Find the nominal rate which is compounded semi-annually, that yields an effectiverate 6%?

Question

Find the nominal rate which is compounded semi-annually, that yields an effectiverate 6%?

Answer 1

The nominal rate is the interest rate that you give a "name" to. For example, 8% compounded quarterly is really 2% interest every quarter.
Thus if r is the nominal rate compounded semi-annually, you are really earning (r/2)% every half year.
If the effective rate will be 6%, then we need to solve the equation:
[1 + (r/2)]^2 = 1.06
Taking square roots:
[1 + (r/2)] = 1.029563
Subtracting 1:
r/2 = .029563
Multiplying by 2:
r = .059126 or 5.91%

It makes sense that this should be lower than 6% because it is being compounded more than once a year, and the compounding gives it a "boost" to be equivalent to 6% compounded annually.

Answer 2

When a nominal rate is compounded semi-annually, a rate equal to half the nominal rate is applied twice.

For example a 10% nominal rate compounded semi-annually yields an effective rate of 10.25%.

half of 10% is 5%. 1.05*1.05 = 1.1025.

So for any n nominal rate, the effective rate i = (1+n/2)^2-1 if it is compounded twice per year.

Now we know that i = 6% so
.06 = (1+n/2)^2 - 1
1.06 = (1+n/2)^2
1.0296 = 1+n/2
.0296 = n/2
n=.0591
n= 5.91%

Answer 3

1.06 = 1(1+r/2)^2
1.06 = (1+r/2)^2
1.06 = 1 + r +r^2/4
4.24 =4 +4r +r^2
r^2+4r -.24 = 0
r = [-4 +/- sqrt(16-4(-.24)]/2
r = [-4 +/- sqrt(16.96)]/2
r = [-4 + 4.11825]/2 = .11825/2 = .059125 = 5.91%