## November 25, 2005

### Arithmetic return is what you see, geometric return is what you get

I am reading a fairly enjoyable book ‘A Mathematician plays the stock market’ (see section ‘currently reading’ under sidebar). Found the discussion on geometric mean v/s arithmetic mean (for returns) fairly relevant for an investor.

Let me explain

Arithmetic mean (or returns) is the simple average of returns over the time period being considered.

For ex: consider a stock X that has the following returns for the year
Week 1 : + 80 %
Week 2 : -60 %

The average return for a two week period is +10%. The ‘average’ return for the year would be 1.1^52 = 142 times the original investment. If average return is what an investor gets, then anyone can be fabulously rich in no time.

Geometric mean of the same stock will be derieved by the following formulae
Total return = (1+0.8)*(1-0.6)*(1+0.8)…….( for 52 weeks) = 0.000195

T
he scenario in the above formulae is that the investor makes a positive return the first week, followed by negative the next and then positive and so on. So the real return he/she actually gets is less than 1 % of capital invested

Another example
An investor is on the lookout for a hot mutual fund. He looks at the mutual fund rankings and sees a fund, which has returned 100 % last year. He invests his money in the fund. The hot fund promptly proceeds to lose 50% next year ( reversion to mean or maybe bad luck ).

The return the fund publicizes is 25 % (average for 2 years ). An investor who was invested for 2 years is lucky to get his money back. The performance chasing investor looses half his money. The fund manager and his company get their asset management fee and are able to show great performance at the same time.

Reminds me of a famous title of a book – ‘where are customer’s yatch?’

There is another
interesting discussion happening on the BRK board on MSN (registration required ) on the same topic.