Nominal vs real returns and the geometric mean trap

You will be able to convert a quoted return into a real, geometric figure and explain why the arithmetic average flatters.

You open your brokerage app and a fund you hold shows an average return of 8% a year over five years. Your friend's robo-advisor shows 6%. Your own account, after a bad 2022, feels like it has gone nowhere. Which number is true? Possibly all three, and possibly none of them means what you think. An analyst's first habit is to refuse a return figure until four questions are answered.

Is it nominal or real? Is it an arithmetic average or a compounded one? Does it include dividends, or only the price change? Is it in the currency you spend? This lesson takes the first two. Lessons 1.2 and 1.3 take the other two.

Start with inflation. A nominal return is the change in what your holding is worth, in dollars, before any adjustment. If a fund goes from S$10,000 to S$10,700 in a year, the nominal return is 7%. But the S$10,700 buys less than S$10,700 did a year earlier, because prices in the shops rose too.

A real return is the nominal return with inflation taken out. The exact formula is one plus the nominal return, divided by one plus inflation, minus one. Take a made-up year with 7% nominal and 3% inflation. The real return is 1.07 divided by 1.03, minus 1, which is about 3.9%, not the 4% you get by subtracting. The gap looks small in one year and grows over a working life. For any goal measured in what money buys, such as a retirement income or a child's university fees, real return is the number that matters. The Department of Statistics publishes Singapore's consumer price index, and MAS publishes a core inflation measure. Look up the current figures there rather than trusting a number someone quoted to you.

Now averages. Most marketing material and many app screens show an arithmetic mean: add up each year's return and divide by the number of years. That works for exam scores. It fails for money, because returns compound, and a loss hurts compounding more than a gain of the same size helps it.

Here is the classic example, with made-up figures. A fund gains 50% in year one and loses 50% in year two. The arithmetic mean is zero: plus 50 and minus 50, divided by two. So you broke even? No. S$10,000 grows to S$15,000, then halves to S$7,500. You lost a quarter of your money while the average said nothing happened.

The geometric mean is the one average that takes you from the starting balance to the ending balance. Multiply the growth factors, 1.5 and 0.5, which gives 0.75. Take the root for the number of years, here the square root, and subtract one. That gives about minus 13.4% a year. Compound minus 13.4% for two years and you land on S$7,500. Fund managers call this the compound annual growth rate, and in a spreadsheet it is one formula: ending value divided by starting value, raised to the power of one over the number of years, minus one.

The gap between the two averages is not random. It grows with volatility. A rough rule analysts use is that the geometric mean sits below the arithmetic mean by about half the variance of the yearly returns. Two funds with the same arithmetic average can leave you with very different sums if one swings much harder than the other. That's your first link between return and risk, and module 2 builds on it.

Why does this matter to you and not only to fund reports? Because the arithmetic number flatters. A brochure that quotes an average annual return without saying which average is telling you less than it seems. A friend who says a stock averaged 20% a year may be describing a ride that left them flat. Your own spreadsheet, if you average a column of yearly percentages, will overstate how you did whenever you had a down year.

So from here on, this course follows one convention. Every return is geometric, real where the goal is spending power, net of fees, and in Singapore dollars. When you see a return quoted anywhere without those terms stated, treat it as unconfirmed until you know which kind it is.

That convention starts with your own numbers. Pull the year-end values of one holding or of your whole account for the last few years, and you're ready to put the two averages side by side.

Take the year-end values of one holding for the last five years and calculate its arithmetic and geometric average returns side by side.

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