Back-of-the-Envelope Calculation

A back-of-the-envelope calculation is a quick mathematical approximation or an informal calculation performed in daily life on a rough piece of paper. The main advantage of using such a method is to arrive at the necessary results quickly to get a brief idea about something, thus hastening the decision-making process.

The back-of-the-envelope calculation is not just a random guess; some amount of mathematical computation is done. This method does not have any scope in itself, as for practical purposes, one requires detailed calculations by weighing many other relevant factors. Nevertheless, it is used frequently in finance and science.

• Back-of-the-envelope calculation or Fermi number is a rough estimation that people use for computational purposes in finance, economics, and science.
• The importance of back-of-the-envelope calculation is that it helps save time by providing required information immediately.
• To perform such computations, a person must have mathematical knowledge and good logical reasoning. They should also be fast in working out calculations.
• There is no specific step-by-step procedure or established rules to perform such calculations. However, it depends on the skills of the person working it out to perform faster and more accurately.

Back-of-the-envelope calculation was first theorized by the physicist Enrico Fermi, known for approximate computing values instead of extensive mathematical calculations and experimentation. These were called the Fermi numbers or Fermi estimates. Its origin thus dates well back to early scientific studies.

Despite such calculations being mere approximations like ballpark figures, the two are not the same. The back-of-the-envelope calculations estimates to a higher degree. So, for instance, if the expected return on an asset is $1119, the ballpark figure will be around $1000, whereas the back-of-the-envelope calculation will show $1100. 

This term became popular in finance when professional managers and accountants started making simple calculations on a scribble pad or scrap paper lying on the table. Finance experts, economists, governments, and researchers commonly use these estimations. A real estate back-of-the-envelope calculation is commonplace during the initial stages of property dealings.

However, it is evident that with technological developments, the scope of such calculations is low.

As many professionals are increasingly becoming familiar with using calculators and computers, they can get exact results in no time with minimal effort. Nevertheless, the importance of back-of-the-envelope calculation is not to be underestimated. It helps make quick decisions using little effort. It forms the basis for many significant projects and decisions. Also, such calculations will provide insights to the person performing them while working them out. For example, in real estate such calculations help get a general idea about property and market values.

Further, it is worth mentioning that although back-of-the-envelope calculations are informal, people also use it formally at times. For example, consider the tweet by Bloomberg, according to which around $9.6 billion of daily marine traffic was halted in the Suez Canal on March 26, 2021. This report took place when the container ship 'Ever Given' blocked the Suez Canal in March of the same year. The point is that formal media also use such computations to communicate an idea.

Here are a few examples of back-of-the-envelope calculations.

Example #1

Consider an electronics company. Owing to the Christmas season, the firm decided to give a bonus to the employees. The HR manager informs the Finance manager that 89 employees of the company shall receive a Christmas bonus of $208. The Finance manager immediately works out a rough calculation of $200 x 90 = $18000.

Here, the Finance manager should be aware that by not considering the $8 from the $208, they are compensating the total amount by considering 90 people instead of the actual 89 people from the HR department. Also, the numbers 90 and 200 facilitate easier calculation than 89 and 208, respectively. Therefore, the manager should know such techniques and ideas to help get immediate results.

Example #2

Here's an article published by Forbes, according to which entrepreneurs must frequently use back-of-the-end calculations to assess the business constantly. For example, entrepreneurs can determine how and when their entity will become profitable. Or they can decide how many people they should hire in various departments as their business scales up while also considering their ability to pay salaries to those hired.

Not just that, an individual running an enterprise has to work out many aspects of the business constantly. Such as expenses, salaries, budgeting, expected revenue during Christmas or Summer season, or anything that concerns their operations. This will allow them to keep track of their business. 

This article has been a Guide to what is Back-of-the-Envelope Calculation. Here we explain its definition and examples along with a detailed explanation. You can learn more about it from the following articles –

• Financial Modeling Tools
• Financial Modeling Process
• Financial Modeling Benefits