Silas is about to begin regular month-end contributions of $500 to a bond fund. The fund's long-term rate of return is expected to be 6% compounded semiannually. Rounded to the next higher month, how long will it take Silas to accumulate $300,000?
To determine how long it will take Silas to accumulate $300,000 with regular contributions and a specified rate of return, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n),
where:
FV is the future value ($300,000),
P is the amount of each contribution ($500),
r is the annual interest rate (6%),
n is the number of compounding periods per year (semiannually, so 2),
and t is the time in years.
First, let's calculate the effective interest rate per period (semiannually) by dividing the annual interest rate by the number of compounding periods per year:
r/n = 6% / 2 = 0.06 / 2 = 0.03
Next, substitute the given values into the formula and solve for t:
300,000 = 500 * [(1 + 0.03)^(2t) - 1] / 0.03
Multiply both sides of the equation by 0.03 to get rid of the denominator:
9,000 = 500 * [(1 + 0.03)^(2t) - 1]
Divide both sides of the equation by 500:
18 = (1 + 0.03)^(2t) - 1
Add 1 to both sides of the equation:
19 = (1 + 0.03)^(2t)
Take the logarithm of both sides of the equation:
log(19) = 2t * log(1 + 0.03)
Solve for t:
t = log(19) / (2 * log(1.03))
Using a calculator, we find:
t ≈ 30.53
Since we need to round to the next higher month, Silas will accumulate $300,000 in approximately 31 months.