terence-lim / actuarialmath Goto Github PK

Hi 👋 Nice to meet you. I'm https://www.linkedin.com/in/carlcorder/ on LinkedIn and connected with you a while back.

I was reading through the docs and noticed a typo here:

https://github.com/terence-lim/actuarialmath-guide/blob/8078954dcd049ef9cf654d09c024886391094b35/docs/udd.ipynb#L25

I believe it should be the following:

$1+j=(1+i)^{2} \implies \big(1+\frac{j^{(m)}}{m}\big)^{m}=\big(1+\frac{i^{(m)}}{m}\big)^{2m} \implies j^{(m)}=\frac{1}{m}(i^{(m)})^{2}+2 \cdot i^{(m)}$

Note that:

$i=i^{(1)} \quad$ and $\quad j=j^{(1)}=\frac{1}{1}(i^{(1)})^{2}+2 \cdot i^{(1)}=i^2+2i$

$\therefore \quad j^{(m)} \cdot {}^{2}A^{(m)}_x = j^{(1)} \cdot {}^{2}A^{(1)}_x = j \cdot {}^{2}A_x \implies \boxed{{}^{2}A^{(m)}_x=\dfrac{i^2+2i}{\frac{1}{m}(i^{(m)})^{2}+2 \cdot i^{(m)}} \cdot {}^{2}A^{}_x}$