I used this example in classes for financial advisors back in the day:

Client gives you $100,000 and it goes up 100% in period 1 so the balance is $200,000.

Client is excited then and gives you $1,000,000 more so the account has $1,200,000.

Account then goes down 25% in period 2 so $1,200,000 becomes $900,000.

You tell the client – we’re doing great! The average annual return has been 22.47%!

[((1+1)*(1-0.25))^(1/2) – 1= (2*0.75)^0.5 – 1 = 1.5^0.5 – 1 = 22.47% (for geometric – i.e. time-weighted – you add one to each return, take the product, then the nth root (the number of periods) then subtract 1)]

The client responds, “Are you smoking something hallucinogenic? I gave you $1,200,000 and you turned it into $900,000 and you have the audacity to tell me you made money?

The problem is the client’s. He/she should have given the advisor all the money up front! Since managers generally don’t have control over cash flows it is unfair to penalize (or reward) them for the timing. So that’s why all standard reporting is CAGR (Compound Annual Growth Rate) which doesn’t account for cash flows.

The *client’s* compound return is -13.40%

[What you got divided by what you paid for it to the nth root, minus 1, or ($900,000/$1,200,000)^(1/2) – 1 = 0.75^0.5 – 1 = -13.40%]

That’s still a compound (geometric) return, but it’s dollar weighted.

The *arithmetic *return is just the average like you learned around fifth grade or so. 100% + -25% = 75% / 2 = 37.5%.

To recap:

- If you want to know how the
*manager*is doing over time (particularly with investments that have different volatilities), you want the geometric return without any cash flows. - If you want to know how the
*client*is doing over time you want the geometric return*with*cash flows. - If you want to know the shape of the annual distribution of returns (the bell curve) as input to an MCS, or to know what to expect in a single year, it is best described as the arithmetic mean and a standard deviation (not getting into higher moments here, but the skewness and kurtosis also matter for investments such as hedge funds).

Over time, the arithmetic return compounds into the geometric return as I demonstrated here: Converting Arithmetic to Geometric Averages Spreadsheet

See the second tab for actual historical data, the market’s arithmetic average from 1926-2023, 98 years, was 12.16% but the geometric average was 10.28%. If you invested $X in 1926 you would have $X*(1.1028^98) at the end. But, assuming the distribution of future returns is expected to be like the past (which is probably erroneous), your expectation of *next *year (or any single year) would be a bell curve with a mean of 12.16% and a standard deviation of 19.72%.

Finally, given a high enough volatility, a positive arithmetic return can be a negative geometric return. This is the problem with extremely levered investments like the 2x, 3x, and -2x, -3x funds (and, again, some “hedge” funds).