Enrico Fermi, the Italian physicist, was renowned for his estimates. With little or no information at his disposal, he would often do back-of-the-envelope calculations to come up with a number that subsequent measurements revealed to be impressively accurate. The Fermi estimation process is frequently used by scientists and engineers in their problem-solving efforts. As a Product Manager, specifically early in the product life-cycle, one has to estimate the market size for their products to make sure pricing is optimal. And Fermi’s estimation process can be an indispensable tool in such situations.
I have tried to simplify the Fermi’s estimation process below and I illustrate this process with an example.
Decompose the problem to as many sub-components (or factors) as possible. Think of a multi-level tree structure. Estimate each sub-component, and then recombine the results. This will become clear when we go through an example.
Now, the question is how do we generate estimates if you have no clue about the topic in question. This is where the art and science of estimation come into play. It’s a two step process, first one is the art, and the second is the science. And more well-read you are, better you will be with the “art” part.
Step 1: Determine the lower bound & upper bound. Since we may not have knowledge about this topic, we have to make some educated guesses by approximating with a concept that resembles the topic you are trying to estimate. For example, if you know the population of Germany, you may make some educated estimates about the lower and upper bound about the population of France. Note, it’s far easier to make a range estimate than a point estimate, hence the need to estimate the lower and upper bound.
Step 2: This is the science (or math) part. Don’t take the average of your lower and upper bound estimates, but Approximate Geometric Mean (AGM). AGM can be calculated by averaging the coefficients and exponents of your estimates. For instance, if we need to compute the AGM of 4 and 800. Then,
4 = 4 x 100 and 800 = 8 x 102
Average of coefficients = (4+8)/2 = 6 and Average of exponents = (0+2)/2 = 1
AGM = 6 x 101 = 60 ( versus average = 402)
Given the demonetization efforts in India to root out black money if somebody where to ask you this question, ” How much black money is there in India?” . Your first answer will be, “I have no clue”. But wait, can we use the Fermi Estimation process to answer this question. Probably, we can.
Let us start by decomposing the problem into factors or sub-components. We could easily get confused and complicate things trying to figure out deposits made by Indian nationals to swiss bank accounts or estimate the percent of transactions that occur in cash or computing the income tax evading population, etc. I think we can keep this simple by just breaking this into two sub-components –
- The size of Indian Economy (GDP)
- Percent of the Indian Economy that occurs in Black (Shadow economy)
Note, we have an only one-level tree in this case, however, for more complex problems you can potentially break a sub-component further into sub-sub-components.
Next, let us try to estimate the lower and upper bounds for each component.
Size of Indian economy
I will make the assumption that one might know the size of US economy, which is $18+ trillion dollars. Given this information, we can make a range estimate. Given that India is still an emerging country and much less developed compared to the US, my range estimate for Indian economy will be from $1 trillion to $4 trillion. And if I follow the AGM formula above, the AGM comes to $2.5 trillion.
Percent of the Indian economy that occurs in black
I don’t want to complicate things but based on my readings and intuition I could come up some “reasonable” range estimate of 10% to 40%. Again, note it’s easier to estimate a range compared to point estimate. Computing AGM, it comes to 25%
Combining the two factors –
Amount of Black Money in India = Size of Economy x % of Economy that occurs in black = $2.5 trillion dollars x 25% = $625 billion
How close are we to reality? Well, we estimate the size of Indian economy to be 2.5 trillion dollars and the actual value is $2.25 trillion. Not bad.
Regarding percent of GDP in black, a bit of googling lead me to this –
Source: A 2012 paper on black money from Indian Ministry of Finance
Note, our estimate was 25% compared to the estimate of 19-21 % from 1984. Since then, the Indian economy has increased rapidly, so has black money since tax-evasion and corruption is still rampant in India. So our estimate might not be far from reality.
In conclusion, I believe this simple estimation tool is very powerful tool when dealing with complex and ambiguous estimation problems.