Determining Whether Your Rewards Spending Outweighs an Annual Fee

In my review of the Blue Cash Preferred, I broke down what level of spending you needed to have in the groceries category to outweigh the annual fee, as compared to various other cashback rates you can attain on groceries. In this post, I would like to explain how you can do this for any card, for any spending. There are any number of calculators that will help you do this, but I think the calculation is easy enough in the general that you don’t need them. Note that this type of analysis only applies to whether you get more cashback from spending in the appropriate categories—it cannot take into account the other benefits of a card, which can significantly change your results.

Background & Results

The calculation of whether the cashback you can from an annual fee card outweighs the cashback from a no-annual fee card (or a reduced annual fee card) is actually a very simple algebra problem. We shall define some variables to make this clear.

p.af is the cashback rate, expressed as a decimal, on the card with an annual fee. p.naf is the cashback rate, expressed as a decimal, on the card with no annual fee. s is your spending in some category that earns the cashback rates of p.af and p.naf. Lastly, a is the annual fee for the card with p.af cashback, and r is the reduced annual fee for the card with p.naf cashback (obviously, r is $0 for a card with no annual fee).

Were this structured as a word problem, the question of determining a break-even point in spending would be: What amount of cashback (spending * percent cashback as a decimal) for a card with an annual fee of a is greater than or equal to the amount of cashback (spending * percent cashback as a decimal) for a card with an annual fee of r, where the annual fee r is less than the annual fee a? In equation form, we would write this as: s * p.af – a ≥ s * p.naf – r.

Grouping like & like (cashback vs. annual fees) can be done by subtracting s * p.naf from both sides of the equation, and adding a to both sides of the equation, which gives us: s * p.af – s * p.naf ≥ a – r. Using the distributive law, we can then write s * (p.af –p.naf) ≥ a – r.

The Final Equation

Since the goal is to find the level of spending that gives a break-even point, we do one last thing to isolate spending on the left-hand side of the equation and divide both sides by (p.af –p.naf): s ≥ (a – r) / (p.af –p.naf). In other words, your spending must be greater be greater than or equal to the marginal annual fee (i.e. the amount of annual fee greater than the annual fee you would pay on another card) divided by the marginal percent cashback (i.e. the percent cashback greater than the percent cashback on another card).

You must be careful to take into account spending caps on categories, as a result that requires spending more than the spending cap means that it is impossible to break even.

Simple Example: Costco Executive Membership vs. Costco Membership

I believe this is the simplest possible example because it is a binary choice without any other factors: you can either pay $55 / year for a Costco Gold Star membership, which comes with 0% cashback, or you can pay $110 / year for an Executive Gold Star membership, which comes with 2% cashback on eligible purchases. This cashback is independent of any credit card rewards. Since you must pay $55 for a Costco membership in any case, the marginal annual fee is $55. Since you get 0% cashback for a Costco membership, the marginal percent cashback is 2%.

$55 / 0.02 ≤ $2,750 is the amount you must spend on eligible purchases at Costco to break even on an Executive membership as compared to a regular membership.

Note that taxes on membership will increase the marginal annual fee (h/t reader Eric).

Complicated Example: Chase Sapphire Reserve vs. Chase Sapphire Preferred

This example is a bit trickier and the actual result will depend on your personal perspective. The Chase Sapphire Preferred has a $95 annual fee after the first year and gives 2x on dining & travel. The Chase Sapphire Reserve has a $450 annual fee and gives 3x on dining and travel. To truly evaluate the earning rate on these cards, we must take into account the consistently cash-able benefits of each card & assign a valuation to Ultimate Rewards points.

First: most would agree that the effective annual fee of the Reserve is only $150 because of the $300 annual travel credit. In order for this to be true, there is an implicit restriction that I will make explicit: this is only the case when you are book travel that you already would’ve booked & would’ve booked with cash. note: the annual fee is canceled by the travel credit for the first two years, so the cashback comparison needn’t be applied at that point. This calculation is for year 3 of card membership and beyond.

Second: the Reserve sets the floor of UR point redemption at 1.5 cents per point while the Preferred sets the floor of UR point redemption at 1.25 cpp. The enhanced cashback rate is thus 4.5% on dining and travel for the Reserve and 2.5% on dining and travel for the Preferred. However, because both cards gives you access to travel partners, it would be equally valid to value UR points the same for both cards—a reasonable value would be 2 cpp, yielding 6% on the Reserve and 4% on the Preferred.

The marginal annual fee for the Chase Sapphire Reserve is thus $55. The marginal percent cashback is 2% for both of the valuations we have used. As you may recognize from the previous example:

$55 / .02 ≤ $2,750 is the amount you must spend on travel & dining to break-even on the Reserve as compared to the Preferred.

Now, if you were to use a valuation of 1.5 cpp for both the Reserve & Preferred, you would get 4.5% on the Reserve and 3% on the Preferred, which would give you:

$55 / .015 ≤ $3,666.67 is the amount you must spend on travel & dining to break even on the Reserve as compared to the Preferred.

Further discussion is warranted if you believe that UR have different valuations and you have, for example, a Chase Freedom. Using an enhanced cashback rate for the Freedom of 7.5% with the Reserve (1.5 cpp * 5x) and 6.25% (1.25 cpp * 5x) with the Preferred:

$55 / .0125 ≤ $4,400 is the amount you must spend in the quarterly categories at 5x to break even. You do not even need to spend any money on travel & dining with the Reserve to offset the annual fee in this scenario.

Were you crazy enough to want to know how different valuations affect the Chase Freedom Unlimited, it’s 2.25% on the Reserve and 1.875% on the Preferred:

$55 / .00375 ≤ $14,666.67 is the amount you must spend on the Freedom Unlimited to break even.

By design, this does not take into account any of the variable cash-able benefits & non-cash-able benefits of the Reserve, which Will has covered here.

Digression: Multiple Cashback Rates & Hyperplanes

The main thrust of this post is above, and you needn’t worry about this section. But in case you’re interested: the break-even points when comparing one card to multiple cashback rates can be calculated, but this quickly spirals out of control because each cashback rate requires a separate variable. Computer scientists & those who’ve taken multivariate may recognize that multiple cashback rates is, in essence, a constraint optimization problem, and its solution is a hyperplane. In simpler terms, the number of cashback rates (i.e. the number of variables) is the number of dimensions it takes to solve the problem, and the solution is in one less dimension. Solving a problem for two cashback rates requires a 2-dimensional space (the Cartesian plane) and the solution is a line. Solving for three cashback rates requires a 3-dimensional space and its solution is a 2-dimensional plane of values. Beyond that, you can’t even visualize the solution.

(For fun, you can plug the following into Wolfram Alpha: .02 * x + .0125 * y ≥ 55 [this article’s thumbnail & solution for spending on the Reserve + Freedom at 1.5 cpp vs 1.25 cpp for the Preferred]; .02 * x + .0125 * y + .00375 * z ≥ 55 [solution for spending on the Reserve + Freedom + Freedom Unlimited at 1.5 cpp vs 1.25 cpp for the Preferred]). note: Wolfram does not graph the solution bounded by x > 0 and y > 0, so the graph looks a little bit wonky; the y-intercept is $4,400 and the x-intercept is $2,750 for the thumbnail.

This does not mean that you can’t calculate which card is better for your spending, however! Instead, you must reduce the dimensions of the problem, by fixing a variable. Instead of allowing your dining & travel spending to be any number, you can set it to a certain value, such as $2,000. (in the Reserve + Freedom example, this leads to the result that 0.02 * $2,000 + 0.0125 * {spending in Freedom 5x categories} ≥ $55, or: you need to spend $1,200 in the 5x categories on the freedom to balance the increased AF).

This is why proper budgeting is, in my opinion, the key to optimizing your credit card rewards. It eliminates any guesswork and tells you right away where your rewards spending is better.

 

Questions? Drop them below.
    sirtheta
  • On January 16, 2017, 10:15

View Comments (26)

  • You can type multiple inequalities to Wolfram Alpha separated by ";"
    try:

    .02 * x + .0125 * y + .00375 * z ≥ 55; x ≥ 0; y ≥ 0; z ≥ 0

    Although in this case it does not draw a shiny chart (at least for me), it does solve the system.

    • Ah, I did that minus semi-colons. It does cut out the incorrect information (-$2,000 in spending!), but the graph looks even harder to read for me :/

  • Excellent work sirtheta.

    I have a question though, shouldn't the metric be CSR (or CSP) vs. Citi DC?
    I think that's the question more people such as myself have a hard time answering. In addition, INK+ keep or cancel without a good way to MS heavily...

    Anyways, I enjoyed this series and the previous one on BCP.

    Thank you!

    • That's a perfectly valid comparison; it just depends on what you're looking for. Many people value the CSR & CSP for their benefits (transferring to travel partners, primary rental insurance, various other insurances, etc.), and since they are (largely) redundant, a common question is whether the CSR or CSP makes more sense. So, a useful starting point (but by no means the whole picture, given the CSR's better benefits such as GE reimbursement and lounge access) is the rate at which they earn UR. Ultimately, a proper evaluation of which card is a keeper will rely heavily on your valuation of the benefits (although if you spend enough on travel & dining, I think the CSR is a clear winner no matter how you value the benefits!).

      Valuing UR at 2 cpp, you can use the method in this post to find that $150 / 0.04 ≤ $3,750 is the amount you need to spend on travel & dining to make the CSR better than the Citi DC from a rewards perspective (but obviously, this doesn't take into account the benefits).

      As for the Ink+, it's not something I have a lot of experience with. It's most valuable for MS, so if you aren't putting enough spend on it, it's not a keeper. Since you aren't comparing it to any card, you would just want to offset the AF: $95 / {your UR cpp valuation * 5x}.

      You're very welcome :)

  • I have the Costco exec membership and was told that I'd be reimbursed for the annueal fee differences if I didn't spend enough to earn in rebates to justify to annual fee difference amount every year. I thought this was only for the initial year, but the people assured me it's every year. I guess I'll find out.

  • Is it safe to conclude that the result of the breakeven point is heavily influenced by the individuals valuation of points/miles and the individuals valuation of the card benefits? I mean, with straight cashback cards, the calculations are easy. But some things you've left out in the CSR vs CSP analysis are:

    $85 (or $100) Precheck (or Global Entry) reimbursment. This would effectively reduce the CSR annual fee by 85/5 or 100/5 depending on which is used.

    Subtracting (up to) $1500 in restaurant spend because you'd be using Freedom for a quarter regardless of which card you have.

    An individuals valuation of Priority Pass membership - I don't think I've ever seen anyone apply a valuation to this but if you're going so far as to start mentioning hyperplanes, I think adding in valuations for this should be done.

    If an individual has Ink Preferred for legitimate business spending then they will inherently get 3x on travel (with the assumption that the annual fee is ignored because it is a business expense).

    • Is it safe to conclude that the result of the breakeven point is heavily influenced by the individuals valuation of points/miles and the individuals valuation of the card benefits?

      Quite definitely. Since individual valuations of the benefits vary so widely, it's hard to accurately compare them on a general level. But the rewards aspect is a good starting point. (for example, while GE is worth $100, would you really get that over TSA Pre if you were paying in cash? for most people, the answer is no, and that benefit is in reality only worth $85 instead of $100. and for those who wouldn't even pay for TSA Pre [me], the benefit is really worth $0—it's a nice perk, but you can't take it into account if you wouldn't pay for it in the first place).

      As for the restaurant spend in conjunction with the Freedom, you would need to subtract that, but it's equivalent to think about it more like "I need to spend $2,750 on travel & dining for 9 / 12 months" instead of having the spend be per year.

      The ultimate determination is going to be a more wallet-wide comparison rather than 1:1, but I like this framework as an initial determination of what you need to get out of a card.

      • "for example, while GE is worth $100, would you really get that over TSA Pre if you were paying in cash? for most people, the answer is no, and that benefit is in reality only worth $85 instead of $100. and for those who wouldn’t even pay for TSA Pre [me], the benefit is really worth $0—it’s a nice perk, but you can’t take it into account if you wouldn’t pay for it in the first place"

        Technically, the perk is worth whatever you would pay for the benefit. If you could pay $10 for TSA Pre, then would you pay cash for it? If so, then the perk is worth at least $10 to you.

        • "Technically, the perk is worth whatever you would pay for the benefit."

          This is a good point, also useful when considering the value of a particular redemption. I wouldn't pay $8000 more to fly first to Asia, but I might pay $400. That dramatically changes the value of redemptions for me.

  • As a math major, I very much appreciate this kind of analysis. I think its a good framework for initial thinking.

    Of course, as a churner, I have to point out that it's even more complicated...fee cards may also have more perks besides higher levels of cashback, and you also can have other cards that could intersect the spending categories at odd times (like changeable categories that Discover offers). And if you acquire new cards during the year you may end up diverting spending to those cards to meet the bonus requirements.

    For me, personally, I've found that I get annual fee cards if:
    1. they offer an annual bonus that is attractive enough for me (like the IHG card)
    2. their bonus is more than enough to cover the cost of the fee and I have no plans to keep the card

    Rarely do I see that real life (aka organic) spend justifes a fee card versus a no fee card, but of course if you manufacture spending, then all bets are off.

    • Thanks, I had intentionally excluded benefits from analysis in my head but forgot to mention it in the post—I've added such a disclaimer to the intro. This is not a good framework for analyzing & comparing benefits, of course, but I agree that it is one of the best starting points for comparing cards (before factoring in benefits). It's also one of the major selling points of the Reserve: it's that good before factoring in benefits that it still makes sense to have one!

  • Costco charges tax on the membership (at least where I live) so your calculation is slightly off. It's not a big deal but it will change your math a bit.

      • Costco also guarantees a $55 return. If your CB is less than $55 from the Exec upgrade, they will up your rebate check to $55.

        Even so, I love this post. This is exactly what I wanted to see on the requests page. Can you go category by category and compare the best no-AF card to the best AF card and determine if card X is worthwhile to have?

        • A few other comments have noted this and I must confess I am a bit baffled. I don't understand how they make money off of Executive Memberships! Maybe that's the point?!

          That would be quite a herculean task, but I will certainly put it in my ideas list—hopefully I can have a post or two about this in the next month or so.

          • There are probably enough people who shift extra spend to Costco for the 2% to more than make up for the people who don't and get their rebate check upped to $55.

  • I love that you took the time to go through this. But, TBH, I had a hard time following this (once you got to "hyperplaning" my mind exploded), and I've got a PhD in biomedical sciences. Maybe you could make some sort of app that people could plug in there numbers to come up with the answer...i.e. a simple plug-n-chug. Don't stop getting into this much detail though. Even if I couldn't/had a hard time following it, I still appreciate it.

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