Stevenson Interiors of Kingston has a $67,500 liability they must pay four years from today. The company is opening a savings account so that the entire amount will be available when this debt needs to be paid. The plan is to make an initial deposit today and then deposit an additional $10,000 a year for the next four years, starting one year from today. The account pays a 5% rate of return. How much does the firm need to deposit today?

Question

Quizplus · Accepted Answer

To find the initial deposit required today, we can use the future value of an annuity formula for the annual deposits and the future value formula for the single deposit today. The future value of the $67,500 liability needs to equal the future value of the initial deposit plus the future value of the annuity of $10,000 deposits. The formula for the future value of an annuity is $FV_{annuity} = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the payment, $r$ is the interest rate per period, and $n$ is the number of periods. The future value of a single sum is $FV = PV \times (1 + r)^n$, where $PV$ is the present value or the initial deposit we're solving for. Rearranging the future value formula to solve for $PV$ gives us $PV = \frac{FV}{(1 + r)^n}$. Given that the future value needed is $67,500, the interest rate $r$ is 5% or 0.05, and the number of periods $n$ is 4 years, we can calculate the present value of the $10,000 annual deposits and then solve for the initial deposit required today to meet the future liability. The correct answer accounts for both the future value of the annuity and the present value calculation to meet the $67,500 future liability, which is option B, $20,072.91. This option correctly reflects the initial deposit needed today when considering the 5% rate of return over four years and the annual $10,000 deposits.

Stevenson Interiors of Kingston Has a $67,500 Liability They Must