Question

You plan to retire in year 20
Your retirement will last 25 years starting in year 21
You want to have $50,000 each year of your retirement.
How much would you have to invest each year, starting in one year, for 15 years , to exactly pay for your retirement ,if your investments earn 6.00% APR (compounded annually)?

Answers:

$21,349

$21,546

$20,930

$20,520

Answer 1

Annuity starting after retirement (P) =   50,000.00  
Time in years (n) =   25  
interest rate (i) = 6% or   0.06  
      
Present Value of annuity formula =( P *(1-(1/(1+i)^n))/i)      
50000*(1-(1/(1+0.06)^25))/0.06)      
$639,167.81      

So before retirement $639167.81 would be required to accumulate at end of 20 years      
Deposit will made upto 15 years      
Future value at end of 20 years=   $639,167.81  
value (at T=15)= FV /(1+i)^5      
639167.81/(1+0.06)^5=   $477,623.37  

This will be accumulated by savings per year.

Future value at end of 15 years = $477623.37

Future value of annuity formula = P *{ (1+i)^n - 1 } / i      
$477,623.37   = P*(((1+0.06)^15)-1)/0.06  
$477,623.37   =P*   23.27596988
P=   $20,520.02  

So amount required to contribute each year upton15 years is   $20,520.02