Okay let's do a couple of examples of customer lifetime value calculations, so we can see how it can relate to pricing decisions. So, here's the first example. Let's consider a customer who pays $10 per year for a subscription to some kind of online service. But that servicing this customer costs a company $5 a year, right? And that cost can include a whole lot of different things, but that's going to mean their margin is $5, all right. That's going to be the total margin for the year for the company, $5. Customers have to sign up for a year and the contract is renewed yearly. The appropriate discount rate is 10%, and again that would be coming from company data. And individual companies would know their discount rates. So, the company data shows that only 30% of customers who sign up stay an additional year. So that's a very low retention rate, right? So, only 30% stay for that second year. But if you do stay for a second year, then 80% of those customers stay for additional third year. So what we're going to do is, what's called a 3-Year CLV. Now CLV is Customer Lifetime Value. And you might think well, shouldn't we be measuring the entire financial value of this individual over their entire lifetime? Well theoretically, yeah, but companies usually take a certain period of time. Like their planning horizon for maybe strategic activities they want to do. And they say look, I don't care what happens with this customer ten years down the road. I just need to figure out how much money I'm going to get in the door over the next say, three to five years in order to figure out what to do. So while it says lifetime in the definition, it usually in practice is done over a finite period of time. So, we're going to do a three-year CLV. So, we start with a five here. Where is that five coming from? That's the amount of money they made in the first year, right. And it's un-discounted and you don't apply a retention rate to it. Because you start this calculation when the customer walks in the door. Either literally or figuratively and spend some money, so it's $5. Now in the next period, this is year two right here. What's this? There's that $5. I'm dividing by 1.1, because that's the discount rate. That's that denominator 1 + .1 from the formula. Now 1 would be coming from the company and then, I'm multiplying it by 0.3 and what is that? That's the retention rate. So in essence, this CLV is saying there is only a 30% probability that this amount of money right here, will ever actually show up at our door. And that's why I'm multiplying it by 0.3. Now analogously, this is year 3 over here. I've got the $5 again and I've got 1.1 squared and I think that should be intuitive to you from the net present value formula. But then, now up front, I have got .3 times .8 which equals, if you do the math, .24. Where is that coming from? That's the original .3. It means that in year three only 30% of those customers even made it to year two. So we know the most we're going to have is 30% of the original customers in year three. That's at most. We also know that only 80% of those customers that bought in year two also bought in year three. So, we have to multiply those two probabilities together. So, this becomes the total probability that they make from the first purchase all the way to year three, right. That's that total probability. I multiply it by this number here and what comes out the other side is 7.35. So, that is my total three year customer lifetime value for an individual buyer of my subscription service. So, let's look at another example that ties are a little bit closely to pricing. So, very similar kind of set up. But in this situation, the company who is charging $10 a year is thinking about price increase and that price increase is to $15 per year. A lot of this other stuff stays the same. The $5 cost stays the same. The discount rates stays the same. The one year renew in the contract stays the same, but some of the retention numbers changed. Retention numbers do change and in particular, now only 20% of customers who sign up to stay an additional year. But of those who stay two additional years, 50% stay a third year, but there's one other effect. So, it changes these retention probabilities that price. It also changes the number of people who would initially buy the thing to begin with. And in particular, the company believes that if they go from $10 to $15, they would reduce the initial customer base by 20%. Where you get numbers like that, numbers like that would come from marketing research. And we are going to talk about some marketing research to gauge price sensitivity. However right now, let's just take the 20% at face value, so how would you evaluate this situation? Well, here is how you'd do it. This is going to look very similar to the example I just did before but with a little bit of twist because of that price change. So, up front, I have $10. Why? That's my new margin, right? The $15 minus 5 that comes in the door the first day. I don't need to discount that and then, during the next year what do I have here? Well there's that $10 again. I have to discount that by the 1.1 and out front here I have to also take into account the attrition rate. On this case, the retention rate which is only 20% with this new price so I multiply by the 20% and finally in year three. Out here I have the 50% minus the 20%. Those are those two retention rates between years one and two and two and three multiplied together. And of course, I take that $10 and I divided by 1.1 squared coming directly from the NPV formula. When I do that, what I get is $12.63, but we're not quite done yet. We're going to do one more thing to that and that is because we believed that when we raise price fromm 10 to $15. That 20% of our customer base is not even going to consider buying us anymore. At 20% fewer than would've considered us under the $10 price scheme, right. So what do we do? So what we do is, we take this $12.63 and we multiply it By 80%. Because 20% is going to go away but 80% is going to remain, multiplied by the 80% which is $10.10. What do we do with $10.10? Well, we can compare it to the customer lifetime value we had in our first example when we were only charging $10. But in that situation, we had a higher retention rate. We had more people coming in the door and buying our product right? But in the second example, we're making more money each time we make a sale. So is it worth it or not? Well it is. The price increase was worth it, because our CLV now stands at $10.10. And before, we were making $7.33 over three years with an individual customer. So this is an important way where customer lifetime value analysis can really begin to inform pricing decisions.