The term “expected return” can be confusing. It not a rate of return that you *actually expect* to get. Rather, it is the probability-weighted average of all possible returns that you might receive over some period of time, such as a year.

A simpler way to understand this would be through a standard six-sided fair die. When you roll a die, possible values are 1,2, …, 6. All possible values differ from the expected value of 3.5.

So, if expected returns are never actually realized, why are they important? They are important because they tell us the centres of the probability distribution of each potential component of a portfolio, which I assume here to be asset classes such as Canadian stocks, U.S. stocks, Canadian bonds, etc.*Note: *As important as expected returns are, they should never be the sole criterion for building a portfolio. If we only went by expected return, we would place 100% of the portfolio in the asset class with the highest expected return, which is usually the riskiest. For most investors, the risk is simply not worth it.**Building Blocks of Portfolio Construction**When building portfolios, we need two additional pieces of information beside expected return:

1. First, we need to know the volatility of the returns of each asset class. This is usually measured by another probability-based measure called

*standard deviation*. The standard deviation of return is the square root of the probability-weighted average of the square of the deviation of each possible return from expected return. It is usually estimated from historical returns using a standard statistical formula in which all returns are treated as being equally likely. In Excel, the STDEV function and its variations, implement the standard statistical formula.

Different asset classes have different standard deviations that indicate their riskiness. A highly volatile asset class, such as emerging market stocks, would have one of the highest standard deviations among the asset classes being considered for a portfolio. Canadian and U.S. stocks would have lower standard deviations that that of emerging market stocks, but significantly higher than those of bonds. Cash would have a very low standard deviation, but also a very low expected return.

2. The final piece of information needed for portfolio construction is the

*correlation*of returns between each

*pair*of asset classes. A correlation coefficient is a number between -1 and +1 that measures the strength and direction of the relationship between two variables. A correlation of 0 indicates that there is no relationship between the variables. A correlation of +1 indicates a perfect positive relationship and a correlation of -1 indicates a perfect negative relationship.

Correlation is important in portfolio construction because it indicates the extent to which there are opportunities to reduce risk through diversification. A correlation of -1 is ideal because it means that the two asset classes can be used to offset each other and eliminate risk entirely. A correlation of +1 means that risk cannot be reduced through diversification.

The formula for correlation is complex so I won’t go into it here, but I will give an example. From Morningstar’s database, I obtained monthly returns over the period 2000-2020, on the Morningstar indices for the Canadian and U.S. stock markets, both in Canadian dollars. I brought the data into Excel and used the Excel function CORREL to estimate the correlation between the two series. The result was 62%. This makes sense because while we expect the Canadian and U.S. markets to be strongly related, the relationship is not exact, giving plenty of opportunity to reduce risk through diversification.

**The Importance of Risk**

In 1952, the economist Harry Markowitz published a paper that for the first time, brought together expected return, standard deviation, and correlation together into a model of portfolio construction. (Markowitz went on the win a Nobel Prize in 1990 for his pathbreaking work.) Ideally, to form a portfolio, an investor should use expected returns in conjunction with standard deviations and correlations as inputs. While in practice, it may not be feasible to use the Markowitz model, any portfolio construction process should take into account both risk and expected return, and the benefits of diversification.

*and*Expected Return**Types of Expected Returns**

One final note on expected returns is that they come in two forms, arithmetic and geometric.

An

*arithmetic*expected return is single-period forecast, typically for a year. These are inputs to the Markowitz model.

A

*geometric*expected return is a forecast of a long-run rate of return.

Consider a game where you repeatedly flip a coin. Each time you flip heads, your money doubles and time you flip tails, your money is halved. Hence, the two possible returns are +100% and -50%. The arithmetic expected return is +25% which seems like a great bet to take. But, if you play the game repeatedly, you’ll end up breaking even with a geometric expected return of 0%. So when you are working with expected returns, it is important to know which type you are dealing with.

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