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in his book "The four pillars of investing", the author describes this simple method to compute the rate of return for a portfolio that has a net inflow of cash:

Without in- or outflow, the rate of return would be the ratio of final value to initial value, minus 1.

His example: I have 12000 at the end, and started with 10500, so the rate of return is 12000 / 10500 - 1 = 14.3%.

Now if I had the same initial and final values, but due to various contributions I had also a net inflow of $300, he says that now you should subtract half of that inflow from the numerator and add half of that inflow to the denominator:

(12000 - 150) / (10500 + 150) - 1 = 11.3%

I can't really find the mathematical reason for this formula. What I personally would have done to compute a portfolio return with contributions or withdrawals is to compute the return for each period in between contributions and compound those. Say that I start with 10500 and then it grows to 11500. Then I contribute 300 dollars so I have 11800, and then it grows to 12000. The return would then be (11000 / 10500) * (12000 / 11300) - 1 = 11.4

So... is that simple method just an approximation that's good enough, or am I just using the wrong definition of total portfolio return?

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3 Answers

up vote 2 down vote accepted

It is a good enough approximation. With a single event you can do it your way and get a better result, but imagine that the $300 are spread over a certain period with $10 contribution each time? Then recalculating and compounding will be a lot of work to do.

The original ROI formula is averaging the ROI by definition, so why bother with precise calculations of averages that are imprecise by definition, when you can just adjust the average without losing the level of precision? 11.4 and 11.3 aren't significantly different, its immaterial.

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Thanks. Now I don't have to go mad looking for a mathematical derivation of the "exact" formula :) –  Lagerbaer Oct 29 '12 at 22:12
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The author is using an approximation to what you have exactly, which is called a "true" time-weighted rate of return.

You have expressed the total time-weighted return for the period in question. In order to express this as an annual rate, you may annualize it by adding one, raising to the 1/y power, and subtracting one again, for a period of y years.

The alternative to a time-weighted return is a money-weighted return, which is actually another name for the internal rate of return.

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The author is using the simple Dietz method, (alternatively the modified Dietz), with the assumption that the net cash-flow occurs halfway through the time period. Let's say the time period is one year for illustration, so the cash-flow would be at the end of the second quarter.

The money-weighted method gives a more accurate return, but has to be solved by trial-and-error or using a computer. The money-weighted return is 11.2718 % and the simple or modified Dietz return is 11.2676 %. When the sums are done backwards to check, the Dietz is half a dollar out with a final value of $11,999.50 while the money-weighted return recalculates exactly $12,000.

It is worth pointing out that the return changes if the cash-flow is not in the middle of the time period. A case with the cash-flow at the end of Q3 is added to illustrate.

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