# Credit card math: Chase Sapphire Reserve or Citi Double Cash?

## 2017/06/12

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I was talking with a friend about how to choose between credit cards, and I realized that the math is sufficiently complicated that it would be nice to write about it here.

I assume that you are a person with good credit who pays your entire credit card bill every month. For people in this category, credit cards are a scheme for just making money, but you need to be smart about it.

Some cards come with a joining bonus. The Chase Sapphire Reserve made a splash for having something like 50,000 bonus points if you spend a certain amount within a certain period. (In general, a “point” is worth $0.01 in cash, travel, or whatever.) This means using the card can get you$500 in free money. After you get the bonus, should you keep the card? In other words, ignoring the bonus, is the card worth it?

This comes down to a few factors:

• How many points do you collect? I call this the collection multiplier. For example, the Reserve gets 3 points per $1 spent on travel and dining and 1 point for every other$1 spent, so the multiplier is 3 for one part of your spending and 1 for the rest.

• How much cash do you get for your points? I call this the redemption multiplier. For example, the reserve says you can get “up to” 50% extra redemption on travel. As I mentioned above, the base rate if $0.01 per point, so the 50% extra redemption means$0.015 per point.

• What’s the annual fee? For the Chase Reverse, this is about $400, but they give you a$300 bonus if you spend money on travel, so I’ll say it’s $100. The number of points $$P$$ that you get depends on the total $$T$$ amount you spend in a year, the collection multiplier $$c_i$$ that you get for purchases in category $$i$$, and the fraction $$f_i$$ of your total $$T$$ that you spend in each category. For example, for the Chase Reserve, you get $$c = 3$$ for the fraction $$f$$ that you spend on travel and restaurants and $$c = 1$$ for the remaining $$1 - f$$, so $$P = T \left[ 3f + (1 - f)\right] = T (1 + 2f)$$. More generally, you would write $$P = T \sum_i c_i f_i$$. The amount of cash $$C$$ that you get depends on the number of points $$P$$ and the redemption multipliers. Most of the cards I look at give you a base rate redemption of$0.01 per point and some higher rate for redemption on travel. I’ll assume that I spend enough of my budget on travel that I can redeem all the rewards at the higher rate. For example, with the Chase Reserve, you get a number of points that’s somewhere between 1% and 3% of $$T$$. Because I spend more than 3% of my budget on travel, I can redeem all those points at the higher rate. So in general you would need to write the cash reward $$C$$ as some weighted sum of the redemption categories, but I’ll just assume you get everything redeemed at the highest rate $$r$$ so that $$C = rP$$. For the Chase Reserve, $$r = 0.015$$.

As an aside, I’ll note that the Reserve lets you transfer your points to airline miles rather than cash. Comparing cash and miles requires assigning a cash value to miles. For me, miles feel almost worthless: I’ve only been able to buy a flight with miles every few years. In general, the value of miles seems to be less than $0.02 per mile. So you might argue that the Chase Reverse redemption multiplier is closer to $$r = 2 \times 0.015 = 0.03$$. Putting this all together, the total cash that you get from a card is $$C$$ minus the annual fee $$F$$. For the Chase Reserve, this could be as high as $$rP - F = r (1 + 2f) T - 100$$. If you don’t spend any money of the card, then $$T = 0$$, and you just end up paying the annual fee $$-100$$. If you spend a lot of money, then $$T$$ is so large that the points overwhelm the fee. The break-even point, when $$rP - F = 0$$, depends on $$r$$, $$f$$ and $$T$$. Let’s say you spend a generous quarter of your money on travel and restaurants (i.e., $$f = 0.25$$). Assuming the very generous $$r = 0.03$$, then the break-even point is$2,222. (If you spend more than this, the card is better than no card at all.) Conservatively, if you get take cash rewards and don’t redeem on travel, then $$r = 0.01$$, and the break-even point if $6,667. This is a pretty manageable number for a graduate student/postdoc making smaller-than-median salary. However, it’s unfair to compare one card to no card at all. The real question is whether this card is better than a different, competing card. I’ll compare the Chase Reserve against the Citi Double Cash. The way they advertise the Double Cash is confusing, but essentially you get 1 point per dollar ($$c = 1$$) and$0.02 per point ($$r = 0.02$$) with no annual fee ($$F = 0$$), so the total cash you get is $$0.02 T$$.

How does this compare to the Chase Reserve? For small $$T$$, the Chase Reserve costs money (the annual fee), and the Double Cash costs nothing. For large $$T$$, the Reserve’s annual fee is negligible and the Reserve’s higher collection/redemption multiplier beats the Double Cash. Where is that break-point $$T$$? Let’s keep $$f = 0.25$$. For the generous $$r = 0.03$$, then the break-even point is the reasonable $4,000. For a more conservative $$r = 0.015$$, then the break-even point is$40,000, which is only a little lower than the NIH postdoc salary.

The break-point is more sensitive to the redemption rate $$r$$ than the fraction $$f$$ you spend on travel/restaurants. For the generous $$r = 0.03$$ and a conservative $$f = 0.05$$, the break-even point is \$7,700. So whereas halving $$r$$ from $$0.03$$ to $$0.015$$ increased the break-point by a factor of 10, dividing $$f$$ by 5 (from 25% to 5%) only doubled the break-point.

What’s the bottom line? If you get good value out of your airline miles, then the Chase Reserve is a good deal. If you redeem travel through the card, then you’re doing better than having no card at all, but you’ll need to spend a lot of money to do better than the Double Cash. If you just use the card to get cash rewards, then the Double Cash is better.