Effective ROI for investments with recurring costs/earnings

Edit: I finally found the existing term for the idea I was exploring in this post. Basically, this post was an exercise in re-deriving Internal rate of return (IRR) and giving it a worse name and acronym.

In my last post, I discussed how to analyze the “investment performance” of buying goods in bulk or on sale, by applying the idea of ROI (or more accurately, CAGR) to those purchases. As a refresher, ROI can be calculated from the follow formula, and traditional investments are typically expected to have an ROI of ~5-15% long term.

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

One point I briefly mentioned in that post was that it’s more valuable to receive payouts from an investment earlier, but for simplicity we treated the payouts from buying in bulk as being received in a lump sum once everything had been used. This wasn’t just for the simplicity of that post, it was also because (as far as I know) there is no version of ROI that accounts for an investment with recurring costs or recurring earnings. If you look for how to calculate ROI/CAGR, the equations all assume a single starting value and a single final value – there’s no room to account for extending or unwinding a position over the course of holding an investment. If you continuously buy and sell shares of a stock, for instance, you can track the ROI of a particular share by pinpointing the date you bought and sold it, but there’s no equation to to figure out the overall ROI that all the shares of that stock have provided you.

Introducing ROIRCE

In this post I’d like to propose a method for characterizing the performance of an investment that has recurring costs/earnings (or more broadly speaking, doesn’t have a single fixed purchase date and a single fixed sell date). I’m sure someone else could come up with a much better name and acronym, but here I’ll call this metric Return On Investment for Recurring Costs/Earnings, or ROIRCE. ROIRCE borrows from how Net Present Value is calculated, in terms of how it discounts costs and earnings in the future.

The ROIRCE value is chosen such that the following equation holds:

 \sum \frac{\textrm{cost}_i}{(1+ROIRCE)^{t_i}} = \sum \frac{\textrm{earning}_i}{(1+ROIRCE)^{t_i}}

where t_i is the time in years from the beginning of the first investment where the associated cost is incurred or the associated earning is collected. The left side of the equation is the effective total cost of the investment, while the right side of the equation is the effective total earnings from the investment, with each discounted based on how long they took to realize.

Unfortunately, this is an implicit function, which makes it inconvenient to solve, and probably precludes ROIRCE from popular use. Here is one algorithm we could use to solve for ROIRCE:

  1. Guess a value, and plug it into the ROIRCE equation
  2. If the left side of the equation is greater, you need to reduce the ROIRCE value, while if the right side is greater, you need to increase the ROIRCE value
  3. Repeat steps 1 and 2 until the two sides of the equation are effectively equal

(For a more specific approach for raising/lowering the ROIRCE value, you could try a bisection method.)

An application of ROIRCE

The reason I was reminded of this idea (which I had intended to write about since I wrote the last post, but never got around to) was a post on the personal finance subreddit about whether after winning a lotto jackpot, one should take the lump sum or monthly payments. Unfortunately I can’t find the exact post anymore, but the options were either to take $60k in a lump sum, or $1k/month for 10 years (which totals to $120k). The actual post was full of good advice about practical concerns like which option is more advantageous from tax and psychological perspectives, but in this post we’ll consider the options in a simple, tax-free world, and approach the choice purely mathematically.

One response argued that it’s better to take the lump sum of $60k, since if you invested it you could expect to have >$120k by the time you would’ve received the last monthly payment. Another pointed out the flaw in this reasoning, which is that with the monthly payments you’re not sitting on them until the 10 years is up, you can invest that money and earn from it in the mean time. I agree with the second poster, since the first poster falls victim to the same oversimplification we made last post.

We can calculate the traditional ROI required for the lump sum to be better than receiving $120k ten years later through the following calculation:

ROI = (\frac{\$120k}{\$60k})^{(\frac{1}{10})} - 1 = 7.2\%

From a simple ROI perspective, if you can earn better than 7.2% returns annually on your investments (which is not a sure thing, but is definitely reasonable), then you should take the lump sum.

However simple ROI doesn’t consider that we could invest each of our monthly $1000 payouts as soon as we receive them. If we invest each monthly payment over the course of the 10 years, we calculate a ROIRCE value of 17%, a very hard investment to beat! The only common reason I can think of where it would be beneficial to take take the lump sum when considering ROIRCE was if you had a significant amount of credit card debt. This is because getting rid of debt is like investing at the debt’s interest rate, and credit card interest rates can exceed 20%.

ROIRCE for buying in bulk

Using ROIRCE doesn’t add much to our analysis of buying goods on sale, since each incremental item will be held for a different amount of time, and that amount of time can be used in the traditional ROI calculation. We can calculate an overall ROIRCE for buying on sale, but we should still use the individual ROI of each increment to decide how much to buy.

Deciding if it’s worth buying in bulk, on the other hand, can benefit from ROIRCE. Take the example we used in the last post where I can buy a year’s worth of toilet paper for $100, which would cost $120 if I bought my toilet paper month by month. Buying toilet paper in bulk here is like buying an investment that pays me $10/month for a year. While the toilet paper itself isn’t money I can use, each month the $10 I would’ve budgeted towards toilet paper is freed up so I can spend it on whatever I want. Using traditional ROI, we found that in this case buying in bulk provided a 20% return. However, using ROIRCE and taking into account that I can do something productive with my freed up $10/month over the course of the year, the return on buying toilet paper in bulk is 41.3%! The return is almost twice as high when considering the effect of our recurring “earnings,” so for buying in bulk we can analyze our opportunities much more accurately with ROIRCE than with traditional ROI.

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

Note: technically the term for this financial concept is Compound Annual Growth Rate (CAGR). I will use “ROI” throughout this post since it’s an acronym that gets used much more often than CAGR – even though CAGR is the better measure and they’re fundamentally trying to measure the same thing: investment performance. Also, I assume Mark Cuban meant CAGR even though he said ROI.

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.