It is easier to see the compounding effect in savings and other financial tools that give continuous growth. To be clear, compounding simply means not taking out the interest or growth. Hence, in FD, it simply means leaving the interest together with the principal for another year.
Similarly, in variable priced UT, there is compounding effect too if its growth is not trim. But of course, it is harder to see this effect since equity funds are volatile and can be having a negative growth as well as positive.
CAGR and IRR are just measurements - giving the same thing, which is the effective rate or annualised rate. Similarly in step up FD rates with different interest rates every month, the effective rate is the actual rate of return at the end of the FD tenure.
CAGR usually means the effective rate of a transaction, ie. a single purchase of a UT fund. While IRR usually means the effective rate of a bunch of purchases, bought at different times.
Check out the simple CAGR formula to calculate the effective or annualised rate. As you can see, when the purchase is held more than a year, its effective rate is not simply derived by dividing the ROI% by the number of years.
Similarly, when there is more than one purchases of the same fund or various funds, with each having a different CAGR rate, we cannot simply added up the CAGR rates and then divided it by the number of purchases to get the effective rate or IRR.
The formula to calculate the IRR is a bit more complicated than the simple CAGR formula. But this is easily done by using a Excel speadsheet, and its XIRR function.
Why don't people just use [(what you have now) - (what you put in) / (what you put in)] x 100 ? is that somehow wrong ?