Question

Glenn and Arlene plan to invest money for their newborn grandson so that he has \$20 000 available for his education on his 18th birthday. Assuming a growth rate of 7\% per year, compounded semi-annually, how much will Glenn and Arlene need to invest today?

Short answer

Glenn and Arlene need to invest approximately \$7,007.04 today.

Step-by-step solution

- Understand the Formula

To find the present value (P) needed to reach a future value (FV) with compound interest, the formula used is \[FV = P(1 + \frac{r}{n})^{nt}\]. Where: \(FV = \$20,000\), which is the future value,\(r = 0.07\), the annual interest rate,\(n = 2\) (since it is compounded semi-annually), and \(t = 18\) years.

- Identify the Variables

Given that the future value \(FV = \$20,000\), the annual interest rate \(r = 7\%\), \(n = 2\) for semi-annual compounding, and \(t = 18\) years: \[FV = \$20,000\]\[r = 0.07\]\[n = 2\]\[t = 18\]

- Rearrange the Formula to Solve for P

Rearrange the formula to solve for P: \[P = \frac{FV}{(1 + \frac{r}{n})^{nt}}\]

- Plug in the Values

Plug the known values into the rearranged formula and solve for P: \[P = \frac{20,000}{(1 + \frac{0.07}{2})^{2 \times 18}}\]\[P = \frac{20,000}{(1 + 0.035)^{36}}\]\[P = \frac{20,000}{(1.035)^{36}}\]

- Compute the Result

Calculate the value: \[(1.035)^{36} \approx 2.854\]\[P = \frac{20,000}{2.854} \approx 7,007.04\]Thus, Glenn and Arlene need to invest approximately \$7,007.04 today.

Key concepts

Future Value

The future value (FV) is the amount of money an investment will grow to over a period of time at a specified interest rate. It's what Glenn and Arlene hope to have for their grandson's education. To find future value using compound interest, the formula is:

\[FV = P(1 + \frac{r}{n})^{nt}\]
Where:

• P = Present Value (initial investment)
• r = Annual interest rate
• n = Number of compounding periods per year
• t = Time in years

Present Value

Present value (P) is the initial amount of money that Glenn and Arlene need to invest today to reach their future value goal. It's important to know this so they can start saving effectively. Using the future value formula, we can rearrange it to solve for present value:

\[P = \frac{FV}{(1 + \frac{r}{n})^{nt}}\]
By plugging in the future value, rate, compounding periods, and time, you can determine how much you need to invest now.

Interest Rate

The interest rate (r) is the percentage at which your money grows annually. For Glenn and Arlene's investment, the rate is 7% per year. Here's a breakdown:

• Annual interest rate means how much money will grow each year.
• A higher rate increases the future value faster, meaning they would need less initial investment.
• In our calculation, 7% is written as 0.07.

Understanding this concept helps predict how your investments will grow over time.

Compounding Period

Compounding period (n) is how often the interest is applied to the investment. Glenn and Arlene's rate compounds semi-annually:

• Semi-annual means twice a year.
• The more frequently the interest compounds, the more the investment will grow.
• For semi-annual, n = 2.

By knowing the compounding period, you understand how often your investment earns interest.

Investment Growth

Investment growth is the increase in value of the initial amount invested over time due to interest. Glenn and Arlene are using compound interest to grow their \(7,007.04 to \)20,000:

• Compound interest means that you earn interest on both the initial investment and the accumulated interest from previous periods.
• This exponential growth can significantly increase the amount over longer periods.
• Using our formula and values, their investment grows approximately 2.854 times over 18 years.

By understanding how these components work together, you can effectively plan your investments for long-term growth.