document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086544>Question 1086544</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.) What rate (%) compounded quarterly is equivalent to 6% compounded semi-annually? <BR>\n" );
document.write( "a. 5.93 <BR>\n" );
document.write( "b. 5.99<BR>\n" );
document.write( "c. 5.96<BR>\n" );
document.write( "d. 5.9\r<BR>\n" );
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document.write( "2.) Which of the following has the least effective annual interest rate?<BR>\n" );
document.write( "a. 12% compounded quarterly <BR>\n" );
document.write( "b. 11.5 compounded monthly<BR>\n" );
document.write( "c. 11.7% compounded semi-annually<BR>\n" );
document.write( "d. 12.2% compounded annually\r<BR>\n" );
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document.write( "3.) A bank offers 1.2% effective monthly interest. What is the effective annual rate with monthly<BR>\n" );
document.write( "compounding?<BR>\n" );
document.write( "a. 15.4%  <BR>\n" );
document.write( "b. 8.9% <BR>\n" );
document.write( "c. 14.4%<BR>\n" );
document.write( "d. 7.9% </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700777 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700777\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Problem 1\r<BR>\n" );
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document.write( "EAR = effective annual rate<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "We have some unknown interest rate. Call it x. This value is compounded quarterly to get some EAR, so,\r<BR>\n" );
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document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+x/4)^4 - 1<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "We have another rate of 6% = 0.06 compounded semi-annually to get the same EAR value\r<BR>\n" );
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document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+0.06/2)^2 - 1<BR>\n" );
document.write( "EAR = (1+0.03)^2 - 1<BR>\n" );
document.write( "EAR = (1.03)^2 - 1<BR>\n" );
document.write( "EAR = 1.0609 - 1<BR>\n" );
document.write( "EAR = 0.0609<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "Set the two EAR expressions equal to one another. Solve for x\r<BR>\n" );
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document.write( "(1+x/4)^4 - 1 = 0.0609<BR>\n" );
document.write( "(1+x/4)^4 - 1+1 = 0.0609+1<BR>\n" );
document.write( "(1+x/4)^4 = 1.0609<BR>\n" );
document.write( "[(1+x/4)^4]^(1/4) = (1.0609)^(1/4)<BR>\n" );
document.write( "1+x/4 = 1.01488915650922<BR>\n" );
document.write( "1+x/4 - 1 = 1.01488915650922 - 1<BR>\n" );
document.write( "x/4 = 0.01488915650922<BR>\n" );
document.write( "4*(x/4) = 4*0.01488915650922<BR>\n" );
document.write( "x = 0.05955662603689\r<BR>\n" );
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document.write( "which rounds to 0.0596 and converts to 5.96%\r<BR>\n" );
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document.write( "So if you have 5.96% compounded quarterly, then it's roughly equivalent to 6% compounded semi-annually.\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "Problem 2\r<BR>\n" );
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document.write( "For each of these, we'll use the same formula as in problem 1<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
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document.write( "A)<BR>\n" );
document.write( "r = 0.12<BR>\n" );
document.write( "n = 4<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+0.12/4)^4 - 1<BR>\n" );
document.write( "EAR = 0.12550881<BR>\n" );
document.write( "EAR = 0.1255<BR>\n" );
document.write( "EAR = 12.55%<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "B)<BR>\n" );
document.write( "r = 0.115<BR>\n" );
document.write( "n = 12<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+0.115/12)^12 - 1<BR>\n" );
document.write( "EAR = 0.12125932813801<BR>\n" );
document.write( "EAR = 0.1213<BR>\n" );
document.write( "EAR = 12.13%<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "C)<BR>\n" );
document.write( "r = 0.117<BR>\n" );
document.write( "n = 2<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+0.117/2)^2 - 1<BR>\n" );
document.write( "EAR = 0.12042225<BR>\n" );
document.write( "EAR = 0.1204<BR>\n" );
document.write( "EAR = 12.04%<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "D)<BR>\n" );
document.write( "r = 0.122<BR>\n" );
document.write( "n = 1<BR>\n" );
document.write( "EAR = (1+r/n)^n - 1<BR>\n" );
document.write( "EAR = (1+0.122/1)^1 - 1<BR>\n" );
document.write( "EAR = 0.122<BR>\n" );
document.write( "EAR = 12.2%<BR>\n" );
document.write( "Note: because of annual compounding, the EAR is the same as the nominal APR.<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "The smallest EAR value is 12.04%, which is from choice C. That's why choice C is the answer. <BR>\n" );
document.write( "================================================================<BR>\n" );
document.write( "Problem 3\r<BR>\n" );
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document.write( "EMR = effective monthly rate<BR>\n" );
document.write( "EMR = EAR/12<BR>\n" );
document.write( "12*EMR = EAR<BR>\n" );
document.write( "EAR = 12*EMR\r<BR>\n" );
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document.write( "The EMR is given to be 1.2% = 0.012, so the EAR is,<BR>\n" );
document.write( "EAR = 12*EMR<BR>\n" );
document.write( "EAR = 12*(0.012)<BR>\n" );
document.write( "EAR = 0.144<BR>\n" );
document.write( "EAR = 14.4%</font> </SPAN>\n" );
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