Find the periodic payment that will amount to the given sum under the given conditions: S=$150,000; interest is 6% compounded semiannually; payments are made at the end of each semiannual period for 11 years
To find the periodic payment that will amount to a given sum under the given conditions, we can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = P * (1 - (1 + r)^(-n)) / r
Where:
PV is the present value or the given sum ($150,000)
P is the periodic payment we are looking for
r is the interest rate per period (6% compounded semiannually, so r = 0.06/2 = 0.03)
n is the number of periods (11 years, so n = 11 * 2 = 22 semiannual periods)
Now we can substitute the given values into the formula and solve for P:
150,000 = P * (1 - (1 + 0.03)^(-22)) / 0.03
To solve this equation, follow these steps:
1. Add 1 to the interest rate raised to the power of -n:
1.03^(-22) = 0.59814
2. Subtract this value from 1:
1 - 0.59814 = 0.40186
3. Multiply the periodic payment (P) by this value:
0.40186 * P = 150,000 * 0.03
0.40186 * P = 4,500
4. Divide both sides of the equation by 0.40186:
P = 4,500 / 0.40186
P ≈ $11,195.48
Therefore, the periodic payment that will amount to $150,000 under the given conditions is approximately $11,195.48.