A Lesson On Double Counting

Kitty PoP doesn’t have a “teacher hat”, so this one will just have to suffice…

Today I’m putting on my “teacher hat”, because lately I’ve been seeing some pretty blatant errors in “double counting” in some other personal finance blogs out there. So pull up your chair, and let’s have a quick lesson.


What Is Double Counting?

Simple answer, it’s counting something twice. If you want to get technical about it, it’s a topic that’s covered in basic combinatorics (the study of counting – yes, there is a study of counting), and probability theory (the study of how likely events are).

Here’s a baby example:
You have a bucket that contains 3 different types of balls, solid white, solid black, and patterned white and black. There are 5 solid black balls, 8 solid white balls, and 4 patterned balls in that bucket. Let’s answer some basic counting questions!

  • How many balls have black on them? 5+4=9. Easy, right?
  • How many balls have white on them? 8+4=12. Too, easy.

So if the bucket only contains balls that are black and white in color, are there 9+12 =21 balls in the bucket? Of course not! You know this! We said at the beginning that there were 5 black + 8 white + 4 patterned = 17 balls in the bucket. What went wrong? We double counted the intersection of the sets of balls that have black and white on them (ie. the balls that are both black and white).

These same types of problems occur all over the place in personal finance, sometimes to your benefit, but more often to your detriment. The thing is, sometimes they’re not laid out there in black and white (pun intended) like our little bucket o’ balls in the example. Let’s look at a few real-life examples.

Example 1 – Sub-Categories of Budgets

For a while we had our Mint account set up so that we had a clothing budget separate from our shopping budget on the “budgets” tab. But on the budgets tab, Mint was actually also including all of our clothing purchases in the “Shopping” budget as well. When the total budget spending was added up on that tab, all the clothing spending ended up being double counted.

As double counting goes, this is pretty harmless. If you’re going to double count anything, double count an expense that you only pay for once! You’ll end up thinking you spent more than you did and have more money in your accounts at the end of the day because of it. Nonetheless, it’s annoying, and worth keeping an eye out for, especially if you’re wondering why your “budgeted spending” isn’t necessarily equaling your total spending on the net income portion of Mint.


Example 2 – Getting Gift Cards For Spending


Buy 2 Cat Products, get a $5 Gift Card this week at Target… I may have to go pick up some litter for Kitty PoP!

Don’t get me wrong, I love Target. But their marketing department is an evil genius when it comes to using psychology and spending pattern analysis to get you to spend more. (Read our review posted yesterday of The Power of Habit for more info on that topic!) One of their marketing tactics is: Buy 2 (or 3 or 4) Get a $5 Gift Card for a future purchase. Target knows that you mentally count that $5 gift card as a discount on the shopping trip during which you earn it, and then again on the shopping trip when you redeem it. (Don’t lie – we all do this. Seriously.)

You have now double counted that $5 savings in your mind. Essentially, you behaved as though you saved $10, when really you only saved $5. You have to admire Target’s ingenuity in this. I haven’t seen many other marketers using this gimmick as widely as Target does, but I wouldn’t be surprised if it starts to get more widespread.

But how about a bigger example where you could be fooling yourself

Example 3 – Calculating “Savings” Using Future Value Formulas

Future value calculations seem to be “hip” to use in personal finance today, but the way many are applying them are ripe with double counting. Here’s a real example I saw on another blog. This guy was cutting back on his $2/day K-cup habit. (This is a K-cup in case you’ve never seen one.) He claimed if he stopped using the K-cups, over 10 years, assuming a 7% yearly market return, his total savings would be $10,792,03! Amazing! Just by cutting out K-cups! Almost $11K!

Here’s the Excel formula he undoubtedly used to get that value. Feel free to type it into Excel yourself.

-1*FV(0.07, 10, 2*365, 0, 1) = $10,792,03

What are the numbers (parameters) that we put into the FV function?

  • 0.07 = 7% yearly market return
  • 10 = number of years
  • 2*365 = $2/day for a year, the amount you add to the account each year
  • 0 = starting value of the account
  • 1 = indicates payments are made at beginning of each period


So where’s the problem with this math?
(1) The 1 in the last entry in this formula indicates that payments are made at the beginning of the period, not the end. What does that mean in real life? Basically that you start your K-cup “savings” account at your brokerage by depositing the first year’s worth of savings (that’s $730 at $2/day) into that account before you have even finished the last K-cup in your cupboard. Totally unrealistic. Okay, you say, change the “1” in last entry of the formula to a “0” for making deposits at the end of each period. Now you have

-1*FV(0.07, 10, 2*365, 0, 0) = $10,086.01

… which is still great, right? But…

(2) If you give up your precious K-cups, you will inevitably substitute something else to get your caffeine fix. Here’s where the bulk of the double counting occurs. Maybe the occasional Starbucks, or a daily dose of Dunkin Donuts coffee, McCafe, heck maybe you are super disciplined and just spend $10/month on a can of Folgers to make coffee at home.
If that substitution costs you ANYTHING, you are double counting your savings at least in part. So, say you’re really disciplined. You never treat yourself, and get your caffeine fix with $10 in Folgers every month. We need to back that out of our formula. $10/month = $120/year. So our formula is now:

-1*FV(0.07, 10, 2*365 – 120, 0, 0) = $8,428.03

Still nothing to sneeze at. Leaving out the other main weakness of the FV formula (are you estimating using an average rate of return or a CAGR? – do you know the difference?), you still have to…

(3) Execute the plan. Until you actually follow through on this plan and have automatic payments into a brokerage account, never buy another K-cup for 10 years, and the price of Folgers never increases, then all these calculations are pretty worthless. If you stop buying K-cups but don’t invest the money, and end up spending it on lifestyle inflation elsewhere, what have you “saved” in reality? And the only person you’re fooling is yourself. If you do stick to the plan for 10-years, congratulations! But you still won’t have $8,428.03 to show for it. That’ll be $8,428.03 – capital gains taxes on the profits.

I’m not trying to make anyone feel bad here, and I’m certainly not saying don’t work on cutting down on expensive habits. I’m just saying to be aware. Double counting is something that humans are practically programmed to do naturally. That’s why probability is one of the least intuitive of all math disciplines, even when we use it everyday.

Basically, don’t count “savings” on future unspent money. Only count “savings” as what is in your savings account. Otherwise, the only person you’re fooling is yourself. (Seriously, ’cause the computers that keep track of your bank accounts know how to calculate this stuff.)


So that’s it. Lesson over. Any questions? Your homework is page 93, all the even numbered problems…  

What have you double counted lately? Coupons? Calories burned?


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19 comments to A Lesson On Double Counting

  • Oh, I’m sure I double count all the time. But you’re right, you aren’t really saving unless it actually makes it into your savings account! I definitely do a good job at that part.

    • We all do it – it’s pretty much second nature! But glad that you’re getting those savings into an account where they can make a difference!

  • The Target gift cards ALWAYS get me. I am definitely guilty of that.

  • I chuckled reading this. Yep, double counting happens for sure. When I spend, I work very hard to simply look at the object in my hand, and decide if it’s worth what I will pay at the register. Forget that it’s on sale, forget that I have a giftcard, forget all that. Is this hammer worth $75? Hm, probably not. Put it down!

    Double counting in our actual books is in my wife’s jurisdiction, but I think she does a pretty good job, we havn’t really run into any big math problems yet!

    • You’re right that it takes work to ignore all the flashy “sale” and clearance signs and decide if it’s worth paying the price that is on it, without thinking about what discount that represents to some possibly fake MSRP. Good for you to do so!
      And a $75 hammer? Is it plated in sterling? =)

  • I used to do this ALL the time. When I bought books while working at the book store, I reasoned that as I was using my discount and earning rewards money that I was really saving a great deal. Because then I could use the reward money to buy books the next month right? It took me a while to realize that even though I was earning those reward bucks-the fact that I earned the same amount every month ($25 at least) and when buying with those rewards, it didn’t count to gain more reward money? That meant that every month I was spending at least double what I had before.

    • Good for you for realizing it before it was too late. I hate all those delayed rewards programs – like CVS with its rewards bucks or whatever they are called.

  • I like how you mentioned what Target does. I never really realized it till now but that is a sneaky way to get make extra money off their customers. I have never gotten one of their cards this way but I will certainly remember this when I shop there.

    • Target’s not the only one that uses similar tactics. I remember Express used to give spend $100 today get $25 off your next purchase. It made it feel like you were getting the discount twice, when all it really accomplished was getting you to spend at least $100 that day AND come back to do the same a couple weeks later (since the coupons expired!)

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  • doubleE

    You said white twice:

    “How many balls have white on them? 5+4=9. Easy, right?
    How many balls have white on them? 8+4=12. Too, easy”

    I was like.. whoa I totally cannot do math..

    • Too right you are! Error corrected now =) I am way better with numbers than with words. And the Mr’s colorblind. So do we have enough excuses for the error?

      Thanks for the catch.

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  • Very perceptive! I hate when my lazy thinking gets exposed like that. :)

    There’s another area we do something similar. It’s gas mileage: we tend to think about miles per gallon as the only way to save money. Trade in that clunker and get a Prius.

    Not so fast. When we lived in California, we traded in a sippy Civic and bought a gas guzzling Ford Crown Vic… and saved money! How? In California you pay more per month for car insurance than for gas. So a 10% saving on insurance puts more in your pocket than a 10% saving on gas. A Crown Vic had (at the time) the lowest insurance rating, so we saved more on insurance than we paid extra in gas. Really.

    And while we were doing this, we realized that by simply eliminating one or two thoughtless trips per week, we ended up paying hardly any extra in gas at all.

    And you won’t believe how many people were on us for wasting all that money by trading in the noisy, small, uncomfortable stick shift for the comfortable, roomy, reliable, safe (although politically way incorrect) old boat! And when we explained the math to them, they all had this look that said they know we’re bamboozling them, but they just can’t say how. A small car just HAS to be cheaper than a big one, it just HAS to.

    These are the people who never look past a single number. And miss opportunities to save some pennies to plant! :)

    • It’s not just car insurance… a friend recently traded in her gas guzzling mustang (on which she was underwater on the loan) for a lease on a gas sipping jetta. All she could see was the $200 she would save each month, not the $5K+ that she was going to have to pay upfront (to make up the difference on the loan) to do so. It’s going to take a long time to make up that money in gas savings!

  • Patrick

    Excellent post, you should forward to MMM 😉

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