Trouble with ratios………. Innumeracy Series(2)

Rama Nimmagadda
4 min readFeb 5, 2022


“You better cut the pizza in four pieces because I’m not hungry enough to eat six.” -Yogi Berra

……………a 2 bhk (bed room, hall, kitchen) on 9th floor with garden facing view. We are fine to sell this also. Expected rent is 32k and sale price is 1.05 crore

Second is 3 bhk on 5th floor back facing. This is for rent only. Expected rent is 45k

Third is 3 bhk on 1st floor back facing. We are fine to sell this also. Expected rent is 40k and sale price is 1.4 crore……….

We received this message in our apartment society’s whatsapp group earlier this week. Seeing such non-trivial rental amounts, my first reaction was that real estate investment is rather lucrative (I pay rent every month and am generally aware of these figures but the whatsapp message brought matter to the fore). To be sure that it wasn’t just me, I asked my wife for her impression on reading this message. She also felt that these were good returns.

While the perception of value is founded in one’s personal circumstances, I would like to believe that amounts such as ₹32,000 (US $425), ₹40,000 (US $533) and ₹45,000 ($600) are non-trivial amounts.

These asking numbers reflect rental yield of about 3%. Capital appreciation has been around 2% (I’m basing this on an actual transaction that I’m aware of). So, roughly the overall annual yield is 5%. Even in today’s relatively low interest rate environment, bank fixed deposits are yielding over 5% annual returns. On the face of it this may not come across as fair comparison given differential tax rates across different asset classes but once you incorporate, transaction and maintenance charges, it more than evens out the lower tax rate of real estate income. So, it is clear that one can better return more easily by investing in FDs (click of a few buttons on your computer), but these non-trivial rental amounts make us feel that they are better investments. Why?

In stock investing, I have come across a common enough misunderstanding that a company with lower stock price is considered cheaper than a company with higher stock price. It is easier for us to see a one ₹2 stock becoming ₹3 given that it has to increase by only ₹1. But it is much harder to see ₹2000 stock growing to ₹3000. Although, assuming both are highly liquid stocks, buying one ₹2000 stock is equivalent to buying 1000 stocks of ₹2.

This made me wonder — why do we default to numbers as opposed to ratios or proportions? As it turns out, certain cognitive factors are at play here — this is what my research into this turned up.

Not that they are mathematically complex but proportions are external abstract concepts for which we have no natural intuition, where as we have good intuition for identifying large numbers and differentiating them from small numbers.

Hence our first reaction is getting swayed by direct quantities (like large-ish rental amounts earlier). This is compounded by our tendency to not go past first order thinking. Second order thinking (like sizing/proportioning returns to the investment amount) is cognitively expensive — so we avoid it if we can get away with it.

There is a well-recognized tendency called “natural number bias” that affects how we evaluate ratios. Without explicitly calculating, try guessing the larger of the following two pairs: (7/8, 2/3) and (3/5, 2/3) (I have taken these examples from this research study). Now, see if you got the right answers — 7/8 and 2/3 respectively.

As opposed to looking at the ratio in its entirety, we tend to compare the numerators and denominators across ratios. Our default intuitive thinking has a bias for natural numbers over ratios. Once we engage our conscious part of mind explicitly, we are quite likely to get the correct answer.

Here is another example — BSE Sensex (India’s largest stock market) fell over 1500 points on 24th January this year. This looks big but it is a fall of just 2.6% — par for the course of a superhot stock market.

From an evolutionary perspective, we did not have a need to delve in abstract mathematics (even rudimentary concepts such as ratios) and probably hence we struggle with them.

Few other terms that describe this bias are: Denominator-neglect bias, ratio bias, numerosity bias.

So, are there ways to address this?

I think the best way to deal with this is practice solving explicit scenarios/problems across different contexts. This way we are more likely to recognize the bigger picture more readily and hence lesser cognitive load to be spent than otherwise — in other words, this increases cognitive efficiency and productivity as we assess real world situations that come our way.

Also, do not stop short of second order thinking. Of course, it is good to take help from experts particularly in matters such as investments etc but given many experts also readily suffer from these biases, it is important to develop an independent personal feel as well.

Few interesting articles I read in this context:

How natural is numeracy?

System 1 and System 2 Thinking

Don’t suffer from Denominator Blindness

PS: This blog is about innumeracy and not about investment merits of real estate as an asset class. Just like with most other asset classes, real estate investments may turn out great or lousy depending on timing, luck and a host of other specific factors.