**Published on** March 15th, 2017 | *by sirtheta*

# Citi Double Cash: 2.00% or 1.98%? A Mathematical Analysis

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:

- [0.02
*x*+ 0.01*y*] / [*x*+*y*] ⇒ [2% of*x*in cashback] + [1% of*y*in cashback], all divided by [*x*plus*y*] yields an aggregate % cash back of… - [0.02
*x*+ 0.01*y*+ 0.01 * (*y*– 0.02*x*)] / [*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* <= 0.02*x*: linear monotone decrease on the interval [2.00%, 1.9803921568627450%]

For 0.02*x* < *y* <= 1.00*x*: linear monotone increase on the interval [1.9803921568627450%, 1.99%]

For *y* > 1.00*x*: 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* > 0.02*x*).

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* = 0.02*x* = $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* > 0.02*x* > $200, your aggregate % cash back is decreased ever so slightly below the aggregate % cash back for *y* = 0.02*x* = $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* = 0.020202*x*, 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* > 0.02*x*).

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!*

#370455My apologies if I missed this in the post, but given that I can direct deposit cash back from my Citi double cash into my bank account in a few business days, why would I ever pick the statement credit option? Thx!

#370482No worries—I covered that in a companion post, at https://www.doctorofcredit.com/get-2-citi-double-cash/

Basically, the cost is so small that it doesn’t really matter at all. You can request a direct deposit or a statement credit without needing to worry about the percentages. Whatever is easiest for you!

#370626Thanks! I have been doing direct deposit since I got the double cash ~18 months ago (very 1st time had to be a physical check, I believe) as that means that I get the money in a few business days. Even though the statement credit percentage is very similar, it doesn’t really change my bank account for 30+ days (as the credit affects the current statement and then there is a grace period before you have to pay the bill).

#370459tldr; If you were generating $100,000 of spend on this card annually, the maximum .02% application would be $20 less from $2,000 … but it is actually less than that Seems like a lot of math for such a small % 🙂

This math applies to Amex with Amex offers as well. If you have an Amex offer for $20 off $100 at Staples and use your Simply Cash card, you will get effectively 4% instead of 5% (.05 * (100-20)). I assume everyone is ok with this as a function of arbitrage.

#370485AmEx Offers are a bit different because you get a straight cash discount too. Using your example, you get $20 + 0.05 * $100 – 0.01 * $20 [+5% at office supply, -1% for the statement credit]. $24 / $100 is a % cash back of 24%, which is great 😀

#370494No doubt! The returns is great on Amex offer. But notice it is not 25% 🙂 The principle is exactly the same as the citi card. A “credit” that is applied is taken into account before the bonus is calculated. With Citi, your credit that is applied reduces the credit your would get for paying cash for the same thing.

My point is that I assume this form of arbitrage is understood and generally accepted as the small cost of the bigger gains.

#370478or just do statement credit when your balance is at $0 and request a refund check. Does anyone know if you use online bill pay to pay the citi double cash, say you pay $3000 when your balance is $5, does that money get 1%?

#370483You can actually just get the money direct deposited to a bank account (or a check cut) without needing to go statement credit @ $0 + refund check. I wrote that up in a companion post, at https://www.doctorofcredit.com/get-2-citi-double-cash/

For your other question: no. Citi states: “You will also earn 1% cash back on payments you make that appear on your current month’s billing statement as long as the amount paid is at least the Minimum Payment Due that is printed on your billing statement and there is a balance in the Purchase Tracker. The balance in the Purchase Tracker is reduced by eligible payments you make. When the Purchase Tracker reaches $0, you won’t earn cash back on payments until more eligible purchases are made.”

By my reading, there is a tracker of how much you’ve put on the card in purchases, and paying that off reduces the tracker. If you had a balance of $5 your purchase tracker would only have $5 remaining and you wouldn’t get more cash back for paying more.

#370501I’m confused. Does it mean that if I have $500 charged to my DoubleCash card but paid in full before statement closes (which means $0 balance), I will lose 1% for pay for the balance?

#370505No – Citi states: “You will also earn 1% cash back on payments you make that appear on your current month’s billing statement as long as the amount paid is at least the Minimum Payment Due that is printed on your billing statement and there is a balance in the Purchase Tracker. The balance in the Purchase Tracker is reduced by eligible payments you make. When the Purchase Tracker reaches $0, you won’t earn cash back on payments until more eligible purchases are made.”

So the 1% you earn is based on paying off the purchases you make, which Citi registers in the purchase tracker. It doesn’t matter when you pay them off 🙂

#370510Wow, that’s a lot of maths. From my understanding, if I purchase $100, then I will get 1% reward which is $1. And if I decided to use this $1 dollar as a credit and pay the rest $99, I will get $.99 in reward. In total, I get $1+0.99=$1.99

#370519Yep, that’s exactly how it works (of course, disregarding the $25 minimum redemption). In this case you get 1.99% instead of 2% 🙂

#370522All that math and then you didn’t even mention that the number gets rounded to nearest cent making anything past the third decimal point irrelevant. Shame! :p

#370536I was so taken with repeating decimal representations of fractions that I completely forgot about that!

#370540This was a needlessly complex exercise. If you redeem as a statement credit you wind up with 1.99% on average. Cash back is 2% on average. The remainder is rounded off.

No need for nth decimal outcomes.

I covered this in a comment on a travel with grant post when the card first came out a few years ago. It was also cited on Freequent-Flyer.

Others also made note of it.

I’m not sure why anyone would ever want a credit anyway unless speed of delivery is an issue.

#370565While the mechanics are somewhat well-known, I wanted to do a deep dive into the math for my own personal edification, plus repeating decimals are cool even if it takes 16 places and doesn’t matter 😉 It’s not meant to be taken

veryseriously!As mentioned in the comment thread on this piece’s companion post (https://www.doctorofcredit.com/get-2-citi-double-cash/#comment-370543), 1.99% won’t be a population-level average for redeeming for a statement credit unless you cash out by some other means, which kind of defeats the point of taking a statement credit 🙂

#370995loved this post, and thank you for the maths. i do however, think we need some new card with a juicy bonus instead.

#370574Lawsuit!!!!. Lol

#370581Wow, hard to believe I came up with the 1.99 by logic a year ago. Of course, my problem is more with the purported 2.something cents per $ travel benefit from Barclays, where the smallest redemption can still leave points left over if you try to do everything right before cancelling the cards.

#370583Yeah, the Arrival+ is more insidious due to the staggering $100 redemption minimum. With the Citi Double Cash, you always get a flat 2% if you don’t redeem it for a statement credit and Citi makes it really easy to do so.

#370590My 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.

#370610Lol

#370675See 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 !

#371992Only 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…

#372027Well, 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.

#370743This is the useless post ever.

#371235I mean, yes, I fully acknowledge that this is very niche and is pretty useless—but I find it interesting 😉

#370767This 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.

#371230That’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 🙂

#370846Thanks for this Sirtheta I found it really interesting !!

#371231I’m glad someone else enjoyed it 😀

#370866Are you kidding me? You have nothing to write about today. Sad

#370890Are 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.

#371197To 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.

#371241Obviously, 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.

#371006This 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.

#371927And it’s up 🙂

https://www.doctorofcredit.com/bank-america-better-balance-rewards-program-rules-cashback-equivalence-opportunity-cost/