How to calculate decrease in client portfolio?

[Henry]

Hello,
I have two ways of calculating same thing, but each way gives different result. Here is the problem:

I want to open an e-shop and I would like to know how many customers I am going to have in one year.
I opened it on 1.1.2012. Each month there are 300 new unique registrations=users.
At the end of year (12 months) I am planning to have 12*300 new customers, but I must count with some customers that will cancel their account during the year.
The expected annual loss of clients is 15%

I would do this: 12*300*(1-0,15) = 3060 (number of customers after one year).

But I found different way - calculate loss of clients for each month by portion of annual loss (15% divided by 12) and sum all results together
month1 = 300
month2 = 300*(1-0,15/12) => it means at the end of month 2.(February) I have 300 new customers starting in February plus 296 from January
month3 = (300*(1-0,15/12))*(1-0,15/12) => 300 From march + 296 from February + 293 from January
...
month12 = 300*(1-0,15/12)^11
This makes 3336 users at the end of the year.

The second approach seems logical to me, but I am not sure if it is correct - the difference between 3060 and 3336 is quite big.
My questions:
1) What is the correct way of calculating this - I would like to understand the difference between both ways and when to use them.
2) If I use the second way, can I still say that the annual attrition is 15%? Because 264 (3600-3336) is not 15% from 3600.
Could anyone help me with this?

 

[tkhunny]

You have discovered the magic of compound interest.

Rather than (1 - 0,15/12), try it with \(\displaystyle (1 - 0,15)^{\frac{1}{12}}\)

This seems likely to lead to a thrid answer.

After that experiment, you need to decide which model meets your needs. The "right" way is the way you understand - that makes sense to you - and helps you understand your business.

You could also use a continuous model if you like. Or a semi-continuous sigmoidal dose response model. Seriously, I'm neither trying to impress you nor trying to discourage you. I'm only trying to emphasize that the "right" way probably doesn't exist unless you use the definition I provided above.

 

[Henry]

This is exactly my problem, I do not understand the difference clearly. I am familiar with compound interest, I just could not match it to my problem, but I will look at it now more from this side. I have no problem reading more, what I need is a hint where to look, so thank you for showing me the way!

First time I see "semi-continuous sigmoidal dose response model" I will try google and this forum, but it would be great if you could give me a tip.

Last thing, is there something that could make it more clear to me about the problem, that I am using annual rate 15%, but the result when i sum all months is different (in the case i used 7%)?

 

[tkhunny]

Why do you expect to get the same answer with different methodologies? If you understand compound interest, you know this is not the case.

For Compound Interest

(1 + 0.15/12)^12 = 1.160754517723

Applying 1/12 of 0.15 each month results in a total for the year of 16.075%

For Your Demographic Decrement Model

(1 - 0.15/12)^12 = 0.859894659249

Applying 1/12 of 0.15 each month results in a total for the year of 14.01%
This is the effect on just January. February is a little different. March is a little different from that. etc...

What you can do is solve for the rate that gives 15% for the year.

(1 - R/12)^12 = 0.85
R = 16.1423364%
Remember, though, that this is perfect ONLY for January.

 

[Henry]

Thank you for detailed explanation!
Maybe it was not clear from my original post - I am not looking for ready-to-use solution, but for explanation that would help me understand each way. In the end I would have to „defend“ my calculation if I would like to explain to someone my plans (the shop is not done yet, I used 1.1.2012 just to make it simple).
I hope I understand the difference now. I say that the annual rate is 15%, but the model 2 result does not say so (after one year). From the explanation I see that if I use model 2 (or any other similar) I should not say that annual rate is 15%, but that monthly rate is 1,25% together with the way how it is used.
I guess I understand examples you gave me, just need to think a bit more about it all.

 

[Henry]

Hello, thanks a lot for explanation. Your explanation make sense to me, I will check my calculation during weekend.