If annual interest rate falls from 12 to 8% per annum, how much more be deposited in an account to have 600000 in 5 years if both rates are compounded at semi annually?
To determine how much more needs to be deposited in order to have $600,000 in 5 years with a decrease in annual interest rate from 12% to 8% compounded semi-annually, 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 (amount initially deposited)
r = the annual interest rate (as a decimal)
n = the number of compounding periods per year
t = the number of years
For the initial interest rate of 12%, substituting the given values into the formula:
600,000 = P(1 + 0.12/2)^(2*5)
600,000 = P(1 + 0.06)^10
600,000 = P(1.06)^10
Now, for the decreased interest rate of 8%, we'll use the same formula:
600,000 = P(1 + 0.08/2)^(2*5)
600,000 = P(1 + 0.04)^10
600,000 = P(1.04)^10
To find the difference in the principal amounts, we can divide the two equations:
(P(1.06)^10) / (P(1.04)^10) = (1.06)^10 / (1.04)^10
Simplifying, we get:
(1.06)^10 / (1.04)^10 = 1.6399...
So, the amount 1.6399... times greater needs to be deposited to achieve the same future value of $600,000 with the decreased interest rate.
Multiplying the original deposit by 1.6399..., we can calculate how much more needs to be deposited:
Additional deposit = P * 1.6399...
Note: Without knowing the specific initial deposit amount, we cannot calculate the exact additional amount required.
To determine how much more needs to be deposited in the account to have $600,000 in 5 years with a change in interest rate from 12% to 8% per annum, compounded semiannually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the initial principal (amount to be deposited)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
Let's calculate the values for the two scenarios and find the difference.
First, let's calculate the future value using an annual interest rate of 12% compounded semiannually:
A1 = P(1 + r/n)^(nt)
A1 = P(1 + 0.12/2)^(2*5)
A1 = P(1 + 0.06)^10
A1 = P(1.06)^10
Now, let's calculate the future value using an annual interest rate of 8% compounded semiannually:
A2 = P(1 + r/n)^(nt)
A2 = P(1 + 0.08/2)^(2*5)
A2 = P(1 + 0.04)^10
A2 = P(1.04)^10
We want to find the difference between A1 and A2, so we have:
A2 - A1 = 600,000 - A1
Now, we can set up the equation:
600,000 - P(1.06)^10 = P(1.04)^10
Simplifying the equation:
600,000 = P[(1.06)^10 - (1.04)^10]
To find the value of P, divide both sides of the equation by [(1.06)^10 - (1.04)^10]:
P = 600,000 / [(1.06)^10 - (1.04)^10]
By evaluating this expression, we can determine the amount more that needs to be deposited in the account.