Percentage contribution over the long-term

Two business partners

Some time ago, I received an unusual request for help from a stranger via email. It was something of a financial nature, and I felt incapable of answering it fully. So I turned to Christopher, my engineer friend who’s also versed in finance, for help. I felt that the stranger’s request was quite personal, and I wasn’t sure if the stranger wanted the request to be publicised here on the blog. So to protect the innocent, I will wrangle the original problem to be most indecipherable. However, the solution given by Christopher was worth mentioning.

On to the problem then. We shall stick only to money as the sole source of contribution. Emotional and effort labour are just as important, but it’s difficult to quantify them mathematically. We’ll even ignore time labour. It’s not fair, but the world sometimes operate that way, so we’ll just accept it as best as we can.

Suppose there are 2 partners in a new business. Each of them contributes a different amount of money each month to the business, and may even contribute variably across each month. The money is used to maintain the business (for example, settle any bank loans, pay salaries, web site hosting costs, marketing expenses, product creation costs, utility bills and so on). After say 3 years, they want to sell off the business and liquidate their assets in it.

How much should each business partner take from that lump sum of money (from selling the business), based solely on their monetary contribution over the years?

The main problem of using a simple summation of their monetary contributions and calculating a percentage based on that, is inflation. $100 contributed in the 1st month is worth more than $100 contributed in the 37th month (4th year, 1st month).

Christopher suggested abiding by a yearly rate for the purpose of inflation, in particular, using government bonds as the rate. To add stability to the rate, he suggested using the 10-year bond rate. The currency also matters. For our example, we’ll use the United Kingdom pound. Where you get that rate is up to you, as long as you feel it’s legitimate and correct, and that everyone agrees to using that rate. I got mine from Bloomberg (suggested by Christopher):

http://www.bloomberg.com/markets/rates/uk.html

Government bond rate
[note: the above was correct as of time of writing this article. You are advised to go visit the Bloomberg site for the updated numbers. Or some other trusted site you know because you’re probably more financially powerful than me.]

The rate is 3.51% for the 10-year yield.

The mathematical formulation

Let’s give our business partners names, Arianna and Ben.

Let s(i) be the “sum” at end of year i
Let x(i) be contribution of Arianna in year i
Let y(i) be contribution of Ben in year i
Let b(i) be bond interest rate in year i (3.51% is changed to 0.0351 without the percent).

Then s(i+1) = [1 + b(i)] * [s(i) + x(i) + y(i)]

Let cx(i) be sum contribution of Arianna, and cy(i) be sum contribution of Ben at end of year i.

Then cx(i+1) = [1 + b(i)] * [cx(i) + x(i)]
and cy(i+1) = [1 + b(i)] * [cy(i) + y(i)]

Then contribution percentage at end of year i of Arianna is
100 * cx(i) / s(i)

Contribution percentage at end of year i of Ben is
100 * cy(i) / s(i)

A short example (with numbers)

And a note. The numbers were picked out of thin air. I am not saying a man is better than a woman in the business arena. I need some numeric variation, and I just took it as the man contributing more. And the fact that I don’t want to search the stock photos for generic businessy settings with 2 people in them for hours. For some reason, it’s very hard to find a photo with 2 business people of the same gender. Then you wouldn’t be able to guess which one was which. I originally had Andy and Ben starring in my example.

Anyway…

Let’s say, in the 1st month, Arianna and Ben contribute £100 and £150 respectively. In the 2nd month, £200 and £100 respectively. In the 3rd month, £0 and £250 respectively. For the 1st year, Arianna and Ben contribute £1200 and £1800 respectively, for a grand total of £3000.

So at the start of the 2nd year, we look at the rate and say we find it’s 3.5%. Then we inflate their contributions by 3.5% to £1242 (1.035 * 1200) and £1863 (1.035 * 1800) respectively. We also inflate the sum total from £3000 to £3105.

Arianna and Ben continue to contribute to the business in the 2nd year. And at the end of the 2nd year, Arianna and Ben contributed £2100 and £1900 respectively (total £4000).

At the start of the 3rd year, we find the rate to be 3.52%. Here’s where it’s different. First we add the respective contributions to the previous inflated amounts. So we have £3342 (£1242 + £2100), £3763 (£1863 + £1900) and £7105 (£3105 + £4000) for Arianna, Ben and the sum total respectively. Then we inflate them by 3.52% to get £3459.64, £3895.46 and £7355.10.

For the 3rd year, Arianna and Ben contribute £2400 and £2600 respectively. And at the start of the 4th year, the rate is 3.51%.

Adding the sum contributions first, we get £5859.64, £6495.46 and £12355.10 for Arianna, Ben and sum total respectively. Inflating the numbers by 3.51%, we get £6065.31, £6723.45 and £12788.76. At this point, our business partners want to sell off their business, and they want to know what’s their percentage contribution.

So Arianna contributed 100 * 6065.31 / 12788.76 = 47.43%.
Ben contributed 100 * 6723.45 / 12788.76 = 52.57%

Further notes

I didn’t take care of rounding issues and decimal places in the example. I’ll leave that to you as an exercise. The most accurate method will be to store the values as they are, and only take them out for calculations at the final stage. Storing intermediate results might skew the accuracy if done over many iterations.

The other point is that you are free to calculate on a monthly basis instead of a yearly basis. The assumption is that the rate stays stable throughout the year, so a yearly rate is usable. If you’re inflating on a monthly basis, the contribution values will shoot up very quickly. They will not have any semblance to their original values. I want you to remember that the values are used to calculate the final percentage contribution, not the absolute contribution.

Contributing $40 out of $100 is 40%. Contributing $673.20 out of $1683 is also 40%. It’s relative.

[note to self: why didn’t I think of using a currency with $, the generic dollar sign? I had to go pick the pound, and then had to replace all the monetary values in the example with the correct currency symbol…]

[UPDATE: the stranger found another of my article on percentage calculations. It must have made sense, because the stranger took that to mean I could help with the problem the stranger was facing. Yes, I’m using “the stranger” so you don’t even know the gender. How’s that for anonymity?]

[leading image by nyul]

Percentage calculation with negative numbers

Suppose you have 2 numbers. You want to sum them up, and calculate the percentage of each number upon that sum. Say, 4 and 6. So 4 contributes 40% and 6 contributes 60% to their sum 10.

What happens when you have negative numbers?

I did some simple research, and the relevant articles have someone trying to calculate percentage changes from one number to another. Like financial growth reports.

My question is more of, how much each component number contributes to the sum, as a percentage. The problem comes when one of the numbers is negative. Consider the trivial case, 1 and -1. The sum is 0. You already get a division by zero error when calculating the percentage (1/0) * 100.

The solution, which is the same as that in my research, is to take the difference between the two numbers and use that as the basis. So difference of 1 and -1 is 2. So 1 contributes (1/2)*100 = 50%.

What about -1? Use the absolute function. ( abs(-1) / 2 ) * 100 = 50%.

The difference method works fine if you have only two numbers. What if you work with several? My friend actually posed this question to me. Suppose you have 6 numbers, and you want to calculate the percentage contribution for each number towards their sum.

My suggestion? Absolute everything. The percentage contribution of n1 is
abs(n1) / ( abs(n1) + abs(n2) + abs(n3) + abs(n4) + abs(n5) + abs(n6) ) * 100

Then my friend posed a killer question. What if all the numbers are zero? What’s the percentage for each number then (even though each number is zero)?

It’s for a reporting application, and my friend was wondering how to calculate the percentages. Now the sum of a bunch of zeroes is also zero. You hit the division by zero error. Even the absolute-everything method fails, since each number is zero, so there’s nothing to “absolute”.

Since there’s no defined way of calculating when all the numbers are zero, I gave the 2 obvious solutions. The first is that, since each number is zero, and the sum is zero, therefore each zero contributed 100% to the zero sum. The second is, since each number is zero, therefore each contributed 0%.

My friend chose the second solution. The most compelling reason for that choice was that it’s easier to explain the logic behind that choice to the user. It’s an edge case. When there’s no right answer, choose the answer which is easier to explain.

UPDATE: Steve has given 2 more alternative solutions.

UPDATE: I wrote an article to explain some of the confusion by some readers. How can a poorly performing product contribute the highest percentage in a company’s bottom line? Read the explanation here.

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