$4700 at 4.65%

Question

Find the future value after 9 years, the effective rate, and the time to reach $10,000.

Answer 1

To solve this problem, we need to use the compound interest formula. We are asked to calculate three things:

1. The future value of $4,700 after 9 years.
2. The effective annual interest rate.
3. How long it will take the investment to grow to $10,000.

One important detail is missing: the question does not specify how frequently interest is compounded. For this explanation, we will assume interest is compounded annually.

Step 1: Identify the information given.

1. Initial investment = $4,700
2. Annual interest rate = 4.65%
3. Interest rate as a decimal = 0.0465
4. Investment period = 9 years
5. Target amount = $10,000

Part A: Find the Future Value After 9 Years

The compound interest formula is:

FV = P(1 + r)^t

Where:

1. FV = Future value
2. P = Initial investment
3. r = Annual interest rate
4. t = Number of years

Substitute the values:

FV = 4,700(1 + 0.0465)^9

Simplify:

FV = 4,700(1.0465)^9

Calculate:

FV = 4,700(1.505404...)

Future Value = $7,075.40

This means that if we invest $4,700 at 4.65% interest compounded annually, the investment will grow to approximately $7,075.40 after 9 years.

The total interest earned is:

$7,075.40 - $4,700 = $2,375.40

Part B: Find the Effective Annual Interest Rate

The effective annual interest rate tells us the actual percentage earned over one year after accounting for compounding.

The formula is:

EAR = (1 + r/n)^n - 1

Where:

1. r = Nominal annual interest rate
2. n = Number of compounding periods per year

Since we are assuming annual compounding:

n = 1

Substitute:

EAR = (1 + 0.0465/1)^1 - 1

EAR = 1.0465 - 1

EAR = 0.0465

Convert to a percentage:

Effective Annual Rate = 4.65%

Because interest is compounded only once per year, the effective annual rate equals the stated annual interest rate.

If interest were compounded monthly or daily, the effective annual rate would be slightly higher.

Part C: Find the Time Required to Reach $10,000

Now we want to determine how long it will take the original $4,700 investment to grow to $10,000.

Start with the compound interest formula:

FV = P(1 + r)^t

Substitute:

10,000 = 4,700(1.0465)^t

Divide both sides by 4,700:

10,000 / 4,700 = (1.0465)^t

2.1276596 = (1.0465)^t

Since the unknown variable is in the exponent, we use logarithms.

Take the natural logarithm of both sides:

ln(2.1276596) = t × ln(1.0465)

Now divide by ln(1.0465):

t = ln(2.1276596) / ln(1.0465)

Calculate:

t ≈ 16.61 years

This means the investment will reach $10,000 after approximately 16.61 years under the mathematical compound growth model.

However, if interest is credited only once at the end of each year, the account will not actually reach $10,000 until the 17th annual compounding date.

Final Answers (Assuming Annual Compounding):

1. Future value after 9 years = $7,075.40
2. Effective annual interest rate = 4.65%
3. Time to reach $10,000 = Approximately 16.61 years (17 annual compounding periods)

Why Does This Make Sense?

Compound interest allows your money to grow because you earn interest not only on your original investment but also on the interest accumulated in previous years.

For example, during the first year:

$4,700 × 0.0465 = $218.55

Your new balance becomes:

$4,700 + $218.55 = $4,918.55

During the second year, you earn interest on $4,918.55 rather than just the original $4,700.

This process continues each year, causing the investment to grow faster over time.

Important Note: These answers assume annual compounding. If the original assignment specifies monthly, quarterly, daily, or continuous compounding, the calculations must be adjusted accordingly. The compounding frequency is necessary to determine the exact answers.