A bank offers a rate of 5.3% compounded semi-annually on its four year GICs(Guaranteed Investment Certificates). What monthly and annually compounded rates should it quote in order to have the same effective interest rate at all three nominal rates?
5.3% annual rate compounded semi-annually is the same as
(1+.053/2)²
=1.05370
=5.370% compounded annually
compounded monthly (6 times for semi-annual),
(1+.053/2) = (1+r)^6
r=((1.0265)(1/6)-1)*6
=5.242409%
$10,000 deposited into savings account day Katie was born 18th birthday 6.5% compounded monthly how much in the account on 18th birthday
To determine the monthly and annually compounded rates that would result in the same effective interest rate, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, we have a four-year GIC with a rate of 5.3% compounded semi-annually. Let's calculate the effective interest rate using this information first.
Given:
Annual interest rate (r) = 5.3% = 0.053 (as a decimal)
Number of compounding periods per year (n) = 2 (semi-annually)
Number of years (t) = 4
Using the formula, we can calculate the effective interest rate for the GIC:
A = P(1 + r/n)^(nt)
A = P(1 + 0.053/2)^(2*4)
A = P(1 + 0.0265)^8
A = P(1.0265)^8
Now, let's find out the monthly compounded rate that would result in the same effective interest rate:
We know that the interest is compounded monthly (n = 12), and the time for calculating the effective interest rate remains the same (t = 4).
Using the formula, we solve for the monthly compounded rate (r) that would result in the same effective interest rate:
A = P(1 + r/n)^(nt)
A = P(1 + r/12)^(12*4)
A = P(1 + r/12)^48
To achieve the same effective interest rate, the equation becomes:
P(1.0265)^8 = P(1 + r/12)^48
Taking the 8th root of both sides of the equation, we get:
(1.0265) = (1 + r/12)^6
Solving for (1 + r/12), we raise both sides to the power of 1/6:
(1 + r/12) = (1.0265)^(1/6)
Subtracting 1 from both sides gives us the monthly compounded rate (r):
r/12 = (1.0265)^(1/6) - 1
Finally, we can solve for r:
r = 12[(1.0265)^(1/6) - 1]
Similarly, we can calculate the annually compounded rate (r) that would result in the same effective interest rate:
We know that the interest is compounded annually (n = 1), and the time for calculating the effective interest rate remains the same (t = 4).
Using the formula, we solve for the annually compounded rate (r) that would result in the same effective interest rate:
A = P(1 + r/1)^(1*4)
A = P(1 + r)^4
To achieve the same effective interest rate, the equation becomes:
P(1.0265)^8 = P(1 + r)^4
Taking the 4th root of both sides of the equation, we get:
(1.0265)^2 = (1 + r)
Subtracting 1 from both sides gives us the annually compounded rate (r):
r = (1.0265)^2 - 1
Finally, we can calculate the monthly and annually compounded rates:
Monthly Compounded Rate:
r = 12[(1.0265)^(1/6) - 1]
Annually Compounded Rate:
r = (1.0265)^2 - 1
Now, you can evaluate these expressions to obtain the monthly and annually compounded rates that would result in the same effective interest rate as the 5.3% compounded semi-annually rate.