Reader Nate had a very interesting suggestion in the comments of my “Citi Double Cash: 2.00% or 1.98%? A Mathematical Analysis” post (warning: that post is quite niche):
How about a post with a table/chart about diminishing rate of returns as spending increases on the BBR [Better Balance Rewards card]?
Common sense would dictate that you want to put as little on a BBR as possible in order to get the rewards, and this is absolutely the case (what is meant by “as little as possible” is covered in the “Better Balance Rewards Basics” section). Even though it’s common sense, I’ll break down the cashback equivalence of the BBR and its opportunity cost in subsequent sections.
Contents
Better Balance Rewards Basics
We’ve covered the Better Balance Rewards card in previous posts, including this one two months ago, but I’ll recap the program rules as an introduction. If you pay more than the minimum amount due and make your monthly payments on time, Bank of America (BOA) gives you $25 per quarter. If you have a BOA (or Merrill Edge) checking or savings account, you get $30 per quarter. To be eligible for the bonus you must have a balance on your account in every month in a calendar quarter, or use your card every billing cycle of every month in a calendar quarter.
Do note that “paying[ing] more than the minimum amount due” (the description BOA uses) is not correct and introduces confusion. If it were true, putting $10 on the card and paying $10 on time would not count since that is not more than the minimum amount due ($10), yet many users do just that. The Better Balance Rewards Program Rules (accessible via this page) give a much clearer picture:
- You obtain a quarterly cash reward ($25 or $30) by making qualifying payments each month of a calendar quarter, where qualifying payments are
- A monthly payment received on or before the due date, which is more than the Minimum Payment due, OR
- A payment in full received on or before the due date.
Since payments in full count, you can charge as little as you wish and still get the bonus. In accordance with this PSA about using $5 charges on the BBR, you don’t want to go too low – BOA has been known to forgive charges up to $4, which would result in you losing out on the quarterly cash reward for a given quarter.
You can learn more about the BBR by reading Chuck’s post, which mentions some other important details (for example, if you receive the quarterly cash reward as a statement credit, you’ll need to make sure to charge more to the card to eliminate the negative balance!). (thanks to reader JP for reminding me about this!)
Cashback Equivalence
Back to Nate’s comment:
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).
Let’s take a look at the % cash back equivalence of the card using various charges per month. (I’ll start low, even though the amount might be forgiven—do keep that in mind!) The $ per year you put on the BBR card can be distributed in any way as long as it results in a payment due ($90 per month for 6 months and $60 per month for 6 months is the same as $75 per month). ($ per month is a terrible way to organize the table for high levels of spend, but I think it’s the most logical way to organize the information given the program rules and the fact that low levels of spend are typically recurring charges.)
| $ / mo ($ / yr) | % @ $25 / qtr ($100) | % @ $30 / qtr ($120) |
| $0.50 ($6) | 1,666.66% | 2,000% |
| $1.00 ($12) | 833.33% | 1,000% |
| $2.50 ($30) | 333.33% | 400% |
| $5.00 ($60) | 166.66% | 200% |
| $7.50 ($90) | 111.11% | 133.33% |
| $10 ($120) | 83.33% | 100% |
| $15 ($180) | 55.55% | 66.66% |
| $25 ($300) | 33.33% | 40% |
| $50 ($600) | 16.66% | 20% |
| $75 ($900) | 11.11% | 13.33% |
| $100 ($1,200) | 8.33% | 10% |
| $250 ($3,000) | 3.33% | 4% |
| $416.67 ($5,000)* | 2% | 2.4% |
| $500 ($6,000) | 1.66% | 2% |
* $416.67 per month sums to $5000.04 but I included this for the break-even point, so the $0.04 is disregarded for nicety of numbers—just consider it $416.66 for 4 months of the year.
Opportunity Cost
At first blush, the results above look very pleasing. 8.33% cash back for on up to $1,200 a year is better than any card out there. You’ve probably sensed that something is missing, though, and that’s the opportunity cost of putting $1,200 a year on a BBR card as opposed to a flat 2% cash back card. Every $0.50 you put on a BBR card over the amount necessary to get your quarterly cash reward costs you $0.01 in cash back (that you would’ve gotten by putting it on a 2% cash back card).
Let’s take $5 as the “safe” minimum amount per month to charge to a BBR card. We can then construct a table that tells us the opportunity cost per year (in $) of putting more or less money on the BBR card as compared to the $5 baseline, with the difference always being put on a 2% card. We can consider this as unbonused spend because if you could get more than 2%, the opportunity cost grows much more quickly! (Amounts less than $5 per month have a positive change; amounts greater than $5 have a negative change.)
(Feel free to skip from here to after the table, if you don’t care for the math!)
We can also find several other quantities:
- Change in % per year
- If positive, this is % cash back gained by putting less than $5 per month on a BBR. In other words, by putting $0.50 per month on a BBR and the other $4.50 on a 2% card, you gain an additional 18% cash back per year in the previous table ($0.50 becomes 1,684.66% or 2,018%).
- If negative, this is % cash back lost by putting more than $5 per month on a BBR. In other words, by putting $10 per month on a BBR instead of $5 on a BBR and $5 on a 2% card, you don’t gain an additional 1% cash back per year—you get 83.33% or 100% (the numbers from the previous table) instead of 84.33% or 101%.
- % Change in % per year
- This is the change in % per year as a percentage increase or decrease of the numbers in the previous table. Equivalently, it is the [BBR cash rewards per year] plus the [opportunity cost per year], all over the [BBR cash rewards per year] (but expressed as a difference in percent).
| $ / mo ($ / yr) | Change $ / yr | Change % / yr | % Change % / yr @ $25 | % Change % / yr @ $30 |
| $0.50 ($6) | +$1.08 | +18% | +1.08% | +0.90% |
| $1.00 ($12) | +$0.96 | +8% | +0.96% | +0.80% |
| $2.50 ($30) | +$0.60 | +2% | +0.60% | +0.50% |
| $5.00 ($60) | $0.00 | 0% | +0.00% | +0.00% |
| $7.50 ($90) | –$0.60 | –0.66% | –0.60% | –0.50% |
| $10 ($120) | –$1.20 | –1% | –1.20% | –1.00% |
| $15 ($180) | –$2.40 | –1.33% | –2.40% | –2.00% |
| $25 ($300) | –$4.80 | –1.6% | –4.80% | –4.00% |
| $50 ($600) | –$10.80 | –1.8% | –10.8% | –9.00% |
| $75 ($900) | –$16.80 | –1.866% | –16.8% | –14.0% |
| $100 ($1,200) | –$22.80 | –1.9% | –22.8% | –19.0% |
| $250 ($3,000) | –$58.80 | –1.96% | –58.8% | –49.0% |
| $416.67 ($5,000)* | –$98.80 | –1.976% | –98.8% | –82.3% |
| $500 ($6,000) | –$118.80 | –1.98% | X** | –99.0% |
* As before, $416.67 per month sums to $5000.04 but I included this for the break-even point, so the $0.04 is disregarded for nicety of numbers—just consider it $416.66 for 4 months of the year.
** You lose more than 100% here, which is why it’s an X. In other words, you make more money by using a 2% card (as is obvious from the previous table).
The two columns labeled “% Change % / yr” probably seem like complete gobbledygook, but what they basically say is: “how optimal is my usage of my BBR card compared to the minimal “safe” usage?”. (You could do this in a number of other ways as well – including comparing “Change $ / yr” to the quarterly or yearly BBR earnings – but I happen to like percentages.)
Let’s say you’re a gambler and you put $0.50 on your BBR every month in the hope that BOA will not forgive the balance. Your gain from this turns out to be very small, at an additional 0.90% to 1.08% in yearly cash back. If BOA forgives your balance one month, you’ve lost an entire quarter’s worth of earnings (25%), which means it would take you 23–27 years to make up the difference, as compared to putting a “safe” amount of $5 per month on your BBR.
Likewise, the table states that you can go up to about $15 on your BBR every month while still being very close to optimal because the loss is small enough that it would take 40–50 years for your extra earnings on a 2% card to equal a BBR’s yearly earnings. So, if you prefer the ease of using a Netflix subscription, or another small, monthly recurring charge, you lose very little—make the BBR work for you!
I find this type of comparative analysis very useful when looking at numbers close to the “safe” monthly charge, as it helps provide context for the amount of cash back you’re losing. This is always the case for numbers below the “safe” monthly charge (which may differ from person to person) because the potential loss is one quarter’s earnings, so comparing how long it would take to make up the loss is the primary question. But, it is only one way of looking at the problem, and unfortunately, it breaks down once you start going “well” above $5 per month.
The reason is that it removes from context the actual cash back amount by only comparing your loss to the yearly earnings on a BBR, which is a pretty large cash back number ($100 or $120). For example, when we compare $75 per month on the BBR to $5 per month on the BBR and $70 per month on a 2% cash back card, it takes 6–7 years for the 2% card earnings to make up the yearly BBR earnings, which doesn’t seem that bad. But when you look at the total earnings, you probably have a strong preference for making an extra $16.80 per year!
Savings/Checking Account
Reader PatWithThePens brought up a good point about the opportunity cost of having an account with BOA in order to get the extra $5 per quarter, if that’s the only reason you have the account (which is probably somewhat rare). We can do a quick calculation for a Savings account based on the minimum balance requirements:
- A Regular Savings account requires a $300 minimum daily balance to waive the $5 monthly fee. You earn 0.01% APY in BOA ($0.03 per year), but could earn 1% APY with many other accounts ($3 per year). The opportunity cost in this case is $2.97 per year of the $20 per year extra you get from having a BOA account. (You would already earn 14.85% of the bonus.)
- If you could earn 6.68% APY, you would earn $20.04 on the $300, resulting in an opportunity cost of $20.01 to earn an extra $20. (You’re better off with the APY than the bonus.)
- We have neglected to take into account that interest is taxable (at your marginal tax rate), but the extra $5 is not.
- A marginal tax rate of 25% ⇒ $2.23 opportunity cost. (You would already earn 11.15% of the bonus.) The break-even APY is 8.90%.
- A marginal tax rate of 28% ⇒ $2.14 opportunity cost. (You would already earn 10.70% of the bonus.) The break-even APY is 9.27%.
- We have also neglected to take into account getting the bonus on multiple BBRs. In this case, the opportunity cost is the same, but you earn more. You can thus divide the “would already earn” % by the number of BBRs you have.
- The opportunity cost for a checking account is not so easy to derive because the fee can be waived by direct deposit or a minimum balance, checking APYs are extremely variable, and timing of removing the direct deposit funds (and whether you remove direct deposit funds) is extremely variable. Thus, examples are left as an exercise to the reader.
Sidenote: Hyperbolas
You may recognize based on the equation Nate used (120 / x) and the values in the cashback equivalence table that the equation describing % cash back is a hyperbola. Unfortunately, pixel limitations make a graph very hard to view, but I’ve prepared two Wolfram|Alpha links so that you can get an idea. This graph is for annual spending from $0 to $60 & this graph is for annual spending from $0 to $6,000 (10000 / x and 12000 / x are used to make the y-axis a %).
The vertices of these hyperbolas correspond to the square root of the numerator. For $25 per quarter, the vertex is ($100 per year, 100% cash back), or $8.33 per month. For $30 per quarter, the vertex is ($109.54 per year, 109.54% cash back), or about $9.13 per month.
For this class of hyperbola, the vertices indicate the point on the hyperbola where the slope of the tangent line is –1, or equivalently the point where the tangent line to the hyperbola is 45°, or equivalently the point where the tangent line to the hyperbola has the greatest slope difference from the vertical & horizontal asymptotes.
I’ve included this as a sidenote because it is interesting, but it has no bearing on the optimal amount to charge to a BBR. The optimal amount is always the lowest amount that BOA will never forgive (which may be an ever shifting target!).
Questions, comments, etc. can be dropped below!
h/t reader Nate
