All right, well we did a couple of examples of this but those were examples that I just made up for the purposes of just showing you the basics of the calculation. Let's look at some real data. This is real data from the retail relay case that's coming up. And, okay, so what are we looking at here? Well, the first column is Purchase Occasion, and this is approximately monthly data in the relay case. So, this tells us kind of month by month what is going on. Now, there's this next column called Transition Probability, and I will talk about where this kind of information can come from in just a couple of minutes. But, let's just kind of hang with the calculation at the moment. So, Transition Probability. What is that 68%? That is the probability that someone who orders from relay in the first month also orders from them in the second month. And then 80%. That's the probability that someone who orders in the second month will also order in the third month, and so forth all the way down the line. So, skipping over to the final column, if we think about the probability of a new customer reaching state t, and that means the probability that a new customer will, let's say, get to month 23 and still be ordering from us, those can be computed by multiplying each of the subsequent months' transition probabilities together. For example, the 54% here, that's the probability that a new customer also makes an order in the third month. Where does that 54% come from? That's just 68%. That's the probability make it two, month two multiply by 80%, the probability that once they get to two they'll also make it to month three. And that's the way you calculate these all the way down the line, right? And, of course, they're decreasing, right? And why are they decreasing? Because as you go farther and farther out, at each stage of the process, more and more people drop out. Once in a while, you do get some months where it's 100%. And essentially, this is going to remain the same because it means that once you get to week 16, or month 16, pardon me, then everybody or almost everybody goes ahead in 17, 18, and 19 and makes an order. They've kind of gotten in the habit of making an order at that point. Finally, what's this average basket size? That is the average purchase amount in dollars of an individual purchase occasion. That's not the amount of money you make per se, because you would have to multiply that by some kind of margin, right? That's just the dollars that they're handing over to you, and from that you would have to subtract your costs. So that's what real data kind of looks like. Here's a nice graph of those retention rates. You see something that's very common in real industry data, in these retention rates, and that is the probability that they stay with you drops off very rapidly at the beginning, but then, over time, it starts to flatten out as you get towards lower numbers. And why? These are your hard core, loyal customers out there. They stuck around for a while and they like you. So it really does tend to flatten out. And what companies really want to do is they would like to really increase retention rates there or kind of get away from that steep decline, right? But, out here, it often gets kind of smoothed with people that are well practiced at ordering from you. Okay, so where do you get those figures that I had in that little Excel spreadsheet looking thing that I just showed you with the data in it? Well, the transition probabilities, pretty straight forward. If you had 1,000 people in your database that made at least one purchase and 850 of them made a purchase in the subsequent month after they made their first purchase, then you've got a transition probability of 85%. So, you're really just taking how many people were there at the beginning, how many people were there the next time around, and you take the number of people that were there the next time around and divide it by that original number of people you have and that's where you get that percentages. That percentage, and from that, you can calculate the retention all the way out through the subsequent years. And then the average basket size that was also in that data, that's going to be obvious from a company's database. Amazon, any direct marketer is going to know what their average purchase size is, so I would just say self evident from the data. Now, what can you do with that kind of information like what we just saw with relay. Well, they can attempt to kind of segment a customer base into very valuable customers, maybe second and third tier customers, or even customers on which they lose money. And there are customers in almost all of these direct marketing situations, there are customers that cost more to service than they bring in in terms of profit, all right? So, they can segment our customer base so they know what kind of money and how well they should treat certain customers to try to retain them because you've got customers to reward, customers to grow and below zero, you could have customers that really, you would rather they don't order from you anymore because they cost more than you make on them. So, the way companies do that with this kind of data is they try to figure out what your likely customer lifetime value is going to be as soon as they can. So, the calculations that I was doing before was really at the aggregate level. That means I took all the data, I bunched it together, and here is kind of the average customer lifetime value. You might realize, of course you do realize that customers vary on that dimension. And to the extent that I can figure that out early on by maybe some of their behaviors, then I can know up front how much to spend on them, how much love and attention to give to them, right? Do I know certain things up front that can help me with that? Yeah, size of the first order. You're going to know the size of their first order. And in several applications that I've seen with customer lifetime value where I've had access to big datasets, the size of the first order is a very good predictor of how valuable that customer's going to be going forward. Very small purchases upfront, they may grow over time. But if someone spends a lot of money with you right away, that is a pretty good predictor of their ability to spend money with you later. You also have early subsequent orders, so if they go on and order from you, do they sign up for certain things that it's likely to make them sticky. Like you go on an online grocery store and say, I want you to deliver to me a box of laundry detergents ever month. Well that's going to make them stickier because they've already signed up for these services going forward. You know the retention rate on that customer's likely to be larger than the average customer. And then, also I'll have mentioned zip code. It's just a demographic fact that people that have varying levels of wealth tend to congregate in different zip codes and you can buy that information, it's relatively easy to buy. This are just a few of the things that you can do. Early on, you might find some other things as a company, ways to segment the customers, but these are good ways to so it. And from that, be able to make some inferences, is the value going to be high or the value going to be low on this one? So, there are some challenges with CLV, it's not a panacea. Some things you need to worry about, in order estimate these probabilities you have to go far back in the data to get accurate purchase probabilities. In order to figure out what's the likelihood that somebody's going to buy three years from now, I have to go back in the data three years to see situations where people did actually purchase three years from the time of their first purchase. So, you have to go back a ways to get that kind of data. And if you think people's behavior is changing rapidly, you might come to the belief that those probabilities are not reflective of what's going on in the market place right now. Now, they're the best probabilities you've got, you probably not going to be able to get any better ones mathematically, but you have to take into account, do you think there's changes in your customer's behavior that makes adjusting those probabilities in one way or another make sense in terms of a look forward kind of a metric? So, that's the primary difficulty with CLV calculations. And that's why, too, the companies often limit CLV calculations to two, three, four, maybe five years instead of really long calculations because those very long calculations require you to look way, way back. So, strategic implications of this, it's a wonderful comprehensive forward looking measure of the customer relationship, and a customer relationship is central to the value of so many companies and it could connect marketing strategies to financial outcomes. And I will tell you in the real world, connecting marketing strategies to real financial outcomes often proves quite difficult. But in this case, this is a pretty good way to do it and it allows us to determine the full financial impact of pricing decisions. Not on just the purchase probability today, what you make on your customer today, but what you're going to make on them today, tomorrow, and the next day. So, it's a really more of a comprehensive tool in your pricing tool kit. It allows for selecting customers, good ones, bad ones, particularly if you have some of that up front data. And, you can look at different strategic alternatives to see, do they improve customer satisfaction? Do they improve customer retention? And if they do improve customer retention, you can then figure out well, what's the financial value on that thing that I'm doing that's increasing customer retention? You can use that in the calculation to do kind of scenario analysis.