Can anyone help explain terminology related to annuities?

Question

I am a bit confused about some terms in regard to annuities.

for example,

My book starts to talk about annuities with differing payment and interest conversion periods and defines

k- # of interest conversion periods per payment

n-# total periods in terms of interest conversions

What does that mean? Moreover,

on a timeline it has $0 , 1 , 2, .....,(n/k)$

why $n/k?$

Can anyone actually clear this up what is going on in these questions.

Answer 1

n is the number of conversion periods in the total term of the loan, so for example, if you consider monthly conversion periods over 15 years, n=180.

k is the number of periods per payment, in your case (annuities) you consider payments to be yearly, so for monthly conversions k=12

Answer 2

Your timeline doesn´t make sense. It must be

$0 \qquad \frac{1}{k} \qquad \frac{2}{k}\qquad \frac{3}{k}\qquad \ldots \qquad \frac{n-1}{k}\qquad \frac{n}{k}$

Let´s say you have quarterly compounding (every 3 months) for $2$ years then $k$ is $4$ and $n=8 (=4\cdot 2)$. Your timeline then is

$0 \qquad \frac{1}{4} \qquad \frac{2}{4}\qquad \frac{3}{4} \qquad \frac{4}{4}\qquad \frac{5}{4} \qquad \frac{6}{4}\qquad \frac{7}{4} \qquad \frac{8}{4}$

Let $i$ be the annual interest rate. Then the corresponding quarterly interest rate is $i_4=\frac{i}{4}$

Now you want to know how much is your initial capital ($C_0$) after $9$ months. It is has to be compounded $3$ times three months.

$C_9=C_0\cdot \left(1+\frac{i}{4}\right)^3$