Question

For $55,500 compounded continuously at an interest rate of 3.7%, determine:

(a) What is the future value after 8 years?
(b) What is the effective rate?
(c) How many years will it take to reach $18,000?

Answer 1

(a) The future value after 8 years for $55,500 compounded continuously at an interest rate of 3.7% is approximately $71,993.15.

(b) The effective rate is approximately 3.79%.

(c) It will take approximately 5.54 years to reach $18,000.

Step-by-step explanation:

In continuous compounding the future value (FV) can be calculated using the formula:

`\[ FV = P \cdot e^(rt) \]`

Where:

P is the principal amount $55,500

r is the interest rate per time period 3.7% or 0.037

t is the time the money is invested or borrowed for (8 years), and

e is the mathematical constant approximately equal to 2.71828.

So for part (a):

`\[ FV = 55,500 \cdot e^(0.037 \cdot 8) \]`

`\[ FV \approx 55,500 \cdot e^(0.296) \]`

`\[ FV \approx 55,500 \cdot 1.345 \]`

`\[ FV \approx 71,993.15 \]`

For part (b) the effective rate can be calculated using the formula:

`\[ \text{Effective Rate} = e^(r) - 1 \]`

`\[ \text{Effective Rate} = e^(0.037) - 1 \]`

`\[ \text{Effective Rate} \approx 1.0379 - 1 \]`

`\[ \text{Effective Rate} \approx 0.0379 \]`

So the effective rate is approximately 3.79%.

For part (c) to find the time it takes to reach $18,000 we rearrange the continuous compounding formula:

`\[ t = (\ln(FV/P))/(r) \]\[ t = (\ln(18,000/55,500))/(0.037) \]\[ t \approx (9.8)/(0.037) \]\[ t \approx 265.14 \]It will take approximately 5.54 years to reach $18,000.`