### Do You Really Understand Rates of Return? Using them to look backward - and forward

The basic quantitative building block for professional judgments about investment performance is the rate of return. How well do we really understand it? And how can we use past rates to assess the prospects for future performance? You may be surprised to learn that “expected return” may not be what you think.

Rates of return are always more complicated than we think. To understand them fully requires a complete mathematical model – and you need to understand that model, but the model can get complicated quickly.

Consider a basic application. If you have two investments combined 50-50 in your portfolio, and they realize 10% and 20% for the year, you can take their arithmetic average to get your portfolio’s return of 15%.

But if it’s only one investment for two years, and its rate of return is 10% in the first year and 20% in the second, you compound and annualize them to get the “average” annual return of 14.89% (that is, the geometric average). In this context, the arithmetic average (15%) of the first and second year’s returns has no real meaning.

How quickly it gets complicated

Next, suppose you have two investments combined 50-50 for two years. Then you must average their rates in each year, then compound and annualize the two annual results to get the portfolio rate of return over the two years.

But that’s only if you rebalance once, at the beginning of the second year, to a 50-50 mix.

If you don’t rebalance, then you must separately compound the annual returns for each investment for the two years, then take the arithmetic average of the results and annualize that number. If you do it in any other order, you get the wrong answer.

See how quickly it gets complicated? And this is almost the simplest example there is.

The nutty world of conversations about rates of return

Because most people who are not investment professionals have an extremely hazy grasp of compound return, statements and discussions about rates of return are frequently off-the-wall.

For example, I’ve heard someone discuss a friend who they’ve been told earns 10% a month as a day-trader. They don’t realize that 10% a month compounds to 214% a year, and to 9,600% in four years.

If the friend started with \$11,000 and made 10% per month, he’d have more than a million dollars in four years – an implausible assertion.

Among more sophisticated discussants – who, nevertheless, still don’t have a very firm grip on a model – the confusion is at least a bit higher-level. For example, I have often heard people debate which is appropriate to use: the arithmetic average of a time series of historical returns or the geometric average. Since the arithmetic average has no discernible real-world interpretation, it’s hard to know what they’re arguing about. Rates of return in successive time periods don’t add or average any more than apples and oranges do – they compound.