Calculating ROI of buying in bulk/on sale

I recently saw a video on YouTube with advice from Mark Cuban on how to get rich. I found one piece of advice particularly interesting: that buying non-perishable consumable goods in bulk or on sale is going to give you a better return on investment than any traditional investment opportunities (stocks, real estate, etc.). Return on investment (ROI) is a financial concept – it’s a measure of how beneficial it is to tie up money in a particular investment. I was curious to calculate the ROI from buying goods in bulk, and the only example I found online of someone attempting the same calculation used a bizarre and almost certainly incorrect method.

First, we can define ROI for traditional investments. ROI is typically given in percent per year: if a certain stock has an ROI of 10%, that means that if you buy $100 of that stock, you could sell it one year later for $110. The simplest ROI calculation can be made as follows:

ROI = (\frac{\textrm{final value}}{\textrm{amount paid}})^{(\frac{1}{\textrm{years invested}})} - 1

We can see that plugging in the numbers from the stock example above (an amount paid of $100, a final value of $110, and being invested for 1 year) indeed gives an ROI of 0.1, or 10%. A slightly more complicated example might be that you buy a rental property for $200,000, the property has a net income of $1,500/month, and after 8 years you sell the property for $300,000.  Plugging in these numbers (final value = 300,000 + 1,500*12*8, amount paid = 200,000, and years invested = 8) gives an ROI of about 10.5%. This would indicate that the rental property was only a slightly better investment than the stock. Most traditional investments that don’t require a large amount of capital will have ROIs from 5 – 15%, so that’s the target that buying in bulk should beat if Mark Cuban’s advice is sound.

It should be noted that this ROI calculation assumes you pay the total cost up front and collect the total value at the end of the investment. In reality, the rental property is more valuable than we’re calculating with this simple formula, since you collect rent over the course of owning the property, not in a lump sum at the end. This means you could be reinvesting that rental income as you receive it, and you could be earning additional returns in the meantime that we’re not accounting for. Even so, this ROI calculation is still useful for comparing different investments, and we’ll ignore this timing shortcoming for now.

ROI of buying in bulk

Buying in bulk isn’t exactly an investment, but it follows the same principle since we’re tying up our money in something (in this case, non-perishable consumables rather than stocks or real estate) because we think we’ll end up with more money in the long run. As an example, suppose that I typically buy a $10 package of toilet paper which lasts me one month. In a year, I’ll go through 12 of these packs, spending $120 on toilet paper. At a bulk supply store, I might have the opportunity to buy a year’s worth of toilet paper for $100, a 17% savings. What would the equivalent ROI be? Here the amount paid is clearly $100. For the simple ROI calculation, we can say that the final value is $120, since I’m getting $120 worth of toilet paper compared to my usual shopping habits, and we can say that I’m invested for 1 year, since I have to pay for the toilet paper 1 year ahead of when I’ll finish using it. In this case, the ROI would be:

ROI = (\frac{120}{100})^{(1/1)} - 1 = 0.2 = 20\%

The equivalent of a 20% return, much better than most traditional investments!

ROI of buying on sale

We can also use the same formula to calculate ROI for situations when a product is on sale. Suppose I go to the store one day and the toilet paper I usually buy is $7 a pack (30% off). What’s the ROI, and how much should I buy? In this case, the ROI of a particular pack of toilet paper depends on how far in the future I’ll end up using it. Each pack costs $7 and gives me $10 of value, but the number of years invested depends on how many packs I’m buying. For example, for my sixth pack it will take half a year before I use it, which yields:

ROI = (\frac{10}{7})^{(1/0.5)} - 1 = 104\%

That sixth pack has an ROI of 104%, an awesome investment! Suppose I really go to town and buy 5 years worth of toilet paper. That means for the last pack, I’ve invested that $7 for 5 years, and the ROI is given by:

ROI = (\frac{10}{7})^{(1/5)} - 1 = 7.4\%

While still a reasonable return, a 30% discounted pack of toilet paper that I don’t use until 5 years later isn’t necessarily beating traditional investments.

Carrying costs and increased consumption

The calculations above neglect some costs which could potentially reduce the ROI on buying in bulk/on sale. The first is carrying cost, which refers to the cost of storing a bunch of stuff that you aren’t using yet. For some goods (like toothpaste) this probably isn’t a big deal, but for the toilet paper example, it likely would be. Suppose I bought 2 years worth of discounted toilet paper, as in that case the last pack would still have an ROI of almost 20%, which is a very solid return, but I only have enough room in my apartment to store 1 year worth of toilet paper. My building rents storage bins for $60/year which would be sufficient to store all the extra toilet paper, so for the first year I have to spend an extra $60 to hold all the toilet paper. In this case, the overall ROI would be given by:

ROI = (\frac{10*24}{7*24+60})^{(1/2)} - 1 = 2.6\%

Since I have to pay extra money to be able to store the toilet paper, it completely kills the savings, and I’m left with a meager 2.6% ROI. Usually carrying cost is more difficult to factor into the calculation than this, but it’s worth keeping in mind that if a large portion of your house is being used for storage, buying in bulk probably isn’t doing you that much good, and you could lower expenses overall by living somewhere smaller without the bulk buying.

Another trap you should be careful of is increased consumption of the good you bought in bulk. In the bulk toilet paper example (12 packs for $100), the value I’m getting is $10/month, since that’s how much I usually spend on toilet paper. If I start using toilet paper to clean up spills in the kitchen since I have so much TP lying around (whereas I used to use reusable sponges), and this causes me to go through 12 packs of toilet paper in 11 months, then my ROI would be:

ROI = (\frac{110}{100})^{(12/11)} - 1 = 11\%

By increasing my consumption by ~9%, the original ROI of 20% is cut almost in half. So while buying in bulk/on sale can provide good ROI, you should be careful about carrying costs and increased consumption, which can negate or even reverse the return.

It’s also worth noting that if you buy something on sale which you normally wouldn’t buy at all, it can’t be treated as having an ROI in the sense we’re looking at here. While you can still get value from it, you’re not saving money since in the absence of the sale you wouldn’t have bought the item, and therefore wouldn’t have spent any money at all.

Some interesting math

There is some interesting math to explore following this treatment of buying in bulk/on sale as an investment. We can rearrange the ROI equation to find how much of a good we should buy when it goes on sale based on the ROI we want to achieve:

 t = \frac{\ln(\frac{1}{1-\textrm{discount}})}{\ln(1+ROI)}

where t is how much of the thing we should buy in years (for example, if the calculation yields t = 2, you should buy enough of the item on sale to last you 2 years). This is plotted for a few target ROI values below:

As an example, if you’re targeting an ROI of 50%, and the laundry detergent you usually get is 40% off, our ROI calculations say you should buy ~1.3 years worth of detergent. If you want an ROI of 100%, you should only buy 9 months worth of detergent, whereas if you are happy with an ROI of 25%, you can buy ~2.3 years worth of detergent. This chart can also be used to decide whether buying in bulk is worth it. In this case, if the bulk point is up and to the left of your target ROI curve, you shouldn’t buy, but if it’s down and to the right, you should. For example, if we can get a 17% discount by buying 1 year worth of toilet paper, that point is located up and to the left of all three of these ROI curves, indicating that it’s a worse ROI than any of those values (we know it’s 20% from calculating it before).

In general is seems like the ROI on buying in bulk/on sale is very good, so why don’t we “invest” more of our money this way? The funny thing about this “investment” is that the better the ROI, the less money we’re able to invest. Looking back at the toilet paper example, for something that has a bulk discount of 17%, the most I’d be able to “invest” for one year is 83% of my typical annual spending. If the bulk discount was instead 33%, corresponding to an ROI of 50%, I’d only be able to “invest” 67% of my typical annual spending. Larger discounts lead to higher ROI, but they also lead to lower investment potential:

 \textrm{investment limit} = t*\textrm{annual spending}*(1 - \textrm{discount})

Typically when you find a good investment opportunity, you want to put as much of your money in it as possible. However with buying in bulk, your investment potential is inherently limited by your normal spending, and is lower the better the discount.

 

Levelized cost of wear

I consider this post a work-in-progress. If there’s any real merit to this concept, I’ll need to come back and clean up the post so that it’s coherent to someone who isn’t already familiar with all the ideas I use to build up the concept.

There’s an idea related to clothes shopping called “cost per wear.” It’s a simple idea for quantifying the cost of different items of clothing that’s (arguably) more useful than their sticker price. To find an item’s cost per wear, you simply divide its cost by the number of times you expect to wear it before getting rid of it:

CPW = \frac{cost}{times\ worn}

You can find plenty of articles about cost per wear (some good, and some which grossly misinterpret it), but in this post I’ll propose an extension to the idea. But before proposing the extension, I’ll recount how I ended up with it. I was first introduced to the idea of cost per wear by my undergraduate research advisor, who claimed that you should never spend less than $2000 on a suit (he always dressed very sharp). The reason he gave was cost per wear: if you buy a cheap suit, you’ll never want to wear it, and you’ll only get a few uses out of it before throwing it away. If you buy a nice suit, you’ll use it as much as you can (for dates, interviews, any vaguely formal event, etc.), and in the long run it will be cheaper – at least in terms of cost per wear. Putting numbers to this, there seems to be some wisdom in his claim. If I buy a cheapo suit for $120, it might last me two years, and I’ll only wear it when I absolutely need to wear a suit (maybe three times a year). This yields:

CPW_{cheapo\ suit} = \frac{\$120}{2 years* 3 wears/year}=\$20/wear

If I buy a primo suit for $2000, it could last me closer to ten years, and I’d want to wear it whenever I had the chance (maybe once a month). Therefore:

CPW_{primo\ suit} = \frac{\$2000}{10 years* 12 wears/year}=\$16.67/wear

Since the nice suit has a lower cost per wear, the argument goes, it’s actually the thriftier purchase. Somehow this idea came up in a discussion with a fellow grad student, and from our perspective it didn’t seem to paint an accurate picture of our situation if we were in the market for a new suit. A point that cost per wear misses is that a dollar in my pocket today is not the same as a dollar in my pocket ten years from now (something that grad students are acutely aware of, as in ten years we imagine our salaries will be at least triple what they are today).

There is another idea, called levelized cost of energy, or LCOE, which is used to calculate the effective cost of electricity from different sources – specifically as a way to compare renewable sources like wind and solar that are “free” once they are installed to conventional sources that required burning fuel which you have to pay for. The piece that’s relevant here is that even if you had a maintenance free solar panel that lasted forever, if you have to pay for the panel initially it’s LCOE would not be zero (i.e., the electricity it generates is not free). This is because even though you get infinite electricity from that panel, it’s spread out over infinite time. And when you consider inflation, the electricity that the panel generates in one or two hundred years has almost no value to you today.

I’m sure this explanation is insufficient for anyone not already familiar with LCOE, but in any case, that’s the idea I borrowed from to come up with “levelized cost of wear” or LCOW, an extension to cost per wear. LCOW is calculated as follows:

LCOW=\frac{cost}{\sum\limits_{t=1}^n \frac{wears/year}{(1+r)^t}}

where n is the number of years you expect the item to last and r is your “personal inflation rate” – nominally we can say that it’s how much you expect your salary to increase annually. LCOW takes into account how the value of money might change over time for an individual, whereas CPW does not. We can revisit the cheap vs. expensive suit using LCOW, using an r of 12% (this would be very high for most individuals, but for the PhD student example, it corresponds to about triple the salary in 10 years):

LCOW_{cheapo\ suit}=\frac{\$100}{\sum\limits_{t=1}^2 \frac{3}{(1.12)^t}}=\$23.67

LCOW_{primo\ suit}=\frac{\$2000}{\sum\limits_{t=1}^10 \frac{12}{(1.12)^t}}=\$29.49

When considering LCOW, the cheap suit is the thriftier purchase. Intuitively this makes sense – my friend and I will have plenty of time to go shopping for nice suits once we have real jobs. In the meantime, it’s a better use of our money to buy the cheapest suit that can get us through graduation.

Making digital art: IM T-shirt design

For my entire college career, I’ve been heavily involved in intramural sports, both in my undergrad studies at Cal and in my graduate studies at MIT. At Cal, I was only involved as an athlete, but at MIT I’ve been more involved on the leadership side (MIT IMs are almost entirely organized by students). Currently we’re running a competition to design the new IM champion T-shirt, and I decided to submit an entry. While I’m no expert on design (just a sometimes artist), I thought it might be useful to document my process here in case other similarly non-expert designers would find it useful for their own work. Here’s the final product, if you’re interested in seeing how I got to this point, read on:

IM-shirt-LWBefore one can start with the design, you need to come up with the concept. For this design, I wanted to riff off the MIT mascot (the beaver, “nature’s engineer”) in a way that’s relevant to intramurals. I wanted a design that addressed that the T-shirt was the prize for winning a championship and that IMs are the most informal manifestation of organized sports in the college environment. The concept I came up with was a beaver holding a championship cup hewn from a log. After coming up with the concept itself, the first step in my design process was to sketch out a few different versions of it:

logo-sketches

Ideally the sketches in this part of the process would have more variation than the ones here. I guess it worked out that they were so similar in this case because I was happy enough with my concept that I didn’t need to iterate much. I did use the sketches to block out the design, and hone down exactly how I would try to draw the beaver (which is a lot harder to draw than a cup). It also helped me nail down a few details, like the cup handles being made from twigs and the neck of the cup having a beaver-chewed pattern. After the sketches, the next step was to draw a cleaner version in ink rather than pencil:

logo-drawing

If I was more handy with vector based image processing programs (programs like Adobe Illustrator or Inkscape), I think I could skip this step: I’d be able to make an iconic, high-contrast image entirely digitally. Since I don’t have a ton of experience with these programs, it’s much easier for me to draw an image the old school way. From the photo of the drawing, I do some post-processing (increasing the brightness and contrast so it looks as close to two tone as possible) and then feed it into autotracer.org a cool, free, online tool that “vectorizes” bitmap images. The output is something like this:

high_contrast_-_darkened

With the output from autotracer, I have a high contrast image more amenable to digital editing. For the T-shirt design, I wanted a white image and text on a black background, so some minimal editing gives the final T-shirt design:

IM-shirt-LW

I hope you like the design and enjoyed reading about the process, and if this inspires you to make some digital art of your own, I’d love to see it!

Why I made a personal website

One of the requirements of the course I’m enrolled in this semester, MIT Massive, is to create and maintain a blog. We were provided with a turnkey blog framework, but I decided to create a custom personal website instead and added this blog to it. Making a custom site was a lot more work than the default option (and I’ve barely added anything to it yet!), but I felt it was an important step in acting congruently with some beliefs I’ve developed about the current state of education related to assessment and GPAs.

The problems with GPAs

Anyone who has been through the U.S. education system is familiar with the grade point average, or GPA. Ostensibly, the GPA is a quantitative measure of students’ knowledge and competency in all of their academic pursuits. It is a convenient way to quickly compare different students, since it boils down their achievement to a single number. GPA is weighed heavily in university admissions decisions (for both undergraduate and post-graduate programs) as well as in hiring decisions at many companies, and I think this is the case primarily because using GPA is convenient.

I think the continued use of GPAs is a mistake and I’ve become increasingly convinced that 1. GPA doesn’t measure overall academic knowledge and that 2. using a single number to report academic achievement is hopelessly flawed.

What GPAs actually measure

First, if GPA doesn’t measure academic knowledge, what does it measure? Tautologically, a high GPA indicates that the student earned high grades, and this could have resulted from a number of reasons: they are smart, they are hardworking, they care deeply about their grades, they have deep academic content knowledge, they cheat on exams or their family has the resources to provide academic support such as tutoring, to name a few. While some of these reasons are aligned with the intended purpose of GPA, some are definitely not. Research suggests that GPA is most strongly correlated to self-discipline, which is undoubtedly a useful characteristic to measure and is important to success. That being said, because most academic environments are “artificial,” in the sense that there is a single correct answer a student is asked to find (a situation not often encountered in real, innovative work), some companies are finding that GPA has little to no correlation to job performance. I would also argue that even if GPA is a good measure of self-discipline, a metric which measured knowledge and competency more directly would be more valuable. In the current system, a lot of time, effort and money (on both the student’s side and the educator’s side) are spent awarding high GPAs and providing evidence of good work ethic without the student actually internalizing the material they are supposed to be learning. Anecdotally, I know this to be true because I took college freshman chemistry alongside classmates who passed AP chemistry (and therefore could have skipped the class had they wanted to), but still struggled with concepts that were covered in my intro high school chemistry class.

The fallacy of distilling student achievement into a single number

Second, why is trying to measure academic achievement with one number flawed? For a very long answer, you could refer to The End of Average by Todd Rose (a book which I thoroughly enjoyed and highly recommend), but even if we take “average” academic achievement as being an acceptable idea, the GPA still has problems. There are two main arguments one could make for the value of using GPA to judge students: that it is a useful tool for comparing the relative achievement of students or that it is a useful tool for comparing the absolute achievement of students.

If we consider GPA as a tool to measure the relative quality of students, it would only work within the same system (e.g., within the same school). This is fine for globally famous institutions, but high school GPAs are purportedly important even though there are very few high schools that a majority of college admissions officers would know by name. I earned a perfect, non-weighted 4.0 from Sir Francis Drake High School, but I think very few people would argue that holds the same weight as a 4.0 from Phillips Exeter Academy, even though Drake is a “California Distinguished School.” My 4.0 would also mean something different than a 4.0 from a rough inner-city high school (their’s would be a better indicator of resilience to adversity, for example). Context is important to communicating student performance and academic achievement, and GPA doesn’t provide any context.

In theory, GPA could be reasonable as a measure of a student’s absolute level of academic achievement, but it doesn’t appear that GPA fulfills this role in practice. If GPA should reflect absolute academic achievement, then we would expect that schools which perform poorly (e.g., on standardized tests, although those have their own slew of issues) would tend to give lower GPAs. In reality, you can find students with perfect or near perfect GPAs at any school, regardless of how well the school is preparing its students for employment or their next step in education. This can also be seen in “grade inflation,” which is the practice of better grades being given with each passing year. The data show that despite college grades improving significantly over the past few decades, students are more disengaged from learning, spend less time studying, and are less literate than in previous decades. I don’t think GPAs could be treated as an absolute measure unless some outside governing body was given control of assigning student grades, a prospect about as unlikely as scrapping the concept of the GPA altogether.

An alternative to GPAs: portfolios

While GPAs have their issues, we still need a way to assess students’ academic achievement, so what’s a better option? I think a reasonably implementable answer is student portfolios. A student portfolio is a collection of the student’s most exemplary work, selected and organized to emphasize the student’s strengths. Since the portfolio includes actual content they’ve generated, it allows the student to provide direct evidence of their mastery of subject matter. Ideally, project content included in a portfolio would be more closely aligned to work that would be performed in “the real world,” and would thus be much more useful than the result of an artificial test. The portfolio would therefore provide a much richer way for admissions officers or potential employers to evaluate the student than a transcript. Having students focus on building a portfolio rather than a GPA could also be pedagogically advantageous, because instead of “teaching to the test,” instructors would be incentivized to design assignments that would result in strong portfolio items, which would map more closely to genuine learning.  The single number provided by a GPA provides very little information, but part of a project included in a portfolio provides direct evidence of that student’s achievement.

Motivation for my personal website

What does this have to do with my making a personal website? So far in my academic career, I’ve benefited heavily from having a high GPA, but if I want to show potential future employers (among other people) that I have real content knowledge and experience, I need a portfolio of my own. As I add more content to the website, I hope it will play that role, and I’ll be able to share work I’ve done which has had outcomes more significant than a grade on a test. And at the very least, the website itself is inherently demonstrative of what I’ve learned about web development.