What is a customer who buys at most once a season worth?

Some customers can only buy on fixed occasions: one pass per season, one donation per annual appeal, one order per catalogue ^[C01]. Without a subscription, nobody announces when they leave: the firm only sees a silence, which may be a departure or a mere pause ^{[C02, C03]}. Two shortcuts prevail: extend each customer's past buying rhythm, or declare lost anyone who stays silent for two seasons. The beta-geometric/beta-Bernoulli (BG/BB) model of Fader, Hardie and Shang settles both questions at once: is the customer still buying, and is the customer still there?

On synthetic data drawn from the BG/BB model itself, which is therefore right by construction, 4,000 customers are observed for six seasons; the methods are judged on the next five. Extending the past rhythm forever, as if nobody left, overestimates future purchases by 43.2% and the value of the customer base by 74.3%; damped at the rate at which the number of buyers declines, this rhythm underestimates the value by 13.7%. The inactivity rule sets aside 2,441 customers, who nevertheless make 707 purchases over the next five seasons. The BG/BB forecasts 5,033.2 purchases against 5,099 observed (−1.3%, a gap of the order of chance) and values the base at €1,227,640, against €1,256,437 with the true parameters: this is the part of customer lifetime value (CLV) still to come. Two caveats: 52.9% of this value lies beyond the five verifiable seasons, and the discount rate weighs heavily.

A ski resort sells season passes. There is no subscription: every autumn, the customer buys a pass again or not. If not, the resort does not know whether the customer has stopped skiing there or is taking a one-year break. In the same season, 4,000 customers bought their first pass. The resort has observed the following six seasons: for each customer, it knows how many passes they bought again and in which season the last one was bought. Each pass brings it a margin of €150.

Management asks two questions: what is this cohort of customers worth, and should it keep contacting those who have bought nothing for two seasons — the in-house rule declares them lost and removes them from the re-engagement mailing list.

The data on this page are synthetic. The probability of buying in a given season, as long as the customer is still there, varies from one customer to another according to a beta distribution (a way of describing how a probability is spread across customers) with parameters 1.0 and 0.8, i.e. 55.6% on average; the probability of leaving for good at the start of a season follows a beta distribution with parameters 0.5 and 2.5, i.e. 16.7% on average. To judge the methods, five additional seasons, seasons 7 to 11, are simulated and then set aside.

For a customer of whom we only know the number of repeat purchases over six seasons and the season of their last repeat purchase, how many purchases should be expected in the following seasons, and what is this purchase stream worth today? The authors call DERT (discounted expected residual transactions) the number of expected future purchases, each discounted to today; multiplied by the margin, it gives the customer's residual value ^[C14]. The difficulty fits in one sentence: the only sign of a departure is an unusually long silence, and departed customers must be told apart from those taking a long break ^[C03].

Three common pitfalls:

• Extending the observed rhythm. A customer who repurchased 4 times in 6 seasons is "worth" 4/6 of a pass per season, forever, as if nobody left. Yet a customer with a perfect record may have been lucky, and may leave ^[C23]; conversely, a rare but recent customer is worth more than their past rhythm. Damping this rhythm at the rate at which the cohort's number of buyers holds up from one season to the next, 91.5% here, brings the overall level closer without correcting the differences between customers.
• The inactivity rule. Two seasons without a purchase, and the customer is declared lost. The rule looks at recency, but it cuts where a probability is needed: a customer who bought one season in two is not in the same situation as a customer who bought every season and then fell silent.
• A continuous-time model. The Pareto/NBD (Pareto/negative binomial) model, the reference for purchases at any moment, assumes purchases follow a Poisson distribution, that of events occurring at random at any moment. When the number of purchases per period is capped, a Bernoulli process — buy or not at each occasion — is more appropriate ^[C04]; on annual donation data, the Pareto/NBD fits poorly ^[C33].

A more sophisticated model accounts for differences between customers but not for departures: the beta-Bernoulli (BB), the natural benchmark when customers are assumed never to leave and simply split their purchases across suppliers ^[C37]. It does not capture the "leakage" of customers ^[C38]; on the authors' data, it overestimates cumulative repeat purchases at the end of validation by 20% ^[C20].

The BG/BB model tells the following story ^[C06]:

1. at the start of each season, a customer who is still "alive" may leave for good, with a probability θ of their own ^[C08];
2. if still there, the customer buys with a probability p of their own, with no memory of past purchases ^[C07];
3. p and θ vary from one customer to another according to two beta distributions ^[C09], independent of each other ^[C10].

A customer's whole history then boils down to two numbers, their frequency (the number of repeat purchases) and their recency (the season of their last repeat purchase) ^[C11]. The story explains two effects that no shortcut reproduces. A customer who bought every season and has been silent for three seasons has probably left: the silence is too long for their rhythm. A customer who bought rarely and has been silent just as long may simply be keeping to a slow rhythm. At equal recency, when the last purchase is old, having bought a lot can therefore be a bad sign ^[C28].

For each customer, over the same series of occasions: their frequency and their recency, recency being the season of the last purchase, not the time elapsed since — the convention differs from that of direct marketing ^[C12]. A table of counts is enough: with six seasons, there are only 22 combinations of frequency and recency, whatever the number of customers ^[C11]. The estimation and the main results can be computed in a spreadsheet ^[C13].

You also need the margin per purchase and a discount rate expressed per occasion: with one occasion per year, the annual rate applies as is, here 10% ^[C16]. Valuing DERT with a single margin assumes that the amount per purchase is independent of the buying rhythm and stable over time ^[C15]; otherwise, a separate model of amounts is needed ^[C39]. The authors themselves model only whether a donation is made each year, and leave its amount aside ^[C46].

This section is for practitioners. Decision-makers can skip it and resume at "The declared calculation": the "In plain words" sentence at its end is enough.

• n: the number of observed occasions, here 6 seasons; h: the forecast horizon, here 5 seasons; k: the rank of a future season, k = 1 for season 7.
• x: the number of repeat purchases over the n seasons; tₓ: the season of the last repeat purchase, 0 if x = 0 ^[C12]. Yₙ₊ₖ equals 1 if the customer buys at occasion n + k, 0 otherwise; X is the number of purchases over the horizon h; E[·] denotes the expectation, the average to be expected.
• p: the probability that a customer who is still there buys at an occasion; θ: the probability that the customer leaves for good at the start of an occasion ^{[C07, C08]}.
• p follows a beta distribution with parameters α and β, with mean α / (α + β); θ a beta distribution with parameters γ and δ, with mean γ / (γ + δ) ^[C09]. B(a, b) denotes the beta function.
• Likelihood of a history: L(x, tₓ) = [B(α + x, β + n − x) / B(α, β)] × [B(γ, δ + n) / B(γ, δ)] + Σᵢ [B(α + x, β + tₓ − x + i) / B(α, β)] × [B(γ + 1, δ + tₓ + i) / B(γ, δ)], where i, from 0 to n − tₓ − 1, counts the seasons the customer spent after their last purchase before leaving. The first term corresponds to a customer who stayed for all n seasons.
• Log-likelihood: LL = Σⱼ fⱼ ln L(xⱼ, tₓⱼ), over the 22 combinations j, where fⱼ is the number of customers in the combination and ln the natural logarithm. We look for the values of α, β, γ and δ that maximize it.
• E[Yₙ₊ₖ] = [B(α + x + 1, β + n − x) / B(α, β)] × [B(γ, δ + n + k) / B(γ, δ)] / L(x, tₓ): the chance that the customer is still there at occasion n + k and buys then, given their history. Over the horizon: E[X] = E[Yₙ₊₁] + … + E[Yₙ₊ₕ].
• P(alive) = [B(α + x, β + n − x) / B(α, β)] × [B(γ, δ + n + 1) / B(γ, δ)] / L(x, tₓ): the share of the likelihood of the history that corresponds to a customer still there at occasion n + 1.
• d: discount rate per occasion. DERT = Σₖ E[Yₙ₊ₖ] / (1 + d)ᵏ, for k ≥ 1: the first future purchase is discounted by one season ^{[C14, C51]}. Residual value: m × DERT, where m is the margin per purchase ^[C15].
• Shortcuts: the extended rhythm is worth m × (x / n) / d; the damped rhythm forecasts (x / n) × gᵏ purchases at occasion n + k, where g is the average rate at which the number of buyers holds up from one season to the next, and is worth m × (x / n) × g / (1 + d − g).

In plain words: each customer keeps their own buying rhythm and their own risk of leaving; a long silence raises the probability that they have left, all the faster as their past rhythm was steady.

1. Checking the code before use. Both programs re-estimate the model on the table of 22 combinations published by the authors — 11,104 first-time donors to a nonprofit, followed for six years ^[C17] — and retrieve their estimates: α = 1.204, β = 0.750, γ = 0.657, δ = 2.783, log-likelihood −33,225.6; for the BB, α = 0.487 and β = 0.826 ^[C18]. They also retrieve the 22 values of their Table 5 ^[C21] and those of their Table 6 ^[C29], and stop with an error otherwise.
2. Synthetic cohort, drawn from the BG/BB model by a pseudo-random number generator written out in full (seed 20261005; x ← 1103515245 · x + 12345 mod 2³¹), so that Python and R draw the same numbers.
3. Calibration on the first six seasons, summarized in 22 counts; seasons 7 to 11 are set aside.
4. Estimation of the BG/BB and the BB by maximum likelihood (Nelder-Mead simplex algorithm, which searches for the maximum step by step, restarted until stable); on the authors' data as on the cohort, a start at 0.01 for all four parameters leads to the same optimum.
5. Five forecasts, customer by customer: extended rhythm and inactivity rule, both extended without end and without departure, the rule giving zero after two seasons without a purchase; rhythm damped at the rate at which the number of buyers declines; BB; BG/BB. Reference: the expectation computed with the true parameters.
6. Values: €150 per pass, discounting at 10% per season, series of 2,000 terms.
7. Uncertainty: parametric bootstrap — 1,000 cohorts simulated from the estimated parameters, each re-estimated; interval from the 25th to the 975th sorted value, i.e. about 95%.
8. Repetition on 100 independent cohorts drawn from the true parameters; the damped rhythm, which requires buyers season by season, is not included.
9. Sensitivity to the discount rate, at 5% and 15%, and share of the value beyond seasons 11, 16 and 26.
10. Invariants: 15 identities, including the equality between DERT computed as a series and the authors' closed form; the programs stop if one fails.

The data. Over six seasons, the 4,000 customers repurchase 8,762 passes. 435 buy again every season; 1,271, i.e. 31.8%, bought no further pass in these six seasons; 2,441, i.e. 61.0%, bought nothing in the last two, and the in-house rule declares them lost.

The estimates. The BG/BB gives α = 0.882, β = 0.744, γ = 0.574 and δ = 3.342, i.e. a mean purchase probability of 54.2% (approximately 95% interval: 51.8% to 56.8%) and a mean departure probability of 14.7% per season (11.8% to 17.8%), against true values of 55.6% and 16.7%. It fits much better than the BB: log-likelihood −12,227.6 against −12,732.6.

The forecast, judged on the seasons set aside.

Chance alone makes this total vary by about 71.2 purchases (one standard deviation, i.e. 1.4%): the gap of the BG/BB cannot be told apart from it. The total of the inactivity rule looks acceptable; its composition does not. Mean purchases per customer in seasons 7 to 11, by season of the last repeat purchase:

The rule sets to zero customers who are still buying, and overestimates those who have just bought.

Six customer profiles. Expected purchases in seasons 7 to 11 and discounted residual value per customer:

The shrinkage works both ways: a customer with a single repeat purchase, but in season 6, is worth more than their past rhythm.

The value of the cohort.

The first two shortcuts value the past rhythm forever, without departure: it is mainly this assumption that inflates their value. Damped, the rhythm underestimates the value: it treats all customers alike and does not see that loyal ones stay. The size of these gaps also depends on the departure rate chosen for the simulation.

The 2,441 customers the rule declares lost still carry, according to the BG/BB, €175,833 of expected value if nothing changes, i.e. 14.3% of the cohort; in seasons 7 to 11, 366 of them come back and buy 707 passes, against 720.9 forecast. The 1,271 customers without any repeat purchase are worth €28.17 each, i.e. €35,805 altogether, 2.9% of the value.

What the history does not verify. 52.9% of the BG/BB value of the cohort lies beyond season 11, the last season set aside, 29.1% beyond season 16 and 9.4% beyond season 26.

The discount rate. At 5% per season, the BG/BB value of the cohort rises to €2,087,835, against €2,164,955 with the true parameters; at 15%, it falls to €881,456, against €897,102. This choice, which belongs to the finance department, weighs far more than estimation uncertainty.

Across 100 independent cohorts, the purchases forecast for seasons 7 to 11 depart from their expectation computed with the true parameters by +44.2% on average for the extended rhythm (standard deviation 0.8%), by +10.6% for the inactivity rule (0.7%) and by −0.3% for the BG/BB (1.9%). The extended rhythm overestimates this expectation in 100 draws out of 100; the mean gap of the BG/BB cannot be distinguished from zero, its simulation error being 0.2%.

• Fixed occasions, at most one purchase per occasion ^{[C01, C04]}. If the customer can buy several times per period, management may choose to count only whether a purchase occurred, as for a rare behavior ^[C05]; the information on the number of purchases is then lost. Do your customers sometimes buy two passes in the same season, for a relative?
• A permanent departure ^[C06]. Applying a permanent-departure model to customers who never quite leave systematically underestimates their value, according to Rust and colleagues, cited by the authors ^[C50]. Do some of your customers stay away for a few seasons, during a stay abroad, before returning?
• A constant individual rhythm, with no memory of past purchases ^[C07]. Do your customers buy more often as they grow attached to the resort, or less often as their children grow up?
• Independence between buying rhythm and risk of leaving ^[C10]. On the authors' donation data, it is violated ^[C34]: the estimated correlation is 0.361 ^[C35]. Allowing for this correlation keeps the forecasts close overall, with some noticeable differences — 3.59 purchases instead of 3.75 for a donor with a perfect record ^[C36] — and clearly improves the fit, the χ² falling from 47.9 to 4.8 ^[C48]; the authors find the cost-benefit trade-off uncertain ^[C49]. Are your most regular customers also the most loyal?
• Future commercial policy similar to the past: the model ignores the effect of marketing actions ^[C40] and assumes they will remain broadly the same ^[C41]. Did you change prices, ski area or sales channel during the observed period?
• A margin per purchase that is constant and independent of the rhythm ^{[C15, C39]}. Do your loyal customers get a reduced rate?
• A representative cohort: the authors recommend estimating the model separately by date or channel of acquisition ^[C44]. A cohort that has had only one purchase occasion since recruitment does not allow the model to be estimated; cohorts can then be pooled under common parameters, at the price of a possibly restrictive assumption ^[C32]. Do your customers recruited online behave like those recruited at the ticket office?
• What the data cannot settle: whether departure is permanent, since a departure and a very long break leave the same silence ^[C03], and the future stability of commercial policy. The other assumptions can be checked: out-of-sample validation for the constant rhythm, an extended model for independence ^[C34], billing records for the margin.

• Out-of-sample validation: on real data, set the last seasons aside and compare, by recency and by frequency, forecast and observed purchases. Here, the BG/BB forecasts the total with a gap of −1.3%, and 1,667.5 customers buying at least once in seasons 7 to 11, against 1,690 observed. On the donation data, its forecasts track cumulative repeat purchases during the six calibration years and the five validation years ^[C19]; their only notable flaw is to underestimate the purchases of donors whose last donation was old ^[C47], part of the segment the inactivity rule sets aside. This validation covers only five seasons, whereas 52.9% of the value lies beyond.
• Prefer an observable quantity: the probability of still being there concerns a state nobody observes; that of buying at least once over a future period can be checked ^[C31].
• The error at the level of one customer: the mean absolute gap between forecast and observed purchases is 0.747 per customer for the BG/BB and 0.745 with the true parameters: individual chance dominates. The inactivity rule obtains 0.689, because this criterion rewards a forecast of zero for customers unlikely to buy; the median forecast by the BG/BB, instead of its mean, obtains 0.653. An expectation is judged by group.
• Fit: the χ² (a measure of the gap between forecast and observed counts) over the 22 combinations is 20.98 for 17 degrees of freedom, a probability of 0.23: a gap that chance easily explains. The likelihood ratio, which measures how much better the BG/BB fits than the BB, is 1,010.1; since the BB is reached only in the limit where γ tends to zero, the usual distribution of this ratio does not apply, but such a gap remains overwhelming.
• Estimation uncertainty, by parametric bootstrap: the approximately 95% confidence interval for the expected purchases in seasons 7 to 11 runs from 4,781.6 to 5,232.2; that of the value of the cohort, from about €1,091,000 to €1,327,000; that of the value of a customer with 6 repeat purchases out of 6, from €847.92 to €1,003.04. These intervals concern the expectation, not what the customers will actually do; they assume the model is right and cover neither model error nor uncertainty about the margin or the discount rate.
• What the simulation does not prove: the data are drawn from a BG/BB model, right by construction. The simulation measures the cost of the shortcuts when this model is true; it does not show that it suits your customers.

The extended rhythm and the BB treat each customer as if still there; they ignore the leakage of customers ^[C38]. From the perfect customer, 6 repeat purchases out of 6, we expect not 5 purchases in the next five seasons but 3.86 on average: they may have been lucky, and they may leave ^[C23]. The authors find the same shrinkage among their donors, 3.75 expected purchases over five years ^[C22].

The inactivity rule looks at the right variable, recency ^[C25], but turns it into a guillotine. Compare two customers: 5 repeat purchases with a last purchase in season 5 give 2.04 expected purchases; 4 repeat purchases with a last purchase in season 6, 2.74. The customer with one repeat purchase fewer, but more recent, is worth more — the authors observe the same reversal between two donors, 1.81 against 2.71 ^[C24]. Conversely, a customer with 3 repeat purchases in a row followed by three silent seasons has most likely left the resort: only a 0.14 probability of still being there remains, against 0.94 for everyone who bought in season 6, whatever their frequency ^[C30]. The extended rhythm still credits them with 2.50 purchases; the BG/BB, 0.32.

Finally, customers without any repeat purchase weigh little one by one, without being worthless together: 2.9% of the value here. Among the authors' donors, those without any repeat donation, over 30% of the cohort, should make over 240 donations in five years, more than about half of the other groups ^{[C26, C27]}.

• Allowed: "For this cohort, if the BG/BB model is right and at a discount rate of 10% per season, the expected residual value is about €1,228,000 of discounted margin, with an approximately 95% confidence interval from €1,091,000 to €1,327,000."
• Allowed: "In this simulation, extending the past buying rhythm overestimates the expected purchases of the next five seasons in 100 draws out of 100."
• Allowed: "Customers silent for two seasons are not lost as a block: if nothing changes, they still carry 14.3% of the expected value."
• Forbidden: saying that a given customer has left, or will come back. The model gives probabilities and expectations; being there and being active are two different things ^[C31].
• Forbidden: concluding that a re-engagement mailing will bring silent customers back. The model forecasts what will happen if commercial policy stays the same ^[C41]; it does not measure the effect of an action ^[C40].
• Forbidden: presenting these figures as a market result; the data are synthetic.

• Do not declare lost, by rule, customers who are still buying. Ranking customers by expected purchases — the ranking by discounted value is the same ^[C52] — tells where they stand, not what a re-engagement mailing would make them do: their expected margin is that of an unchanged policy. To know whether contacting silent customers brings in more than it costs, an experiment is needed, with a control group that is not contacted. If actions are then targeted on the model's outputs, it must be re-estimated on the updated data; its forecasts then offer a low-cost baseline for following the effect of these actions ^[C42], and a model that includes marketing actions will have to deal with the endogeneity of targeting, that is, the fact that targeted customers already differ from the others ^[C45].
• Value the base with caution. About €1,228,000, but more than half of this value lies beyond the verifiable horizon, and depending on whether the rate chosen is 15% or 5%, it ranges from €881,456 to €2,087,835, more than twofold. The amount remains a decision of the finance department.
• Monitor. Re-estimate the model every season and compare its forecasts with what is observed ^[C43]; estimate it separately for each acquisition cohort and each channel ^[C44].

The decision remains human and commits the sales and finance departments.

• Use: purchases tied to fixed occasions — a season pass, an annual fundraising appeal, a response to a catalogue, registration for an annual event ^[C01] — or rare behaviors one chooses to count per period ^[C05], with no subscription and no declared departure ^[C02].
• Do not use with a subscription, where the firm sees its customers leave: see the page on subscriber value, or the one on retention and survival analysis.
• Do not use when purchases can occur at any time and in any number: the BG/NBD model is then the right tool — see the page on estimating noncontractual CLV, which also deals with amounts per purchase.
• Do not use alone to target an action: the model ignores the effect of marketing ^[C40]; to identify at-risk customers from explanatory variables, see the page on predicting churn with logistic regression.
• To judge a forecast on seasons set aside, see the page on validating a marketing forecast.

• Python 3.14.5, standard library only: msc-p044-reference.py, run with python msc-p044-reference.py
• R 4.6.1, base only: msc-p044-reference.R, run with Rscript msc-p044-reference.R
• Both programs print exactly the same lines; their sha256 fingerprint, after line-ending normalization, is recorded in the dossier manifest. Declared seed 20261005. No third-party library: the software does not validate the method, it reproduces it.

The table of the 22 combinations of frequency and recency, the four estimated parameters, the validation by recency and by frequency on the seasons set aside, the value of each combination and of the cohort with its interval, its share beyond the verified horizon, its sensitivity to the discount rate and the assumptions to check before deciding.

• Fader, P. S., Hardie, B. G. S. & Shang, J. (2010). Customer-Base Analysis in a Discrete-Time Noncontractual Setting. Marketing Science 29(6), 1086–1108. ↗ — published version, distributed by INFORMS as a courtesy to the authors, full text verified ^[C01–C52].
• Level of evidence: demonstration on synthetic data drawn from the model itself, one detailed draw, a bootstrap of 1,000 cohorts and 100 repeated draws, reproducible in Python and R; code checked against the values published by the authors, from their donation data. This is not a market result.