Confused with "nominal" "convertible" rates; How to calculate a rate for an $n^{th}$ of a year?

The terms and how they're calculated is very unclear to me. My understanding of "nominal" is that this is a rate which isn't in unit time. i.e. $5\%$ per annum "is" in unit time (year) but $5\%$ nominal would be some other time unit(say every 4 months, 6 months etc).

Convertible rates, I don't know, I don't see a clear distinction with the said "nominal" rate. As often it is in financial context, perhaps it's the language that makes it vague.

Say, I am given

an annuity of $\$100$ payable quarterly for 10 years at the end of each year. The rate is $8\%$ "convertible quarterly"(Okay, does this mean a. $8\%$ per quarter of a year b. simply $8\%$ per year(i.e. effective interest rate)? If "b.", then why even bother saying "convertible quarterly"? I should be able to convert it to whatever period I want, a month, 3 months, 5 months etc, no? I mean,why not just say "the effective interest rate is..." and I'll take that rate and convert it for my convenience.) Find the present value of this annuity.

I assumed "b." by the way, so then I want to find the "nominal" rate i.e. the rate per quarter of a year given effective interest rate $8\%$. Let this nominal rate be $i$. Then,

$(1+i)^4=1.08$ is what we need to solve; $4$ since this is quarterly, and RHS is simply equal to the effective interest rate $1+0.08$. Solving gives me $i=1.9426\%$

Problem is, the solution is much much simple-minded. All it does is $\frac{8}{4}\%=2\%$. So the "per year" percentage" divided by the period we want. And yes, I've seen this way of calculating "nominal" rates too.

So, the "sum" of the quarterly rate gives the effective rate. But, while it's regretful I can't find it immediately, I am definitely positive some other questions have calculated "nominal" rate the way I did from the effective rate. In fact, my value is very close to the one in the solutions($1.94\%$). Is this just coincidence?

So in a nutshell, when

An effective interest rate of $r\%$ is given and we wish to find the interest rate effective for $n$ months, then

Let $i$ be the rate we wish to find, then when would you use $(1+i)^n=1+\frac{r}{100}$ and solve for $i$ and when would you use $\frac{r}{n}=i$? The former seems to use "compounding" to the rate $r$ and the latter "summing" to $r$.

I really am lost in the forest of so many different types of interest rates and words and names that go with it which some sound similar, some are suggestive but not really clear(just like the word "convertible" say) so it would be great to get some clarification.

Thanks so much

A nominal rate of interest is always specified with two quantities: the annualized rate (typically expressed as a percentage), and the compounding frequency per year. So for example, if I say that a loan is repaid at a nominal rate of $i^{(6)} = 9\%$ compounded every two months (i.e., 6 times a year), then what this means is that every two months, interest of $i^{(6)}/6 = 1.5\%$ is accrued, and this is an effective two-month rate of interest.

The equivalent effective annual interest rate is therefore $$i = \left(1 + \frac{i^{(6)}}{6} \right)^6 - 1 = (1.015)^6 - 1 = 9.344\%.$$

At this same effective annual rate, what is the nominal rate compounded monthly; i.e., what is $i^{(12)}$? It is the solution to $$\left(1 + \frac{i^{12}}{12}\right)^{12} = \left(1 + \frac{i^{6}}{6}\right)^6,$$ or $i^{(12)} = 8.9665\%$. What is the nominal rate compounded daily (365 days/year)? $$i^{(365)} = 8.93426\%.$$ What is the nominal rate compounded continuously? it is $$\delta = i^{(\infty)} = 6 \log 1.015 \approx 0.0893317.$$ This is what we call the force of interest.

To answer your first question, 100 payable quarterly for 10 years at the end of each year, at a rate of $8\%$ convertible quarterly tells you that the $8\%$ figure is a nominal rate of interest because of the "convertible quarterly" phrase; that the effective quarterly rate is $i^{(4)}/4 = 2\%$; and the effective annual rate is therefore $i = (1.02)^4 - 1 = 8.243\%$. But we don't need the effective annual rate for our calculations because the payments are also made quarterly. If the payments were made for a period that doesn't correspond to the nominal conversion frequency, then we need to convert it to the equivalent effective rate for the period of payments.

Your question, "why even bother saying 'convertible quarterly,'" reveals exactly why the rate of $8\%$ is a nominal rate. If it were effective, as you observed, then there's no point in mentioning the conversion frequency.