Swan Furnace Cleaners wants to add 0.50% to the effective rate of interest on its credit card. If it currently charges a nominal rate of 4.25% compounded semi-annually, at what value should it set the new nominal rate?
The answer for this question is 4.74%,
the formula for this question is
f=(1+i)^m
=total - 1
I know that j=4.25%
m=2
i=j/m=.0425/2=.02125=2.125%
f=(1+.02125)^2 -1
=0.042951562
=4.30%
but the right answer is 4.74%, i cant seem to figure out how to get that same answer.
To find the value at which Swan Furnace Cleaners should set the new nominal rate, you need to add 0.50% to the existing effective rate. Here's how you can calculate it step by step:
1. Convert the existing nominal rate to the effective rate. Since the nominal rate is compounded semi-annually, you can use the formula:
Effective rate = (1 + (nominal rate/number of compounding periods))^(number of compounding periods) - 1
In this case, the number of compounding periods is 2 (semi-annually) and the nominal rate is 4.25%. Plugging in these values:
Effective rate = (1 + (4.25%/2))^2 - 1
= (1 + 0.0425)^2 - 1
= (1.0425)^2 - 1
= 1.08628125 - 1
= 0.08628125 or 8.63%
So the existing effective rate is 8.63%.
2. Add 0.50% to the existing effective rate:
New effective rate = existing effective rate + 0.50%
= 8.63% + 0.50%
= 9.13%
3. Convert the new effective rate back to the nominal rate. Rearranging the formula used earlier:
New nominal rate = (1 + new effective rate)^(1/number of compounding periods) - 1
In this case, the number of compounding periods is still 2. Plugging in the values:
New nominal rate = (1 + 0.0913)^(1/2) - 1
= (1.0913)^(1/2) - 1
= 1.04567022 - 1
= 0.04567022 or 4.57%
Therefore, the value at which Swan Furnace Cleaners should set the new nominal rate to add 0.50% to the effective rate is approximately 4.57%.