Upon attempting to solve a problem regarding annuities, I am a little puzzled regarding the following strategy of investment.

$10,000$ can be invested so that one can purchase an annuity-immediate with "$24$ level annual payments at an effective annual rate of $10\%$.The payments are deposited into a fund earning an effective annual rate of $5\%$."

The bolded part is what I am not sure what is going on.

Is $10,000 \over 25$ deposited each year for $25$ years? Under which interest rate, $5$ or $10$?

Or, is it the case that $10,000 \over 24$ is deposited into $24$ intervals during a year which earns $10\%$ effective annual interest, and then that payment is the present value of the annuity that earns $5\%$ annual interest?

Thanks so much for your help.


I see you're familiar with actuarial notation, so I will be using it. Suppose $X$ is the present value of the payments at $10\%$. Then $$Xa_{\overline{24}|10\%}=10,000$$ since in order to receive the annuity-immediate $X$ for 24 years, you have to pay 10,000 (and we assume everything is "actuarially equivalent" in the FM/2 world). The $5\%$ rate has nothing to do with the pricing of the annuity-immediate. This distinction is something that people studying for FM/2 consistently struggle with.

For each payment, you throw the payment of $X$ in a new account with interest of $5\%$ (I assume immediately after receiving the payment of $X$). This new account has nothing to do with the pricing of the annuity-immediate. Overall, all that is happening in this situation is you get a payment of $X$ and then you throw that payment you receive into a new account with interest so that you can get even more money.

  • $\begingroup$ Oh, I think I get it. One purchases an annuity whose present value is $10,000$. The payments are made through out the 24 years spread out evenly, therefore each payment with value $X=\frac{10,000}{a_{\overline{24}\rceil 10\%}}$. And this is invested immediately to the $5\%$. Am I on the right ball park? $\endgroup$ – hyg17 Oct 21 '14 at 6:57
  • $\begingroup$ @hyg17 - Indeed. $\endgroup$ – Clarinetist Oct 21 '14 at 12:25

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