Question

How much must you deposit now on January 1, 1999 to have a balance of 10,000 on December 31, 2003? Interest is compounded at an 8% annual interest.

Answer 1

To find out how much you need to deposit now to reach a target amount in the future with compounded interest, you can use the formula for compound interest:

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

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of years the money is invested or borrowed.

In this case, we want to solve for \( P \) (the initial deposit):

\[
P = \frac{A}{(1 + r)^n}
\]

Given:
- \( A = 10,000 \)
- \( r = 0.08 \) (8% as a decimal)
- The time from January 1, 1999, to December 31, 2003, is 5 years.

So, \( n = 5 \).

Now, plugging in the values:

\[
P = \frac{10,000}{(1 + 0.08)^5}
\]
\[
P = \frac{10,000}{(1.08)^5}
\]
\[
(1.08)^5 \approx 1.469328
\]

Now, calculate \( P \):

\[
P = \frac{10,000}{1.469328} \approx 6805.97
\]

Therefore, you must deposit approximately **$6,805.97** on January 1, 1999, to have a balance of $10,000 on December 31, 2003.