How large an amount would you have to place at the end of 2000 in a bank account, yielding 7% interest, so that you can make the following withdrawals: at the end of 2001, 2003, 20... How large an amount would you have to place at the end of 2000 in a bank account, yielding 7% interest, so that you can make the following withdrawals: at the end of 2001, 2003, 2004, 2007, 2009, and 2011 until the account is depleted? Read more

Steps to Solve

1. Identify the Problem Requirements
   The problem asks for the amount needed to deposit at the end of the year 2000 in an account that yields 7% interest, so withdrawals can be made at specified times.

2. List the Withdrawals and Their Timing
   The withdrawals occur at the following times with specified amounts:

   • End of 2001: € 300
   • End of 2003: € 500
   • End of 2007: € 6,000
   • End of 2009: € 6,000
   • End of 2011: € 7,000

3. Calculate the Present Value of Withdrawals
   To find the present value (PV) of each withdrawal, apply the formula for PV: $$ PV = \frac{C}{(1 + r)^n} $$ where ( C ) is the cash flow (withdrawal amount), ( r ) is the interest rate (0.07), and ( n ) is the number of years until the withdrawal.

   • For the withdrawal at the end of 2001:
     $$ PV = \frac{300}{(1 + 0.07)^{1}} $$
   • For the withdrawal at the end of 2003:
     $$ PV = \frac{500}{(1 + 0.07)^{3}} $$
   • For the withdrawal at the end of 2007:
     $$ PV = \frac{6000}{(1 + 0.07)^{7}} $$
   • For the withdrawal at the end of 2009:
     $$ PV = \frac{6000}{(1 + 0.07)^{9}} $$
   • For the withdrawal at the end of 2011:
     $$ PV = \frac{7000}{(1 + 0.07)^{11}} $$

4. Calculate Total Present Value
   Add up all the present values calculated from the withdrawals to find the total amount you need to deposit at the end of 2000.