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Homework Help: Calculating compound interest

  1. Feb 14, 2014 #1
    1. The problem statement, all variables and given/known data

    I have borrowed $100. The interest rate is 5% per year compounded semiannually. How much will I have to pay the bank after 1 year?

    2. Relevant equations

    ##F=P(1+i)^{ n }##, where i is the effective interest rate per period and n represents the number of periods

    3. The attempt at a solution

    How do I get the effective interest rate per period?
    Last edited by a moderator: Feb 14, 2014
  2. jcsd
  3. Feb 14, 2014 #2


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    Shouldn't that just be the nominal annual rate (5%) divided by the number of compounding periods? What's the number of periods here?
    Last edited by a moderator: Feb 14, 2014
  4. Feb 14, 2014 #3
    The textbook says that 5% interest every year compounded semiannually is the same as 2.5% interest every 6 months. But I don't understand how they're equivalent. How can you just divide by two?
  5. Feb 14, 2014 #4


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    Because this is apparently how economists and finance people think. :yuck:

    The 5% is called a 'nominal' *annual* rate. Dividing it by the number of compounding periods gives you an 'effective' *period* rate. Applying an exponential formula to it will give you the 'effective' *annual* rate.

    That 5% quoted to you is actually a "fake" rate, meant to lull you into a false sense of security. The 2.5% every 6 months is a real thing, and it's simple to work out the actual compound interest you'll be paying at the end of the year. When you do the calculation, you'll find it's actually a little higher than 5%. That's basically how they pull the wool over your eyes when you take a loan without realising how much you'll actually end up owing. Of course, it works in your favour if you're actually investing and those are returns rather than a debt you're repaying.

    You can look all this up easily on the web. It's just a matter of getting used to the (dumb) terminology.
  6. Feb 14, 2014 #5
    Ah, that explains why there were no formulas that dealt with nominal interest rates - only effective interest rates. Thanks for the clarification.
  7. Feb 14, 2014 #6

    Ray Vickson

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    You are right to be confused: 2.5% compounded twice produces a true annual interest rate of about 5.0625%. Nevertheless, the convention suggested in your problem is the one that is used throughout the banking and finance industries. So, a mortgage with 12% annual rate, compounded monthly, yields 1% per month---by the definitions of the financial world---which amounts to an 'actual' annual rate of about 12.6825%.
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