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Question 1086544: 1.) What rate (%) compounded quarterly is equivalent to 6% compounded semi-annually?
a. 5.93
b. 5.99
c. 5.96
d. 5.9
2.) Which of the following has the least effective annual interest rate?
a. 12% compounded quarterly
b. 11.5 compounded monthly
c. 11.7% compounded semi-annually
d. 12.2% compounded annually
3.) A bank offers 1.2% effective monthly interest. What is the effective annual rate with monthly
compounding?
a. 15.4%
b. 8.9%
c. 14.4%
d. 7.9%
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Problem 1
EAR = effective annual rate
EAR = (1+r/n)^n - 1
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We have some unknown interest rate. Call it x. This value is compounded quarterly to get some EAR, so,
EAR = (1+r/n)^n - 1
EAR = (1+x/4)^4 - 1
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We have another rate of 6% = 0.06 compounded semi-annually to get the same EAR value
EAR = (1+r/n)^n - 1
EAR = (1+0.06/2)^2 - 1
EAR = (1+0.03)^2 - 1
EAR = (1.03)^2 - 1
EAR = 1.0609 - 1
EAR = 0.0609
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Set the two EAR expressions equal to one another. Solve for x
(1+x/4)^4 - 1 = 0.0609
(1+x/4)^4 - 1+1 = 0.0609+1
(1+x/4)^4 = 1.0609
[(1+x/4)^4]^(1/4) = (1.0609)^(1/4)
1+x/4 = 1.01488915650922
1+x/4 - 1 = 1.01488915650922 - 1
x/4 = 0.01488915650922
4*(x/4) = 4*0.01488915650922
x = 0.05955662603689
which rounds to 0.0596 and converts to 5.96%
So if you have 5.96% compounded quarterly, then it's roughly equivalent to 6% compounded semi-annually.
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Problem 2
For each of these, we'll use the same formula as in problem 1
EAR = (1+r/n)^n - 1
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A)
r = 0.12
n = 4
EAR = (1+r/n)^n - 1
EAR = (1+0.12/4)^4 - 1
EAR = 0.12550881
EAR = 0.1255
EAR = 12.55%
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B)
r = 0.115
n = 12
EAR = (1+r/n)^n - 1
EAR = (1+0.115/12)^12 - 1
EAR = 0.12125932813801
EAR = 0.1213
EAR = 12.13%
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C)
r = 0.117
n = 2
EAR = (1+r/n)^n - 1
EAR = (1+0.117/2)^2 - 1
EAR = 0.12042225
EAR = 0.1204
EAR = 12.04%
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D)
r = 0.122
n = 1
EAR = (1+r/n)^n - 1
EAR = (1+0.122/1)^1 - 1
EAR = 0.122
EAR = 12.2%
Note: because of annual compounding, the EAR is the same as the nominal APR.
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The smallest EAR value is 12.04%, which is from choice C. That's why choice C is the answer.
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Problem 3
EMR = effective monthly rate
EMR = EAR/12
12*EMR = EAR
EAR = 12*EMR
The EMR is given to be 1.2% = 0.012, so the EAR is,
EAR = 12*EMR
EAR = 12*(0.012)
EAR = 0.144
EAR = 14.4%
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