*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.

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:

where 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:

- Guess a value, and plug it into the ROIRCE equation
- 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
- 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:

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.