Why And How The Math Behind Indexing Adds Up

Buying a fixed percentage of the free float of each stock in a particular market segment is the fundamental basis of index investing. The fact that index investing works is a matter of mathematical fact. But most investors do not understand the math behind index investing. Understanding this aspect is essential to properly appreciate the logic behind index investing. Therefore in today's post I am going to explain the math behind index investing in detail.

When an investor holds a fixed percentage of the free float of the stock of every company in a particular market segment, he is said to hold a market portfolio. Investors holding a market portfolio are guaranteed to outperform the overwhelming majority of active investors around them. This has a lot to do with the inherent costs of investing. We will first understand how index investors perform versus active investors before accounting for costs. We will then see how performance is impacted after accounting for costs.

The Story Before Costs

For ease of understanding, let us assume an index of two stocks ITC and HUL. Look at the data given below for the two stocks.

Now let us say that there are 3 investors A, B and C. Investor A is a passive investor. Investors B and C believe in active management. Investor A buys 10% of the available free float of both ITC and HUL. This implies that investor A buys 100 stocks of ITC and 150 stocks of HUL. His returns would therefore be as given in the graphic below.

Given that investor A holds 10% of the free float of each stock, it is understood that the remaining 90% is held by investors B and C between them. The combined results for investors B and C are given in the graphic that follows.

There is a chance that either of investors B or C may outperform investor A. But if one outperforms, the other must necessarily underperform. Let me explain.

Between the two stocks ITC and HUL, HUL has a higher return. This means that if investors B or C wish to outperform investor A, they must have greater exposure to HUL in their portfolio. Investor A already holds 150 of the 1500 available free float stocks of HUL. He also holds 100 of the 1000 available free float stocks of ITC. That means there are 1350 stocks of HUL and 900 stocks of ITC available for investors B and C. Let us say that investor B holds 300 stocks of ITC and 900 stocks of HUL. His results would then be as follows.

Investor B's return of 27.5% is clearly higher than investor A's return of 26%. But now there are only 600 stocks of ITC and 450 stocks of HUL left. And investor C must necessarily hold these quantities. Investor C's results would therefore be as follows in the graphic below.

It is clear that investor C underperforms in comparison to investor A. But ultimately investors B and C must necessarily hold the remaining 90% of the free float of each stock between them. This remains true regardless of which of them out performs investor A. The collective results for investors B and C are given in the graphic that follows.

Notice that the collective return of investors B and C is the same as that of investor A. This is because just like investor A holds a fixed 10% of the free float of each stock, investors B and C also hold a fixed percentage between them (90% of the free float of each stock).

The inferences are clear. When one active investor outperforms, another must necessarily underperform. The aggregate portfolio weights of index investors and active investors in any market must be the same. And therefore before costs, actively managed money and passively managed money earn the same returns.

The Story After Costs

Anything paid to acquire, operate or sell a particular investment product may be termed as an investment cost. All investment products come with inherent costs (for instance brokerage, expense ratios, taxes and so on). These costs have a major impact on our long term returns. Costs as seemingly harmless as 1% per annum can severely dent our long term returns. Take this quote from Nobel laureate William Sharpe for proof.

Actively managed products carry significantly higher costs in comparison to passive products. Actively managed products may cost anywhere between 0.6% and 3% (or more) per annum. Direct plans of large cap index funds cost 0.2% per annum. These costs must be deducted from the gross return earned to arrive at the net return.

We have seen by now that actively managed money and passively managed money earn the same return. We also know that actively managed products cost significantly more than passive products. Therefore it is a matter of mathematical fact that after costs are deducted, passively managed money would outperform actively managed money. There may be sporadic occasions or periods where actively managed products outperform passive ones. There are two major reasons for this. The first of them is the fact that there are other active investors in the market along with managers of active products.

Secondly, most actively managed large cap funds are benchmarked to the Nifty 100 Total return Index (TRI). Most actively managed midcap funds are benchmarked to the Nifty Midcap 150 TRI. These are inexact benchmarks for these categories. Also actively managed large cap funds are allowed to invest a small portion of their assets outside the Nifty 100. Actively managed midcap funds are allowed to invest a small portion of their assets outside the Nifty Midcap 150.

This may see actively managed mutual funds in aggregate outperform passive funds over the short term. But any incremental return generated by active funds would be eaten up by taxes and higher costs over the long term. Therefore passive funds would necessarily outperform active funds over the long term.

The Last Word

Active management strategies occupy the majority of the mindspace of most Indian investors. Active management strategies provide a lot of intellectual stimulation. They make us feel like we are learning a lot and becoming better investors. But they demand two things from us in significant proportions. These are knowledge and time. Virtually all of us are employed in extremely challenging and demanding jobs. Our personal lives are no less demanding. This means we may not be able to offer the kind of knowledge and time required for effective active management of the portfolio. I firmly believe that if we are unable to meet the demands of a particular approach, it is best not to follow it.

Indexing mitigates the need for product selection and frequent portfolio management. This keeps things simple and allows us to pretty much put our equity portfolios on autopilot. This allows us to focus on pursuing more important things in life such as our profession and passions. With minimal fine tuning whenever required, our portfolios would take care of themselves. And as seen today, it is a mathematically proven fact that index investing works. It therefore deserves serious consideration from us when we pick our investment approach.