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What is the annual nominal interest rate of a $\$500$ loan payable in two weeks at $\$675$? What is the effective annual interest rate? Assume it compounds $26$ times in a year, how much money would be owed at the end of the year?

I think if I could determine the nominal interest rate the rest of the problem would flow well from there. I'm aware that it's a 35% interest rate compounded every two weeks but I'm unaware of how translate this to an annual nominal rate.

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    $\begingroup$ Welcome to MathSE. When you pose a question here, it is expected that you share your own thoughts on the problem. For an exercise such as this, you should indicate what you have tried and where you are stuck so that you receive responses appropriate to your skill level. $\endgroup$ – N. F. Taussig Apr 16 '17 at 22:57
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    $\begingroup$ Okay, well, I think if I could determine the annual nominal interest rate the rest of the problem would flow well from there. I'm aware that it's a 35% interest rate compounded every two weeks but I'm unaware of how to translate this to an annual nominal rate. $\endgroup$ – The who slips Apr 16 '17 at 23:05
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Hint.

Since $675/500 = 1.35$ you've correctly calculated the $35\%$ rate for two weeks. After two more weeks you'd owe $$ 1.35\times 1.35 \times \$500 = 1.35^2 \times \$500 = 1.8225 \times \$500 = \$911.25 $$ for a four week interest rate of $82.25\%$.

Can you finish?

Don't be surprised when the amount owed and annual interest rate seem unimaginably large.

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  • $\begingroup$ I think I understand you'd end up at the amount owed with 500x1.35^26, but how would I find the nominal annual interest rate with this information. Wikipedia defines the nominal interest rate as 'periodic interest rate multiplied by the number of periods per year' which seems very odd because it'd be 420% in this scenario, not high enough. $\endgroup$ – The who slips Apr 16 '17 at 23:28
  • $\begingroup$ I think the nominal annual rate is just $26 \times 35\% = 910\%$. My calculation says you owe more than $\$1.2$ million so the effective rate is much much more. That's what the compounding does. $\endgroup$ – Ethan Bolker Apr 16 '17 at 23:54
  • $\begingroup$ Ah, the 26x35 comment really guided me through this. I was conflating months and periods. The nominal rate is 910%, the effective rate is 2447.25%, and the amount owed after one year is 1,223,624. You kind of snapped me out of a road block there. thanks for that $\endgroup$ – The who slips Apr 17 '17 at 0:09
  • $\begingroup$ You're welcome. The right way to thank folks here is to accept an answer (check mark). You can also upvote (up arrow). You can in fact upvote more than one answer should there be several that are useful. $\endgroup$ – Ethan Bolker Apr 17 '17 at 0:14

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