Spoiler: it’s anywhere between 1.98% and 2.00%!
This was originally part of “How Do You Get 2% on the Citi Double Cash?”, but it’s so niche that I decided to split it out entirely. As mentioned in that article, there’s one little quirk with redeeming cash back as a statement credit for the Citi Double Cash: the redemption does not count as paying for the purchase and does not give you an additional 1% cash back on the amount you redeem. Head over to that article to see details on other redemption options.
What follows is a mathematical analysis of the % cash back rate of the Citi Double Cash in the statement credit case.
Contents
1.98%?
I’ve seen 1.98% in various places and never gave it much thought. It makes intuitive sense: if you charge and pay off a set of purchases that cost x, you will have 2% of x in cash back. Redeeming that cash back as a statement credit means that you are forsaking 1% of the amount redeemed, r; 1% of 2% is 0.02%, which when subtracted from 2.00% yields 1.98%.
Observant readers may have noticed that I glossed over a variable there. When you forsake 1% of r, you aren’t forsaking 1% of 2%, but 1% of 2% of x (a set of purchases that cost x)—on some set of future purchases that cost y. This changes the numbers, and it turns out that you always get more than 1.98%—in aggregate. More on this in the next section!
What about not in aggregate? In other words, what about the cash back rate just for that future set of purchases that cost y? If y is less than or equal to the cash back redeemed, you earn 1%. If y is equal to x and you redeem the cashback from x (2% of x) and the cashback from y (1% of y), you earn 1.97%. If y is equal to x and you only redeem the cashback from x (2% of x), you earn 1.98%. This brief analysis is entirely useless (since the only number that matters is the aggregate cash back), but it’s mildly interesting that this is the question that the intuitive value of 1.98% is answering.
Analysis of Aggregate Cash Back Percentage
So, moving on from a naïve analysis, what’s our aggregate % cash back? We’ll continue to keep things simple by only considering two sets of purchases at a cost x and a future cost y, and express y in terms of x. Recall the following: you always earn 2% of x in cash back and you always earn 1% of y in cash back. If y is greater than the amount you are redeeming, r, you additionally get 1% of (y – [2% of x]) cash back from paying off the remaining balance of the purchase not covered by your redemption amount.
You may find this easier to follow by substituting a nice big whole number for x, such as $10,000. That would make y = $100; $200; $1,000; $5,000; $10,000; $40,000; $990,000; and $9,9990,000.
As for the equations, they can be interpreted as follows:
- [0.02x + 0.01y] / [x + y] ⇒ [2% of x in cashback] + [1% of y in cashback], all divided by [x plus y] yields an aggregate % cash back of…
- [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] ⇒ [2% of x in cashback] + [1% of y in cashback] + [1% of (y – [2% of x])], all divided by [x plus y] yields an aggregate % cash back of…
| y | r | Base % Eq. | Simplified % Eq. | % Cash Back |
| 0.01x | 0.01x | [0.02x + 0.01y] / [x + y] | [(0.02 + 0.01 * 0.01) * x] / [(1 + 0.01) * x] | 1.9900 |
| 0.02x | 0.02x | [0.02x + 0.01y] / [x + y] | [(0.02 + 0.01 * 0.02) * x] / [(1 + 0.02) * x] | 1.9803921568627450 |
| For y > 0.02x we need to include an additional term in our base equation to account for the 1% from paying off y. | ||||
| 0.10x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 0.10 + 0.01 * (0.10 – 0.02)) * x] / [(1 + 0.10) * x] | 1.98181 |
| 0.50x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 0.50 + 0.01 * (0.50 – 0.02)) * x] / [(1 + 0.50) * x] | 1.986666 |
| 1.00x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 1.00 + 0.01 * (1.00 – 0.02)) * x] / [(1 + 1.00) * x] | 1.99 |
| 4.00x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 4.00 + 0.01 * (4.00 – 0.02)) * x] / [(1 + 4.00) * x] | 1.996 |
| 99.00x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 99.00 + 0.01 * (99.00 – 0.02)) * x] / [(1 + 99.00) * x] | 1.9998 |
| 9999.00x | 0.02x | [0.02x + 0.01y + 0.01 * (y – 0.02x)] / [x + y] | [(0.02 + 0.01 * 9999.00 + 0.01 * (9999.00 – 0.02)) * x] / [(1 + 9999.00) * x] | 1.999998 |
note: as reader Aahz mentions, all these decimal places are unnecessary since currency units only go to cents. I’ve included them because repeating decimal expansions are cool & they provide some differentiation in the table.
That’s a lot of math but the conclusion is pretty simple: if you make a set of purchases that cost x and redeem the 2% cash back from x against a future set of purchases y, the minimum aggregate % cash back is 1.9803921568627450% (which occurs when y is the exact amount of cash back you have to redeem) and, disregarding y > x, the maximum is 1.99% (which occurs when y = x). Summarizing:
For 0 < y <= 0.02x: linear monotone decrease on the interval [2.00%, 1.9803921568627450%]
For 0.02x < y <= 1.00x: linear monotone increase on the interval [1.9803921568627450%, 1.99%]
For y > 1.00x: linear monotone increase on the interval [1.99%, 2.00%)
Below the Bounds?
It is theoretically possible to dip slightly below these numbers.
So far, we have limited our analysis to redeemed cash back only from the set of purchases that cost x, without considering redeeming the 2% cash back from x and the 1% cash back from y then paying off the remaining balance of y. If y = x, this puts your aggregate % cash back at 1.985% instead of 1.99% and smoothly decreases your percentage elsewhere (for y > 0.02x).
This also creates a discontinuity in the intervals as compared to our previous analysis. Taking actual numbers this time, let us proceed from the minimum cash back rate in the previous section. Set x = $10,000 which means you have $200 in cash back. If y = 0.02x = $200, you can redeem all your cash back and your aggregate % cash back is 1.9803921568627450%—as expected, since you can’t redeem the $2 in cash back from y because y isn’t greater than the cash back from x.
However, for values of y > 0.02x > $200, your aggregate % cash back is decreased ever so slightly below the aggregate % cash back for y = 0.02x = $200 (for a little while)—due to the 1% of y that you can now redeem. I believe the following to be true: x = $10,000, y = $202.02; cashback = $200 from x + $2.02 from y. Redeeming $202.02 in this scenario yields a total cash back rate of 1.980196% (10101 / 510101 — a rational that starts to repeat 510,100 digits behind the decimal place).
Going back to variables, this minimum is y = 0.020202x, which can be conceived as [2% of x] + [1% of 2% of x] + [1% of 1% of 2% of x]. Due to cents being the minimal unit of currency, you can’t go lower than that.
So What Does It All Mean?
Applying this to the real world, you can of course consider 1% cash back deriving from many purchases as one set of purchases y and then 2% cash back deriving from many purchases as another set of purchases x and come to the same conclusion as above. (Though you’ll obviously need to shift some of x to y in the case that y > 0.02x).
But, realistically, there’s very little point in doing any such analysis. Mostly, I found the specifics of the aggregate % cash back very interesting and wanted to write up my thoughts. Hopefully, someone aside from me will find the copious amounts of digital ink I spilled here interesting as well!
Questions, comments, etc. can be dropped below!
h/t forgotten commenter who asked me to cover this months ago.
drop a line below and I’ll credit you!
View Comments (45)
spend = pay + .01*spend + .01*pay
pay = .99/1.01*spend
cashback is 2%/1.01
DOC - PSA : If you pay more than the balance you lose the 1% on the payment. There is a purchase tracker that tracks how much your net purchases are (Previous payment tracker + Current settled Purchases - Current Returns). However, there is no payment tracker as such. Say, you make a big purchase ($1000) and pay your credit card to keep you debt to limit ratio low, and the purchase actually settles next cycle. Your payment will be tracked against purchase tracker and you'll get 1% of it. So, the $1000 'overpaid' doesn't get 1% back. However, Citi conveniently doesn't track payments so the next month, you'll get 1% of your net purchases that month + 1% of your payment that month (tracked against the purchase tracker). You'll be left with the $1000 on your purchase tracker that you can never use.
@will @sirtheta this is essentially what I am asking in my other post, and much more important of a situation than the difference between accepting a statement credit over a check. Thanks in advance if you choose to do the math.
Rajansv- If I understand what you're saying, at the most, I should only pay the statement balance every month? Is that correct, or is there more to the timing of when my purchase and payment settles? TIA
Seriously this one makes me hate math for the first time. I didn't read the article at all.
Doesn't read article, still hates post.
Just had my Citi account shut down yesterday and no one in customer service can tell me why. I have to write a letter to their Presidential Office lol. Might lose about $150 in cash back now because of it. I don't care that they closed my account because I'll just switch back to my Quicksilver account that never gave me a problem.
I've had credit cards for about twenty years and this has been the most frustrating card to have when it comes to customer service.
If you don't care, why are you posting?
This suggestion should be easy, based on the complexity of your DC post: how about a post with a table/chart about diminishing rate of returns as spending increases on the BBR.
Most people are just putting $5 (or less if they can get by without having the small amount forgiven) each month, making it 200% cash back. I calculated the indifference point of using a 2% card at $6000 in a year (if a BoA member getting the extra $20 per year).
The equation I used was 120/(total amount spend per year). I'm sure you can spice it up and make it more interesting. Readers might not find the post very interesting, though, judging by the reaction of this post.
And it's up :)
https://www.doctorofcredit.com/bank-america-better-balance-rewards-program-rules-cashback-equivalence-opportunity-cost/
Are you kidding me? You have nothing to write about today. Sad
Are you kidding me? You have nothing constructive to say. Sad. We wrote like 15+ new posts today, sorry if you didn't love all of them. This sort of stuff actually matters to people doing significant volume.
To anyone doing volume, they already know this. To everyone else, this is just pointless math that is done to look impressive. As other readers have said, rounding negates the majority of the posted figures.
Obviously, I disagree that it's "pointless math that is done to look impressive". I think it's pointless math that happens to be interesting to me (and hopefully a few other people), and so I wrote about it. I don't think there's anything wrong with writing up an extremely niche subject like this.
Thanks for this Sirtheta I found it really interesting !!
I'm glad someone else enjoyed it :D
This post could be improved by starting from scratch... Saying.
Warning to readers! Do Not use the DC's Cash back as a statement credit or it will Yield 1.98% because you don't earn that extra 1% on the cashback used as a statement credit. Instead have it transferred to your bank account and pay it with that... or a discover checking account to yield a free .10$. The end.
That's what the companion post mentioned in the introduction is for, which you can find at https://www.doctorofcredit.com/get-2-citi-double-cash/
I split this post from that companion post precisely so that people could ignore this one if they weren't interested in a stupidly deep mathematical analysis :)
This is the useless post ever.
I mean, yes, I fully acknowledge that this is very niche and is pretty useless—but I find it interesting ;)
My analysis concluded that this article is a huge waste of bytes and everyone's time, and your talents would be better applied towards other subjects.
See the irony.. cool people will like Sirtheta's this post which is full of his math madness as a one off post, but people who count each byte of space and second of time will dislike it inspite of his 17th decimal level calculations !
Only if you use Alanis Morisette's definition of "irony".
You'd hope it'd be one off, but he just did another one today on BBR...
Well, I see the posts as distinct in purpose (but obviously, various readers may have a different point of view!).
This post has no real usefulness aside from being a theoretical exercise. Some people may find it interesting (I did, which is why I wrote it :D), but there's absolutely no practical implication aside from the fact that you don't maximize cash back if you use the statement credit option (a trivial result that doesn't require any real analysis).
On the other hand, I felt that the BBR post had some non-trivial implications in terms of what amount of recurring charge (or spend) is "optimal" and at what level it doesn't make sense to worry about optimality. In that sense, I felt that the BBR post presented a result that people may find useful & that they can actually use.
Lol