Question

Prove the benefits of diversification mathematically

Answer 1

Solution:-

There is no formula or mathematical theory that sets a certain criteria for measuring benefits of diversification. So, there is no standard way to measure benefits of diversification in all cases. However, for the sake of understanding let's take a look at the benefits that are attributable to diversification and a numerical example to understand it.

How does diversification help?

Diversification helps using the basics of probabilistic behaviour. Every stock, company and industry are exposed to some risks that are onlhy for those stocks/companies/industries. For e.g.: Death of a CEO would only impact the company or a fall in oil prices will only have a major impact on oil companies and so forth. If an investor invests his entire portfolio in one stock (or multiple stocks of same industry), he exposes himself to the huge potential adverse impact of risks that a particular stock or industry is exposed to.

Lets say an investor invests his entire portfolio in oil companies, his portfolio can take a major beating if oil prices see a crash. So, in order to avoid these high risks attributable to one stock or industry, the concept of diversification steps in. It involves the concept of investing in multiple stocks across various industries. So, even if unfavourable and unforseen events happen in a certain industry, only a small part of his portfolio gets impacted while the remaining part stays away from that risk.

This works because as per the laws of probability, it is unlikely that many number of companies or industries will see their specific bearish events happen at the same time. For e.g. it is unlikely that the CEOs of 20 company die at the same time or that 30 companies are found doing accounting fraud at the same time, etc etc. In short, it is unlikely that the type of risks which are specific to a particular company or industry will occur at the same time for many companies or industries in a portfolio. This is why the concept of diversification helps in reducing risk.

Numerical example of diversification's benefits:

Let's compare two situations, one with diversification and one without diversification. Let's say that Mr. X invest the 100% of his portfolio in company ABC Inc. while Mr.Y invested in 50 different companies including ABC inc. such that every company out of 50 constituted 2% of the portfolio share.

Due to announcemenmt of an accounting fraud at the company ABC Inc., the stock fell by 90% which resulted in Mr. X's portfolio falling by 90% whereas it had a negative impact of just 1.8% on Mr.Y's portfolio (i.e. 2% of portfolio*-90%). Thus, we can see that diversification signifcantly reduced those risks related to stocks that are specific to those stocks or industries.