May 1, 2025.

I know that novices have a difficult time figuring out when to use “one plus” formulas and when not. If you are one of them, maybe this little webpage helps, maybe it doesn’t.

First, realize that you have almost surely encounted this issue in your life many times before. This is because the concept is the same as when you say ==I just doubled my money (= 2x) and this is 100%, not 200%==. It has to become second nature for you to be able to coexist with these two different ways of working with gains and losses.

I always find analogies and numerical examples more helpful. You can’t just want to work with one or the other. It’s like asking when you use inches and when you use feet. You can’t live being able to deal only with one but not the other measure of gains and losses. Like inches and feet, rates of returns and “one plus” (“multiples”) are just different ways of measuring the same thing. You will encounter both in the real world many times. It’s also not enough never to learn about inches, and expect to use chatGPT whenever someone refers to inches in a conversation. You are expected to understand both without reference. These are not some obscure metrics. They are at the heart of all businesses.

Examples:

  • If you earn a 50% rate of return, you will have 1.5 times (1+50%) as much money at the end. (You can think of the “1” as representing your original stake, if it helps.)

  • If you end up with 1.2 times as much money, you have earned a rate of return of 20%.

  • If you end up with 1.1 times as much money after one year, and then again with 1.1 times as much money the following year, you end up with 1.21 times as much money.

  • If you end up with 1.21 times as much money after two years, it’s the same as if you earned 10% every year. This gave you 1.1 times as much as you had before.

  • If you end up with a 140% rate of return, every $1 will have turned into $2.4. (It did not turn into $1.4, because this would only be a 40% rate of return.)

  • If you end up with a 140% rate of return after 30 years, every $1 will have turned into $2.40 after 30 years.

  • What would be an annual rate of return to accomplish this? Well, (1+x)^(30)= (1+140%) = 2.4. This turns out to be x=(1+1.4)^(1/30)-1= (2.4)^(1/30)-1= 0.02961= 2.96%.

    • It’s the same problem that we had above with two years, but there we only had (1+x)^2= 1+21% = 1.21 (and we got x=1.21^(1/2) -1 = 10%).

  • When in doubt, check your answer. For example, here calculate (1+0.0296)^(30). This is about 2.399, i.e., 2.4. You would have ended up with 2.4 times as much money. In turn, a 2.4 times end result is a rate of return of 140%.

  • These calculations tend to be a lot more intuitive for short horizons and small interest rates. No one stepping back for a moment and reflecting would confuse an investment of $100 that ends up with $110 as a 110% rate of return. It is a 10% rate of return. Anything 100% and larger means at least doubling. Conversely, a 10% rate of return gives you the $10 extra, plus the $100 that you started with. It doesn’t turn your $100 into $10 – which would be a loss of $90. Answers that are obviously and intuitively wrong are good hints with fenceposts.

    Alas, if the time interval is very long, it is not unusual for projects to return 300%. At this point, the intuition becomes more difficult, because you don’t get obvious wrong answers. A 300% rate of return means you are left with 4x as much money — each $1 became $4. It means that if this took two years, you would have done the same if you had doubled your money in each year. The first year, your $1 became $2 and the following year it turned from $2 into $4. Your rate of return: 300%, not 400%.

You have no choice. You have to learn how to work with rates of return. You cannot skip over the issue. This is a concept that you have to understand.

I think is easiest for you to internalize it if you work lots of examples; and when you are confused on a particular problem, if you make a simpler version of it. (For example, instead of working with 300%, pretend for a moment that you only get 30%. Do your calculations come to a sensible intuitive result, or does it look like your $1 turns into $0.0001, a non-sensible answer for a positive rate of return.)?

I cannot give you a “cookbook recipe” when to use plus-one, vs. when not to use it. You have to internalize the whole spiel. Yes, I could give you an artifical recipe for the sake of a midterm, where I always formulate a question in the same way, so you can learn to pick up my cue. But then I would be training monkeys for my midterm. I don’t care about my midterm. I care that I want to train executives that can sit at a table in the real world and discuss gains and losses intelligently. Being able to work with rates of returns and knowing when you need to use a “1+” formula (giving you x times as much as you started with), and when not, is simply a non-negotiable skill. You have to acquire this intuition, and if the skill is not yet obvious to you, then you must work many, many problems until it becomes intuitive to you.

I am sorry — I get no sadistic pleasure out of this.