Let's assume $r_{360,sa}$ is a zero coupon rate compounded semi-annually, under act/360 day count. I'd like to convert it to zero coupon rate compounded annually, under act/365. Is the below conversion correct?
$$r_{365,a}=\Big(1+\frac{r_{360,sa}\cdot365}{2\cdot360}\Big)^2-1$$
I found it in many textbooks. Nevertheless, some people claim that I should use
$$1+r_{365,a}=\Big(1+\frac{r_{360,sa}}{2}\Big)^{2\cdot \frac{365}{360}}$$
to get $r_{365,a}$. I'm kind of confused now...
Note that your second formula is equivalent to $$ r_{365,a}= \left(\left(1+\frac{r_{360,sa}}{2}\right)^{365/360}\right)^2 - 1 $$
So the only disagreement between the two formulas is that $$ \left(1+\frac{365}{360}\cdot\frac{r_{360,sa}}{2}\right) \neq \left(1+\frac{r_{360,sa}}{2}\right)^{365/360} $$
For small values of $r_{360,sa}$ these formulas are approximately equal. Compounding over many years will magnify the discrepancy.
The second formula appears to be a slightly better predictor of the value of the $180$-day compounding over the course of very many years; for example, over the course of $180$ periods of $365$ days there will be $365$ compounding periods of $180$ days each, which agrees with the fact that the second formula predicts a value of $\left(1+\frac{r_{360,sa}}{2}\right)^{365}$ over that period of time.
The first formula appears to be a lot easier to calculate using pencil and paper or the kinds of electronic (or even mechanical) calculators that were in common use in years past (before practically everyone had an $x^y$ key). Moreover, unless you have an extremely high rate or a very long time period, the discrepancy with the second formula will be a tiny fraction of a percent of the balance. Perhaps that explains why it became so popular.
