PS....

On Jan 7, 2:59*pm, I wrote:
The [annual] XIRR result is 17.4522% (approximately).

If we apply XIRR to each of the quarters, set up as above for the
"endBalance/startBalance" analysis, we get the following quarterly
return rates [3]: *-0.7%, 5.5%, 3.0% and 8.5%. *Those are the same as
the hypothetical quarter rates that I used to derive the example.
[....]
However, (1-0.7%)*(1+5.5%)*(1+3.0%)*(1+8.5%)-1 [2] is 17.0762%
(approximately), not 17.4522%. *This is also true if I "normalize" the
quarterly results [4], which results in an annual rate of 16.9760%.

I have not given any thought to why this "discrepancy" exists. *I
believe there is a mathematical explanation -- perhaps something
similar to the fact that the average of averages of different size
groups is not equal to the average of the whole.

Looking at the two problems algebraically, I think it is "obvious"
that that is exactly the explanation, at least analogously.

The quarterly rates of return are the solutions to the following
equations:

0 = SUM(CF[k]/(1+i1)^k, k=0,...,2)
0 = SUM(CF[k+3]/(1+i2)^k, k=0,...,2)
0 = SUM(CF[k+6]/(1+i3)^k, k=0,...,2)
0 = SUM(CF[k+9]/(1+i4)^k, k=0,...,2)

The key thing to notice is that the exponent of (1+i[q])^k is never
more than 2.

But the annual rate is the solution to the following equation:

0 = SUM(CF[k]/(1+i)^k, k=0,...,11)

Thus, for the example, for cash flows in the 2nd quarter, the exponent
of 1/(1+i)^k is 3 through 5, not 0 through 2, significantly reducing
the influence of those cash flows on the total solution. So off-hand,
I see no reason to expect that (1+i1)*...*(1+i4) would equal 1+i.

Yet we know that the quarterly formulas correctly find the quarterly
rates of return. Recall that they match the rates that I used to
construct the hypothetical example.

So I think it is fair to say that we should, in fact, not expect that
there is some simple function of the quarterly rates -- much less
their product -- that would equal the annual rate, because the two
sets of rates are based on very different assumptions. But there is
nothing wrong with either the quarterly or annual rates computed using
XIRR in the manner that I prescribed.

Similarly, there is no simple function of subgroup averages -- like
(a1+...+a4)/4 -- that equals the average of the whole, in the case
when the subgroups differ in size. We have to compute (n1*a1+...
+n4*a4) / (n1+...+n4). That does not inhibit from reporting the
individual subgroup averages along with the average of the whole.

That is hardly a rigorous proof. But I hope it is a convincing
argument.

Then again, I could be completely wrong <wink.


PS: I have a little off-by-one error in my previous XIRR
constructions. But I think you can get the idea.