RCM Managed Asset Portfolio: How to understand performance returns geometrically

RCM Managed Asset Portfolio

By Christopher Chiu, CFA

April 2026

How to understand performance returns geometrically

It is often easy for investors to misjudge fund performance, especially when they evaluate it over multiple periods. One error they often make is relying solely on intuition and misapplying the math they learned for other kinds of problems. For example, their intuition may tell them to view performance returns arithmetically when they should actually view performance returns geometrically. 

The difference between arithmetic and geometric returns

There is a difference between arithmetic and geometric returns. If a fund made -10% in the first year and then grew +10% in the next year, some people would say it had a 0% return for the two years. But that is simply to see returns incorrectly in arithmetic terms. 

This is revealed when accounting for these returns in dollar terms. Suppose there is $100 to start the account. It loses 10% and becomes $90 after the first year. Then working off this base of $ 90 it gains 10% and becomes $99 in year 2. Yet this is not $100, which would have been equivalent to a 0% return.

Some will say it’s no big deal. Arithmetic return thinking just fails to account for a $1 difference. While accounting for returns arithmetically is a useful heuristic, that $1 accounting error becomes much larger when performance is carried to extremes, like a crack that is never fully patched and expands over time.

A crack that is not fully patched 

This error becomes clearer if we consider a more volatile strategy than one that varies +/-10%. In a more volatile fund, we would easily see a greater difference. For example, let’s say a fund incurs a -50% loss and the $100 becomes $50 after year 1. Then gaining +50% off of the $50 base in year 2 only gets you to $75. This is now very far from the original $100, but using arithmetic thinking, you would think the return was just 0%.

A crack that grows over time

Arithmetic returns also underestimate the long-term effects of compounding. If a fund had +10% in year 1, +10% in year 2, and +10% every year for the remainder of a decade, using simple arithmetic accounting simple arithmetic accounting would suggest a cumulative 100% return. But the actual total using geometric returns are somewhat higher at 159%--a big difference from 100%. (1+r)ⁿ-1 = (1+0.10)¹⁰-1=159%. Using geometric returns, this means the 10% return in the final year is much greater than the 10% in the first year. This is because an arithmetic return, where you simply add up the totals, neglects the effect of growth on growth.

Compounding with zombies

What does growth on growth mean? An example from pop culture illustrates its effect. Imagine we are in a city of a million people under attack by 1,000 zombies. The residents have barricaded themselves in their homes to prevent zombies from easily infecting the population. But the zombies still grow by 20% every year after getting through to some residents barricaded in their homes.

If we apply arithmetic accounting, 1,000 zombies will infect 200 people after the first year, 200 additional people in year 2, and 200 people in year 3. This is because arithmetic accounting assumes that only the initial army of 1,000 zombies is infecting people.

But as we know from watching any zombie movie, once one person gets infected and becomes a zombie that zombie will not remain inactive but will maniacally try to infect others as well.  This is what happens with geometric returns. The new zombies are added to the original zombie army and they also grow at the same growth rate, 20%. This is the meaning of growth on growth. 

After the first year there are indeed 200 newly infected. But in year 2 it is not simply 200 more. It is bound to be more. At the start of year 2, there are 1,200 zombies growing at a 20% rate, not the original 1,000 zombies. This results in 1,440 zombies at the end of year 2, who will then also go on to grow at 20%. (And 20% of 1,440 is more than 20% of 1,000.)

The geometric infection rate of Covid-19

We know from our own experience that infections grow geometrically. The spread of Covid proceeded at an even greater growth than any of the examples we provided above, where the R naught (the growth rate of the infected) was between 140 to 240 percent. In other words, the newly infected went on to infect 1.4 to 2.4 other people, who in turn infected 1.4 to 2.4 other people, and so on. So the rapid spread of Covid was inevitable until there was a developed immunity that could limit it.  This spread seemed small enough to count at the very beginning as would any infectious disease. But when large numbers are involved, driven by volatile growth or long spans of time, it is necessary to utilize a method that tracks the progress of these geometric effects.