Calculate Annual Interest Rate for Deposits

Explanation

On the BAII Plus, press 20 N, 0 PV, 500 PMT, 18000 +/- FV, CPT I/Y. On the HP12C, press 20 n, 0 PV, 500 PMT, 18000 CHS FV, i.

To solve this problem, we need to find the annual interest rate that will cause a series of 20 deposits of $500 to accumulate to $18,000, assuming the first deposit is made one year from today.

Let's break down the problem and solve it step by step:

1. We have a series of 20 deposits, each amounting to $500. Therefore, the total amount deposited over the 20 periods is 20 * $500 = $10,000.

2. We want to determine the annual interest rate that will cause this series of deposits to accumulate to $18,000. Therefore, we need to find the interest earned on the initial $10,000 deposit.

3. To find the interest earned, we subtract the initial deposit from the final accumulated amount: $18,000 - $10,000 = $8,000. This $8,000 represents the interest earned on the $10,000 deposit over the 20 periods.

4. Now, we can use the formula for compound interest to find the annual interest rate. The compound interest formula is given by:

   A = P(1 + r)^n

   Where: A = final accumulated amount P = principal amount (initial deposit) r = annual interest rate n = number of compounding periods

   Plugging in the known values: $8,000 = $10,000(1 + r)^20

5. Divide both sides of the equation by $10,000 to isolate (1 + r)^20:

   $8,000 / $10,000 = (1 + r)^20

6. Simplify the equation further:

   0.8 = (1 + r)^20

7. Now, we need to find the 20th root of both sides to isolate (1 + r):

   (0.8)^(1/20) = 1 + r

8. Calculate the 20th root of 0.8:

   (0.8)^(1/20) ≈ 0.9694

9. Subtract 1 from the result to find the value of r:

   r ≈ 0.9694 - 1 ≈ -0.0306

   Note: The negative sign indicates a discount rate, but we're looking for an interest rate, so we take the absolute value.

10. Convert the interest rate to a percentage: