How to convert an instantaneous mortality rate to a weekly mortality rate

Question

I want to convert an instantaneous mortality rate that is reported per year (actual value = $0.58y^{-1}$) into a weekly mortality rate.

This answer gives the formula as $j=(1+i)^{1/12}-1$ where $j$ is monthly interest rate and $i$ is the annual rate.

But this answer gives the formula as $j=1-(1-i)^{1/365}$ where $j$ is daily mortality rate and $i$ is the annual rate again.

If I use both of these to calculate my weekly rate I get different answers.

   x = 0.58
    ((1 + x)^(1/52)) - 1
# Versus
    1 - (1 - x) ^ (1 / 52)

Where am I going wrong?

Answer 1

Let's break this down a little. Assume you had a normalized population of $1$ at the beginning of the year. The annual death rate is given to be $0.58$, i.e., $0.42$ population will be alive by the end of the year. To calculate the weekly mortality rate, let's go week by week. Assume the weekly death rate is $x$.

End of week $1$:

$1-x$ people are left.

End of week $2$:

$(1-x)\times(1-x)$ people are left.

And so on till End of week $52$,

$(1-x)^{52}$ people are left.

Now it must happen that:

$$(1-x)^{52} = 0.42 = 1-0.58$$

This will give you $x$.