Questions: How much must be deposited today into the following account in order to have 50,000 in 8 years for a down payment on a house? Assume no additional deposits are made. An account with quarterly compounding and an APR of 5.6% should be deposited today. (Do not round until the final answer. Then round to the nearest cent as needed.)

How much must be deposited today into the following account in order to have 50,000 in 8 years for a down payment on a house? Assume no additional deposits are made. An account with quarterly compounding and an APR of 5.6% should be deposited today. (Do not round until the final answer. Then round to the nearest cent as needed.)

Transcript text: How much must be deposited today into the following account in order to have $50,000 in 8 years for a down payment on a house? Assume no additional deposits are made. An account with quarterly compounding and an APR of 5.6% $ should be deposited today. (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution

Solution Steps

Step 1: Calculate the Effective Annual Rate (EAR) if necessary

To account for compounding more frequently than annually, we calculate the EAR using the formula: $EAR = (1 + \frac{APR}{n})^n - 1$, where $APR = 5.6\%$, and $n = 4$. Thus, $EAR = (1 + \frac{5.6\%}{4})^4 - 1 = 0.0572$.

Step 2: Calculate the Present Value (PV)

Using the formula $PV = \frac{FV}{(1 + r)^{nt}}$, where $FV = $50000$, r = 0.0572$ (the interest rate per period), $n = 4$ (the number of compounding periods per year), and $t = 8$ years, we find the present value. Thus, $PV = \frac{$50000}{(1 + 0.0572)^{32}} = $8435.52$.

Final Answer:

To achieve a future value of $50000$ in 8 years with an APR of 5.6% and a compounding frequency of 4 times per year, an initial deposit of $8435.52 must be made.

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