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?
for the rate compounded monthly equivalent to a rate of 5.3% compounded semi-annually :
let the monthly rate be i
then
(1+i)^12 = (1 + .053/2)^2 , take √
(1+i)^6 = 1.0265 , now take the 6th root
1+i = 1.004368675
so i = .004368675
so the annual rate is 12(.004368675)
= .052424
So 5.2424% compounded monthly is equal to 5.3% compounded semi-annually
for the annual rate,
let the annual rate be j
(1+j) = (1 + .053/2)2
1+j = 1.053702
j = .053702 ---> an annual rate of 5.3702%
if grandpa payed an insurance policy weekly payments of $1.02 at 3 and 1/2 percent interest compounded annually for 49 years(588 months) how much would it payout after his death to granddaughter? Based on Commissions 1961 Standard Industrial Mortality Table.
To calculate the monthly and annual compounded rates that would give the same effective interest rate as the given semi-annual compounded rate, we can use the formula for effective interest rate and solve for the monthly and annual rates.
The effective interest rate formula is given by:
(1 + r/n)^n - 1
Where:
r = nominal interest rate
n = number of compounding periods per year
Let's first calculate the effective interest rate for the given semi-annual compounded rate of 5.3%.
r = 5.3% = 0.053
n = 2 (since semi-annual compounding)
Effective interest rate for semi-annual compounding:
(1 + 0.053/2)^2 - 1 = 0.0540249 = 5.40249%
Now, let's solve for the monthly compounded rate:
Let r_m be the monthly rate.
n_m = 12 (since monthly compounding)
Effective interest rate for monthly compounding:
(1 + r_m/12)^12 - 1 = 0.0540249
To solve for r_m, we can rearrange the equation:
(1 + r_m/12) = (1 + 0.0540249)^(1/12)
(1 + r_m/12) = 1.0044672
r_m/12 = 0.0044672
r_m = 0.0536064 or 5.36064%
Therefore, the monthly compounded rate that would give the same effective interest rate as the given semi-annual rate of 5.3% is approximately 5.36064%.
Next, let's solve for the annual compounded rate:
Let r_a be the annual rate.
n_a = 1 (since annual compounding)
Effective interest rate for annual compounding:
(1 + r_a)^1 - 1 = 0.0540249
To solve for r_a, we can rearrange the equation:
1 + r_a = 1.0540249
r_a = 0.0540249 or 5.40249%
Therefore, the annual compounded rate that would give the same effective interest rate as the given semi-annual rate of 5.3% is approximately 5.40249%.