Published on April 7th, 2017 | by sirtheta44
When Stacking Goes Wrong: The Danger of Percentages & Framing a Problem
Yesterday, I went through an example of how stacking can misleading in response to US Credit Card Guide’s article “Hacking the Cost of an Engagement Ring”, which involved stacking two disparate deals. The other recent example that inspired that response was the (now dead) Office Depot/OfficeMax (OD/OM) deal, which involves stacking one deal. Stacking deals is an area where percentages and fungible money sources can be misleading, so here is a self-contained example using the OD/OM deal. (You can view an instance of stacking gone wrong in this post about the OD/OM deal at Laptop Travel; scroll down to the scenarios where the deal is leveraged further.)
Do note that there are any number of ways to stack deals, and many of them are not prone to “going wrong”—for example, stacking a cashback portal with a Visa Checkout offer with an Amex Offer, or stacking Hilton promotions for 52x Hilton points (some of the promotions mentioned in that article no longer exist).
This article is extremely long and detailed, but I believe it should be fairly easy to skim without losing the point. It’s still probably not for the faint of heart, though! It also makes any number of assumptions that likely won’t hold in the real world; they should all be explicitly stated and shouldn’t affect the math.
This article’s featured image comes from Wikimedia. CC BY-SA 3.0, etc.
You can read our full article for the details, but I’ll recap to keep everything in one place. The deal was simple: save $10 when you buy $50 or more in gift cards at OD/OM, restricted to American Eagle Outfitters, Catherines, Fandango, Groupon, MasterCard (MCGC), and Toys R Us. The discount was automatically applied at the register and buying $100 or more actually resulted in two discounts (–$20). Importantly, the limit was two discounts per transaction and mixing and matching gift cards worked (a $25 Fandango and a $25 Groupon gift card could be bought for $40).
The deal was especially lucrative if you used a card that earns a high rate at office supply stores. For the sake of this article, we’ll assume the use of a card that gives 5% cash back.
Initial Outlay – MCGC
The best percentage returns came from buying $100 gift cards, as anything above that still only earned a –$20 discount and they have a $5.95 purchase fee. Since OD/OM $20–$200 variable load MCGC with a $6.95 purchase fee, it made more sense to buy $200 MCGCs – easier to unload.
We’ll consider both possibilities, though, and it works out as follows. A $100 MCGC costs $85.95 and earns $4.30 in cashback – a savings of 18.35%. A $200 MCGC costs $186.95 and earns $9.35 in cashback – a savings of 11.2%.
Stacking the Deal – More GC
Once the initial outlay of MCGC occurs, you could use them to buy more discounted gift cards, whether MCGC or otherwise. Here are some examples:
- $100 MCGC to $100 MCGC. $85.95 cost, $4.30 cashback, $14.05 left on original MCGC; $118.35 in total. Savings of 27.38%.
- $100 MCGC to $100 any other gift card. $85.95 cost, $4.30 cashback, $20.00 left on original MCGC; $124.30 in total. Savings of 30.85%.
- $200 MCGC to 2× $100 MCGC in separate transactions. $186.95 cost, $9.35 cashback, $28.10 left on original MCGC; $237.45 in total. Savings of 21.27%.
- $200 MCGC to $200 MCGC. $186.95 cost, $9.35 cashback, $13.05 left on original MCGC; $222.40 in total. Savings of 15.94%.
- $200 MCGC to 2× $100 any other gift card in separate transactions. $186.95 cost, $9.35 cashback, $40.00 left on original MCGC; $249.35 in total. Savings of 25.03%.
Theoretically, you could’ve stack this infinitely. An example with $200 MCGC proceeds as follows:
|Iteration||Buy $200 MCGC with..||..for (c total)..||..with Cashback x..||..giving $200 MCGC +|
y left on n MCGC (z total)
|0||5% earning credit card||$186.95 ($186.95)||$9.35||$0, 0 ($209.35)|
|1||$200 MCGC (from 0)||$186.95 ($186.95)||$0||$13.05, 1 ($222.40)|
|2||$200 MCGC (from 1)||$186.95 ($186.95)||$0||$13.05, 2 ($235.45)|
|3||$200 MCGC (from 2)||$186.95 ($186.95)||$0||$13.05, 3 ($248.50)|
We can quickly construct an equation and find that an initial outlay of $186.95 yields, after each iteration, $200 (MCGC) + $9.35 (Cashback) + $13.05 × n (MCGC), where n is the number of times you’ve bought a $200 MCGC with a $200 MCGC. (It works the same for $100 MCGC except an initial outlay of $86.95 yields, after each iteration, $100 (MCGC) + $4.35 (Cashback) + $13.05 × n (MCGC).) Ignoring the feasibility of infinitely stacking, we can find the total profit and the % savings by claiming that all MCGC can be transformed into cash at face value with no opportunity cost.
|Iteration||$200 MCGC + x Cashback + y left × n MCGC||% Savings|
|5||$200 + $9.35 + $13.05 × 5 = $274.60||1 – $186.95 / $274.60 = 31.92%|
|10||$200 + $9.35 + $13.05 × 10 = $339.85||1 – $186.95 / $339.85 = 44.99%|
|25||$200 + $9.35 + $13.05 × 25 = $535.60||1 – $186.95 / $535.60 = 65.10%|
|50||$200 + $9.35 + $13.05 × 50 = $861.85||1 – $186.95 / $861.85 = 78.31%|
|100||$200 + $9.35 + $13.05 × 100 = $1,514.35||1 – $186.95 / $1,514.35 = 87.65%|
When Stacking Goes Wrong – MCGC
Any time you can theoretically manufacture infinite money out of thin air, there’s probably a flaw somewhere. The flaw happens to be the same as that from the Churning & Engagement Rings article: independent events and opportunity cost. Perhaps I’ll refer to this by the fancy term “dependence of relevant alternatives“. Independent events come into play when there is an alternative to using your $200 MCGC to buying further $200 MCGC in a chain—for example, liquidating it to a money order (MO). And opportunity cost comes into play when the alternative involves earning a different amount of rewards.
Let’s proceed with a few more assumptions and then do an example of buying many $200 MCGC with a credit card instead of MCGC. We’ll continue to ignore the feasibility of infinitely stacking. Crucially, we’ll assume the end goal is to convert MCGC to some form of cash. And we’ll assume that you can liquidate all the MCGC from the previous section (the 1 × $200 and the n × $13.05) at full value while you can only convert the $200 MCGC in this section to individual MOs of $199.30. Extremely generous assumptions for the previous section, much less generous assumptions for this section; I introduce this dichotomy to drive home the point.
|Iteration||Buy $200 MCGC with..||..for (c total)..||..with Cashback x..||..giving n $199.30 MO|
|1||5% earning credit card||$186.95 ($186.95)||$9.35||1 ($208.65)|
|2||5% earning credit card||$186.95 ($373.90)||$9.35||2 ($417.30)|
|3||5% earning credit card||$186.95 ($560.85)||$9.35||3 ($625.95)|
We can again quickly construct an equation and find that each iteration yields a cost of $186.95 × n and $9.35 × n (Cashback) + $200 × n (MCGC), where n is the number of $200 MCGC you bought. (The numbering difference brings the iteration number in line with the variable n; it arises due to the previous section requiring only an initial outlay while this section requires a continuous outlay.) (Also, it again works the same for $100 MCGC.) Ignoring the feasibility of infinitely stacking, we can find the total profit and the % savings by claiming that all MCGC can be transformed into $199.30 in cash. We cut the table short here because the % savings is always the same.
|Iteration||x × n Cashback + $199.30 × n MO||% Savings|
|5||$9.35 × 5 + $199.30 × 5 = $1043.25||1 – $186.95 × 5 / $1046.75 = 10.40%|
|10||$9.35 × 10 + $199.30 × 10 = $2086.50||1 – $186.95 × 10 / $2086.50 = 10.40%|
So…what gives? Why has stacking gone wrong when the % savings is much worse than in the previous section? Why is this article subtitled “The Danger of Percentages & Framing a Problem”? Because percentages are dangerous when presented without context, or when the problem is not framed correctly.
Our crucial assumption from above (the end goal is to convert MCGC to some form of cash) means that money is fungible, and so is the outlay – even though our brains don’t automatically perceive the situation that way. The previous section makes everything seem simple because you just keep buying $200 MCGCs with $200 MCGCs in a chain. In this section, it seems like each $200 MCGC requires an additional outlay, which destroys your % savings. But because the money is fungible, you could easily consider the new $200 MCGC (costing $186.95) to be purchased with the funds from already-liquidated or to-be-liquidated MCGC. And you can in fact consider this to be in a chain: the initial outlay is $186.95 for a $200 MCGC that is converted to $199.30 cash (at any point now or in the future), and each successive iteration is bought with the $199.30 in funds from liquidating the MCGC from the previous iteration. In this case, instead of the chain leaving a string of $13.05 in MCGC, the chain “leaves” $12.35 in “MO” (cash) ($199.30 – $186.95).
|Iteration||Buy $200 MCGC with..||..for (c total)..||..with Cashback x..||..giving $199.30 MO +|
y left on n MO (z total)
|0||5% earning credit card||$186.95 ($186.95)||$9.35||$0, 0 ($208.65)|
|1||5% earning credit card||$186.95 ($186.95)||$9.35||$12.35, 1 ($230.35)|
|2||5% earning credit card||$186.95 ($186.95)||$9.35||$12.35, 2 ($252.05)|
|3||5% earning credit card||$186.95 ($186.95)||$9.35||$12.35, 3 ($273.75)|
Now we’re getting somewhere! I won’t bother producing a table of % savings; the z total in this table is directly comparable to the z total in the first table of the previous section. The zeroth iteration (the initial outlay) is –$0.70 compared to the previous section’s zeroth iteration due to the assumption that all MCGC in the previous section can be liquidated at face value. Under this scenario, you gain an additional profit of $8.65 × n over the previous section, where n is the number of times you’ve bought a $200 MCGC after the initial outlay.
The basic thrust here is that % savings can be very misleading because the natural inclination is to frame the two situations in different ways that makes them incomparable. For this reason, it’s a much better idea to focus on the actual numbers.
This also applies to the decision between buying a $100 MCGC or a $200 MCGC (whether for the initial outlay, or otherwise). The 18.35% savings for a $100 MCGC seems better than the 11.2% savings for a $200 MCGC. But the actual profit for a $100 MCGC is $18.35 ($20 discount – $5.95 purchase fee + $4.30 cashback) while the actual profit for a $200 MCGC is $22.40 ($20 discount – $6.95 purchase fee + $9.35 cashback). Because the difference in profit arises from the cash back and the difference in cost is exactly $101, the break-even rate for which one to buy is about 1% cashback, at which point you make $0.01 more on the $200 MCGC. (Note that this occurs due to the limit of two discounts per transaction. It would be more profitable to buy 2 × $100 MCGC, but that requires two transactions, in which case you could buy 2 × $200 MCGC.)
When Stacking Goes Wrong – GC Reselling
The exact same analysis can be done for gift card reselling, and we will use the exact same assumptions.
Iteration 0: buy a $200 MCGC with a 5% earning credit card for $186.95, earning $9.35 in cashback.
Iteration 1: Buy 2× $100 gift cards in separate transactions, yielding $200 in gift cards, $40 on the MCGC, and $9.35 in cashback (a total of $249.35 against a cost of $186.95).
Under the assumption that everything is worth its face value, you “stacked” the deal to get a savings of 25.03% on the gift cards, which would theoretically make you a nice profit if sold at 80%+ of face value. Let’s say you sell them at 90% of face value, netting $180 in cash. Your total is then $229.35 (cash) against $186.95 (cost), which puts your total % savings at 18.5%. Your total profit is $42.40.
Once again the dependence of relevant alternatives comes into play, though. You could liquidate that MCGC and use those funds to buy the gift cards with a 5% card.
Iteration 0: buy a $200 MCGC with a 5% earning credit card for $186.95, earning $9.35 in cashback and resulting in $199.30 in cash.
Iteration 1: Buy 2× $100 gift cards in separate transactions for a total of $160, yielding $200 in gift cards and an additional $8.00 in cashback. Subtracting $160 (cost) from $199.30 (cash) yields $200 in gift cards, $39.30 in cash, and $17.35 in cash back (a total of $256.65 against a cost of $186.95).
Under the assumption that the gift cards are worth their face value, you “stacked” the deal to get a savings of 27.16% on the gift cards, which would again theoretically make you a 7.16%+ profit if sold at 80%+ of face value. Again, let’s say you sell them at 90% of face value, netting $180 in cash. Your total is then $236.65 (cash) against $186.95 (cost), which puts your total % savings at 21.00%. Your total profit is $49.70.
Because we’re using a high % face value for reselling, the problem with % savings does not immediately stand out here, but it still exists. Buying the MCGC and buying the gift cards are independent events, and rolling them into one % savings number cumulates data that should be kept separate. In the second example, if you were to liquidate the gift cards for 80% of face value, your profit would be $29.70 and your % savings would be 13.71%—a clear profit on its face. But $21.70 of that profit comes from the MCGC – the only profit you made from the gift cards was $8.00 from the 5% cashback; this is perfectly fine, but 5% profit is very different from 13.71%. And if you were to do liquidate the gift cards in the first example for 80% of face value, you wouldn’t make any additional profit!
If you actually read all the words I typed up above, you already know my concluding thoughts (and I’m impressed!). First, be careful about using percentages; ofttimes the actual monetary value of the profit reveals much more information. Second, it’s essential to be careful about how you frame the problem when you stack deals—money is fungible and the way you conceive of the funding matters. And third, you have to keep independent events separate. The last two points deserve particular emphasis.
Questions, comments, etc.? Drop them below.