SOLUTION: 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 annu

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
-----------------
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
-----------------
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
-----------------
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.
================================================================
Problem 2

For each of these, we'll use the same formula as in problem 1
EAR = (1+r/n)^n - 1
-----------------
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%
-----------------
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%
-----------------
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%
-----------------
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.
-----------------
The smallest EAR value is 12.04%, which is from choice C. That's why choice C is the answer.
================================================================
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%
RELATED QUESTIONS
1. What nominal rate compounded semi-annually is equivalent to a 10% effective rate?... (answered by ikleyn)

what annual rate compounded semi-annually is equivalent to an effective rate of... (answered by ewatrrr)

1. How long will it take for 74,500 to accumulate to 101,000 if the interest rate is... (answered by ikleyn)

An amount of K27, 600 is invested for nine years in a credit union which pays an annual... (answered by Boreal)

Topic: Financial Mathematics Is it A? I want to check my answer. Which account has (answered by MathLover1,ikleyn)

I have a few similar questions, so I preferred submitting them in a single question... (answered by ankor@dixie-net.com)

What is the effective annual rate of an investment that pays 6% for 5 years, compounded... (answered by fractalier)

1. Sajid has a goal of saving $20 000 in 8 years. What principal invested for 5 years at... (answered by josmiceli)