9. FIND THE EFFECTIVE RATE THAT IS EQUIVALENT TO:
i) 4 1/2 percent compounded semi annually
ii) 12 percent compounded daily
(1+.045/2)^2 - 1 = 4.55%
(1+.12/360)^360 - 1 = 12.75%
is there any formula to solve this problem/effective rate.?
you know that the total amount for an interest rate of r compounded n times for t years is
(1+r/n)^(nt)
In this case, for #1, n=2 and r = .045, so the ending amount is
(1+.045/2)^(2*1)
then subtract off the original amount (1) to get just the interest.
(1+r/n)^(nt) - 1
To find the effective rate that is equivalent to the given rates, we can use the formula for compound interest:
\(A = P(1 + r/n)^(nt)\)
Where:
A = Final amount (including interest)
P = Principal amount (initial deposit)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
Now, let's calculate the effective rates for the given rates:
i) 4 1/2 percent compounded semiannually
The annual interest rate is 4 1/2 percent, which can be written as 0.045 (converted to decimal form). Since interest is compounded semiannually, we have n = 2 (twice a year).
To find the effective rate, we need to solve for r in the formula above. We rearrange the equation to solve for r:
\(A = P(1 + r/n)^(nt)\)
\(A/P = (1 + r/n)^(nt)\)
\( (A/P)^(1/nt) = 1 + r/n\)
Now, we substitute the values:
\( (A/P)^(1/nt) = 1 + r/2\)
We are looking for the effective rate, so we need to find (A/P)^(1/nt) - 1:
\( (A/P)^(1/nt) - 1= r/2\)
Since we are given the annual interest rate, we need to find the effective annual rate. So we plug in the values:
\( 0.045/2 = 0.0225 \)
Hence, the effective rate that is equivalent to 4 1/2 percent compounded semiannually is 0.0225, or 2.25 percent.
ii) 12 percent compounded daily
Similarly, for the second part, the annual interest rate is 12 percent, which is 0.12 in decimal form. Since interest is compounded daily, we have n = 365 (number of days in a year).
Using the same steps as above, we calculate the effective rate:
\( (A/P)^(1/nt) = 1 + r/365\)
\( (A/P)^(1/nt) - 1= r/365\)
Substituting the values, we have:
\( 0.12/365 = 0.00032877 \)
Hence, the effective rate that is equivalent to 12 percent compounded daily is 0.00032877, or 0.032877 percent.