Two variants of the geometric mean or TWRR


I was solving some PM questions and one thing I cant wrap my head around is when to use the geometric mean formula under the square root and when to use the other one. What type of things am I looking for in the question? What are immediate giveaways?

Please find below the question under question. Thank you for your time all.

Q. The quarterly returns on a portfolio are as follows:

Quarter 1 . 2 . 3 . 4
Return . 20% −20% 10% −10%

The time-weighted rate of return of the portfolio is closest to:

  1. −5.0%.
  2. −1.3%.
  3. 0.0%.

Greetings friend! On time-weighted returns, you treat each quarter equally, compound the quarters and then take the root of 1/4 (aka 0.25) to get the average return… and calculate as follows:

[(1.2 x 0.8 x 1.1 x 0.9)^0.25] -1 = -0.126 so the answer is B

Cheers - good luck you got this :+1:

Hello greybeard,

Thanks for replying. You actually made the same mistake I did. The answer is actually A.

A is correct. The time-weighted rate of return of this portfolio

= (1 + r 1)(1 + r 2)(1 + r 3)(1 + r 4) − 1,

where r 1 = holding period return (HPR) for the first quarter, second quarter, and so on:

= (1 + 0.20)(1 − 0.20)(1 + 0.10)(1 − 0.10) − 1

= −4.96% (or ~ −5.0%).

B is incorrect because it is calculated as

= [(1.20)(0.80)(1.10)(0.90)]1/4 − 1 = −1.26% and confused with

rTW = [1 + r 1)(1 + r 2)(1 + r 3)(1 + r 4)]1/N − 1

I’m trying to understand the reasoning behind this but can’t seem to figure it out. I can almost swear that I’ve seen identical problems where I did the square root and the answer was correct. However this one is different. Why? I don’t know.

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Greetings friend! I guess that they are saying the quarterly returns here are not annualized. That’s why they are just multiplying them together. If the quarterly returns were annualized here, then you’d take the 1/4 root. But I guess here since they are 4 equal time distances away within the same year, you just multiply them together to get one year’s holding period return overall. If they gave you 8 quarters then you’d multiply 4 together then the second 4 together to get 2 years of annualized holding period returns and then multiply them together and take them to the 1/2 power.

Thanks for your clarification response! I just refreshed on something with you today lol.

For the CFA exam I’d focus on the feature of quarters then, thats the key to their TWRR answer.

Cheers - good luck - you got this👍