Problem 56 - College Algebra

Question

To get the best loan rates available, the Riches want to save enough money to place \(20 \%\) down on a \(\$ 160,000\) home. They plan to make monthly deposits of \(\$ 125\) in an investment account that offers \(8.5 \%\) annual interest compounded semiannually. Will the Riches have enough for a \(20 \%\) down payment after five years of saving? How much money will they have saved?

Step-by-Step Solution

Answer

They will not have enough saved; they will have about $14,493.40.

Step 1: Calculate the Down Payment Needed

To determine the amount needed for a 20% down payment, we calculate 20% of the home's total price of \(160,000. \[ \text{Down payment} = 0.20 \times 160,000 = 32,000 \] Thus, the Riches need \)32,000 for the down payment.

Step 2: Determine the Interest Rate per Compounding Period

The annual interest rate is 8.5%, compounded semiannually. We need to convert this annual rate to a semiannual rate. \[ \text{Semiannual interest rate} = \frac{8.5}{2} = 4.25\% = 0.0425 \] So, each compounding period accrues an interest of 4.25%.

Step 3: Calculate Total Number of Compounding Periods

Since the compounding is semiannual and they plan to save for 5 years, there are:\[ \text{Number of periods} = 5 \times 2 = 10 \text{ periods} \]

Step 4: Calculate Total Savings with Regular Deposits

We use the future value of an annuity formula to calculate their savings: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] where \( P \) is the regular deposit of $125 each month, \( r \) is the rate per period (0.0425), and \( n \) is the number of periods (10). Substituting the values:\[ FV = 125 \times \left( \frac{(1 + 0.0425)^{10} - 1}{0.0425} \right) \approx 125 \times 11.5947 = 1449.34 \text{ per compounding period} \]Since there are two periods each year and they save for 5 years:\[ \text{Total savings} = 10 \times 1449.34 = 14493.40 \]

Step 5: Conclusion

After 5 years of saving, the Riches will have approximately $14,493.40. This is less than the $32,000 needed for the 20% down payment. Therefore, they will not have enough saved for the down payment.

Key Concepts

Future Value of an Annuity

The future value of an annuity is the total sum of money accumulated over time when making regular payments into a savings account or investment plan. This concept helps individuals plan financially for future goals. Without understanding the future value of an annuity, it's challenging to know how much a series of regular deposits will grow.

This involves calculating how much these regular deposits will be worth at a specific point in the future, considering the interest they accrue over time. The formula for this calculation is:

\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]

where:

• \( FV \) is the future value of the annuity.
• \( P \) is the amount of each deposit.
• \( r \) is the interest rate per period.
• \( n \) is the total number of deposits.

This formula enables you to project how much a series of consistent payments will grow, taking into account the interest rate applied at each compounding period.

Semiannual Compounding

Semiannual compounding is a method where the interest is compounded twice a year. This is different from annual compounding in that it considers the effect of interest rate at two distinct points in the year, rather than just once.

This frequent compounding results in interest itself earning interest another time within the year. As a result, the investment grows at a slightly faster rate compared to annual compounding.

To find the semiannual interest rate, divide the annual rate by two. For example, with an annual rate of 8.5%, the semiannual rate is \( \frac{8.5}{2} = 4.25\% \).

This interest methodology influences how quickly savings grow, and understanding it is crucial for evaluating investment returns effectively over time.

Regular Deposits

Regular deposits refer to a steady stream of payments into an account. In this scenario, the Riches plan to make monthly deposits of $125.

Even though deposits are made monthly, due to semiannual compounding, the effect of the deposits is calculated every six months.

When these regular contributions are made consistently, they can increase overall savings substantially via compound interest. This requires commitment to saving a fixed amount of money periodically, aligning with long-term financial goals.

It emphasizes how disciplined savings, paired with the right investment vehicles, can help achieve future financial milestones with compound interest acting as a growth catalyst.

Loan Rates

Loan rates define the cost of borrowing money, typically expressed as a percentage of the amount borrowed (principal). When the Riches aim for good loan rates, they usually want lower interest rates.

Improving their chance at better loan rates starts with a solid down payment. This 20% down payment strategy not only reduces the amount to be financed but can also mitigate the bank's risk, potentially leading to more favorable loan terms.

Lenders consider various factors while setting rates, including credit score, loan size, and down payment amount. Hence, having a substantial down payment prepared can afford better loan rates, ultimately lowering the monthly repayment burden and total loan cost.