Published on March 15th, 2017 | by sirtheta44
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.
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]
|0.02x||0.02x||[0.02x + 0.01y] /|
[x + y]
|[(0.02 + 0.01 * 0.02) * x] /|
[(1 + 0.02) * x]
|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]
|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]
|1.00x||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]
|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]
|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]
|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]
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!