Almost everyone believes that you should diversify your assets – but there are different explanations about why.
It is sometimes said (though how anyone would know is another story) that fish are not aware that they swim in water.
Similarly, many investors today live in a world shaped by many of the assumptions underlying Modern Portfolio Theory. However, relatively few people have had the time or resources, or perhaps the inclination, to study those background assumptions.
Our goal in this and associated posts is to examine those assumptions, the reasoning behind them and the main conclusions those assumptions lead to.
Diversification – The Foundational Idea
One basic idea of Modern Portfolio Theory is the idea of diversification. Diversification is the spreading of risk over more than one investment. Technically, it means spreading the investments among assets whose returns are less than perfectly correlated.
Example of Diversification
Modern Portfolio Theory involves some fairly technical math. I will try to explain the concepts, without going much into the math. Much of the math concerns risk.[1]
Let’s start with a $1000 bet on a coin flip. For illustration, consider a coin flip with the following payoffs:
- Heads you get $2100,
- Tails you get $0
The risk in this coin flip is that you’ll lose. There’s a 1 in 2 (or 50%) probability that you’ll lose $1000.
One single bet of $1000 is quite risky.
But if you could bet the same $1000 by placing 1000 bets of $1, each with a payoff of $2.1 for heads and $0 for tails, your risk would be much less.
You still wouldn’t be guaranteed to come out ahead, though there is about a 95% probability that you would not lose money.[2]
It can be easily shown that the expected return (expected return is another mathematical concept) is exactly the same whether you make a single bet of $1000 or 1000 bets of $1.
Thus (ignoring transactions costs) the benefits of diversification (i.e. making multiple smaller, independent bets) are free, in the sense that you reduce risk without reducing expected return.
This ability of diversification to reduce risk without reducing expected return is perhaps the fundamental idea that underlies all of Modern Portfolio Theory.
Real World Investments
If in the real world, investments were like our hypothetical coin flips, we would not have to do any further work. The world would be a much simpler place.
But in the real world, actual investments differ from idealized coin flips in many important ways.
Let’s consider the stock market. Investing in individual stocks might feel like betting on coin flips. But unlike coin flips, the outcomes of which are independent of each other, historical stock returns have been correlated.
Correlation is another statistical concept that means that two variables (in this case the returns from owning two or more different stocks) tend to move together. For example, history tells us that if, on any given day, Apple stock increased in price, Microsoft stock probably also went up. There are different ways to quantify this relationship.
The most common quantification is a statistic called the correlation coefficient.[3]
Correlation
Because of the way the statistic is constructed, a correlation coefficient must be between negative one, and one, inclusive.
Perfect Correlation
A correlation of 1 between two values indicates that the two values move in perfect lock-step. In the real world, two values are rarely if ever perfectly correlated, unless they are so by definition.
An example of values in the real world that are perfectly correlated would be the daily temperature at the same time in the same spot, reported in Fahrenheit and in Celsius. The two numbers are perfectly correlated, because they are really just two different ways of saying the same thing.
Perfect Negative (or Inverse) Correlation
Perfectly negative correlations are similarly rarely if ever seen in the real world, unless that negative correlation is the result of measuring the same thing two different ways.
Consider an example from the stock market. Ignoring taxes, dividends, transactions costs, and institutional frictions, the return from being long a stock will be exactly inversely correlated with the returns from being short the stock.
Zero Correlation
Zero correlation occurs when the outcomes of two different events are independent of each other. For example, the outcomes of coin flips are independent. If you have two coins, knowing that one coin came up heads tells you nothing about how the other coin will come up.
Asset Market Correlations
In the US stock market, over the long run, the returns on individual stocks tend to be correlated with each other, and with the market as a whole.
Correlations between and among asset returns vary over time. Over periods of less than year, the correlation of returns tends to be relatively high, while over multi-year periods the correlation decreases.
Here is a table of historical correlations among some of the major markets or regions of the world. These data are based on monthly returns from the period roughly 1990 to 2024.[4]
Note that all of the lowest correlations involve Japan. That may be largely due to the fact that during this particular time period, Japan experienced a steep and protracted bear market, while most of the rest of the world experienced a long period of growth.
From a diversification-only point of view, including Japan seems like it would have reduced risk over the period. But we know that the period was a period of very low Japanese returns, and so including Japan would have significantly reduced overall return.
The overall average of these correlations (removing the own-market correlations which are by definition 1) is about .73. That average might be more useful than the individual specific correlations, which might or might not predict future correlations.
These data are backward looking. But to invest, we need to look forward. We’ll address that in an upcoming post.
Next steps
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[1] There are lots of ways of defining and measuring risk. For much of Modern Portfolio Theory, risk is measured as standard deviation (or variance) of return. For now, you don’t have to worry about the mathematical definition of either of these terms. We’ll illustrate using some examples.
[2] We explain the math in detail in our book Investor’s Dilemma Decoded. It’s available online. If you’d like an excerpt explaining the math, please email us at [email protected] to request it.
[3] The Pearson correlation coefficient is the correlation most frequently seen in finance. It is named after Karl Pearson, who was a British mathematician in the late 1800s and early 1900s.
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