Is this relation, linking equivalent interest rates, correct?

Question

Within my lecture notes for a previous module on actuarial mathematics, I have written the following relation linking the EPV of an $n$-year term insurance payable at the end of the month of death, to the EPV of an $n$-year term insurance payable at the end of the year of death: $$ A_{ \hspace{6mm} x : \overline{n}|}^{(m) \hspace{1mm} 1} = \frac{i}{i^{(m)}} A_{x : \overline{n}|}^1 $$

However, the following screenshot shows part of the solution to a problem which involves making this conversion.

I am assuming that the term $(1+i)^\frac{11}{24}$ implies that $$ \frac{i}{i^{(m)}} = (1+i)^\frac{m-1}{2m} $$

Is this correct?

Answer 1

Please note that it has been years since I've seen this material, but I believe that the equation you've stated $$ A_{ \hspace{6mm} x : \overline{n}|}^{(m) \hspace{1mm} 1} = \frac{i}{i^{(m)}} A_{x : \overline{n}|}^1 $$ is only true under uniform distribution of deaths. If memory serves, this entails uniform distribution of deaths within every single year. I believe you should be able to find a proof in Actuarial Mathematics for Life Contingent Risks, 2nd edition, by Dickson et al. The slides at http://users.math.msu.edu/users/valdezea/stt455f14/STT455Weeks6to8-F2014.pdf discuss this as well:

I am not as familiar with the claims acceleration approach, but I believe that the claims acceleration is just an approximation; one should not equate an approximation with an exact result from a different assumption.

Answer 2

No, this is not correct. In general, using the actuarial notation, you have:

$(1+i) = (1+\frac{i^{(m)}}{m})^m$, which gives that $ \frac{i}{i^{(m)}} = \frac{i}{m\left((1+i)^{\frac{1}{m}}-1\right)}$.