Question

Suppose you deposit $1,151.00 into an account 4.00 years from today. Exactly 19.00 years from today the account is worth $1,706.00. What was the account's interest rate?

Answer 1

What values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

Answer 2

To determine the interest rate of the account, we can use the formula for compound interest, which is given by:

\[
A = P(1 + r)^t
\]

where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial amount of money).
- \(r\) = the annual interest rate (in decimal).
- \(t\) = the time the money is invested for, in years.

In this situation, the account has been deposited with an initial amount of $1,151.00 at \(t_0 = 4.00\) years. The account reaches a value of $1,706.00 at \(t_1 = 19.00\) years. The total time the money is invested is \(t = t_1 - t_0 = 19.00 - 4.00 = 15.00\) years.

So we have:
- \(P = 1,151.00\)
- \(A = 1,706.00\)
- \(t = 15.00\)

Plugging these values into the compound interest formula, we get:

\[
1,706 = 1,151(1 + r)^{15}
\]

Next, we will solve for \(1 + r\):

1. Divide both sides by 1,151:

\[
\frac{1,706}{1,151} = (1 + r)^{15}
\]

Calculating the left-hand side:

\[
\frac{1,706}{1,151} \approx 1.4805
\]

Now we have:

\[
(1 + r)^{15} \approx 1.4805
\]

2. Take the 15th root of both sides:

\[
1 + r \approx (1.4805)^{\frac{1}{15}}
\]

Calculating \( (1.4805)^{\frac{1}{15}} \):

\[
1 + r \approx 1.0287
\]

3. Now solve for \(r\):

\[
r \approx 1.0287 - 1 = 0.0287
\]

4. Convert to a percentage:

\[
r \approx 0.0287 \times 100 = 2.87\%
\]

Thus, the interest rate is approximately **2.87%**.

Answer 3

To solve the problem using a financial calculator, you need to identify the values for the following parameters:

- **N** (Number of periods): This is the total number of compounding periods.
- **i/y** (Interest rate per period): This is the annual interest rate you're solving for.
- **PV** (Present Value): This is the initial deposit amount.
- **PMT** (Payment): This is the amount of money added or withdrawn each period (in this case, it will be $0 since there are no additional payments).
- **FV** (Future Value): This is the amount of money accumulated after the investment period.

For your specific problem:

1. **N**: Since you are calculating over 15 years (the time between the deposit at year 4 and the final value at year 19), \( N = 15 \).

2. **i/y**: This is what you are trying to find out (the interest rate).

3. **PV**: This is the initial deposit amount, which is \( PV = -1,151.00 \). (The negative sign indicates a cash outflow.)

4. **PMT**: There are no additional payments made after the initial deposit, so \( PMT = 0 \).

5. **FV**: This is the future value of the investment after 15 years, which is \( FV = 1,706.00 \).

Putting it all together, you would enter the following values in the financial calculator:

- **N = 15**
- **i/y = ?** (this is what you will solve for)
- **PV = -1,151.00**
- **PMT = 0**
- **FV = 1,706.00**

After inputting these values, you would compute \( i/y \) to find the annual interest rate.