This blog post is adapted from my French column with the newspaper Les Affaires.


We have recently learned of the death of British mathematician Edward H. Simpson (1922-2019). Simpson first distinguished himself during World War II as a code breaker for the Royal Navy, when he enabled the interception of numerous ships carrying supplies for the troops led by Germany’s General Rommel. After the war he completed his PhD at Cambridge and then, in 1947, published a paper called “Measurement of Diversification” in Nature. The simple, remarkably elegant formula that Simpson came up with was designed for studying biodiversity.

The formula was subsequently taken up by economists to measure industrial concentration, with a view to combatting monopolies. Generally speaking, concentration and collusion in sectors of the economy are undesirable, as they allow a handful of companies to rake in huge profits at consumers’ expense, by setting whatever prices they like, shielded from competition.

Concentration is just as toxic for investment portfolios. For example, during the 2008 financial crisis, unfortunate investors with large holdings in one or more companies that went under or were restructured lost a major chunk of their investments, for good. Think of the late Bear Stearns and Lehman Brothers. The same applies to organizations like Citigroup, Fannie Mae and AIG, which still exist (having been recapitalized by the U.S. government), but shed 95% of their value during the meltdown. These losses will probably never be recouped; the damage is permanent.

That brings me back to the diversity index, which helps us measure portfolio diversification and avoid the perils of concentration. Just because your portfolio contains a large number of securities doesn’t mean that it is as diversified as you may think. Let’s look at the example of Portfolio A, with 100 securities, but with 99% of value concentrated in a single asset and the rest distributed equally between the 99 others.


Portfolio A:

Security Weight
Stock 1 99%
Stocks 2 through 100 1% of total


The following formula lets us adjust the number of securities in our portfolio taking into account the weight of the individual assets:


Diversity index = 1 / sum (weight of securities2)


The diversity index for Portfolio A is 1.02 securities. So even though Portfolio A may seem diversified because it contains 100 securities, it is in fact equivalent to a portfolio with a single security.

Now let’s look at Portfolio B — which also contains 100 securities but is equally weighted.


Portfolio B:

Security Weight
Stock 1 1%
Stocks 2 through 100 1% each


This time, the diversity index is exactly 100 securities. That means the number of securities accurately mirrors the portfolio’s diversification.

To illustrate the diversity index in more tangible terms, I have calculated the index for three exchange traded funds available in Canada.


Table 1: Security diversity index

ETF Category Number of securities Diversity index
iShares S&P/TSX60 Canadian Equity 60 securities 30 securities
BMO S&P/TSX Capped Composite Canadian Equity 246 securities 48 securities
Vanguard All Equity Global Equity 12,186 securities 323 securities
Sources: PWL Capital, iShares, BMO ETFs, Vanguard Canada


So although the iShares S&P/TSX60 ETF (ticker XIU) holds 60 securities, once we factor in major concentration in certain individual stocks we can see that in terms of diversification, this ETF holds the equivalent of just 30 securities. Diversification is better with another Canadian equity ETF, BMO S&P/TSX Capped Composite (ticker ZCN), which holds close to 250 stocks and has a diversity index of 48 securities. That is about the best diversity index available for a Canadian equity index portfolio, seeing that our market is highly concentrated. However, when we look at Vanguard All Equity, a global equity ETF, we are struck by the fact that the diversity index for a portfolio made up of Canadian, U.S., international developed markets and emerging markets equities is over 300 securities. That means a substantial reduction in the idiosyncratic risk. With this type of diversification, one or more of the companies included in your portfolio could experience severe problems, without a serious impact on the global value of your investments.

We can also calculate sector diversification for each of these three ETFs, as shown in Table 2:


Table 2: Sector diversity index

ETF Category Number of sectors represented Sector diversity index
iShares S&P/TSX60 Canadian Equity 11 sectors 4.9 sectors
BMO S&P/TSX Capped Composite Canadian Equity 11 sectors 5.9 sectors
Vanguard All Equity Global Equity 11 sectors 7.1 sectors


Once again, we find that an international portfolio has substantially greater sector diversification. Canadian stock exchanges are known for their concentration in the financial services, energy and natural resource sectors, and the virtual absence of the healthcare and technology sectors. Adding U.S. and international equities to a portfolio does much to make up for this weakness.

To wrap up, security and sector diversity indices are a good indicator of a portfolio’s soundness. In my opinion, many investors with a portfolio made up of just a handful of securities expose themselves to risk without any additional expected return. Diversifying your portfolio with ETFs containing foreign equities greatly increases your security and sector diversity indices. And the most surprising thing is that international diversification is only a bit more expensive than investing exclusively in Canadian equities. You can invest in a Canadian equity ETF like BMO S&P/TSX Capped Composite and add complementary ETFs for U.S. and international equities, or choose a global equity ETF. In either case, you will pay fees of 0.25% or less. So the next time you talk to your financial advisor, you should ask him or her about your portfolio’s diversity index.