Question about EAY

This has been bothering me for a little while:

I have a 90 day holding period yield of 1.3%. The annual rate would be:

(1 + 0.013)^(365/90) -1 = 0.0537787

The BDY would be:

2 * [(1 + 0.013)^(365/180) -1] = 0.0530745

along the same lines, if I take the annual rate and try to get back to the BDY:

2 * [(1 + 0.0537787)^(.5) -1] = 0.0530745

HOWEVER, my question is why do I have to use 0.5 as the exponent to get back to the semi-annual rate instead of 180/365? If I use 180/365 I get a different number:

2 * [(1 + 0.0537787)^(180/365) -1] = 0.0523380

you could use either i guess but 180/365 does not = .5

180/360= .5

Just confused as why going from 1.3% 90 day HPY using 365/90 for an annual or 365/180 to a semi-annual gets me to certain numbers, but going back from the annual rate to the semi-annual, I can’t use 180/365, and instead need to use 0.5 to get to that number?

I feel like it should be consistant and since it isn’t, I must be thinking of something the wrong way?

After thinking about this, I guess it is just a matter of convention. The BDY is 2 * the semi-annual yield so when you have the annual rate of 0.0537787 and you want to go back to semi-annual, you have to think in terms of 1/2 a year so the exponent is 0.5, not 180 days.

Let me know if my reasoning is incorrect, thanks.