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The annual rate of return on an investment can be calculated as CAGR of individual investment

And the top response to this question suggests that the return on a portfolio of such purchases is simply the weighted average of returns on individual transactions, where the weighting coefficient is the number of shares bought:

CAGR across portfolio of purchases

However, by this definition, it seems like the calculated portfolio return can be positive, despite losing money.

Consider the following example: I purchase two shares of stock, one for $100 and one for $10, and sell both for $50, the first held for 1 year and the second held for 0.5 years.

So I've spent $110 and received $100, but the calculated return is +1175% [ (-50% + 2400%)/2 ].

What am I doing wrong?

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1st equation

R = (Pt/Po)^(1/t) - 1

annualised returns for shares 1 & 2 are

r1 = (50/100) - 1 = -50 %

r2 = (50/10)^(1/0.5) - 1 = 2400 %

2nd equation

R = Sum[mi*Ri]/n

Calculating half-year returns corresponding to the holding periods:

"weighted average of returns on individual transactions" for each half-year

hr1 = (100*(1 + r1)^(1/2) + 10*(1 + r2)^(1/2))/110 - 1 = 9.73698 %

hr2 = (1 + r1)^(1/2) - 1 = -29.2893 %

giving the return over the year as

(1 + hr1)*(1 + hr2) - 1 = -22.4042 %
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  • Got it, thanks. For a collection of purchases with different holding periods, it looks like this calculation could become quite complex. Is internal rate of return (IRR) the best approach in this case? – Michael Boles May 6 '20 at 23:22
  • Calculation of CAGR is an IRR problem, e.g. 0 = 10 - 50/(1 + r)^0.5r = -50% but if you mean put all the cash flows in one IRR like so: 0 = 110 - 50/(1 + r)^0.5 - 50/(1 + r)r = -11.875% you get a quite different result because you are omitting certain information. The closest to reality is a time-weighted return, which you could use if you knew the value of share 1 when you sold share 2. Let's say it was $72. Then hr1 = (72 + 50)/(100 + 10) - 1 = 10.9% and hr2 = 50/72 - 1 = -30.56% so r = (1 + hr1)*(1 + hr2) - 1 = -22.98%. So the approach in the answer is closer to reality. – Chris Degnen May 7 '20 at 9:07
  • However, if you have numerous transactions IRR (or money-weighted return) is usually a fair enough approximation. It gets less accurate compared to time-weighted return when there are large mid-period cash flows. – Chris Degnen May 7 '20 at 9:17

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