IRR works simply on periodic amounts, so you need to annualise the result yourself, e.g.
for your simple example where one month periodic IRR = 150%
∴ annual IRR = (1.5 + 1)^12 - 1 = 59603.6 = 596.036 % per annum
This is the same as your one month CAGR result. In these examples the months of the year are treated as equal periods of time.
They are slightly different from your XIRR return because XIRR annualises on a daily basis to more accurately account for mid-period cashflows. The calculation to annualise on a daily basis calculates the daily rate first, then annualises:
From 01/01/2015 to 01/02/2015 is 31 days
annualised XIRR = ((1.5 + 1)^(1/31))^365 - 1 = 48463.4 = 484.634 % per annum
The XIRR rate is less than the IRR because January is a longer month than the average.
In your more complicated example the difference is the same. IRR is calculating a periodic return (your periods are years), and XIRR is calculating an annualised return based on daily resolution.
The calculations can be reproduced as follows. Discounting each amount to net present value (NPV) the sum of all the amounts is zero; solve for r:
First, calculating IRR (for the whole 4 year period) - ref. link
-50000/(1 + r)^(0/4) + 5000/(1 + r)^(1/4) + 5000/(1 + r)^(2/4) +
5000/(1 + r)^(3/4) + 50000/(1 + r)^(4/4) = 0
∴ r = 0.349104
annualised IRR = (r + 1)^(1/4) - 1 = 0.0777334 = 7.77334 % per annum
XIRR
From 01/01/2009 to 01/01/2010 is 365 days
From 01/01/2009 to 03/01/2011 is 732 days
From 01/01/2009 to 04/01/2012 is 1098 days
From 01/01/2009 to 05/01/2013 is 1465 days
-50000/(1 + r)^(0/1465) + 5000/(1 + r)^(365/1465) +
5000/(1 + r)^(732/1465) + 5000/(1 + r)^(1098/1465) +
50000/(1 + r)^(1465/1465) = 0
∴ r = 0.349174
annualised XIRR = ((r + 1)^(1/1465))^365 - 1 = 0.077472 = 7.7472 % per annum
