alternative improvements:

• Latin Hypercube sampling for variance reduction.
• Model correlation across years or non-normal benefit distributions.
• Sensitivity analysis on mu, sigma, cost, and discount rate.

Output metrics to include in business case:

• Mean, median, standard deviation of NPV
• Percentiles (5th, 10th, 25th, 75th, 90th, 95th)
• Probability NPV > 0 (breakeven probability)
• Value at Risk (e.g., 5% VaR) and Conditional VaR (CVaR)
• Histogram / density plot and cumulative probability curve
• Tornado/sensitivity chart showing drivers (mu, sigma, cost, discount)
• Expected payback time distribution (if needed)

Situation & objective: The procurement team needs a sensitivity-based vendor selection and contract plan where SLAs directly affect payments. As a data scientist I’d produce a probabilistic financial model that quantifies expected costs, upside rebates, and downside penalties under realistic performance distributions and stress scenarios.

Approach (steps):

1. Define metrics & contract rules — clear SLAs (uptime, latency, delivery CT), thresholds, stepwise or continuous payment adjustments (e.g., 1% rebate per 0.1% above 99.9% uptime; 2% penalty per 0.5% below 99.5%).
2. Data & parameterization — gather historical vendor performance, industry benchmarks; fit distributions (beta for rates, lognormal for delays) and estimate correlations across metrics/vendors.
3. Build probabilistic forecast — Monte Carlo simulation (100k runs) that samples performance metrics, computes per-run payment adjustments via contract logic, and produces distributions of total cost, rebates paid, penalties incurred, and cashflow timing.
4. Sensitivity analysis — compute partial derivatives or use Tornado charts to identify which metrics and vendors drive variance/expected loss. Run scenario sweeps for key parameters (mean performance shifts, increased variance).
5. Stress-testing worst-case — define severity scenarios (e.g., correlated failure across vendors, sustained degradation for N months). Use tail-risk metrics: 95/99 percentile cost, Expected Shortfall (CVaR), and simulate cumulative liquidity impact.
6. Integration & outputs — produce dashboard with expected cost, distribution, V@R/CVaR, contribution to risk by vendor/metric; produce recommended SLA/payment structures that align incentives (caps, collars, minimum & maximum payments, clawbacks).
   Modeling notes:

• Encode payment rule algebraically so simulation applies piecewise linear/threshold logic deterministically per sample.
• Account for recovery/lags and dispute probabilities (reduce realized penalties by dispute rate).
• Validate with backtesting on historical incidents.
  Decision use: choose vendor(s) with acceptable expected cost and tail-risk; negotiate SLA shapes that minimize CVaR given business tolerance; include triggers for contingency sourcing.

Approach (brief): build param distributions for adoption rate, per-user revenue (ARPU), and retention; model their dependence via a copula (Gaussian or t) to generate correlated draws; compute 5-year cashflows per draw, discount to NPV; analyze joint effects with conditional slices, sensitivity indices, and scenario clustering.

Modeling correlations:

• Fit marginals from data or use informed parametric forms (Beta for adoption/retention bounded [0,1], Lognormal for ARPU).
• Estimate rank correlations (Spearman) from historical or expert-elicited pairwise estimates.
• Use a Gaussian copula to impose those rank correlations: convert marginal CDFs → standard normals, apply multivariate normal with target correlation R, then invert.

Monte Carlo (Python sketch):

import numpy as np
from scipy.stats import norm, beta, lognorm, multivariate_normal

N = 200_000
# Marginal params (example)
a_adopt, b_adopt = 2, 8   # Beta
s_ret, a_ret = 2, 5       # Beta params
mu_arpu, sigma_arpu = 3.0, 0.4  # lognormal
# target Spearman -> convert to Pearson for Gaussian copula via approximation
R = np.array([[1, 0.6, 0.4],
              [0.6, 1, 0.5],
              [0.4, 0.5, 1]])
Z = multivariate_normal.rvs(mean=np.zeros(3), cov=R, size=N)
U = norm.cdf(Z)  # uniform(0,1) with correlation
adopt = beta.ppf(U[:,0], a_adopt, b_adopt)
retention = beta.ppf(U[:,1], s_ret, a_ret)
arpu = lognorm.ppf(U[:,2], s=sigma_arpu, scale=np.exp(mu_arpu))
# compute 5-year NPV per draw (example revenue model)
discount = lambda t, r: 1/(1+r)**t
rate = 0.12
npv = []
for i in range(N):
    users = initial_market_size * adopt[i] * (retention[i] ** np.arange(5))
    cashflows = users * arpu[i]
    npv.append(np.sum(cashflows * discount(np.arange(1,6), rate)))
npv = np.array(npv)

Summarize joint effects:

• Report mean, median, 5/95 percentiles, downside VaR (e.g., 5th percentile).
• Tornado-style conditional sensitivity: fix one variable at percentiles and re-run to see NPV shifts.
• Use partial rank correlation coefficients (PRCC) to quantify monotonic joint influence while preserving correlation structure.
• Compute Shapley value or Sobol-like contributions via variance decomposition on the correlated sample (or use conditional variance decomposition).
• Visuals: joint density contour plots (e.g., adoption vs retention colored by NPV), scatter hexplots, and cumulative distribution vs scenarios.
• Highlight tail dependencies (if heavy tails, prefer t-copula) and perform robustness checks (different copulas, sample sizes).

Key reasoning:

• Copula separates marginals and dependence so you respect bounds and shapes.
• Correlated draws ensure joint scenarios (e.g., high adoption often with higher ARPU) are realistic.
• Use both global (variance/PRCC/Shapley) and local (conditional slices, scenario stress tests) summaries to inform decision-makers about expected value and tail risk.

Approach: simulate monthly cash flows for each Monte Carlo draw. Model adoption as a Beta(a,b) prior → sample final adoption probability p_adopt, convert to monthly adoption path (diffusion or constant). Revenue per adopter per month drawn from a distribution (e.g., LogNormal to ensure >0). Account for staged costs: Year0 immediate $100k, Year1 conditional $50k if adoption > threshold or if any adopters by month 12, Year2 $20k at month 24. Discount monthly at annual rate r (convert to monthly). Repeat N simulations to compute NPV and estimate P(NPV>0).

Pseudocode + example Python sketch:

import numpy as np

def simulate_one(a,b,rev_mu,rev_sigma,annual_r):
    months = 36
    monthly_r = (1+annual_r)**(1/12)-1
    # sample adoption probability
    p_adopt = np.random.beta(a,b)
    # model cumulative adoption fraction over 12 months (e.g., linear ramp) then plateau
    adop_frac = np.minimum(1, np.linspace(0,1,12)*p_adopt)
    adop_frac = np.concatenate([adop_frac, np.full(months-12, p_adopt)])
    # revenue per adopter per month
    rev_per_user = np.random.lognormal(mean=rev_mu, sigma=rev_sigma)
    monthly_revs = adop_frac * rev_per_user * expected_user_base  # set user base
    # costs: t0 at month0, t1 at month12 if adopters > 0, t2 at month24
    cash = -100_000.0  # month 0
    if adop_frac[11] > 0:
        cash -= 50_000.0 / ((1+monthly_r)**12)  # discount to month0
    cash -= 20_000.0 / ((1+monthly_r)**24)
    # add discounted revenues
    for m in range(months):
        cash += monthly_revs[m] / ((1+monthly_r)**(m+1))
    return cash

# run simulation
npv_samples = [simulate_one(a,b,mu,sigma,r) for _ in range(N)]
p_positive = np.mean(np.array(npv_samples) > 0)

Key choices & rationale:

• Beta(a,b) for adoption probability: flexible, conjugate to Bernoulli observations, encodes prior/confidence.
• LogNormal revenue per user: positive, right-skewed to reflect high-value outliers.
• Monthly discounting: convert annual rate to monthly factor for precise timing.
• Staged cost logic: Year1 cost contingent on adoption presence; model as conditional payment.

Outputs to present:

• P(NPV>0) with 95% CI
• Histogram and KDE of NPV samples
• Expected NPV, median, percentiles (5th, 25th, 75th, 95th)
• Tornado/importance plot showing sensitivity to a,b, rev_mu, user_base, discount rate
• Scenario tables: conservative/base/optimistic parameter sets
• Recommendation: go/no-go thresholds and expected payback months

Edge cases:

• Correlate revenue and adoption if appropriate
• Model monthly churn or retention if relevant
• Increase N for stable tail estimates (e.g., 100k+ draws)