Question

Your family has given you 54820 dollars and today you have decided to put that money in a savings account that yields 7% per year. However, you already know that one year from today you will have to use part of that money to pay for a 15735 dollars expense, and another expense of 23582 dollars two years from now. a. If you are planning to collect whatever is left in your savings account 4 years from today, how much money will you be able to withdraw at that time (maximum value)? (note: round your answer to the nearest cent) b. Given the future value found in (a), compute the equivalent 4-year annuity.'

Answer 1

Let's understand this stepwise:

Year	Beginnning Balance	Deposit	Withdrwal	Interest	Ending Balance
0	$                                  -	$ 54,820.00	$                  -	$                  -	$         54,820.00
1	$                  54,820.00	$                -	$ 15,735.00	$     3,837.40	$         42,922.40
2	$                  42,922.40	$                -	$ 23,582.00	$     3,004.57	$         22,344.97
3	$                  22,344.97	$                -	$                  -	$     1,564.15	$         23,909.12
4	$                  23,909.12	$                -	$                  -	$     1,673.64	$         25,582.75

• In year 0, there was nothing in the account, so the beginning balance was 0
  • You deposited 54820
  • You withdrew nothing
  • The interest is calculated as 7% x Beginnning balance
  • Ending balance = beginning balance + deposit - withdrawal +interest
• In year 1, there was 54820 in the account, so the beginning balance was 54820
  • You deposited 0
  • You withdrew 15735 at the end of the year
  • The interest is calculated as 7% x Beginnning balance, so the interest was
    • 7% x 54820 =  $3,837.40
  • Ending balance = beginning balance + deposit - withdrawal +interest
    • 548420 +  $3,837.40 -15735 =  $42,922.40
• In year 2, there was 42922.40 in the account, so the beginning balance was 42922.40
  • You deposited 0
  • You withdrew 23582 at the end of the year
  • The interest is calculated as 7% x Beginnning balance, so the interest was
    • 7% x 42922.40 =   $3,004.57  
  • Ending balance = beginning balance + deposit - withdrawal +interest
    • 42922.40+ 3004.57-23582=   $22,344.97
• Now as there was neither a deposit or a withdrawal in years 3 and 4 so the 22344.97 remaining in the account is compounded as follows:
  • 22344.94 x (1.07)² =  $25,582.75
• So the account will have $25,582.75 at the end of the 4 years

Equivalent annuity is calculated as follows:

We are given the following information

Annual payment	PMT	To be calculated
rate of interest	r	7.00%
number of years	n	4
Future value	FV	$          25,582.75

We need to solve the following equation to arrive at the required PMT

So if you deposit 5761.95 each year, you will have the same amount in your account at the end of the 4 years as in the above case