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Convert the interest rate $j_{2}=9\%$ to $j_{12}$ equivalent

$j_{m}:$ nominal (yearly) interest rate which is compounded (payable, convertible) $m$ times per year

$\$1$ at $j_{12}=12i$ will accumulate to $(1+i)^{12}$; $\$1$ at $j_2=9\%$ will accumulate to $\left(1+\frac{\frac{9}{2}}{100}\right)^2=1.092025$

\begin{equation} \label{eq1} \begin{split} (1+i)^{12} & = 1.092025 \\ i & = 1.00736-1 \\ i & = 0.00736 \end{split} \end{equation}

Hence, $j_{12}=12\times 0.00736 = 0.08832\%$

But the solution is $j_{12}=1.5\%$. Where I did wrong? Any help will be appreciated.

I made mistake when wrote $j_{12}=0.08832\%$, which should be $0.08832$ only or $8.83\%$. Moreover, I consider that exercise as a compound interest which is incorrect. It should be considered as a simple interest.
falamiw
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  • Please explain your notation, what does $j_{12}$ actually mean? – 5201314 Jun 25 '21 at 17:17
  • Added that, thank you to point that @5201314. – falamiw Jun 25 '21 at 17:51
  • The answer doesn't make a lot of sense here, normally we'd expect $j_2$ to be pretty close to $j_{12}$ – 5201314 Jun 25 '21 at 17:52
  • Actually, the topic is asking for the equivalent rates (two nominal rates with different frequencies of conversion are said to be equivalent if they yield the same accumulated value at the end of the year). – falamiw Jun 25 '21 at 17:57
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    No, @5201314 is right. When I did the math, I got essentially the same answer you got: one has the relationship $$\left(1 + \dfrac{j_2}{2} \right)^2 = \left(1 + \dfrac{j_{12}}{12}\right)^{12}\text{.}$$ From some algebra, we obtain that $$j_{12} = 12\left[\left(1 + \dfrac{j_2}{2} \right)^{1/6}-1\right] \approx 8.83%\text{.} $$ – Clarinetist Jun 25 '21 at 17:58
  • Your answer helps me to understand my error. Here is the solution provided by them

    "$J_m = i \times m$ Nominal interest rate = 9% per annum. $J_2=9%$

    $J_1 = J_2 \times 2= 9%×2=18%$

    $J_{12}= \frac{J_1}{12}=\frac{18%}{12}=1.5%$." Which concept is corrent? @Clarinetist

     – falamiw Jun 25 '21 at 18:16
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    @falamiw I think the excerise is about $\textrm{simple}$ interest and not compound interest. – callculus42 Jun 25 '21 at 18:19
  • Aha, now I see. Yes. I consider it as a compound interest. @callculus. Sorry for that. – falamiw Jun 25 '21 at 18:21
  • @falamiw No problem. I`m happy to clarify the situation. – callculus42 Jun 25 '21 at 20:06

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