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Chapter 8: Problem 18

How much money should be deposited today in an account that earns \(7 \%\) compounded semiannually so that it will accumulate to \(\$ 12,000\) in four years?

Short Answer

Expert verified
The required present value or initial deposit is approximately \$8,781.95

Step by step solution

01

Understand the formula

The formula we need to use for future value when interest is compounded semiannually is given by \(FV = PV(1 + r/n)^{(nt)}\), where FV is the future value, PV is the present value, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. Since we are tasked with finding the present value needed to reach a specific future value, we need to rearrange this equation to solve for PV: \(PV = FV / (1 + r/n)^(nt)\).
02

Identify and plug in the given variables

Now, we have to identify the given variables to solve the problem. We are told that: the future value, FV, is \$12,000, the annual interest rate, r, is 7% or 0.07 as a decimal, the number of periods per year, n, is 2 since interest is being compounded semiannually, and the time in years, t, is 4 years. So, by substituting these values into the equation: \(PV = 12000 / (1 + 0.07/2)^(2脳4)\).
03

Perform the calculations

To get the present value, perform the calculations: \(PV = 12000 / (1 + 0.035)^(8)\), and calculate the value inside the parentheses, then raise it to the power of 8, and finally divide 12000 by the obtained result.

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Most popular questions from this chapter

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