Business Metrics and Unit Economics Questions

Evaluate a candidates ability to analyze the financial drivers and per customer economics that determine business sustainability and growth. Core concepts include revenue streams and pricing, gross margin, contribution margin, operating margin, customer acquisition cost, lifetime value per customer, lifetime value to customer acquisition cost ratio, payback period, average revenue per user, churn and retention rates, and metrics for subscription or recurring revenue models such as annual recurring revenue, monthly recurring revenue, expansion revenue, and contraction effects. Candidates should be able to perform back of the envelope calculations and sensitivity analysis, interpret trade offs between growth and profitability, link marketing product and channel activities to financial outcomes, explain how metrics vary by customer segment or acquisition channel, and make strategic recommendations such as pricing adjustments, segmentation strategies, acquisition channel shifts, or investment versus efficiency decisions. Interviewers may request simple calculations, scenario analysis, and prioritized actions grounded in metric changes.

EasyTechnical

91 practiced

Sensitivity question: If ARPU increases 10% from $50 to $55 but monthly churn increases from 4% to 5% at the same time, compute original and new simple LTV (LTV = ARPU / churn) and state which effect dominates and why. Discuss implications for deciding whether to roll out a product change that raises price but increases churn slightly.

MediumTechnical

59 practiced

Write Python-style pseudocode for a Monte Carlo sensitivity analysis of LTV using 1,000 simulations. Inputs: ARPU ~ Normal(mean=50, sd=5), monthly churn probability ~ Normal(mean=0.04, sd=0.01) truncated to (0,1), monthly discount rate = 0.5%. For each simulation, compute discounted LTV as sum_{t=1..60} ARPU * survival_prob(t) / (1+rate)^t assuming constant ARPU and churn. Describe assumptions and key outputs you'd present to stakeholders.

Sample Answer

Approach: sample ARPU and monthly churn from given distributions 1,000 times, compute monthly survival = (1 - churn)^(t-1), then discounted LTV = sum_{t=1..60} ARPU * survival / (1+rate)^t. Aggregate results and report distribution, uncertainty, and sensitivity.

python

import numpy as np

N_SIM = 1000
HORIZON = 60
rate = 0.005  # monthly discount rate

np.random.seed(0)

def truncated_normal(mean, sd, low, high, size):
    x = np.random.normal(mean, sd, size)
    return np.clip(x, low, high)

# draw samples
arpu_samples = np.random.normal(50, 5, N_SIM)
arpu_samples = np.clip(arpu_samples, 0, None)  # ARPU cannot be negative

churn_samples = truncated_normal(0.04, 0.01, 0.0001, 0.9999, N_SIM)

ltv_results = np.zeros(N_SIM)

for i in range(N_SIM):
    arpu = arpu_samples[i]
    churn = churn_samples[i]
    surv = 1.0
    ltv = 0.0
    for t in range(1, HORIZON+1):
        # survival probability that customer is active in month t
        surv = (1 - churn)**(t - 1)
        discounted = arpu * surv / ((1 + rate)**t)
        ltv += discounted
    ltv_results[i] = ltv

# summary stats
mean_ltv = np.mean(ltv_results)
median_ltv = np.median(ltv_results)
p5, p25, p75, p95 = np.percentile(ltv_results, [5,25,75,95])

Key concepts & assumptions:- ARPU and churn drawn independently; ARPU truncated at 0 to avoid negatives.- Constant monthly churn and ARPU over 60 months (no cohort effects, no promotions).- Survival prob at month t = (1 - churn)^(t-1).- Discounting uses fixed monthly rate 0.5%.

Key outputs for stakeholders:- Distribution visualization: histogram + kernel density of LTV.- Summary: mean, median, 5/25/75/95 percentiles and probability LTV > target.- Sensitivity: correlation or partial-rank correlation of inputs (ARPU, churn) with LTV; show tornado plot (churn typically largest driver).- Scenario table: conservative (p95 churn, p5 ARPU), base, optimistic.- Recommendations: expected LTV with confidence intervals and which levers (reduce churn vs increase ARPU) move LTV most.

HardTechnical

56 practiced

Design an LTV model using survival analysis to estimate time-to-churn. Explain how you'd choose between non-parametric (Kaplan-Meier), semi-parametric (Cox), and parametric models; how you'd incorporate time-varying covariates; how to handle left truncation and right-censoring; how to discount future cash flows; and how to compute confidence intervals for LTV estimates.

Sample Answer

High-level approach: treat LTV as expected discounted future revenue until churn -> integrate survival function S(t) with expected revenue rate r(t): LTV = ∫_0^T r(t) S(t) e^{-δ t} dt (discrete sum if periodic).

Choosing model:- Kaplan–Meier (non‑parametric): use for exploratory analysis, estimating S(t) by cohort when covariates are few/nonexistent or proportional hazards assumptions suspect. No covariates, simple and robust but poor extrapolation beyond observed window.- Cox PH (semi‑parametric): good when you need covariate effects (age, acquisition channel, engagement) and proportional hazards roughly holds. Provides hazard ratios, handles right‑censoring naturally, but baseline hazard is unspecified — need baseline survival estimation for LTV extrapolation.- Parametric survival (Weibull, Exponential, Log‑logistic, Gompertz): choose when you need long‑term extrapolation and parametric shape is plausible. Fit several families (AIC/BIC, visual fit) and use bootstrapping for uncertainty; parametric models give closed‑form S(t) so easy to integrate for LTV.

Time‑varying covariates:- Use extended Cox model with time‑dependent covariates (counting process format: start, stop, event) or joint models if covariate is endogenous (e.g., engagement driven by latent health). For parametric case, include time‑varying covariates via piecewise constant intervals or state‑space / joint modeling.

Left truncation & right‑censoring:- Right‑censoring: handled by KM, Cox, and parametric likelihoods using censored likelihood contributions.- Left truncation (delayed entry): use counting process input (enter time >0) so risk set includes only subjects at risk at that time; all standard packages (survival in R, lifelines in Python) support this.

Discounting cash flows:- Choose δ based on company WACC or risk‑adjusted discount rate. Compute discounted expected revenue per customer: for discrete periods t=1..T, LTV = Σ r_t * S(t) * (1+ρ)^{-t}. If revenue is conditional on survival in interval, use interval survival differences.

Confidence intervals for LTV:- Delta method: approximate variance using gradients if closed form and parametric model.- Bootstrapping: resample customers respecting censoring/truncation (cluster by customer), refit model and recompute LTV distribution — robust and handles complex pipelines.- For Cox: combine variance of β estimates and Breslow estimator for baseline to simulate S(t) and LTV; use parametric bootstrap (simulate β ~ N(β̂, Σ̂)) and draw baseline via Greenwood/Breslow variance.

Validation & practical notes:- Compare model fits with time‑split calibration, holdout cohorts, and visual inspection of survival vs empirical KM.- Report scenario LTVs (best/worst), and sensitivity to discount rate and extrapolation horizon.- Communicate assumptions (PH, parametric shape) and uncertainty to stakeholders.

EasyTechnical

45 practiced

Compute a simple LTV. Given ARPU = $20/month, monthly churn rate = 5% (0.05), and gross margin = 70%, compute the simple (non-discounted) LTV and the contribution LTV (i.e., LTV * gross margin). Show formulas and numeric answers. State assumptions and limitations of this simple approach.

Sample Answer

Approach & formula:- Simple (non-discounted) customer lifetime (in months) = 1 / churn- Simple LTV (revenue basis) = ARPU * lifetime- Contribution LTV = Simple LTV * gross margin

Given:ARPU = $20/monthmonthly churn = 0.05gross margin = 0.70

Calculations:- Lifetime = 1 / 0.05 = 20 months- Simple LTV = 20 * $20 = $400- Contribution LTV = $400 * 0.70 = $280

Python (replicable):

python

ARPU = 20.0
churn = 0.05
margin = 0.70

lifetime = 1 / churn
ltv = ARPU * lifetime
contribution_ltv = ltv * margin

lifetime, ltv, contribution_ltv
# (20.0, 400.0, 280.0)

Assumptions & limitations:- Assumes constant ARPU, constant monthly churn, and independent exponential churn (geometric survival). - No discounting of future cash flows (non-discounted). - Ignores acquisition costs, upsells, downgrades, seasonality, cohort differences, and cohort size effects. - For long lifetimes or strategic decisions, use discounted LTV, cohort analyses, or retention curves for more accuracy.

MediumTechnical

64 practiced

Describe how you'd compute cohort-based retention rates by signup month for the first 12 months (cohort analysis). Provide a high-level SQL or pandas approach/pseudocode, and explain how you'd handle reactivations, users with intermittent activity, trial-to-paid conversions, and users who have incomplete observation windows.

Sample Answer

Goal: compute month-by-month retention for cohorts defined by users' signup month, for months 0–11 after signup.

High-level approach1. Define cohort_month = date_trunc('month', signup_date).2. For each user, compute month_index = months_between(activity_date, signup_date) where month_index ∈ [0..11].3. Mark a user as “retained in month k” if they have at least one qualifying activity in that month (define qualifying = any session, paid event, or other business-specific action).4. Aggregate unique users per cohort and month_index to produce retention rate = retained_users / cohort_size.

SQL pseudocode

sql

WITH events AS (
  SELECT user_id, signup_date, date_trunc('month', signup_date) AS cohort_month,
         date_trunc('month', activity_date) AS activity_month,
         DATE_PART('year', activity_date)*12 + DATE_PART('month', activity_date)
         - (DATE_PART('year', signup_date)*12 + DATE_PART('month', signup_date)) AS month_index
  FROM user_events
  WHERE activity_date IS NOT NULL
),
filtered AS (
  SELECT user_id, cohort_month, month_index
  FROM events
  WHERE month_index BETWEEN 0 AND 11
  GROUP BY user_id, cohort_month, month_index
),
cohort AS (
  SELECT cohort_month, COUNT(DISTINCT user_id) AS cohort_size
  FROM users
  GROUP BY cohort_month
)
SELECT f.cohort_month, f.month_index,
       COUNT(DISTINCT f.user_id) AS retained,
       c.cohort_size,
       COUNT(DISTINCT f.user_id)::float / c.cohort_size AS retention_rate
FROM filtered f
JOIN cohort c USING (cohort_month)
GROUP BY f.cohort_month, f.month_index, c.cohort_size
ORDER BY f.cohort_month, f.month_index;

Pandas sketch

python

df['cohort_month'] = df.signup_date.dt.to_period('M')
df['month_index'] = ((df.activity_date.dt.to_period('M') - df.cohort_month).apply(lambda x: x.n))
df = df[df.month_index.between(0,11)]
retained = df.groupby(['cohort_month','month_index'])['user_id'].nunique().reset_index()
cohort_size = users.groupby('cohort_month')['user_id'].nunique().reset_index()
merge and compute rates

Handling special cases- Reactivations: treat any activity in month k as retention; reactivated users count in months after inactivity. If you want "consecutive retention" vs "ever-active", compute separate flags.- Intermittent activity: use dedup by (user, month) to avoid overcount; consider thresholds (≥N events or ≥T minutes) for meaningful engagement.- Trial-to-paid conversions: compute separate retention curves for trial-only, paid-converted cohorts, or add segmentation dimension (is_paid_at_month_k). For revenue retention use MRR cohorts.- Incomplete observation windows (censoring): for recent cohorts, mark months beyond data export as "unobserved" and exclude them from cohort averages or use right-censoring techniques. When reporting, show cohort sizes per month and use shaded/NA cells where observation missing.- Edge considerations: timezone/alignment, user timezones, user merges, multiple signups—use first signup as cohort anchor. Validate with counts and visual heatmap.

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