Mastering Average Return Measurements for CFA Level 1

The CFA Program is notorious for its comprehensive curriculum—and let’s be honest—they excel at crafting questions and answer choices designed to trip you up. If you believe Quantitative Finance for CFA Level I is just about memorizing formulas, you’re setting yourself up for disappointment. Success requires deeply understanding why and when to use each formula. Even when you think you’ve mastered a topic, you’ll inevitably encounter questions that make you pause, reread, and then read once more.

I had one such “aha!” moment myself with question about return measurements.

Today’s Problem

This particular question is one I’ve crafted myself, aiming to capture exactly the kind of ambiguity that stumped many CFA candidates—only about 1 in 7 students got it right on their first attempt (in the learning provider I use)!

As you can see from this example, CFA questions often don’t explicitly state, “Calculate the geometric mean,” or ask directly, “What’s the time-weighted rate of return here?”. Instead, you’re presented with a scenario, and the only path to the correct answer is to implicitly recognize which type of return measurement is the right tool for the job. It’s subtle, almost sneaky.

You can now try to select right answer by yourself. I’ll provide the solution at the end of this post. But first, let’s dive into these different return measurements.

I won’t get too deep into the calculations here (perhaps in future posts), but I do want to highlight the nuances and explain clearly when to use each one. Let’s jump in!

So we have five types of averages for return measurement: Arithmetic Mean, Geometric Mean, Harmonic Mean, and Winsorized Mean. Let’s start with most obvious: Arithmetic Mean.

Arithmetic Mean (Simple Average Return)

The arithmetic mean is the simple average of a set of returns. Mathematically, it’s the sum of all return values divided by the number of periods or number of observations.

For example, if a portfolio had annual returns of $8\mathrm{\%},12\mathrm{\%},$ and $15\mathrm{\%}$ over three years, the arithmetic mean return is $\frac{(8\mathrm{\%}+12\mathrm{\%}+15\mathrm{\%})}{3}=11.67\mathrm{\%}$

This measure is straightforward to calculate but only applicable if you want to calculate expected return for a single period or calculate average of independent results!

Examples:

• Calculating average returns across different stocks for the same period (cross-sectional analysis).
• Determining the average annual return based on historical performance, provided each year’s result is treated independently and in isolation.

NEVER USE IT FOR COMPOUNDING.

The arithmetic mean is NOT appropriate for calculating the average growth rate of an investment over multiple periods if you want to know your actual compounded return. Why? Because it ignores the effect of compounding. Let’s illustrate with an example.

Suppose you invest $\mathrm{\$}100$.

In Year 1, your investment increases by $50\mathrm{\%}$, so it grows to $\mathrm{\$}150$.

In Year 2, your investment decreases by $50\mathrm{\%}$. A $50\mathrm{\%}$ loss on $\mathrm{\$}150$ is $\mathrm{\$}75$, so your investment value drops to $\mathrm{\$}75$.

What’s your average annual return?

The arithmetic mean would be $(50\mathrm{\%}+(-50\mathrm{\%}))/2=0\mathrm{\%}$.

This suggests that, on average, you broke even. But did you? You started with $\mathrm{\$}100$ and ended with $\mathrm{\$}75$. You clearly lost money! The arithmetic mean of $0\mathrm{\%}$ does not reflect the actual investment performance. This is because the base for the percentage change is different each year due to the previous year’s performance.

For accurately measuring compounded growth over time, we need a different tool – the geometric mean, which we’ll discuss in a moment.

Important to remember: the arithmetic mean tends to overstate the true growth rate when returns are volatile (i.e., not consistent across periods). Think of it this way—a 50% gain is not as beneficial as a 50% loss is harmful. Why? Because each 50% increase grows from a smaller base, whereas each 50% decrease cuts from a larger base. The more volatility you have, the greater the discrepancy between the arithmetic mean and the geometric mean.

Additionally, the arithmetic mean is sensitive to outliers – a single extremely high or low return can skew it significantly - we will talk about this too.

Geometric Mean (Compound Annual Growth Rate)

Let’s start with definition (in context of return measurements ofc!)

Definition: The geometric mean is the average rate of return per period that would result in the same cumulative performance as the sequence of actual returns.

For returns: $r_1,r_2,\dots ,r_n$, the geometric mean $r_{geo}$ satisfies:

$(1+R_{geo}{)}^n=(1+R_1)\ast (1+R_2)\ast \cdots \ast (1+R_n)$

This is equivalent to:

In practical terms, the geometric mean tells you the constant annual growth rate your investment would need to match the actual multi-period performance.

The geometric mean is appropriate for measuring investment performance over multiple periods – it captures the effect of gains and losses compounding over time.

This is often referred to as the compound annual growth rate (CAGR) or the time-weighted rate of return (TWR). Use the geometric mean when you want to know the true average growth per period for a time series of returns – for example, “What was the average annual return of this fund over the last 5 years”.

The geometric average return is often called the time-weighted return (TWR) because it effectively removes the impact of cash inflows/outflows and focuses only on the returns themselves. It’s worth noting, however, that to accurately calculate TWR when external cash flows are present, you’ll often need to first process this cash flow information to isolate the periodic returns before applying the geometric mean.

Now, let’s apply the geometric mean to our previous simple scenario: $\mathrm{\$}100$, $+50\mathrm{\%}$ in first year, $-50\mathrm{\%}$ in second. To calculate the geometric mean, we first convert the percentage returns to growth factors:

Year 1: $1+0.50=1.50$

Year 2: $1-0.50=0.50$

Then, we multiply these growth factors and take the nth root, where n is the number of periods. Finally, we subtract 1.

$(1.50\ast 0.50{)}^{\frac{1}{2}}-1=(0.75{)}^{\frac{1}{2}}-1=0.866025-1=-0.133975$

or approximately $-13.4\mathrm{\%}$.

This $-13.4\mathrm{\%}$ geometric mean annual return accurately reflects that your $\mathrm{\$}100$ investment, after two years, ended up at $\mathrm{\$}75$. If you apply a $-13.4\mathrm{\%}$ loss for two consecutive years to $\mathrm{\$}100$ (compounded), you’ll see the ending value is indeed close to $\mathrm{\$}75$. This is the power of the geometric mean: it tells you the constant annual rate of return that would have produced the same cumulative result.

Always remember that geometric mean ≤ arithmetic mean for any set of returns, with the difference growing as volatility increases. This gap (arithmetic minus geometric) effectively represents the drag of volatility on growth.

Harmonic Mean (equal inflows)

Let’s start right with formula:

for values $R_1,R_2,\dots ,R_n$, the harmonic mean:

This might seem completely abstract. The arithmetic mean is intuitive, and the geometric mean also makes sense—it’s about compounding. But what’s going on with the harmonic mean?!

The essential trait of the harmonic mean is that it gives greater weight to smaller values in the dataset. Because it involves taking reciprocals (calculating $\frac{1}{R_i}$), smaller numbers end up having a larger influence, while larger numbers have less impact. After summing these reciprocal values, we then “flip” the result back — giving us the harmonic mean.

Let me give you maybe simple example: $1,1,100$.

Average mean: $\frac{102}{3}=34$

Harmonic mean:

$H=\frac{3}{\frac{1}{1}+\frac{1}{1}+\frac{1}{100}}$

$H=\frac{3}{2.01}\approx 1.4925$

Ok. Hope you now see it’s power! But now the most important question in the context of CFA. Where it is used? Fortunately CFA Curriculum gives us these two applications straight!

They specifically note that the harmonic mean should be used to find the average share price when equal amounts are invested periodically - concept known as cost averaging.

It’s also the correct mean for financial ratios like the average price/earnings (P/E) ratio across multiple stocks (since P/E is a ratio where the “per 1” part is in the denominator; averaging such ratios is done by harmonic mean to avoid bias toward high values).

So where you see: calculate average ratio (most likely they will explicitely mention harmonic mean) or something like this:

Imagine an investor buys $1,000 worth of a certain stock each month for three months, at share prices of $\mathrm{\$}10$, $\mathrm{\$}15$, and $\mathrm{\$}20$ in months 1, 2, and 3 respectively. What was the investor’s average cost per share?

Plugging into the formula:

$HM=\frac{3}{(1/10)+(1/15)+(1/20)}$.

This evaluates to:

$\frac{3}{(0.1+0.0667+0.05)}=\frac{3}{0.2167}\approx \mathrm{\$}13.85$

We can verify this by another method: over three months, $\mathrm{\$}3,000$ was spent in total.

The number of shares purchased:

$\mathrm{\$}1,000/10+\mathrm{\$}1,000/15+\mathrm{\$}1,000/20=100+66.67+50=216.67$ shares.

Total money / total shares = $3,000 / 216.67 = $13.85 per share, confirming the harmonic mean result.

The harmonic mean is a specialized tool – misapplying it can lead to confusion. It should not be used in place of arithmetic or geometric mean for typical return sequences.

Another pitfall is that the harmonic mean requires all inputs to be positive; if you tried to use it with returns that include zero or negative values (e.g., –5%), the formula breaks down.

In practical terms, use the harmonic mean when averaging values like prices, costs, or ratios where each observation’s weight is inversely proportional to its magnitude, and each observation is equally important in a reciprocal sense. In most investment return analyses, this situation is uncommon except for the equal investment scenario we described.

Finally, remember that when returns (or any positive numbers) vary, this holds true:

Now let’s move on to the final member of our “mean family”—the Winsorized mean!

Winsorized Mean (when I want my outliers to be normal)

Well, this is something CFA examiners have recently taken a liking to. In fact, it’s so uncommon in everyday use that even my text editor underlines the word “Winsorized” in red! But what exactly is it?

A Winsorized mean is an average that limits the impact of extreme outliers by capping data points at specified percentiles instead of completely removing them. Essentially, the lowest and highest values in the dataset are replaced with less extreme values at defined percentile thresholds.

By doing this, all observations remain part of the analysis (unlike the trimmed mean, where data points are discarded entirely), but the influence of extreme outliers is significantly softened.

Winsorized means become particularly valuable when analyzing return data with extreme outliers — the type of data that can significantly skew a simple average. If you’re dealing with investment returns (for instance, monthly or yearly performance) that include dramatic market events, like sudden crashes or extraordinary rallies, a Winsorized mean helps identify the “core” or representative average return without letting extreme tail events dominate the calculation.

This brings us full circle to the tricky question I posed at the start! That question required calculating a “representative average IRR,” specifically instructing you to reduce distortion by capping values “above the 90th and below the 10th percentile.”

Notice, it didn’t explicitly tell you to “calculate a Winsorized mean.” Instead, it described the process and relied on you to recognize it implicitly. This subtle approach is classic CFA Institute—testing your conceptual understanding and your ability to identify when and how to apply specific methodologies, rather than simply recalling formulas.

So let’s solve the problem!

Values range is limited to reduce distortion from extremes - this phrase suggest we need to do something wit outliers. The catchy part is that you can right away start to just identify outliers and throw them. The important part is: “using all 15 observations (…) values range is limited”. When you will have something telling you to limit range of values but still use ALL observations, you need to use winsorized mean.

Therefore, using these parameters and the linear interpolation method for percentiles, the Winsorized mean IRR is 20.6%

Comparing the Means and Their Appropriate Uses

Below I summarized when each mean should be applied: