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Discussion => Silk Road discussion => Topic started by: LawlessLucy on March 06, 2012, 12:05 pm

Title: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: LawlessLucy on March 06, 2012, 12:05 pm
====WARNING=====

====WARNING=====

===MATH AHEAD===

===MATH AHEAD===

After doing some mathematical meditation in the shower, I thought of a way to calculate a fair refund rate for customers based 3 major components:
1)What % of your sales ask for refunds. (info gathered from your sales data)
2)What % of ALL SR sales ask for refunds. (compares the two to adjust accordingly to expected vs actual)
3)What you think is a fair refund rate.

Adjusted Refund Rate =
(0.03/[R/N])*0.5

> 0.03 is the expected % of nondeliveries (I've heard SR has an overall 97% success ship rate, though there may be a more exactly accurate number floating around)
> R is the # of Refund Requests you've gotten
> N is the # of Transactions you've processed
> R/N is the chance that any given transaction will request a refund based on your past sales data.
> therefore 0.03/[R/N] is the ratio of expected:actual.
i)When this ratio = 1, Actual = Expected (in this case, 0.03 = R/N)
ii)When this ratio > 1, Actual < Expected (less than expected refund requests; this suggests smooth vendor services and honest buyers)
iii)When this ratio < 1, Actual > Expected (more than expected refund requests; this suggests bad vendor practices and/or dishonest buyers)
>0.5 is the proposed refund rate (50%), what you think is fair. Based on my previous post I'll leave it at .5 for now.

This gives you the new refund rate so that you don't get completely screwed when refunds get out of hand.

WHAT THIS EQUATION SAYS: If there have been more refund requests than expected, adjust my refund rate to be lower. If there have been less refund requests than expected, adjust my refund rate to be higher. If there are exactly as many refund requests as expected, offer the proposed, fair, refund rate.

---META STUFF---
Let's take a look at what this formula implies and how we can manipulate/change it to our liking.

First, let's define Expected Total Loss (in dollars) as:
R*(D/N)*ARR

>R is # of Refund Requests
>D is total $ (or BTC) value of all transactions
>N is total # of transactions
>D/N is avg price for a transaction
(english: #ofRequests * AvgPrice * RefundRate%)

because of the way ARR is set-up, Total Loss is now independent of R, (ARR has R in the denominator) so with this formula your total loss should be stable.

Now let's take a look at Expected Total Loss %, and we can start making adjustments accordingly.

Expected Total Loss % = Total Loss/D = [R*(D/N)*ARR]/D = .03*.5 = .03 (expected) * .5 (fair) = .015 = 1.5% Expected Total Loss.

Our Total Loss is dependent upon what we choose for our expected and fair values in the ARR equation. With these sample #s, a vendor using this equation can expect to only lose 1.5% of his/her total sales to refunds. If you're willing to lose more in order to offer better service, you can see excellent results while appearing extremely generous.

My Example: Instead of 1.5% Expected Total Loss, I'll aim for 5%, since I think that's a fair amount to be allocated to mail mishaps and whatnot. to find my values for this the expected % of refund requests multiplied by the fair refund rate must equal .05.

I'll choose 1 (100%) for the fair refund rate
and .05 (5%) for the expected # of refund requests

Total Loss % = 5%. And with this I'm able to offer 100% refunds guilt-free as long as my %Chance of Refund Request is <= 5% (R/N) and offer only slightly lower discounts when there are more refunds than expected (can offer 50% refunds even when 10% of transactions are refunds)

Obviously every vendor situation is different (harder or easier to hide products, international vs domestic, etc), so adjusting this scenario should allow you to find a fair, adjustable refund rate.

**NOTE: ARR can possibly be greater than 1. Mathematically ARR would be replaced by 1 whenever it exceeds 1, as the max refund you can (and should) give is 100%. Whenever ARR is greater than 1, you provide a buffer of size (ARR-1) that you must reduce refund rate by before giving any less than 100% refunds. When ARR exceeds 1 and the "fair" value chosen is 1, expected total loss and %loss are overestimates.

ARR = (0.03/[R/N])*0.5

Please feel free to use this yourself and/or publish it for others to use.

Any thoughts/concerns with the equation? (maybe my math isn't 100% sound, I'm not perfect)
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: jpisbetterthanme on March 06, 2012, 04:51 pm
Interesting. Two things I thought I'd point out, though:
1- The success rate varies based on geographic data to a point where I think your formula might fall apart how it is :(  ..  You need some kind of table with alpha values comparing country-to-country success rates. Of course it also depends on vendors and such.
2- How about factoring in conditional probability?
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: LawlessLucy on March 06, 2012, 05:11 pm
Interesting. Two things I thought I'd point out, though:
1- The success rate varies based on geographic data to a point where I think your formula might fall apart how it is :(  ..  You need some kind of table with alpha values comparing country-to-country success rates. Of course it also depends on vendors and such.
2- How about factoring in conditional probability?

Though I'm not able to point out the flaw mathematically, I also feel that there's some reason that this doesn't work. Almost seems too simple. However, I think you can easily solve (1) by adjusting the expected % of refund requests based on where you're sending. If you're really organized, you could even separate your sales data into regions and analyze that way. However, I am personally only a domestic vendor, so those complications don't arise as much.

Also, you obviously can't calculate the exact probability for every geographic point in the world, which is why you use regions & statistics to generalize (I overgeneralized, taking the overall 97% success rate of SR sales)

As for 2, I had conditional probability on the mind during the whole derivation of the formula. However, one does not simply factor in conditional probability because it makes it seem more credible. I also sense the need for it as well. Any idea where I'd put it?

Thanks for the input.
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: LawlessLucy on March 06, 2012, 05:19 pm
the major flaw that I can identify:
ARR is the Adjusted Refund Rate for all refunds. However, if this decreases (through numerous refund requests most likely), past refund rates don't decrease, so the overall average refund rate is higher then the ARR. The only way this is accounted for is when refund rates increase (opposite behavior), so this will mostly only work with rates that are more fluctuating rather than going in one direction.

Also, b/c avg refund rate > ARR(of full set), the Expected % Total Loss is also higher, so 5% is no longer a safe estimate if you want to guarantee that you stay under 5% loss. You'd probably find this value by doing .05-2*stddev(all refund rates), which will hold true in 95% of cases. (68, 95, 99.7 rule)
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: jpisbetterthanme on March 06, 2012, 06:51 pm
Thinking about it more, my #1 and #2 are really the same point. I stand by what I said about alpha tables - conditional probability would be A|B as normal, but in this case one of the factors would be the alpha (representing the success rate based on geographic probabilities). And I'm not exactly sure whether the A or the B would involve the alpha factor (or if A|B =  alpha, for that matter).

Developing a chart is actually a lot more simple than you'd think if you only factor in geographic variation and exclude vendor by vendor discrepancies. And even if you do include vendor by vendor discrepancies it shouldn't add too much more variable calculus.

I'm not a total math nerd; my real area of expertise is in social science mathematics, which is almost entirely concerned with statistical analysis of samples/populations and correlation/regression analysis. A lot of that is done with tables, even though the tables generally sum up data from complicated integration (sometimes so complicated there literally is nearly no "real" explanation - see ERF [the "error function"] as it applies to integration of the normal curve).

For comparison check out the z-score table used in the integration of the normal curve, or the "r" tables used in a lot of social sciences to discuss strength of relationships/correlations.

By the way, doing this by hand is almost counterproductive beyond the theoretical construct. In school I had to do more than my fair share of simple population regression and it would take HOURS and HOURS with even the smallest of sample sizes =/

Hope this helps - I'm interested to see what you come up with since you sound like you know way more than I do about this. Always willing to learn from fellow geeks :)
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: LawlessLucy on March 06, 2012, 09:23 pm
Thinking about it more, my #1 and #2 are really the same point. I stand by what I said about alpha tables - conditional probability would be A|B as normal, but in this case one of the factors would be the alpha (representing the success rate based on geographic probabilities). And I'm not exactly sure whether the A or the B would involve the alpha factor (or if A|B =  alpha, for that matter).

assuming that each country (or location "x") has an alpha factor, which represents the probability of success from that location.
P(A|B) means the probability that A is true, given B.
(i.e. probability that:
 Delivery is a success(statement A), given the delivery is to location "X"(statement B))
=> the alpha value is P(A|B).

All this gives you is a bunch of constants. And if you have no data to compare that against, it's pretty useless (knowing that 99% of shipments to chile arrive doesn't help you adjust your refund rate when you're a domestic US dealer). If you did take the time to gather and compare all the data and construct the tables you speak of, you'd probably spend at least 10-20 minutes every time you recalculate, which is impractical. With my formula, plug, chug, and go, in 1 min you have your new refund rate, which is the point.

Even if your method (which I'm still not even sure what it is) kept your actual loss within 1% of your expected loss, mine would still keep my actual loss within some % of my expected loss (probably greater than yours), but unless the discrepancy is enormous, there's no need to over complicate things with all these tables.

practicality is an important feature.

I'm not a math nerd either, I'm just trying to come up with a practical equation for adjusting my refund rate based on those factors, to keep my losses stable, and to let me know when I can comfortably offer full refunds.

This all just stemmed from when people replied to my Product Offers thread (USDomestic LSD) claiming that I couldn't do math b/c I had a miscalculation in a hypothetical scenario. Needless to say I went on to show them I know math :P
Title: Re: ATTN SELLERS: Adjusted Refund Rate formula [mathemaGics]
Post by: jpisbetterthanme on March 06, 2012, 09:52 pm
I was suggesting an alpha factor for success from one country to another. So you'd have a list of countries along the X and Y axes of a table and have a different success rate in each spot.

I don't know where you'd get all this data from, though :-/