And this is a challenge that often comes up. So, when we talk about customer lifetime value. Customer lifetime value is, by design, intended to look at the value of the customer's entire relationship. From the time they are acquired, how much are they worth to the company? And that's fine as a metric and may be helpful in terms of rank ordering potential prospects for whom do we want to approach. But what about the existing customers that we have? If I've had a customer around for a year, looking at his lifetime value is a little bit misleading. Because, well, we've acquired this customer and we've already collected a year's worth of his value. What we really need to take into account is how much revenue remains for us to collect from this customer. That is, rather than looking at the lifetime value, one thing that we can look at is the residual lifetime value of the customer. That is, how much remains for us to collect over his remaining lifetime. So, similar idea, but what it means is there's a little bit of math that we're going to have to do. Because what we have to take into account is, not only how long has he been around currently. But we also have to take into account the fact that our customer base may have shifted in some way. So let me go through an example with you to drive home that last point. Suppose we've got customers spending $100 with us monthly, but our customers are divided into two groups. We've got a high retention group and a low retention group. So one group is more prone to churn, the other group is less prone to churn. If we look, as time passes, which of those customers do we have more of still in our customer base? All right, well, even if they started out in equal numbers, let's take the first period. Who's more likely to churn, the high churning group or the low churning group? Well, by definition the high churning group is more likely to churn. So what's that going to mean is we're going to lose more of the high churners than we do of the low churners. So in our second period, we no longer have a 50/50 mix of customers. We've got more low churners than high churners. Go a period out after that. Again, the high churners are going to drop out at a faster rate than the low churners. And so, as more and more time passes, the customers who remain are disproportionately the low churners. And so, if we're trying to calculate a remaining or residual lifetime value, we have to recognize that it's not just the average customer lifetime value. Because we now know that the individuals who still remain after 3, 6, 12 months fall into this low churner group. And so they may actually be more valuable than we initially thought. All right, and so if we're looking at that contractual setting, just to keep things easy for us for a little bit. Our formula for customer lifetime value, all right, this top formula. Capital M, let's assume we get a constant margin. S(t), that's the survival probability. The probability that the customer is still around at time t. The denominator, 1 plus delta, taking into account time value of money. The reason for the infinite summation, we're going to sum over all periods. So for each time period, what's the margin? How much do I get? That's the M. I only get that money if the customer is still around. That's what the survival probability capital S of T is. And then, how much do I have to discount that because we're looking into the future? We're summing up that term for every single product from zero to infinity. That's going to give us an estimate for customer lifetime value. So from the start of the relationship, how much is this customer worth to us, all right? Well, what changes when we're looking at residual lifetime value? All right, well, residual lifetime value as of time T. Well, the first thing that's going to change is notice the difference in the summation. We're not summing from zero to infinity, we're starting at time T. And we're not interested in what happened between time zero and time T minus one. We've already collected all that reference. So, we're starting at time T, all right. In terms of what we get each period, we get the margin, and that's going to be discounted based on how far out we're looking. And then, what we also have to take into account is, is this customer still around? And that's what this updated survival probability is taking into account. So what's the probability that the customer is still around at time little t, given that we know they've lasted until time capital T, right? And so, what can actually happen in this scenario, is if I learn based on the fact that a customer has survived a long time with us, chances are he's more likely than not to continue sticking around. Well, compare that to the fact that at the beginning of his relationship, he was mixed in with the low churners, the high churners. We didn't know who he was. So because of that mixing that's going on initially, the fact that he sticks around longer might reveal that he's a low churner. His residual lifetime value can actually be higher than his initial lifetime value, than his customer lifetime value. All right, so just to give you an example of this. This is work taken from Bruce Hardie's website based on a presentation he had given at the ART Forum. I've included links to those references for those of you who are interested in gathering more details about this particular example. But in this work, three different groups were identified, low, medium, and high churners. And notice the increasing order of the churn probabilities. And looking at the fraction of the customer base, initially the majority of customers are in that low risk group. Some customers in that high risk group, some in that medium risk group. Well, what happens over time? Who are the customers more likely to drop out quickly? It's going to be those customers in that medium and high risk groups. And so, if we look at a two segment example, where Segment 1 consisted of one-third of our customers, Segment 2 consists of two-thirds of our customers initially. But let's look at the differences in the retention rates. Difference in the retention rate. We've got a retention rate for one group of 0.9, retention rate for another group of 0.5. All right, well, the Segment 1, those are the people more likely to stick around. Segment 2, those are the ones that are going to drop out much more quickly. Well, if we take a look at the retention behavior for Segment 1. Here's our focus on Segment 1, where we retain 90%, so we lose 10% of our customers in this segment each period. In Segment 2, what's happening? Higher churn rate, so we're going to lose 50% of our customers in each time period. All right, well, when we look at the total number of customers that remain, notice that by the time we're out into the fifth period, 2,187 from segment 1, 2,600 total. So even though this is the smaller segment, it's those customers who are going to be represented later on just because they have insurance. So you're losing your fast churners early on. The ones who remain are probably the ones who are low churners to begin with. They may be one who are inherently more loyal to your particular company. All right, so we do want to draw that distinction between customer lifetime value for new prospects. Residual lifetime value when it comes to looking at your current customers. And for a given investment, where do we put our money? Do we put that money into acquiring prospects? Well, we don't know that much about the prospects. We know a lot more about the people who've stuck around for a while. But the down side is putting money into customer retention. If these are the customers who are so loyal to you already, you may not have the opportunity to increase that retention rate very much. And so, that's going to factor in when we're trying to think through, is the dollar best spent on acquisition or best spent on retention?